The Argument from the Implicity of Ultimacy in Reality per se

The argument in a nutshell: the reality of any sort of real thing presupposes, prerequires, and supervenes the reality of the proper formal ultimate of its sort. E.g., 5 presupposes, prerequires, and supervenes infinity. No infinity → no 5. The reality of its proper formal ultimate is logically implicit in any real. There are reals. So there is a real Ultimate.

Along every dimension of formal configuration spaces there is some ultimate; some value that is its outmost bound, and that helps to define it as a formal dimension. Were it otherwise, the dimension would be inadequately specified, and so could not operate as a real formal dimension. For, if a formal dimension is not itself limited, it cannot limit; so, it cannot form. In that case, it is not a form to begin with. It is rather nothing at all.

Consider for example a dimension consisting of the set of natural numbers from 0 to n. Is 5 a member of that set? No way to tell, unless n is specified. Thus “the set of natural numbers from 0 to n” is not a definite set at all. It is a string of characters with no definite denotation, and thus with no meaning. It is noise.

Excursus: Implicit in this consideration is that the Logos has a complete and coherent logical structure, along all dimensions; or else, he is not a logos in the first place, let alone the Logos himself; but rather just a mish mash; a chaos; not a he; not even an it; not at all.

But obviously there must be a Logos, for we see formed things all about us, and we are ourselves formed things.

If there is no form of things, then there are no things; for, to be a thing is to be in a certain specific way; to be is to be formed.

If there is no form of things, then there are no things; for, to be a thing is to be in a certain specific way; to be is to be formed.

There are things. So is there is a form of things.

There being a form of things, ergo there must be a former of things, a conditioner; a creator. For, the former of things is implicit in the form of things. No former, no formation. A form that does not form is not a form. Ideas don’t have themselves; so likewise forms don’t themselves form. No former, no form.

But I’m getting ahead of myself.

In respect to the most tractable, familiar sorts of formal dimensions, such as those that are characterized by maxima (e.g., the possible interior angles of a triangle), it is easy to see that for each formal dimension there must be some ultimate (viz., you can’t adequately specify triangularity without implicitly specifying the limit of the interior angles of triangles). There seems to be a definite maximum of velocity within our cosmos, e.g.; which is to say, a maximum rate of causation (this maximum being implicit in the minimum rate of causation (in a coherent cosmos, could it be otherwise?)). But even when that outmost bound of a formal dimension is itself boundless, as is the case with mathematical infinity, there must be such an outmost bound. And infinity is just such a bound. Boundlessness then is a precise specification of the outmost bound of some sorts of form.

Excursus: How big can a given sort of thing be? In the absence of other constraints, such as the bearing capacity of bone per pound thereof (a limit on the size of terrestrial endoskeletal animals), there would seem to be no such limit. A thing might be infinitely extensive. Indeed, never mind this or that thing in this or that extensive cosmos; there might be an infinitely extensive system of extensive systems, e.g.; secula seculorum might not be finite. This is to say that the thing constituted of the created order in toto might not be finite.

If the creator is infinite, it is rather hard to see how his act of creation might possibly be finite.

But, again, I’m getting ahead of myself.

Given any thing, the set of which it is a member must be real – not itself concretely actual qua set, mind you, but itself real, at least formally – a real set, that is to say – or there could be no actual or formal instance of it, nor therefore of any of its members. No x, no instance of x.

Here, then, is the argument.

Along every properly specified formal dimension, there is necessarily a thing than which there is no greater along that dimension. Infinity along all dimensions capable thereof, e.g., is then the indispensable forecondition of any lesser value of any such dimension: ∞ ∋ x; ∴ ¬∞ → ¬x. By extension, the reality of the maximum of x along any formal dimension (whether it is capable of infinity or not) is the forecondition of the reality of any x along that dimension.

It does not matter whether we are arguing about actual infinity or potential infinity. Either sort of infinity might be real. It is that reality we are concerned to demonstrate.

  1. ∞ ∋ x; ∴ ¬∞ → ¬x
  2. x
  3. ¬¬∞

So likewise for all sorts of ultimacy.

The concrete actuality of any subultimate entails the concrete actuality of its proper formal ultimate. Likewise, the merely formal potentiality of any subultimate entails the merely formal potentiality of the ultimate. But, NB: a merely formal ultimate would be subultimate to its concrete actual. For, since actuals must all have forms, actuality is ipso facto also formality; whereas potentiality is only formality. So, actuality is greater than potentiality.

The actual infinite is ultimate to the merely formal infinite; so the merely formal infinite presupposes, prerequires and supervenes the actual infinite.

Thus a merely formal ultimate could not be obtained except insofar as it had been actualized concretely. Then even to obtain the mere form of 5, e.g., and whether or not there are actually five of anything, the proper formal ultimate of numbers such as 5 – infinity – must be concretely actual.

If there is anything at all, then, the Ultimate than which no greater can be conceived is therefore necessarily real and actual.

44 thoughts on “The Argument from the Implicity of Ultimacy in Reality per se

  1. Pingback: The Argument from the Implicity of Ultimacy in Reality per se | Reaction Times

  2. Does the integer 5 have a proper formal ultimate? Neither mathematician nor philosopher, I cannot answer with any authority, but my intuition is to say, “No”.

    It seems to me that a more accurate assertion is: no 5, no ∞. ∞ is not a thing, but the description of a process. The infinite set of positive integers has twice as many elements as the infinite set of positive even integers. How does that work? I think that the the set of integers is defined in terms of a process to get from 0 to 1, and the observation that this process can be re-applied indefinitely to the result, where this notion “indefinitely” is key.

    • It is true that no 5 → no ∞. If you delete 5 from the set of integers, the set is broken, as it were; you no longer have the set of integers, but something different. But to delete ∞ is to delete the set altogether. The limit of the set is boundlessness; that the set does not terminate upon some specific number does not mean that it is inadequately defined, and thus not a set in the first place. Putting the same notion differently, boundlessness is a perfectly definite notion.

      Generalizing from the fact that deleting 5 from the set of integers breaks the set, we may say that insofar as the members of a set share certain characteristics – those that qualify them as members of the set – they are implicit in each other. The set of the integers cannot be whole without both 5 and 13. But this means that 5 depends for its membership in the set – i.e., its character qua integer – upon 13, and vice versa. The integers are implicit in each other, as well as in ∞; and ∞ is implicit in each of the integers.

      ∞ is the proper formal ultimate of 5 whether we characterize it as a function or as a limit of a function or as a thing or as a mere idea. If the function that we can use to generate the integers can be iterated endlessly – that is, an infinite number of times – then if we were to delete the endlessness that forms the outer limit of the iteration of the function, we would break the function, as it were; we’d no longer have the function that generates the set of integers, but something different. But to delete the function that generates the integers by breaking it is to delete the integers per se; for they are implicit in – indeed, they are specified by – the function that generates them, and vice versa.

      In practice, functions are real – are, i.e., truly functions in the first place – only insofar as they are somehow implemented. Functions are forms. And like any other form, any other idea, functions can’t generate themselves; cannot vouchsafe their own reality. Ideas subsist only in the concretes they inform, either cognitively (as when I think of a recipe) or not (as when the recipe is actually iterated, and generates a cake).

      Furthermore, concretes are all generates of functions; the complete specification of a concrete entity is among other things the specification of the conditions of its actual existence, among which are always the procedure that generated that actual existence.

    • The infinite set of positive integers has twice as many elements as the infinite set of positive even integers. How does that work?

      Er no. The size (cardinality) of the set of positive integers is equal to the size of the set of even positive integers, because they can be 1:1 mapped to each other:
      1 2
      2 4
      3 6

      This is pretty basic and needs to be understood before diving into philosophy of mathematics.

      • The size (cardinality) of the set of positive integers is equal to the size of the set of even positive integers

        One can always reject the conclusion of equality by rejecting trichotomy for the ‘size’ of infinite sets. This is equivalent to rejecting the axiom of choice, and is therefore a reasonable option.

    • The infinite set of positive integers has twice as many elements as the infinite set of positive even integers. How does that work?

      By redefining size. Intuitively if set A is a proper subset of finite set B, then naturally A is smaller than set B. But by the redefinition to tackle infinite sets, this intuition is judged false.

  3. Consider for example a dimension consisting of the set of natural numbers from 0 to n. Is 5 a member of that set? No way to tell, unless n is specified. Thus “the set of natural numbers from 0 to n” is not a definite set at all. It is a string of characters with no definite denotation, and thus with no meaning. It is noise.

    What a weird statement, if it were true it would make all of mathematics meaningless noise. But of course “the set of natural numbers from 0 to n” is perfectly meaningful, even if it doesn’t denote a specific concrete set. Its meaning depends on context, like every other utterance.

    • You have a point. I wrote a little too freely. I think nevertheless that the idea I meant to convey still stands. Given the algorithm for generating the natural numbers implicit in it, the string, “the set of natural numbers from 0 to n” is indeed meaningful as a denotation of a function – so that it is meaningful in the discourse of purely abstract mathematics – but it is not meaningful as a denotation of a set. The use of the term “set” in the string is therefore inapposite: there is no such thing as the set of natural numbers from 0 to n, except insofar as n is specified. No set of natural numbers could be specified, or therefore generated, unless n had been specified.

      Another way of saying this: you can *talk* about the function without specifying n – you can *discuss* the math – but you can’t actually *run* it – can’t actually *do* the math – until you specify a value for n.

      The argument of the post is interested, not so much in the functions that might be used to determine membership in the set, as rather in the set itself, that set being the specification of a dimension in formal configuration space. Such dimensions just are the set of points of which they consist.

      • I’m not sure why you think a function of integers to sets is any less real or more abstract than a set of integers. They are both mathematical objects, with identical metaphysical natures (whatever that may be).

        But I’m also not sure what any of that has to do with your larger argument, which I don’t really understand. Things like ultimateness or actuality or “foreconditions of reality” are not mathematical concepts. Which is not to say they aren’t valid, but unlike mathematics they don’t have clear, crisp definitions that pretty much everybody can agree upon.

      • A.morphous, I gotta hand it to you: these last two comments of yours are the best you’ve ever given us. They are substantive, important, and interesting; and best of all, they are not too snarky. Plus they are smart, and they don’t miss the point, but rather per contra identify real difficulties any honest and well educated reader might naturally discover in grappling with the notions I have here presented. My thanks to you, sir.

        I don’t think that a function of integers to sets is less real or more abstract than a set of integers. On the contrary, I think both functions and sets are concrete actual reals. I think only that a function of integers to sets is neither the same thing, nor yet the same sort of thing, as a set of integers.

        I think that, like all other forms, mathematical functions are concretely real only insofar as they are ideas eternally contemplated in, as, and by the mind of God (their contemplations by creaturely minds then deriving from their prior contemplation by God). I’m a Neoplatonist, in that respect, and an Augustinian. I also think that they are concretely real in that their archetypal concrete creaturely instantiations are angelic – are real beings, with minds. Does that help?

        Functions and sets are indeed both mathematical and logical objects, and they both therefore share the characteristics of such objects. But that does not mean that they are in every respect the same sorts of thing. If they were the same sorts of thing, we would not have discerned their differences, and so named and characterized them differently. A function of a set is not the same thing as that set. It’s that simple. Likewise, an algorithm is not the same thing as its output.

        Things like ultimateness or actuality or “foreconditions of reality” are not mathematical concepts. Which is not to say they aren’t valid, but unlike mathematics they don’t have clear, crisp definitions that pretty much everybody can agree upon.

        They do have clear, crisp definitions. Everyone could agree upon those definitions, if they knew them. As, at one time, all educated thinkers did know them. But – as with the mathematical definitions that are clear and crisp – even the most educated thinkers nowadays generally don’t know them.

        I do not excuse myself from that criticism. Much of my philosophical education over the last decade has amounted to a rediscovery of what was forgotten and lost in and by the modernist rejection of classical philosophy. I have still much to learn – and to unlearn.

  4. Thanks, well, snark is unrewarding and mathematics is an area where I actually know something.

    A function of a set is not the same thing as that set….I think only that a function of integers to sets is neither the same thing, nor yet the same sort of thing, as a set of integers

    Of course they are not the same thing, but they are the same sort of thing in that they are both mathematical objects. Formally a function is also a set, a set of pairs of elements from the domain and codomain (range). Eg, the function f(x) = 2x is also the set {(0,0), (1,2), (2,4), (3,6)…}.

    I also think that they are concretely real in that their archetypal concrete creaturely instantiations are angelic – are real beings, with minds. Does that help?

    Are you saying that the set of even integers (eg) is a being with a mind? Interesting, but unintuitive. What does such a mind *do*?

    Mathematical objects, for all their beauty, are static. Minds, at least the ordinary human ones we know about, are dynamic, they interact with their environment, learn, are born and eventually die, and these qualities are essential to their nature. An unchanging mind is not a mind.


    • Of course [a function of a set is not the same thing as that set], but they are the same sort of thing in that they are both mathematical objects.

      Cats and fungi are the same sort of thing. Both sorts of creature belong to the sort of biological organisms, and in that sense they are the same sort of thing. But they are quite different sorts of things, and this is why we treat them differently. Likewise, men and angels are the same sort of thing in some ways – both belong to the sort of rational creatures – but in others they are quite different sorts of things.

      A mathematical operator – and by extension any function composed of operators, of values, and of variables – is just not the same sort of thing as its operands, or as its operations, or as its outputs, even though these all belong to the sort of mathematical objects. 3, 4 and 12 are different sorts of things than * and =, even though they all belong to the sort of mathematical objects.

      Formally a function is also a set … E.g., the function f(x) = 2x is also the set {(0,0), (1,2), (2,4), (3,6) …}.

      No. The function specifies the set, but is not itself *exactly the same thing* as the set that it specifies. As you have just written: “Of course [a function of a set is not the same thing as that set], but they are the same sort of thing in that they are both mathematical objects.”

      The set is implicit in the function, but that does not mean that the function *just is* the set. To get the set, you must *explicate* the function that specifies it. You must perform it. Likewise, to get an actual instantiation of Bach’s Toccata & Fugue in D Minor, you need more than the score of that composition, and more than the ideas formally encoded in that score. To get real music, you need to *perform* the score.

      Likewise you perform the function f(x) = 2x when you write (or think) {(0, 0),(1, 2), (2, 4), …}. To complete the set either by writing or thinking – an impossibility, of course, at least for finite minds – you’d have to reiterate the function for all x. You’d have to actually perform the multiplication of which the function is the mere form – the formula.

      Let me try to nail this down a bit more precisely: iff x = 2 → f(x) = 4. You have to actually go ahead and specify the value of x in order for the function to *do anything* – to generate any real product.

      Mathematical objects, for all their beauty, are static. Minds, at least the ordinary human ones we know about, are dynamic, they interact with their environment, learn, are born and eventually die, and these qualities are essential to their nature. An unchanging mind is not a mind.

      Are you quite sure that the only sorts of minds are those that change? Are you quite sure that change is that basic? I mean, sure, change is basic to the sorts of lives that men lead, for we are embodied, and as such, temporal. But are you sure it is basic to every sort of life whatever?

      If you knew everything all at once that it is possible for a mind to know, nothing that happened could change your knowledge. But it would be odd to say in such a case that your omniscience was therefore not mindful. For, surely, your omniscience would simply *consist* of your infinite mindfulness.

      • A mathematical operator – and by extension any function composed of operators, of values, and of variables – is just not the same sort of thing as its operands, or as its operations, or as its outputs, even though these all belong to the sort of mathematical objects. 3, 4 and 12 are different sorts of things than * and =, even though they all belong to the sort of mathematical objects.

        I agree with this and have lost track of what the point of this particular subargument was supposed to be.

        The function specifies the set, but is not itself *exactly the same thing* as the set that it specifies. …The set is implicit in the function, but that does not mean that the function *just is* the set.

        In modern mathematical practice, functions are indeed sets. See here http://mathworld.wolfram.com/Function.html (functions are a particular type of relation, a relation is a subset of a Cartesian product, which is a set of pairs).

        To get the set, you must *explicate* the function that specifies it. You must perform it.

        I understand what you are saying, but that is not the case when doing formal mathematics. A function is a relation; the act of computing that relation (what you are calling explication or performance) is only possible on a small subset of functions. The theory of computabilty (and the existence of uncomputable functions) is one of the key insights of 20th century mathematics (Gödel and Turing) https://en.wikipedia.org/wiki/List_of_undecidable_problems

        Are you quite sure that the only sorts of minds are those that change? Are you quite sure that change is that basic?

        I don՚t think I am sure in the sense that you are asking, but the notion of an unchanging mind seems obviously oxymoronic to me, given my concept of what a mind is.

        If you knew everything all at once that it is possible for a mind to know, nothing that happened could change your knowledge. But it would be odd to say in such a case that your omniscience was therefore not mindful.

        Minds do more than know things; they do things with their knowledge (or else the Library of Congress would be one of the best minds around).

      • I agree [that functions are not the same sorts of things as their arguments, their operands or their products] and have lost track of what the point of this particular subargument was supposed to be.

        You had been arguing that functions and their operands and products were the same sorts of things; that the sets related by functions are the same sorts of things as those functions. Specifically, you had argued contra my statement:

        Consider for example a dimension consisting of the set of natural numbers from 0 to n. Is 5 a member of that set? No way to tell, unless n is specified. Thus “the set of natural numbers from 0 to n” is not a definite set at all. It is a string of characters with no definite denotation, and thus with no meaning. It is noise.

        To that, you said:

        What a weird statement. If it were true it would make all of mathematics meaningless noise. But of course “the set of natural numbers from 0 to n” is perfectly meaningful, even if it doesn’t denote a specific concrete set. Its meaning depends on context, like every other utterance.

        Exactly: its meaning depends on the context that is sufficiently provided only in the event that n is specified with a definite value. If n is not specified, then the codomain of the function is not fully specified. In that case, the function will not denote a relation between sets, because the sets in question will not be adequately specified.

        This question related to the argument of the post only because I argued there that:

        … if a formal dimension [in configuration space] is not itself limited, it cannot limit; so, it cannot form. In that case, it is not a form to begin with. It is rather nothing at all.

        Which was to say no more really than that if you don’t adequately define a form, then it can’t function as a form – can’t inform anything – and thus can’t be a form in the first place. And that was important to the argument of the post only because that argument relied upon the premise that for every proper form there is some proper formal ultimate.

        You write:

        In modern mathematical practice, functions are indeed sets.

        But then you have also in this thread written:

        Of course [a function of a set is not the same thing as that set] …

        Which is it?

        [Functions] are a particular type of relation, a relation is a subset of a Cartesian product, which is a set of pairs.

        Whether we define sets in terms of functions or vice versa, the fact remains that we can’t define sets and functions in terms of each other unless they are really different sorts of things.

        And, again: a relation between insufficiently specified sets is not a relation in the first place, because it has no definite relanda.

        A function is a relation; the act of computing that relation (what you are calling explication or performance) is only possible on a small subset of functions.

        Doesn’t matter.

        If you can’t compute a relation, it is not really a definite relation. If you can’t compute a function between sets, then it is not a veritable relation between those sets. If you can’t compute a relation – i.e., a function – then it is not a proper form in the first place. If it is incomputable by any procedure whatsoever, then it *absolutely* cannot relate sets, and so *it cannot be the form of any particular real thing, or set of things.* It can’t do anything whatever.

        In that case, it cannot be a function that defines a set which constitutes a dimension in configuration space. Such cases then are irrelevant to the argument of the post.

        … the notion of an unchanging mind seems obviously oxymoronic to me, given my concept of what a mind is.

        An omniscient mind would be unchanging. Yet, as omniscient, it would by definition be a mind. So your problem with the notion of an unchanging mind would seem to lie with your concept of what a mind is.

        Minds do more than know things; they do things with their knowledge (or else the Library of Congress would be one of the best minds around).

        What makes you think that an unchanging mind could not do things?

        That analogy to the Library of Congress is exactly correct. Ideas can’t have themselves, can’t know themselves, and cannot themselves do anything. For ideas – forms, functions, formulae – to *do* anything, to get any fire into the equations, you need at least one mind. The LOC is not a mind.

  5. Sorry, there seems to be some confusion going on. In modern math, pretty much everything is a set. The integers are a set, and a function is also a set. For example, the function f(x) = 2x (on the nonnegative integers) is the set of pairs:
    {(0,0), (1,2), (2,4)…}

    So a function is a set, and it also maps a different set (the domain) onto yet another (the range). In this case both the domain and range happen to be the same, the set of nonnegative integers, but they don՚t have to be. So when talking about a specific function there are generally three sets involved.

    This is just basic modern mathematics, which is founded on set theory. It՚s not the only way to do mathematics and it is not without philosophical problems, but it is the underpinnings of pretty much everything else.

    If you can’t compute a relation, it is not really a definite relation.

    Gödel and Turing showed otherwise.

    Or I suppose it depends on what you mean by “definite relation”, which is not a mathematical term. Nor is “veritable relation”. But they demonstrated the existence of functions which are perfectly well-defined (“definite”?) but are not computable.

    Note: I՚m not particularly interested in arguing about any of the above; it՚s all very standard stuff that any undergraduate class in abstract mathematics would cover. The philosophical or theological implications, on the other hand, are not at all standardized, so happy to argue about those.

    I suppose you could object to modernism in mathematics just as you object to it in culture or values. Imagine a reactionary mathematics, that refused to recognize the modernist revolutions of Gödel and Turing, that yearned for a simpler time of sturdy values and solid foundations.

    What makes you think that an unchanging mind could not do things?

    Seems pretty self-evident to me. When a mind performs an action, it changes the world and also necessarily changes its own state, if only to register that what was once a proposed action is now an accomplished action. That is to say. I don՚t see how an atemporal mind could have a causal connection to the world, it just doesn՚t make any conceptual sense.

    It may not be quite as obvious, but the concept of an omniscient mind is equally problematic. A mind that doesn՚t learn is not a real mind, and if an if an omniscient mind already knows everything, it can’t learn anything.

    • If you can’t compute a relation, it is not really a definite relation.

      Gödel and Turing showed otherwise.

      Or I suppose it depends on what you mean by “definite relation,” which is not a mathematical term. Nor is “veritable relation.” But they demonstrated the existence of functions which are perfectly well-defined (“definite”?) but are not computable.

      A definite relation is a defined relation between defined objects. If any of the related objects are not specifically defined, the relation is not specific enough for us to tell what it is relating. It is not defined.

      If you have a relation that cannot be computed – that cannot in principle be mapped to any set – then you have a relation that does not in fact relate to any sets. It relates nothing.

      A relation that relates nothing is not a relation.

      To repeat:

      And, again: a relation between insufficiently specified sets is not a relation in the first place, because it has no definite relanda.

      Where your relanda are not specified, your functional relation is a map of something or other to … something or other. What are the somethings it relates? Dunno; can’t tell, given the information on them so far supplied to us. Define that relationship of we don’t know what to … we don’t know what as carefully as you like; as long as it is a relation between denotations that don’t denote anything, it still isn’t a relation between any definite things, or sorts of things; which is to say, that it isn’t a relation in the first place, but rather only something that might possibly be a relation, if its domain and codomain were to be specified.

      I don’t see how an atemporal mind could have a causal connection to the world, it just doesn’t make any conceptual sense.

      That’s because you are taking time as basic. You are taking temporal causation as the only sort. It’s not. Eternity is basic to time. Eternal causation is prior to temporal causation. Eternity doesn’t push and pull things around in time, the way that temporal things push and pull each other around. It causes events formally and finally, rather than efficiently as mundane events cause each other.

      A mind that doesn’t learn is not a real mind …

      What you are really saying here is:

      A mind that doesn’t learn is not a mind like mine …

      That’s a natural thing to think.

      But, again, in so thinking, you are taking your sort of mind as the only sort there might be. That’s an unwarranted presumption. We learn only because our knowledge is imperfect. A perfect mind, that knows all knowables perfectly, would not need to learn. That would not make it stupid, or mindless. On the contrary: that would make it capable of simultaneously and completely comprehending an infinite number of things, all possible things. It would make it an intelligence than which no greater could be conceived.

      Consider two minds, A and K. Suppose there are only two things that either of them might possibly know: T1 and T2. A knows one thing, T1. K knows two: T1 and T2. Is K less mindful than A because he knows more than A? Is K less mindful than A because there is nothing more that K might learn?

      • If you have a relation that cannot be computed – that cannot in principle be mapped to any set – then you have a relation that does not in fact relate to any sets. It relates nothing.

        As I՚ve said repeatedly, the reality of uncomputable functions has been an accepted part of mathematics for almost 100 years. You haven՚t responded to that, instead you are just restating your intuitions over and over again in imprecise language. I don՚t see any point in continuing this particular discussion.

        That’s because you are taking time as basic. You are taking temporal causation as the only sort. It’s not. Eternity is basic to time. Eternal causation is prior to temporal causation.

        Yes time seems pretty basic; I don՚t think I have a very unusual position there.

        “Eternal causation” is an interesting concept. I suppose it is a reference to the Aristotelian theory of causation, one of history՚s really stupid ideas, since it conflates entirely different things under the idea of causation. That is to say, even if there is some reality behind the idea of a final cause, it is something that has absolutely nothing to do with causation in the normal sense, and should be called something else.

        you are taking your sort of mind as the only sort there might be. That’s an unwarranted presumption

        I am not, I am stating that minds are definitionally things that can act, learn, and change. If there is something else that is eternal, unchanging, and not causally connected to the universe except through some mysterious phlogiston, then that thing is not a mind, and no useful purpose can be served by trying to call it a mind – nothing but confusion results from the attempt. There՚s an awful lot of variation possible within those constraints.

      • As I’ve said repeatedly, the reality of uncomputable functions has been an accepted part of mathematics for almost 100 years. You haven’t responded to that, instead you are just restating your intuitions over and over again in imprecise language.

        I’ll lay the eternal truth of the Law of Noncontradiction against your 100 years of mathematics any day of the week. Explain to me how a thing that cannot relate sets is nevertheless by definition a relation of sets, as you seem to be insisting (please correct me if I have misinterpreted you) and we’ll have something to talk about. Or else, I’ll be forced to conclude that you are raving incoherently, and buttressing your hand waving with appeals to authority: “Because Gödel! And Turing! So there!”

        See, here’s the thing, a.morphous. We can talk about uncomputable functions all day long, and our discussions can be most interesting and fruitful. We can treat such functions as real (on the proper sense of “real”). But what we can’t do is treat them as relations of sets. Because why? *Because, as being uncomputable – which is to say, unintelligible – qua relations between sets, they obviously can’t relate sets.* Not in any intelligible way, that is. Not in any way that (if they are in fact absolutely uncomputable) any mind whatever could understand.

        … time seems pretty basic; I don’t think I have a very unusual position there.

        Indeed, your position is quite usual. And it is quite natural, and understandable. It’s a normal way to feel about things. That does not make it correct.

        After all, it’s normal to think that the sky is up, and not down; when really it’s in every direction. It is also normal to think that space is not curved, when of course really it is. It is normal to think that the cosmos is continuous, when really it’s a bunch of discrete quanta.

        There are all sorts of things that it is normal to think, but that on examination have turned out to be quite wrong.

        Observe:

        If there is anything necessary – as it seems there must be, given the necessary truths of mathematics, logic and metaphysics, which cannot possibly be false under any circumstances whatever (that’s what we mean when we call something necessary) – then there is eternity. For, what is necessary cannot possibly fail to be, in any state of affairs; and, that which is in every state of affairs is eternal. It is not contingent. What is eternal is prior to any contingent event. Whether or not any contingent event comes to pass, what is eternal necessarily obtains. Temporal events are all contingent. That which is eternal is therefore prior to all temporal events. That which is eternal is therefore prior to time. Time is therefore not basic. Eternity is basic.

        To say that eternity is basic is not, NB, to say that time is illusory. To think that, one must still be thinking about eternity and time in the wrong way. When you think about time and eternity properly, you can see that neither one rules out the other. On the contrary. But to see this, you have to think about them properly. And that takes a lot of intellectual work.

        … even if there is some reality behind the idea of a final cause, it is something that has absolutely nothing to do with causation in the normal sense …

        It would be more accurate to say that final causation has nothing to do with causation in the modern, post-Cartesian sense *that seems normal to a.morphous,* and that comprehends nothing other than material and efficient causation. The problem with the modern, post-Cartesian account of causation as amounting only to efficient and material causation is that, as Hume so clearly saw, it demolishes the notions of causation, and of causal regularity, and thus of cosmic order. Material and efficient causation by themselves make any inference to causal order from observed regularities in events unwarranted – indeed, unwarrantable. The impoverished Cartesian view of causation therefore renders science impossible.

        This is why scientists are always smuggling final and formal causation into their accounts of the world, generally without even realizing that they are so doing. That’s what’s happening with all this latter day talk of emergence, holism, strange attractors, and so forth. Those notions are not wrong. They are just Aristotelianism dressed up in Cartesian costumes.

        … minds are definitionally things that can act, learn, and change. If there is something else that is eternal, unchanging, and not causally connected to the universe except through some mysterious phlogiston, then that thing is not a mind …

        Under your definition of mind, the more you learn, and the closer you approximate to omniscience and omnipotence, the less mindful you get. Speak for yourself.

        It’s your definition of mind against mine. I’ll take mine, which is also the definition in common use: mind is the faculty of knowledge; of experience. But on your definition of mind, a subject of experience who knows and suffers and comprehends all things is not a mind; and, so, knows and experiences and comprehends nothing. On your definition of mind, nothing = everything.

        That pesky Law of Noncontradiction is just a bitch, no?

        Your devotion to your parochialism is preventing you from learning. If you take time as basic, and your own sort of mind as the limit of what mind might be, why then you are simply bound by the self-imposed limitations of your own thought to beg the questions of eternity and omniscience. By those self-imposed limitations, you prevent yourself from any honest consideration of what those notions might mean. In other words, you prevent yourself from even beginning to try to understand them.

        So doing, you doom yourself to talking about them without knowing what it is that you are talking about.

      • I’ll lay the eternal truth of the Law of Noncontradiction against your 100 years of mathematics any day of the week.

        Oh, you really are going to advocate a reactionary mathematics. Splendid.

        Or else, I’ll be forced to conclude that you are raving incoherently, and buttressing your hand waving with appeals to authority: “Because Gödel! And Turing! So there!”

        Citing Gödel and Turing is not an “appeal to authority”. Maybe you don՚t understand how mathematics works. Both those individuals constructed proofs, and I cite them not because of their names or reputations but because they proved certain things about mathematical and computational systems. Mathematics is perhaps the only area of human thought where authority and rank means nothing.

        There are plenty of accessible popularizations of their work (eg Hofstadter՚s Gödel Escher Bach, or David Foster Wallaces՚s book on Gödel). Or read Wikipedia: https://en.wikipedia.org/wiki/Halting_problem

        You are the one raving incoherently, I՚m afraid. And as I already said, there՚s no point in continuing this part of the conversation.

        Under your definition of mind, the more you learn, and the closer you approximate to omniscience and omnipotence, the less mindful you get. … mind is the faculty of knowledge; of experience.

        A timeless being can՚t have any experience. And no, I don՚t think mind is “the faculty of knowledge”, that is a ridiculously naive view of mind that I thought nobody really believed in.

        Let՚s turn to Wikipedia again (since you are claiming to own the common definition): https://en.wikipedia.org/wiki/Mind

        The mind is a set of cognitive faculties including consciousness, perception, thinking, judgement, language and memory. It is usually defined as the faculty of an entity’s thoughts and consciousness. It holds the power of imagination, recognition, and appreciation, and is responsible for processing feelings and emotions, resulting in attitudes and actions.

        Note that the mind here is not a store of knowledge, but a collection of abilities and activities. That is much closer to both common usage, and reality.

        But on your definition of mind, a subject of experience who knows and suffers and comprehends all things is not a mind;

        We are talking about an eternal timeless “mind”, which by definition is not a subject of experience (and hence can՚t really suffer).

        By those self-imposed limitations, you prevent yourself from any honest consideration of what those notions might mean.

        On the contrary, I am considering what those notions mean and pointing out that they are contradictory and lead to absurdities.

        To be clear – I am trying hard to grasp your notion of the eternal, the absolute, something that is necessary and foundational to everything else. It՚s a suspect notion, but I՚m trying to go with it, if only for the mental exercise.

        My problem is that in conceptualizing something like that, I find it absolutely impossible to map the human concepts of “mind” or “person” to it. That seems absurd, or worse, presumptuous – it seems to be reducing the infinite to the merely human.

      • I’ll lay the eternal truth of the Law of Noncontradiction against your 100 years of mathematics any day of the week.

        Oh, you really are going to advocate a reactionary mathematics. Splendid.

        I take it that you reject the Law of Noncontradiction. In that case, then, so far as you are concerned, both you and I are right about everything; and we are also both wrong about everything; and, also, we are neither both right about everything, nor are we both wrong about everything. Also in that case, we can understand each other, and have something to talk about, and can reach conclusions, and can gain knowledge; and, also, we can’t do any of those things; and, we both can and can’t do any or all of those things.

        You see the problem. Get back to me on the philosophy of math – or on anything else for that matter – when you’ve got something better.

        I’m actually pretty familiar with Gödel, having read several technical philosophical books on his arguments (some of the popular books you mention are on my shelves, but somehow I’ve never had time for them). And, what’s more, I’m a convinced Gödelian. And, what’s even more, I’ve written a fair bit about his arguments and his theorems, at this site. One of the interesting implications – logical implications, NB – of the Incompleteness Theorems is that God – or, at least, an infinite being than which no greater can possibly be conceived by any mind, and which is the forecondition of everything whatever (including itself) – necessarily exists. But you wouldn’t want to hear about that, I suppose.

        My quarrel with your adduction of Gödel and Turing as if the mere mention of their names was somehow dispositive of our argument is that *it isn’t.* It is furthermore not at all the case that the actual arguments of either Gödel or Turing are thus dispositive.

        The difficulty you must overcome is this: you have defined functions as relations between sets, but you have not actually explained how a function – which we both agree is in fact a function – that cannot even in principle relate sets is nevertheless by definition a function as you have defined that term: namely, a relation of sets. Mention of the names of Gödel and Turing can’t help you with that. Nor can their arguments. The problem you face is that you have asserted a proposition of the form x = ¬x. That proposition violates the Law of Noncontradiction. It is therefore *necessarily* false. Indeed, it is meaningless nonsense.

        Now, you can to be sure resort at this point to a rejection of the Law of Noncontradiction. But, if you do reject that Law, you’ll be forced as a consequence to admit that I am just as correct in my assertions as you are in yours; and, by the same token, that you are just as incorrect in your assertions as I am in mine. You will be forced to agree (and in the same breath to disagree) to all propositions whatever; as, e.g., “a.morphous is a reactionary Christian.”

        The mind is a set of cognitive faculties …

        Surely you are familiar with the aetymology and meaning of “cognition,” no? I suppose perhaps not; for, had you been, you would have recognized that it means, “ability to comprehend, mental act or process of knowing … In 17c., the meaning was extended to include perception and sensation.” Like I said:

        … mind is the faculty of knowledge; of experience.

        The definition of mind that you quote to criticize mine turns out to support it. The “ridiculously naïve view of mind that [you, a.morphous] thought nobody really believed in,” is the one that – if what you write is anything to go on – you yourself credit.

        Note that the mind here is not a store of knowledge …

        Note that I did not write that the mind is a store of knowledge. I don’t know where you got that, but it wasn’t from me. A faculty is not a store. Mind is not a batch of knowledge; it is the faculty of knowing, in all its departments (viz., sensation, perception, feeling, ratiocination, apprehension, and so forth).

        We are talking about an eternal timeless “mind,” which by definition is not a subject of experience …

        You here beg the very question at issue, which is whether temporality is a forecondition of experience. By your peremptory definition of what a timeless mind must be – namely, just like a temporal mind – you forestall all your thought on the topic.

        Excursus: Pro tip: if your definition of x forestalls thought on x altogether, there’s probably something wrong with your definition. Materialists fall into that error all the time. They begin by defining (the obsolete 19th Century Rutherfordian notion of) matter as the only sort of real thing; then they deduce from that definition that there are no other sorts of real things. I.e.: they *totally beg the question* of the truth of their materialist definition.

        My problem is that in conceptualizing something like [an eternal mind], I find it absolutely impossible to map the human concepts of “mind” or “person” to it. That seems absurd, or worse, presumptuous – it seems to be reducing the infinite to the merely human.

        This is the most constructive remark in your comment. You are absolutely right that it is profoundly misguided – and, as you suggest, absurdly impertinent – to try to understand omniscience by mapping it to human experience. As you say, that is an impossible project. Human experience is obviously partial, whereas obviously omniscience is by definition complete and total and perfectly accurate. Whatever that must be like, it can’t be like our own experience; rather, our own experience might possibly be a little like that of omniscience in some few respects.

        Excursus: The notion that our experience might possibly be a little like that of omniscience in some few respects is part of what is meant by the Christian doctrine that man is imago dei; which doctrine is in turn the basis of the Christian doctrine that man is therefore capax dei.

        Hell, we can’t even know from our own experience as men what it is like to be a bat. It’s just stupid to think we might from that same human experience know what it is like to be God.

        What we can know, however, is that in logic it is contradictory to assert of omniscience that it knows nothing – as would have to be the case if omniscience were mindless. “Omniscience is mindless” is an instance of x = ¬x. We can’t coherently think that “omniscience is mindless” is true. In fact, we can’t coherently think “mindless omniscience,” any more than we could coherently think “square circle.”

        So, no matter how boggled we might feel at the notion of an eternal mind that knows everything, if we want to think or talk about it, we have no alternative but to wrestle with it.

        I applaud you for doing so.

        But I suggest that you stop trying to understand eternal omniscience under the categories of our partial knowledge. It can’t be done. So you are absolutely right to think that such methods are wrongheaded. You would be wrong also therefore to think that any conclusions you reached using that method were well founded.

        Let me also say that I totally sympathize with your difficulties in grappling with the notion of eternity. Ten years ago, I was in the same boat. Hell, I guess I still am. I still struggle. It’s like wrestling with an angel or something.means

  6. As I՚ve said repeatedly, the reality of uncomputable functions has been an accepted part of mathematics for almost 100 years.

    Yes, uncomputable functions is an accepted part of non-constructivistic ‘standard’ mathematics, . But one can be constructivist and thereby validly not accept many ‘accepted’ theorems in ‘standard’ mathematics.

    From what I see Kristor is a type of constructivist, with some older worldview that you’re not familiar with.

    I suppose you could object to modernism in mathematics just as you object to it in culture or values.

    A non-trivial minority of mathematicians do in fact have a non-modern view of mathematics.

    • First off, almost nobody is a mathematical constructivist. Second, Gödel’s incompleteness theorem is 100% in accordance constructivist mathematics (it involves *constructing* an unprovable statement). Third, constructivism is a mode of mathematics that refuses to employ proof by contradiction and the law of the excluded middle; somehow I doubt Kristor is ready to give those up.

      • I would not characterize myself as a constructivist, because – as you say, a.morphous – I am not willing to give up the Law of the Excluded Middle. As I understand constructivism, though – not far – it would not require me to abandon the Law of Noncontradiction.

        I am not much interested in philosophy of math. So I don’t have strong opinions about most of it. The only things I’m pretty sure of are my Platonic realism in respect to mathematical objects, and my commitment to Gödelian Incompleteness. Turing is interesting to me, and I can see the value of his work, but it doesn’t move me much. Not his fault.

        Re the middle that sometimes seems plausible, but that the Law of its Exclusion would exclude, I have found through long practice that the Scholastic maxim almost always suffices to the predicament: neither deny nor affirm, but rather distinguish. Actually, that policy often helps with propositions that seem prima facie both true and untrue, thus violating the Law of Noncontradiction; in fact, such propositions are *always* (I have no proof of this, just loads of empirical anecdoty) true in certain senses but not in others. So that the application of the aforementioned Scholastic maxim ends by satisfying all parties – or, at least, dissatisfying them all minimally.

      • First off, almost nobody is a mathematical constructivist

        Likewise, almost nobody in the West is not a modern. This means nothing.

        Second, Gödel’s incompleteness theorem is 100% in accordance constructivist mathematics (it involves *constructing* an unprovable statement).

        A diagonalisation that is an infinite procedure does not actually construct anything.

      • I take it that you reject the Law of Noncontradiction.

        No, I didn՚t say that. You are the one who brought up that law based on what seems like a complete misunderstanding of computability theory.

        My quarrel with your adduction of Gödel and Turing as if the mere mention of their names was somehow dispositive of our argument is that *it isn’t.* It is furthermore not at all the case that the actual arguments of either Gödel or Turing are thus dispositive.

        Gödel demonstrated the existence of unprovable yet true propositions, and Turing demonstrated the existence of well-defined yet uncomputable functions. (In both cases, given particular formal definitions of provability and computation). I can՚t tell if those proofs are “dispositive” to whatever it is you think you are talking about.

        The difficulty you must overcome is this: you have defined functions as relations between sets, but you have not actually explained how a function – which we both agree is in fact a function – that cannot even in principle relate sets is nevertheless by definition a function as you have defined that term: namely, a relation of sets.

        I don՚t know what you are talking about. A function relates sets by definition, so I have no idea what you mean by “a function that cannot even in principle relate sets”, which makes about as much sense “a circle which cannot in principle be round”.

        We are talking about an eternal timeless “mind,” which by definition is not a subject of experience …

        You here beg the very question at issue, which is whether temporality is a forecondition of experience.

        This too seems definitionally true. The normal meaning of experience is a set of events that one is personally involved with. Events are temporal (by definition), they involve changes in the world and in the participants. If you want to postulate a being that is unchanging yet still has experiences, I think the onus is on you to make sense of that. Given your devotion to the Law of Noncontradiction, this might be challenging.

        You are absolutely right that it is profoundly misguided – and, as you suggest, absurdly impertinent – to try to understand omniscience by mapping it to human experience. As you say, that is an impossible project. Human experience is obviously partial, whereas obviously omniscience is by definition complete and total and perfectly accurate. Whatever that must be like, it can’t be like our own experience; rather, our own experience might possibly be a little like that of omniscience in some few respects.

        For someone who trumpets the Law of Noncontradiction as loudly as you, you don՚t seem to shy from contradicting yourself in the very same sentence. That is – “like” is a symmetric relationship, and you are saying that human experience and omniscience are both alike and not-alike. Oops!

        I think what you mean to say is that we can՚t take our own experience as a model for God՚s, but we can do the inverse – that՚s what imago dei means. But as you say just below, we have no ability to understand the experience of God, so that doesn՚t make a whole ton of sense. How are we supposed to understand human experience (the only thing, really, which we have direct access to) in terms of God՚s experience, which is entirely inaccessible to us?

      • I’m glad to hear that you don’t want to abandon the Law of Noncontradiction. It means we can proceed to help each other ascertain the truth. It means also that, if you are thorough enough, and honest, and careful, and persistent, why then in plain obedience to that Law, you cannot help but abandon sooner or later your atheism and your liberalism, and become a Christian and a reactionary, and then perhaps even begin to work out your salvation, in fear and trembling.

        Which would be cool. I’d love to see you in Heaven.

        To be fair, I should here mention that whenever we do engage in these colloquies, I learn a lot in the process of responding to you. It’s fruitful. So I appreciate your patience and interest, and the work you obviously perform in generating your part of the conversation. My thanks.

        A function relates sets by definition, so I have no idea what you mean by “a function that cannot even in principle relate sets,” which makes about as much sense “a circle which cannot in principle be round.”

        Given that a function is a relation of sets, a string of operations that cannot generate a set of products – that cannot map a domain to a codomain, and thus cannot relate two sets – might appear to function, but cannot function in fact the way that functions by definition must function. Just as “square circle” appears to denote meaningfully but in fact does not, so such a string of operations might appear to function, but really does not. That’s all I meant.

        Excursus:; Character strings in languages (both formal and informal) can be treated as recipes for the conception of formations of ideal compositions of forms; of composite concepts. The character string “square circle” appears to be such a recipe. But it isn’t. The two forms it appears to compose cannot be composed. It isn’t a recipe for a thought. It is a string of characters that mean just “incoherent nonsense,” and no more.

        It would perhaps have been better if I had used scare quotes:

        … a “function” that cannot even in principle relate sets …

        Most numbers are not computable (e.g., π). They can of course be denoted, and are certainly real, but there is no finite sequence of steps that can be used to complete their specification. There may be a *relation* of such an uncomputable number to values on other domains, but that relation cannot be implemented – cannot be concretely formalized and then carried out – by a finite series of steps. It cannot be completely functionalized.

        So: not all relations are functions. An uncomputable relation between sets can be real enough, but if it cannot be carried into practice – cannot, i.e., function – then it may not be a function, properly so called.

        Alright then: in respect to uncomputable relations, there is no finite algorithm that can complete the calculation of the output of the formalization of the relation. The formalizations of such relations cannot therefore be used as specifications for an algorithm that could complete the mapping of their domains to their codomains. So such formalizations of relations cannot actually *function.*

        So, there can be no concrete instance of their operation; for, to be concrete is to have been completed.

        And this is just to say that such formalizations cannot completely specify the relations of sets that, by definition, functions must all specify. So, we can *call* them functions, insofar as they formalize true relations of sets – C/d = π is certainly a relation of sets – but they can’t be completed, can’t be carried into practice (other than by an approximation sufficient to current pragmatic purposes).

        All functions are relations of sets. But not all relations of sets are functional. Whether we choose to dignify the latter sorts of relations with the term “function” then is a matter of rhetoric and method and convention, rather than of substance.

        Now, this is all extremely cool, because it furnishes the fodder for another theistic proof. To wit:

        1. The set of real numbers that cannot be computed cannot be related to any other set by any finite computable function.
        2. The finitely uncomputable numbers are real – are, i.e., logically and mathematically real, formally real, ergo real *in any way at all* – only in virtue of a computation that is not finite.
        3. Computations that are not finite are infinite.
        4. The real numbers that cannot be finitely computed are real.
        5. There is an infinite computation.

        Thanks, a.morphous! That was excellent!

        Note that Gödelian Incompleteness too entails infinity; in this case, an infinite complete stack of logical calculi. As iff ∞ → 5, so likewise if and only if the entire Gödelian stack is complete can any proposition in any calculus thereof be true.

        A.morphous: We are talking about an eternal timeless “mind,” which by definition is not a subject of experience …

        Kristor: You here beg the very question at issue, which is whether temporality is a forecondition of experience.

        A.morphous: This too seems definitionally true.

        I acknowledge, of course, that it seems true prima facie. The question at issue is whether in fact it is true. No matter how intuitively obvious its accuracy might seem, if a definition is not working, it stands in need of rectification. If a definition of “mind” entails that omniscience – i.e., perfect mindfulness – is mindless, it is incoherent. It is an instance of x = ¬x. It isn’t working, and needs to be fixed. If a definition of “experience” or “knowledge” entails that omniscience – i.e., perfect, exhaustive knowledge of all that can be known – cannot know, it is incoherent. It doesn’t work, and our intuitive understanding of what experience is, which gave rise to that definition, must be wrong somehow.

        But if you stick with that definition anyway, despite its incoherence, you lock yourself into mistaken categories, and thus into error.

        Events are temporal (by definition), they involve changes in the world and in the participants.

        Certainly events have a temporal aspect. But do they also have an eternal aspect? If so, then how do we reconcile the stillness of their eternal aspect with the flux of their temporal aspect? Good questions! Can they be answered? Let’s find out! Are our intuitive definitions of the terms we shall need in order to answer them correct, sufficient, adequate to the discussion? Well, if we say that events do not have an eternal aspect *by definition,* we foreclose ab initio any consideration of such questions. We say, in effect, that there are no such questions. But if we are after the truth, that’s a crazy thing to do. So often have our intuitive categories misled us, that it would be no exaggeration to say that rectification of the names that seem at first intuitively obvious to us is the first and foremost step in the acquisition of knowledge.

        If you want to postulate a being that is unchanging yet still has experiences, I think the onus is on you to make sense of that.

        To be sure. My position is the one that, to the modern way of thinking we all inhabit from earliest childhood, is counterintuitive.

        … you don’t seem to shy from contradicting yourself in the very same sentence. That is – “like” is a symmetric relationship, and you are saying that human experience and omniscience are both alike and not-alike. Oops!

        Close, but no cigar. You would be correct in saying that I had violated the Law of Noncontradiction if I had written:

        … [the experience of omniscience] can’t be like our own experience, period full stop. Rather, our own experience must be like that of omniscience, period full stop.

        But I didn’t write that. I wrote:

        … [the experience of omniscience] can’t be like our own experience; rather, our own experience might possibly be a little like that of omniscience in some few respects.

        NB: alike wholly and simpliciter ≠ a little alike in some few respects. E.g.: Men and angels are alike in many ways, but they are radically different sorts of creatures; ditto for cats and fungi. These examples of the distinction between alike wholly and simpliciter and alike in some ways appeared upthread.

        I think what you mean to say is that we can’t take our own experience as a model for God’s, but we can do the inverse – that’s what imago dei means.

        Yes. Not, however, “model,” exactly. “Image” is really the best term I have encountered.

        But as you say, we have no ability to understand the experience of God, so … How are we supposed to understand human experience … in terms of God’s experience, which is entirely inaccessible to us?

        I don’t think it is true – and I did not mean to convey – that we have no ability to understand the experience of God, or that his experience is entirely inaccessible to us. The best analogy I can think of to convey what I’m getting at is the relation between infinity and 5. Infinity works lots better than 5 as the archetype of quantity, and thus as the basis or substrate of the number line. It makes more sense to say that 5 participates infinity than to say that infinity participates 5. Infinity is logically implicit in 5, and vice versa, of course. If you get one of them, you have the other as well; they come as a package deal, so that neither one of them is prior to the other in order of operation or sequence. Nevertheless infinity is far superior to 5, insofar as it denotes a set comprised of all quantities whatever, whereas 5 denotes a set far smaller.

        Interestingly, the set of values denoted by 5 is itself infinite, if we are counting all the real numbers between 0 and 5. So when we say that 5 participates infinity, we are not saying only that 5 is one of the members of the set denoted by infinity. We are saying also that 5 is an *instance* of infinity. In this sense, 5 is an image of infinity.

        This is one of the reasons that Nicholas Rescher has insisted that there are infinitely many true statements we might make about any particular real.

        Our minds cannot comprehend infinity. But we can understand quite a few things about it. It is not wholly inaccessible to us; on the contrary, all finity participates infinity, just as 5 does.

      • @GJ — Gödel’s proof does not involve an infinite diagonalization. Gödel himself emphasized the constructive nature of his proof.

  7. To be fair, I should here mention that whenever we do engage in these colloquies, I learn a lot in the process of responding to you. It’s fruitful

    Thanks for the kind words. I too feel like I learn something from these exchanges. Probably not what you would like to teach me, but it forces me to articulate my assumptions, which is valuable.

    Most numbers are not computable (e.g., π).

    Pi is considered to be a computable number https://en.wikipedia.org/wiki/Computable_number (although this is slightly different sense of “computable” than we have been discussing, so this probably just confuses matters).

    So: not all relations are functions. An uncomputable relation between sets can be real enough, but if it cannot be carried into practice – cannot, i.e., function – then it may not be a function, properly so called.

    Not according to standard mathematical definitions and language. If you want to make up your own definitions for terms, well, feel free, but I՚m not going to bother engaging with them – it՚s tedious. This is much different from the discussion below about what “mind” means. The latter is a concept on which there is plenty of room for honest disagreement, but the mathematical concept of a function is a universally agreed upon technical term.

    It is true, btw, that not all relations are functions, because functions have to uniquely map values and relations don՚t (I see you mention this yourself later on). But that has nothing to do with computability.

    Infinity works lots better than 5 as the archetype of quantity, and thus as the basis or substrate of the number line.

    It most certainly does not. I can՚t imagine why anyone would think that. For instance, ordinary quantities can be compared – infinity can՚t be. Ordinary quantities can be added to each other to form new ones – infinity can՚t be.

    Of course it depends what you mean by “archetype”, a non-mathematical term which could mean anything I suppose. The usual meaning of “archetype” is “an extremely typical member of a set”, but infinity is hardly typical of the set of integers, of which it is not even a member.

    It makes more sense to say that 5 participates infinity than to say that infinity participates 5

    Neither of those make an iota of sense. “5 participates *IN* infinity” makes no semantic sense but is at least grammatical, “5 participates infinity” isn՚t even that.

    Nevertheless infinity is far superior to 5, insofar as it denotes a set comprised of all quantities whatever, whereas 5 denotes a set far smaller.

    Mathematical entities are not superior or inferior to one another. Some of them can be compared, some have metrics defined on them, and some are composed from others, but “superiority” is mathematically meaningless. If you just mean infinity is *bigger* than 5, well, sure, it is definitionally bigger than any finite number, so what?

    Interestingly, the set of values denoted by 5 is itself infinite, if we are counting all the real numbers between 0 and 5

    Argh, “the set of values denoted by 5” is {5}, if you mean “the real interval between 0 and 5” you have to say that. And that set happens to have the same size (cardinality) as the entire real number line, so your talk of “superiority” above makes even less sense.

    So when we say that 5 participates infinity, we are not saying only that 5 is one of the members of the set denoted by infinity. We are saying also that 5 is an *instance* of infinity. In this sense, 5 is an image of infinity.

    You can say that all you want but it still won՚t make a particle of mathematical sense.

    OK, I give up. You aren՚t speaking mathematics, but some perverted parody of it. If it makes sense to you, feel free to think that way, but I can՚t do it.

    Forget math, onto mind:

    If a definition of “mind” entails that omniscience – i.e., perfect mindfulness – is mindless, it is incoherent.

    It՚s the concept of omniscience that is incoherent. I feel perfectly happy with a definition that excludes it. When I talk about “minds”, I talk about real-world things. My mind, your mind, perhaps animal minds or future artificial software minds if we want to stretch the category. Omniscient minds are not found in the real world, and there is nothing interesting to say about them.

    Certainly events have a temporal aspect. But do they also have an eternal aspect? If so, then how do we reconcile the stillness of their eternal aspect with the flux of their temporal aspect?

    Events definitionally involve something happening, at a particular time and place. Now, if you are a physicist, you can look at spacetime from an “eternal aspect” – that is, they construct a theoretic point of view in which time is just another dimension and the imaginary observer is outside of it. They put themselves in the position of God, in other words. They have a theoretical omniscience, in the sense that their equations can describe any point in spacetime.

    That is a lot different from being able to *actually observe* every point in spacetime. Real observations involve the interaction of two temporal systems (the one under observation, and that of the observer, who has to change in some way in response to what is happening.). An eternal being omnisciently knowing every aspect of the universe couldn՚t perform any sort of actual observation. It just knows by virtue of – what?

    If such a being was possible, It would have static, unchanging knowledge of a static, unchanging world. Such a conception of God is incoherent, and even worse, deathly dull.

    • Pi is considered to be a computable number (although this is slightly different sense of “computable” than we have been discussing, so this probably just confuses matters).

      Yeah.

      So: not all relations are functions. An uncomputable relation between sets can be real enough, but if it cannot be carried into practice – cannot, i.e., function – then it may not be a function, properly so called.

      Not according to standard mathematical definitions and language. If you want to make up your own definitions for terms, well, feel free, but I’m not going to bother engaging with them – it’s tedious.

      I don’t mean to invent definitions, but rather to engage with them. It seems to me prima facie that algorithms that cannot in principle ever finish mapping domains to codomains, by any finite series of steps, *absolutely cannot* be said to relate sets. Because why? Because, since one of the sets they operate upon cannot ever be by them completely defined, or therefore ever be by them definitely specified qua set.

      There is a difficulty in categorizing an uncomputable function as a relation of sets when it can’t relate sets on account of the fact that it can’t compute them. There must be a way around this difficulty. Stamping your feet and insisting that all mathematicians define functions as relations of sets, and that’s the end of the matter, is not it.

      The thing that occurs to me is that a function might conceivably and truly relate sets even though it is not finitely computable, in virtue of the fact that *it is infinitely computable,* and has by some infinite procedure that – as infinite and therefore eternal, and, as such, eternally complete – been in fact somehow or other eternally completed. This would account for the a priori reality of the real numbers that cannot be finitely computed.

      In response to your extended mathematical critique of my analogy, I would say only this: well, sure –but my gosh, it was an *analogy,* dude. Not an equivalence. I was speaking *analogically.* Got it?

      Having said that, I’ll respond to some of your statements, because they raise interesting questions.

      Infinity works lots better than 5 as the archetype of quantity, and thus as the basis or substrate of the number line.

      It most certainly does not. I can’t imagine why anyone would think that. For instance, ordinary quantities can be compared – infinity can’t be. Ordinary quantities can be added to each other to form new ones – infinity can’t be.

      Yes. Exactly! That was one of the main points of my analogy. Infinity – and omniscience, omnipotence, and so forth – are *radically incommensurate* with anything less. Infinity and 5 are alike in that both are quantities, but apart from that they are radically different sorts of things, and it is a category error to treat 5 as if it were in every way the same sort of thing as infinity. Likewise, although omniscience and our partial science – our “partiscience” – are alike in that both are sorts of knowledge, they are radically incommensurate, and it is a category error to treat omniscience under the terms of partiscience. To say then that omniscience is an incoherent notion because it doesn’t involve change the way that partiscience does is like saying that infinity is an incoherent notion because you can’t add infinities the way that you can add lesser quantities.

      Of course it depends what you mean by “archetype,” a non-mathematical term which could mean anything I suppose.

      I grant that “archetype” is a bit fuzzy. It might have been better if I had written “proper formal ultimate” than “archetype.” Infinity is the proper formal ultimate of quantity, thus of all finite quantities such as 5. Whereas 5 is obviously not the proper ultimate of any formal dimension other than quintuplicity.

      “5 participates *IN* infinity” makes no semantic sense but is at least grammatical, “5 participates infinity” isn’t even that.

      I grant that it’s an obscure and archaic usage. But it’s both grammatical and syntactic. Consider the statement, “Joe took Communion.” Likewise, and as that notion is not uncommonly expressed, “Joe partook Communion.” “Partake” is “participate.” So we could equivalently – and with equal correctness and precision – say that “Joe participated Communion.”

      To participate a thing is to take some aspect of its form as one’s own.

      And that set [of real numbers between 0 and 5] happens to have the same size (cardinality) as the entire real number line, so your talk of “superiority” above makes even less sense.

      Likewise the cardinality of the set of real numbers between 0 and 6 is the same as the cardinality of the set of real numbers between 0 and 5. Nevertheless, 5 < 6 < ∞.

      OK, I give up. You aren’t speaking mathematics, but some perverted parody of it. If it makes sense to you, feel free to think that way, but I can’t do it.

      Dude, again: it was an *analogy.* Relax.

      When I talk about “minds,” I talk about real-world things. My mind, your mind, perhaps animal minds or future artificial software minds if we want to stretch the category. Omniscient minds are not found in the real world, and there is nothing interesting to say about them.

      Well, in saying this, I am repeating myself, but: you are begging the question. If your definition of mind rules out anything other than the sorts of mind you are accustomed to think about, you’ll remain stuck in your accustomed way of thinking. You won’t be able to think about omniscient minds at all. And the result will be, that you will never have anything constructive to say about them. You’ll be able to say only, in effect: “I don’t understand.”

      If you are going to rule out omniscience cogently, you first have to take it on board, and consider it honestly, *so as* to understand what it is that you are ruling out, and *so that* you can then proceed to rule it out cogently. If your definition of mind prevents your doing any of that stuff, why then you’ve ruled yourself out of the conversation ab initio, and have designated yourself an utterly ignorant and therefore completely useless interlocutor. You’ve rendered yourself irrelevant.

      Not that I think you are any such thing, really. I take you to be far more intellectually honest and careful than that.

      [Physicists can] put themselves in the position of God … They have a theoretical omniscience, in the sense that their equations can describe any point in spacetime.

      Yes. Like I said, omniscience is not commensurable to us, but nor is it altogether inaccessible to us.

      Real observations involve the interaction of two temporal systems (the one under observation, and that of the observer, who has to change in some way in response to what is happening).

      Here again, your definition of observation as only and not possibly more than the sort of thing that we do when we observe prevents you from considering that the “omniscient” perspective that physicists take – that they participate – when they consider the cosmos as a whole (and, I might add, when they presuppose that physical law is pervasive throughout our cosmos, so that the physical law thereof is indeed lawful rather than random and adventitious, and so that our cosmos is therefore indeed a cosmos, an ordered whole, rather than a mere chaos (so that physics is possible in the first place)) might be concretely real for some sort of being that is quite different from us, and far greater. And that prevents you from even considering whether there might be such a thing as observation that is not temporally constrained. And that prevents you from thinking intelligently about the question, and so from responding aptly to what I have had to say on the subject.

      An observation that was not temporally constrained by our cosmos would be outside the causal nexus thereof. It would *necessarily* comprehend all the events of our cosmogony, regardless of their temporal address, not seriatim, as we who are within our cosmos must apprehend them, but all at once, and in one fell swoop: simultaneously.

      That such an intelligence saw all those events at once would not at all entail that he was himself lifeless.

      For one thing, he might even be such a temporal creature as we, within his own causal nexus, his own world, that enclosed, conditioned, and subvened our own; in which case, all our adventures might be to him as the adventures of the characters in a novel or movie of our world are to us.

      For another, even our own intramundane and temporally constrained experience involves at every moment simultaneous apprehensions of events that have different temporal addresses within our causal nexus. That they arrive to us all at once, and are integrated in one coherent experience – as, e.g., an experience of the starry sky above the back lawn as we savor a sip of whisky and hear the crickets in the bushes and the distant hooting of the owl – does not at all vitiate their character as temporally disparate events. The events in the distant stars and in the nearby owl and in our yet more proximal taste buds are far from each other in time, but they are together in us, all at once.

      And this is so of every one of our mundane experiences, without exception. *All* of them involve inputs from events near in time, and from far away. But our phenomenal integrations of temporally disparate events deprive neither the events in themselves nor the phenomenal event in and by which their apprehensions are composed in us of their own internal vividness, or liveliness. The experience of the owl as he hoots is not at all troubled by the fact that we hear him at the same time that we see stellar events that occurred millions of years before. That we hear him hooting nearby, and can tell that his hoot is recent compared to the music of the stars we apprehend contemporaneously, does not at all ruin the temporal disparity between the stars and the owl. And that we apprehend these two temporally disparate events simultaneously does not sap the liveliness of our apprehension.

      Temporal disparity does not at all vitiate phenomenal integrity; phenomenal integrity does not at all vitiate temporal disparity.

      By a straightforward extension, then, a single omniscient observation of all events of our cosmos wheresoever and whensoever, which integrates them in a single infinite and eternal phenomenal experience, is not thereby obviously less lively than our own phenomenal integrations of different events of different times in single coherent phenomenal experiences.

      Indeed – judging only by the observed intensity and liveliness of my own experiences of the distant, the sublime and the superb juxtaposed intimately and wonderfully with the humble, the humdrum, the merely drab and normal – on the contrary. Nothing so ennobles a day of stupefying ennui as the translucent smile of a toddler. A sunbeam at evening can redeem, and indeed illumine, a devastating defeat.

      After all, and when push comes to shove – try to think of it this way – does the fact that you experience the stars and the lawn and the owl and your taste buds all at once lead you to think that nothing changes, or that it is dull, or that you yourself are dull and lifeless?

      An eternal being omnisciently knowing every aspect of the universe couldn’t perform any sort of actual observation. It just knows by virtue of – what?

      Look at what you’ve just said here. To know *just is* to observe; to apprehend. So, you’ve said that an eternal being omnisciently knowing every aspect of the universe couldn’t omnisciently know any aspect of the universe. You’ve contradicted yourself.

      That’s not a good way to start. It’s a good way to prevent starting.

      We ourselves know anything at all by virtue of … what? Answer me that, and I’ll take up your challenge. Answer me that, and – I have demonstrated this to myself, but am short of time on demonstrating it to you – you shall have shown yourself how I shall answer that challenge.

      If such a being was possible, it would have static, unchanging knowledge of a static, unchanging world. Such a conception of God is incoherent, and even worse, deathly dull.

      Again, you are trying to comprehend infinite awareness within the terms of your own understanding and awareness – such as it is, or can for any of us be: the finite, puny sort. It’s like trying to comprehend infinity by treating it as if it were exactly the same sort of thing as 5. It can’t be done.

      • It seems to me prima facie that algorithms that cannot in principle ever finish mapping domains to codomains, by any finite series of steps, *absolutely cannot* be said to relate sets

        That՚s not what you said (or if you said it, you said it confusingly imprecise language).

        There is perfectly good mathematical language for talking about functions and computability. If you don՚t use it, and use it properly, then nobody will understand what you are talking about.

        hat was one of the main points of my analogy. Infinity – and omniscience, omnipotence, and so forth – are *radically incommensurate* with anything less. Infinity and 5 are alike in that both are quantities, but apart from that they are radically different sorts of things, and it is a category error to treat 5 as if it were in every way the same sort of thing as infinity

        Then how can infinity be an archetype for 5?

        It might have been better if I had written “proper formal ultimate” than “archetype.”

        I don՚t see how infinity is a “proper formal ultimate” for 5 either. Not that I speak the language of formal cause very well, but I thought the idea is that there is some formal object in Platonic space that real objects approximate. So 5 and infinity are both formal objects, in contrast with say a set of 5 apples that instantiates the “archetype” of 5 in the real world.

        likewise, although omniscience and our partial science – our “partiscience” – are alike in that both are sorts of knowledge, they are radically incommensurate, and it is a category error to treat omniscience under the terms of partiscience.

        I՚m not. I՚m in agreement with you, that they are radically different kinds of things. Where we differ is that I think is a confusing category error to call them both “minds”, whereas you think its just fine to call radically incommensurate things by the same name.

        To say then that omniscience is an incoherent notion because it doesn’t involve change the way that partiscience does is like saying that infinity is an incoherent notion because you can’t add infinities the way that you can add lesser quantities.

        As you said, finite and infinite numbers are radically incommensurable kinds of entities. If I grant that when you say “omnisicent mind” you mean something radically incommensurable with the actually existing finite minds we know of, well, then I also will drop the charge of incoherence. It՚s nothing like what I would call a mind, but it might be a coherent concept nonetheless. Again, I don՚t choose to call something like that a mind, but you may feel free to.

        you are begging the question. If your definition of mind rules out anything other than the sorts of mind you are accustomed to think about, you’ll remain stuck in your accustomed way of thinking. You won’t be able to think about omniscient minds at all.

        I՚m cool with that, honestly.

        …And that prevents you from even considering whether there might be such a thing as observation that is not temporally constrained.

        No, I am considering the idea, and discarding it as incoherent.

        That such an intelligence saw all those events at once would not at all entail that he was himself lifeless.

        Such an intelligence doesn՚t change, by definition, correct? That sounds pretty lifeless to me.

        For one thing, he might even be such a temporal creature as we, within his own causal nexus, his own world, that enclosed, conditioned, and subvened our own;

        Well if God lives in some kind of meta-time that՚s a different story. That is an interesting idea which I haven՚t seen discussed very much. There are lots of popularizations of extra spatial dimensions (eg the Flatland/Sphereland books) but it՚s harder to imagine an extra dimension which is both temporal in nature and orthogonal to normal time.

        That also implies that God might have a meta-God outside meta-time. Also interesting, but not very orthodox.

      • For another, even our own intramundane and temporally constrained experience involves at every moment simultaneous apprehensions of events that have different temporal addresses within our causal nexus. That they arrive to us all at once, …. does not at all vitiate their character as temporally disparate events. The events in the distant stars and in the nearby owl and in our yet more proximal taste buds are far from each other in time, but they are together in us,

        all at once….The experience of the owl as he hoots is not at all troubled by the fact that we hear him at the same time that we see stellar events that occurred millions of years before.

        So?

        By a straightforward extension, then, a single omniscient observation of all events of our cosmos….

        We just got through agreeing that infinity is radically different from finitude, so such an extension would be anything but straightforward.

        After all, and when push comes to shove – try to think of it this way – does the fact that you experience the stars and the lawn and the owl and your taste buds all at once lead you to think that nothing changes…

        I don՚t experience them “all at once”. I don՚t know about you, but when I experience things they unfold in time, and even if they occur simultaneously my attention can only focus on one thing at a time so in some sense I still experience them serially. In any case nothing about my experience leads me to think “nothing changes”, of course.

        An eternal being omnisciently knowing every aspect of the universe couldn’t perform any sort of actual observation. It just knows by virtue of – what?

        Look at what you’ve just said here. To know *just is* to observe; to apprehend. So, you’ve said that an eternal being omnisciently knowing every aspect of the universe couldn’t omnisciently know any aspect of the universe. You’ve contradicted yourself.

        No I have not. I don՚t believe in the reality or conceptual coherence of “an eternal being omnisciently knowing every aspect of the universe”. And part of the reason for that is that it is so easy to derive contradictions from assuming such a thing is real. That is a major method be which mathematicians prove things – assume the opposite and prove a contrdiction from that assumption. So if I “contradict myself” please consider it the last step in a proof.

        We ourselves know anything at all by virtue of … what?

        By virtue of being material biological beings structured so as to internalize and utilize discovered regularities about the world.

        Although science is beginning to understand how this works, many of the details of how we know things remain mysterious. However – it isn՚t mysterious about *what it is* to know something. We understand what it means to say a man knows the meaning of “tenebrous” or which is the fastest route to the 7-11, in that he can produce an answer to the question, or not. But unless a photograph can be said to know a landscape, an omniscient but static image of the entire universe doesn՚t know anything, because it can՚t do anything. It՚s more like a photograph than a mind – a photograph of infinite resolution and unbounded dimension still doesn՚t really know anything, because photographs are not structured as minds are.

      • It seems to me prima facie that algorithms that cannot in principle ever finish mapping domains to codomains, by any finite series of steps, *absolutely cannot* be said to relate sets.

        That’s not what you said (or if you said it, you said it [in] confusingly imprecise language).

        It’s what I’ve been trying to say, as clearly as I know how.

        I note that you still have not addressed my substantive question:

        There is a difficulty in categorizing an uncomputable function as a relation of sets when it can’t relate sets on account of the fact that it can’t compute them. There must be a way around this difficulty.

        Instead, you’ve criticized my usage of mathematical terms. Which is fair enough. But it is not the point at issue. I’m genuinely curious to hear what you have to say on the topic. If I’m missing something, it would be great to find out what it is.

        [If infinity and 5 are radically incommensurate, then] how can infinity be an archetype for 5?

        Pretty good question. Infinity and 5 are incommensurate in some ways, but commensurate in others. As I have repeated several times, they are alike – and commensurable – in that both are quantities, and that they are both perfectly definite; but they are totally unlike – and, so, incommensurable – in most other respects. Wherever they are incommensurable, they are radically incommensurable, so that it is impossible to understand infinity by recourse to our understanding of 5.

        It is hard to see how a thing could be an archetype of a set of other things if it was in every respect commensurable with the members of that set; for, in that case, it would itself be a member of the set. It would, that is to say, be a type of some other thing that was the true archetype of the set.

        Infinity is not a member of the set of numbers. It is not a number. Likewise, God is not a member of the set of created things, and omniscience is not a member of the set of partiscient minds.

        I don’t see how infinity is a “proper formal ultimate” for 5 either. Not that I speak the language of formal cause very well, but I thought the idea is that there is some formal object in Platonic space that real objects approximate. So 5 and infinity are both formal objects …

        Another doggone good question. Infinity is not a number, but the numbers and infinity are all members of the set of quantities. Infinity is the outer limit of the set of quantities.

        The fact that infinity is a member of the set of quantities raises a question: what is the archetype of that set? What is the archetype of quantity? That question prompts another: what is the archetype of the set of archetypes? What, i.e., is the Form of all forms? What is the archetype of all formal limits, that is not itself limited (and is not therefore a member of the set of formal limits, of which it is the archetype)? The mind very quickly blows, and one realizes that the regress of questions implicit in these questions is infinite. Their infinite regress can terminate only upon limitlessness. All the religions have a name for that limitlessness. The Cloud of Unknowing, the Tao that cannot be named, ain sof, the Suprapersonal Godhead, nirvana, and so forth.

        Spooky stuff. To it, we simply cannot approximate by any formal language – for rather obvious reasons. Thus the wisdom of the ages: Let all mortal flesh keep silence, and stand in fear and trembling; and lift itself above all earthly thought.

        Where we differ is that I think is a confusing category error to call [omniscience and partiscience] both “minds,” whereas you think it’s just fine to call radically incommensurate things by the same name.

        Like infinity and 5, omniscience and partiscience are radically incommensurate in some respects, and in others they are the same sorts of things, and therefore commensurable.

        It’s getting a little tiresome to repeat this notion again and again. I’m not sure why you are having so much difficulty taking it on board.

        It is indeed a confusing category error to treat omniscience and partiscience as exactly alike, and then to try to understand omniscience as if it worked the same as partiscience. That would be like trying to understand infinity as if it were a number. It is not a category error to treat partiscience and omniscience as different types of the same sort of thing: mind.

        If I grant that when you say “omniscient mind” you mean something radically incommensurable with the actually existing finite minds we know of, well, then I also will drop the charge of incoherence. It’s nothing like what I would call a mind, but it might be a coherent concept nonetheless.

        Yet again: omniscience is radically incommensurable with partiscience in some respects, and commensurable with it in others. Omniscience and partiscience are both the same sorts of things in that both involve experience, knowledge, action, will, intension, intent, and so forth. They differ radically in other ways. So radical are their differences, that it is incoherent to try to understand omniscience as if it were a type of partiscience. It is a fortiori incoherent to suggest that omniscience is incoherent because it is not exactly like partiscience. That would be like suggesting that infinity is incoherent because it is not a number.

        If your definition of mind rules out anything other than the sorts of mind you are accustomed to think about, you’ll remain stuck in your accustomed way of thinking. You won’t be able to think about omniscient minds at all.

        I’m cool with that, honestly.

        … And that prevents you from even considering whether there might be such a thing as observation that is not temporally constrained.
        No, I am considering the idea, and discarding it as incoherent.

        No, you are not considering the idea. Your categoreal scheme ruled it out before you had a chance to consider it. Omniscience is incoherent only on your presupposition that mind is necessarily partiscient. If you begin your investigation by defining mind as partiscient, you thereby exclude omniscience from your investigation ab initio. The result is that you come up with incoherent notions such as that omniscience would be total ignorance, simply because your own categoreal scheme cannot comprehend the possibility that there could be such a thing as omniscience.

        You might indeed be cool with that degree of fundamental incomprehension in yourself, and with your consequent intellectual incapacity. If so, there’s no way I can help you out of your self-imposed predicament. You can lead a horse to water …

        [Omniscience] doesn’t change, by definition, correct? That sounds pretty lifeless to me.

        God changelessly lives his one omniscient moment of life.

        … even our own intramundane and temporally constrained experience involves at every moment simultaneous apprehensions of events that have different temporal addresses within our causal nexus.

        So?

        So the mere fact that a moment of your apprehension integrates inputs from disparate events that occurred at different times does not render that moment of apprehension lifeless. That you know the light of the star and the hoot of the owl all at once does not make you less lively than if you had known only one of them. Nor by a straightforward extension would your moment of apprehension be less lively if it managed to know all things that happen, whensoever. We are not rendered less lively by increase of knowledge. On the contrary; the more we know, the more expansive and vivid is our life.

        … when I experience things they unfold in time, and even if they occur simultaneously my attention can only focus on one thing at a time …

        So you can’t taste and see at the same time? Can’t smell and hear all at once? If a woman is looking into your eyes and talking to you, you can’t see her as well as hear her? You are not able to juxtapose apprehensions at all? Juxtaposition is foreclosed to you? So, you can’t compare things? And, not being able to compare, you can’t measure? And, not being able to measure, you can’t gauge the distance between the cup and your lip? Meaning that you can’t get food or drink into your mouth?

        Weird. OK, bad example in your case, I guess.

        I don’t believe in the reality or conceptual coherence of “an eternal being omnisciently knowing every aspect of the universe.” And part of the reason for that is that it is so easy to derive contradictions from assuming such a thing is real.

        Easy such derivations may or may not be; but, you haven’t shown us that you have derived any such assumptions, because you haven’t postulated the reality of omniscience even for the sake of argument, so that you could then proceed to derivations. On the contrary: as you have repeatedly averred, you have ruled out the possibility of omniscience ab initio, on account of the fact that it can’t be partiscient, as we are. So you have chosen to fail to grapple with the concept of omniscience.

        It is admittedly easy to see that omniscience can’t be like partiscience in many important respects. Omniscience contravenes partiscience, and partiscience contravenes omniscience; this, in the sense that a partiscient being could not be omniscient, and an omniscient being could not be partiscient. But it is a different thing altogether to demonstrate that the notion of omniscience in and of itself entails contradictions. It would be cool if you would take a shot at some such demonstrations.

        We ourselves know anything at all by virtue of … what?

        By virtue of being material biological beings structured so as to internalize and utilize discovered regularities about the world.

        That is a dormitive virtue account – as in, “morphine puts us to sleep because of its dormitive virtue.” It sounds like an informative explanans, but really it just restates the explanandum with flowery language. What you have written is in effect that we ourselves can know by virtue of our being structured so as to know. We know because we can know.

        I don’t want to beat up on you about this, because epistemology is awfully hard, and big. I don’t really expect you to provide an account of knowledge. All I mean to do is make the point that it is no easier to understand human knowledge than it is to understand omniscient knowledge.

        … an omniscient but static image of the entire universe doesn’t know anything, because it can’t do anything.

        What reason do you have for thinking that an omniscient mind can’t do anything? Is it because you cannot but think of dynamic action as temporal?

        It seems to me that at bottom you are struggling because it is impossible to understand eternity under the terms of temporality, and the latter sort of terms are the only ones you’ve got. Believe me, I sympathize. I struggled the same way for 20 years.

        It helped when I dropped the presupposition that God is unchanging while creation changes around him. That presupposition doesn’t work, because it reduces God to a temporal thing, as if he were a rock on the bottom of a river that never changes while everything else does change, like the river flowing above the rock; and that’s a category error. To reduce God in that way is to stop talking about God, properly so called.

        It is horribly hard to see that this is so. It is even harder to conceive of how God could act dynamically at every instant of creation, all via one single act on his part. But the difficulty is as easy to overcome as the difficulty of understanding how God could apprehend events with different temporal addresses in one integral apprehension.

        I have already suggested that his integrated apprehension of disparate temporal events can be understood as formally similar to our own integrations in singular moments of experience of perceptual inputs from widely disparate events, such as the burning of a distant star and the hooting of a nearby owl and the even nearer taste of whiskey. Even your apprehension of a point such as the period of a sentence involves composition and integration of photonic effects arriving from slightly different events with slightly different addresses in time and space. How these different effects are composed in a unified experience is not an easy thing to understand. Indeed, it is, famously, one of the Hard Problems in philosophy of mind. But it is no harder to understand how God might do such a thing than it is to understand how we do it. And we certainly do it. That we do does not at all mean that we are static, or dead. On the contrary.

        By rather the same sort of similation, we may begin to understand God’s action at disparate temporal loci by considering our own ability to act in multifarious ways at the same time. We can walk and think and breathe and talk and chew gum and aim our eyeballs and hold an umbrella, all at the same time. The outputs of a moment of our life are as disparate as their inputs. That this is so does not mean that any such moment of our life is static, or dead. On the contrary.

      • But it is a different thing altogether to demonstrate that the notion of omniscience in and of itself entails contradictions. It would be cool if you would take a shot at some such demonstrations.

        I think I have, you just don՚t accept them. It՚s not worth going round in circles about it.

        By virtue of being material biological beings structured so as to internalize and utilize discovered regularities about the world.

        That is a dormitive virtue account – as in, “morphine puts us to sleep because of its dormitive virtue.” It sounds like an informative explanans, but really it just restates the explanandum with flowery language.

        No. A thermostat or a camera creates something sort-of-like models of the world by virtue of their actual structure – not because they contain some magic. So do human minds, although the structure and knowledge is vastly more complicated. But in neither case is it magic, it՚s due to the structure of the knower and its physical relationship to the world.

        What reason do you have for thinking that an omniscient mind can’t do anything? Is it because you cannot but think of dynamic action as temporal?

        Yes, I can՚t think of dynamic action as anything but temporal, because *that is part of its definition*.

        All the religions have a name for that limitlessness. The Cloud of Unknowing, the Tao that cannot be named, ain sof, the Suprapersonal Godhead, nirvana, and so forth.

        Yes. And actually I have no problem with the Taoist take on that kind of limitless, because it is modest and agnostic:

        Imagine a nebulous thing
        here before Heaven and Earth
        silent and elusive
        it dwells apart and never varies
        it travels everywhere and isn՚t harmed
        it could be the mother of us all
        not knowing its name
        I call it the Tao
        forced to name it
        I name it Great

        (Lao-tse, 25, Red Pine translation)

        The Tao is beyond all binaries and beyond knowledge, and is not the same as an omniscient deity, which is a lesser concept. To call the Tao a mind would diminish it. The Tao does not know, the Tao does not exist, it is beyond knowing and not knowing, beyond existence and non-existence. But frankly once we are in this realm, argument becomes somewhat ridiculous.

        Those who know it do not speak about it.
        Those who speak about it do not know it.

        Let me say that this kind of talk (Platonic or Taoistic) is a form of extreme abstraction, and so is mathematics, but they play by very different rules. Taoism is necessarily nebulous because it is an attempt to point to the indescribable; mathematics is always extremely precise and in some sense is *about* formal precision of concepts. In mathematics, you can *prove things* (given starting axioms), but you won’t see Taoists loudly proclaiming a new proof of the nature of the Tao or some such, because that would be ridiculous and inappropriate.

      • I have addressed [the difficulty in categorizing an uncomputable function as a relation of sets when it can’t relate sets on account of the fact that it can’t compute them], by pointing out that the existence of well-defined but uncomputable functions (e.g., the halting problem as a function of (program, input) pairs to the Booleans) was demonstrated by Turing long ago. You seem to not grasp the basic distinction between a function and a computation that computes that function, but they are different things, and not all functions are computable. In fact, there are only countably (ℵ₀) many possible computations, but the number of possible real valued functions is ℵ₂ I believe, which is vastly greater, thus a very small fraction of all functions are computable.

        With that, you’ve restated the difficulty at some length, without beginning to address it. You seem not to grasp the basic distinction between explanans and explanandum. An uncomputable function cannot completely specify the sets that it relates, precisely because it is uncomputable. And an incomplete specification fails to specify. In what sense then can we coherently call a function a relation of sets when one of the sets it relates cannot be completely specified? Surely there is such a sense. What is it?

        … in normal usage an archetype *is* a member of the set it describes. I.e., when we say “John is the archetype of an engineer,” we mean John is an engineer, and one who is extremely typical of the set of engineers.

        Sure, but that’s a colloquial way of speaking; like saying that Joan is the soul of discretion, when obviously she is not the Platonic essence or form of discretion, but rather an instance of that form, which embodies and expresses it to an exemplary degree. Strictly speaking, John is not the archetypal engineer, for if he was, then all engineers would have his black hair and blue eyes. He is rather an exemplary engineer.

        Infinity is not a member of the set of numbers. It is not a number.

        You just said it was a “quantity.” I think you are suffering from imprecise language.

        On the contrary. Not all quantities are numbers. All, some, none, many: none of these quantities are numbers. A gallon is a quantity, but it is not a number. A quorum, a majority, a bunch, a pinch, an inch; a distance, a volume, a length, an area, an acre: none are numbers.

        No, you are not considering the idea. Your categoreal scheme ruled it out before you had a chance to consider it. Omniscience is incoherent only on your presupposition that mind is necessarily partiscient.

        It’s true … Minds to me are entities that think, and thinking is an action that takes place in time and involves change, so an eternal unchanging mind is a contradiction in (my) terms.

        OK. So you agree that your categoreal scheme prevents you from thinking coherently about omniscience, and so prevents you from deriving any contradictions inherent in the notion of omniscience itself. Your contradictions all derive from your insistence on misconstruing omniscience as if it were exactly like partiscience, when of course the whole point of distinguishing between them in the first place is to notice that they are quite different in important ways.

        All that you’ve shown with your contradictions is that omniscience can’t be exactly like partiscience. But that was not ever in question. *Of course* omniscience is not exactly like partiscience!

        Kristor: Consider a tree of infinite tallness …

        A.morphous, you keep making the same mistake over and over:

        A.morphous: Consider an omniscient partiscience … or a finite infinity …

        Don’t you see what you are doing here? It’s an example of Gaunilo’s category error in responding to Saint Anselm, of proposing to refute his Ontological Argument by the counterexample of an island than which no greater can be conceived. *Of course* it’s impossible to conceive of an island than which no greater can be conceived, because by definition islands are finite. But when we are talking about God or his omniscience, we are talking about something that by definition is infinite. God and his Omniscience are not subject to Gaunilo’s counterexample of the island, or yours of the tree, because those counterexamples are examples of things that are not infinite; they don’t actually counter infinity. They are inapt.

        I suppose you don’t see this distinction, or you wouldn’t keep confusing it. I’m not sure I can explain it to you any better than I have.

        By virtue of being material biological beings structured so as to internalize and utilize discovered regularities about the world.

        That is a dormitive virtue account – as in, “morphine puts us to sleep because of its dormitive virtue.” It sounds like an informative explanans, but really it just restates the explanandum with flowery language.

        No. A thermostat or a camera creates something sort-of-like models of the world by virtue of their actual structure – not because they contain some magic. So do human minds, although the structure and knowledge is vastly more complicated. But in neither case is it magic, it’s due to the structure of the knower and its physical relationship to the world.

        I suppose you didn’t notice that your attempted refutation of my claim that you had provided a dormitive virtue account merely repeated that dormitive virtue account at somewhat greater length. All you’ve said is that the structure of the mind creates something sort of like models of the world by virtue of the structure of the mind. OK, fine. But: how? In virtue of what? Again, I don’t expect you to furnish an explanation of epistemology – that’s beyond the capacity of a comment on a blog. But it is important to recognize when one is falling prey to the dormitive virtue fallacy.

        It is good to state philosophical problems as carefully and as fully as we can, for doing so will more properly constrain and inform the search for their solutions. Indeed, such careful and expansive descriptions often help us see the right solution rather immediately. And it can be comforting to feel the increase in understanding of the problem provided by its more careful and complete statement. It is easy to mistake that increase in understanding of the problem for the understanding of its solution. They are nevertheless two different sorts of understanding.

        Yes, I can’t think of dynamic action as anything but temporal, because *that is part of its definition.*

        Really?

        Dynamic: by 1812, “pertaining to mechanical forces not in equilibrium, pertaining to force producing motion” … from Greek … dynasthai “to be able, to have power, be strong enough” …

        Where is time in that definition? Where is it written that force producing motion must be itself in motion? A stationary mass would generate gravitational force. It would be dynamic. Remember, too, that “motion” does not mean only movement in space and time. It means action or change or process of any sort: the movement from potential to actual.

        Let’s take a different definition:

        Dynamic: Marked by continuous and productive activity or change.

        God, properly so called and properly conceived, is eternal. He does not change, but his action on the created order is continuous and productive. It is dynamic.

        The Tao is beyond all binaries and beyond knowledge, and is not the same as an omniscient deity, which is a lesser concept. To call the Tao a mind would diminish it. The Tao does not know, the Tao does not exist, it is beyond knowing and not knowing, beyond existence and non-existence. But frankly once we are in this realm, argument becomes somewhat ridiculous.

        If you think that the Tao is beyond an omniscient God, then you are misconstruing “omniscient God.” You are falling prey to a category error. You keep taking God, who *just is* ain sof, to be something less than ain sof.

        It is interesting to me that while you reject the God of your own presuppositions – which you mistake for the God of Abraham, Isaac and Jacob, of Plato, Aristotle and Plotinus, of Anselm, Aquinas, and Augustine – you are open to the Tao as the Taoists describe it, when the Church has always described God in exactly the same way.

        There is a lot to say on that score. Allow me to recommend Christ the Eternal Tao, a fascinating book. One section translates the Tao te Ching into the terms of Christian metaphysics. The Greek for Tao is Lógos. Taoism and Christianity map to each other perfectly; so much so, that some theologians aver that, like the works of Plato, Aristotle and Philo, the Tao te Ching is a protoevangelion.

        It seems to me that if you want to understand the idea of God so as to critique it, your first step ought to be to question your presuppositions about God.

      • a.m: in normal usage an archetype *is* a member of the set it describes.
        k: Sure, but that’s a colloquial way of speaking

        Actually no, it՚s one of the standard accepts meanings, as is its use your way to mean “Platonic form”. I was unaware of the latter usage.

        Not all quantities are numbers.

        All, some, none, many: none of these quantities are numbers.

        Those are quantifiers, not quantities.

        A gallon is a quantity, but it is not a number.

        True! It is a unit of measure, which is how numbers are mapped to the physical world.

        A quorum, a majority, a bunch, a pinch: none are numbers.

        The first two aren՚t quantities, the latter are indefinite measures, that is, they are
        a probabalistic distribution which is a generalization of the idea of number.

        I have no idea what the point of this is.

        Don’t you see what you are doing here? It’s an example of Gaunilo’s category error in responding to Saint Anselm, …

        I՚m ignorant of such classics so doomed to reinvent them.

        *Of course* it’s impossible to conceive of an island than which no greater can be conceived, because by definition islands are finite. But when we are talking about God or his omniscience, we are talking about something that by definition is infinite.

        I think minds, trees, and islands are all things of the natural world, and so an infinite mind makes exactly as much sense as an infinite tree. You seem to take it as axiomatic that they are different – that while an infinite tree is ridiculous, an infinite mind is just how things obviously are, and I am too stubborn to admit it or something.

        I suppose you didn’t notice that your refutation of my claim that you had provided a dormitive virtue account merely repeated that dormitive virtue account at somewhat greater length.

        It didn՚t. Now you seem to be being deliberately obtuse.

        Dynamic: by 1812, “pertaining to mechanical forces not in equilibrium, pertaining to force producing motion” … from Greek … dynasthai “to be able, to have power, be strong enough” …Where is time in that definition?

        Hiding right there inside “motion”, for one thing. Here՚s another random dictionary definition:

        1. (of a process or system) characterized by constant change, activity, or progress. “a dynamic economy”
        PHYSICS
        relating to forces producing motion.
        LINGUISTICS
        (of a verb) expressing an action, activity, event, or process.

        I can՚t believe we are arguing over the definition of such a basic term. What՚s the point?

        Remember, too, that “motion” does not mean only movement in space and time. It means action or change or process of any sort: the movement from potential to actual.

        Er, no it doesn՚t, except perhaps metaphorically. Also it seems irrelevant. The point was that a mind that can՚t change cannot learn, cannot think, and cannot act, since those are all inherently temporal, and so if it can be said to know things, let alone everything, it can՚t actually do anything with that knowledge so it is not knowledge in any meaningful sense.

        God, properly so called and properly conceived, is eternal. He does not change, but his action on the created order is continuous and productive. It is dynamic.

        Yeah OK. I՚m starting to get an image of your God as something not very much like a mind, but more like a mass exerting an invisible force on creation, like a star exerts forces on everything in its vicinity. Or more like the abstract idea of gravity. That sort of provides a model for a force that acts without itself changing (although of course gravity is a symmetric force and the star is acted upon as well as acting). Or like an attractor in a dynamic system, which is sort of like aristotelian formal or final cause.

        That is an attractive idea (haha), but I՚m afraid I can՚t imagine something like that that is also mindlike. Perhaps its a failure of my imagination.

        If you think that the Tao is beyond an omniscient God, then you are misconstruing “omniscient God.” You are falling prey to a category error. You keep taking God, who *just is* ain sof, to be something less than ain sof.

        Look, there isn՚t much point arguing about it. Here՚s how I see it: Ein sof and Tao indicates a particular set of ideas to me, basically they are near-synonyms. “God” mixes that meaning with human-like characteristics, such as love, mercy, knowing, and judging, all of which are actions that imply change, and so are inaccessible to a changeless being. Thus the words have different meanings, and whether they actually refer to the same thing behind their separate senses, we will have to agree to disagree.

        More fundamentally, Tao and Ein Sof are essentially negative concepts, they indicate something about which we can know nothing becasuse they are beyoind knowledge. I՚ve already quoted Lao-tse, here is David ben Judah Hehasid:

        Ein Sof is a place to which forgetting and oblivion pertain. Why? Because concerning all the sefirot, one can search out their reality from the depth of supernal wisdom. From there it is possible to understand one thing from another. However, concerning Ein Sof, there is no aspect anywhere to search or probe; nothing can be known of it, for it is hidden and concealed in the mystery of absolute nothingness.

        These are apophatic or mystical images of the divine – something about which all we can know is that is unknowable. You are trying to sell me on something else: that god *knows things*, and furthermore that we humans can know with certainty that god knows these things. That seems very different from the mystical point of view. It strikes me as presumptuous and somewhat crude in comparison. But I don՚t see any point arguing about it, since everyone has their own metaphysical esthetics.

        And thanks for the pointer to Christ the Eternal Tao, that is interesting. But not very surprising to me, since all forms of mysticism tend to converge. Thus I don՚t have much of a problem with mystical Christianity; it՚s the non-mystical parts that bug me. But maybe we can leave that for another time.

      • I can’t believe we are arguing over the definition of such a basic term [as “dynamic.”] What’s the point?

        Remember, too, that “motion” does not mean only movement in space and time. It means action or change or process of any sort: the movement from potential to actual.

        Er, no it doesn’t, except perhaps metaphorically.

        A.morphous, it begins to seem to me that much of the difficulty you are having with the notions I am expressing is due to the fact that you are (naturally enough, given your candid profession of your ignorance of the classical tradition in philosophy) interpreting terms – and reality – under the modern, nominalist, post-Cartesian metaphysic that has dispensed with the ancient formal and final sorts of causation, and has tried (and failed) to reduce all causation to the efficient and material sorts; and that has furthermore tried (and failed) to dispense with the Platonic Forms, and with universals. Whereas I am not. Thus when I use terms such as “archetype,” “action,” or “motion,” I mean to refer to their broader more precise classical meanings – which include their modern meanings as special cases or analogues.

        The point of debating the meanings of such terms is that by doing so we can make our ideas more clear and precise. That clarity and precision will help us avoid errors of thought: such as the hilarious error of treating John as *literally* the archetypal engineer, of whom all other engineers are instances.

        When it comes to such terms as “dynamic,” “action,” and “motion,” correct understanding of their meanings will likewise help us avoid errors. One such error is to think of time as basic, so that all things must be temporal; when, of course, time *cannot* be basic, since every temporal event derives from other events; so that, the entire temporal extent being derivative, time *cannot* be a First Thing.

        If we think of dynamism, action, and motion as essentially temporal, then, we cannot see how the temporal extent came to be, or rather comes to be. We cannot see how it might derive, or from what it might derive. The only way we can see how it might derive, and from what, is to understand dynamism, action and motion more broadly, so that temporal dynamism, action and motion are then seen as special cases of dynamism, action and motion understood more comprehensively and thus more properly.

        The only way to understand time is as a derivate of eternity; and the only way to understand the derivation of time from eternity is to understand dynamism, action and motion as basically atemporal, with their temporal instances derivate from and supervenient upon their eternal instance.

        Yeah OK. I’m starting to get an image of your God as something not very much like a mind, but more like a mass exerting an invisible force on creation, like a star exerts forces on everything in its vicinity. Or more like the abstract idea of gravity. That sort of provides a model for a force that acts without itself changing (although of course gravity is a symmetric force and the star is acted upon as well as acting). Or like an attractor in a dynamic system, which is sort of like Aristotelian formal or final cause.

        That is an attractive idea (ha ha), but I’m afraid I can’t imagine something like that that is also mindlike. Perhaps it’s a failure of my imagination.

        Here at last you seem to be understanding my meaning – and the meaning of all religious philosophers, of all faiths – in our talk of Omniscience. Here at last, then, you and I may perhaps begin to talk about the same thing.

        I grant that it is very difficult for us to see how an attractor in a dynamic system could be a mind. The best way I can suggest you begin to wrap your head around the notion is to recur to that same analogy of a star. The star is an attractor in a dynamic system, and so is the planet in its orbit. The planet acts upon the star; this action is *the very same action* in virtue of which the star acts on the planet. In the gravitational relation between the star and the planet, there are not two acts of attraction, but only one. It is not as though the star first attracts the planet, and then the planet responds by attracting the star.

        We can construe the gravitational relation between the two masses as an exchange of information, in virtue of which they inform each other – in virtue of which, in fact, they literally *deform* each other, so that they are both shaped a bit differently than they would be in the absence of each other’s gravitational fields. Gravitationally, the star would not be quite what it is if the planet were not there; and, a fortiori, vice versa.

        The star informs the planet, and vice versa. As each informed by the other, each *knows* the other. The star knows the planet, and the planet knows the star. But, again, this exchange of information is mediated, not by two acts – one of the star, and one of the planet – but by only one.

        The only thing that needs to be tweaked in order to improve the analogy star : planet :: God : creation is to stipulate that in the analogy, the star defines an absolute frame of gravitational reference. Thus the planet is in the analogy definitely orbiting the star, and not vice versa – this, despite the fact that the star knows all about the planet.

        The star in the analogy is, i.e., an unmoved mover. I’m sure you know that Aristotle characterized God with the same term.

        The star does not need to move in order to be informed by the planet in its orbit; the planet by contrast orbits in the first place only in virtue of the proximity of the star (in the absence of a star, the planet would not be a planet at all, but a mere mass in a void, not orbiting anything, but rather quite stationary and static).

        OK, so here is where it gets a bit spooky.

        As the star in the dynamical system of our analogy is an unmoved mover, so is it an unknown knower.

        As God is the Unmoved Mover, so is he the Unknown Knower.

        This is where we arrive at the apophatic mystical theology of Christianity, Judaism, and Taoism – indeed, of all religion.

        All right then: God cannot be moved; he cannot be known. Yet he moves; he moves *us;* and because we are by him moved, and in his gravitational field live and have our being, from his motions upon us we can know something about him, insofar as he is manifest in us his creatures.

        Thus the distinction between, on the one hand, apophatic theology, which boils down to an analysis of the basic and incorrigible epistemological incapacity of finite minds to infinity as it is in and to itself, and on the other hand cataphatic theology, which boils down to an analysis of what Ultimate Perfection would look like *as manifest* to lesser beings.

        It is in the latter sort of analysis that we find ourselves involved in metaphysics, and in analyzing empirical data from mystical experience and revelation. What does the configuration of the planet, and of the dynamical system in which it orbits, tell us about the Utterly Strange Attractor that lies at the center of the system, and that defines it qua system? That’s where we start using terms like love, knowledge, will, mercy, judgement, and so forth. These are terms that, as applied to God, derive from his effects upon us. Because he is the Unknown Knower, we can’t know what they mean to him.

  8. I note that you still have not addressed my substantive question: There is a difficulty in categorizing an uncomputable function as a relation of sets when it can’t relate sets on account of the fact that it can’t compute them. There must be a way around this difficulty.

    I have addressed it, by pointing out that the existence of well-defined but uncomputable functions (eg, the halting problem as a function of (program, input) pairs to the booleans) was demonstrated by Turing long ago. You seem to not grasp the basic distinction between a function and a computation that computes that function, but they are different things, and not all functions are computable. In fact, there are only countably (ℵ₀) many possible computations, but the number of possible real valued functions is ℵ₂ I believe, which is vastly greater, thus a very very small fraction of all functions are computatble.

    It is hard to see how a thing could be an archetype of a set of other things if it was in every respect commensurable with the members of that set; for, in that case, it would itself be a member of the set

    As I already pointed out, in normal usage an archetype *is* a member of the set it describes. Ie when we say “John is the archetype of an engineer” we mean John is an engineer, and one who is extremely typical of the set of engineers.

    Infinity is not a member of the set of numbers. It is not a number.

    You just said it was a “quantity”. I think you are suffering from imprecise language. There is really no such thing as “the set of numbers”. There is the set of reals, which does not include infinity although it is of infinite size, and the set of transfinite cardinals (ℵ₀…). which includes many different infinite numbers, and many other sets of numbers (integers, complex numbers, etc). “Number” is imprecise.

    No, you are not considering the idea. Your categoreal scheme ruled it out before you had a chance to consider it. Omniscience is incoherent only on your presupposition that mind is necessarily partiscient.

    It՚s true, and my categorical scheme also rules out round squares and colorless green ideas, indicating I have a sadly limited intellect. Minds to me are entities that think, and thinking is an action that takes place in time and involves change, so an eternal unchanging mind is a contradiction in (my) terms.

    Let՚s say instead of talking about minds (one of the chief attributes of humans and other animals) and instead were talking about tallness (one of the chief attributes of trees).

    Kristor: Consider a tree of infinite tallness…

    Me: OK, I՚ve considered it, and I find the notion kind of useless. Such a tree could not exist, and what՚s more, the reality of trees – how they grow, why they grow – presumes a host of mechanical limits. Redwoods are very tall and we admire them more than most trees because they are an extreme solution to a set of conditions. But an infinite tree is not real, and not a solution to a real-world problem. It՚s just something you made up. If it gives you pleasure to think about an infinite tree, knock yourself out, but it՚s not a concept that appeals to me. It՚s not interesting.

    Neither is the idea of an omniscient mind interesting, because it has nothing to do with and says nothing about real minds.

    • At any rate: I think I will have to bring this interesting discussion to a close, since my own time is quite finite and I feel we are going over the same ground repeatedly.

      One last point: Sometimes I feel like this reads as though I՚m pro-time and anti-eternity, while you take the opposite position. That՚s not really the case; what I am opposed to is *inappropriate mapping of temporal concepts onto the eternal*, because that tends to be dangerous and/or fraudulent. In other words: While it seems pointless to be opposed the eternal in and of itself, because by definition it is the matrix in which we exist, it is quite pointed to be opposed to various attempts to enlist the eternal on one side or another in human affairs.

      If eternity, which for humans can only be an abstraction, must have an enemy, it is the concreteness of the present, of the now, of the body, of the specific. All of which are irreducibly temporal. We are embedded in time in such a way that the present moment is distinct from all others, and all our knowledge, even the most abstract concepts, is tied to this embedding. Time is at the core of our Being, and the Now is the cure for being trapped in these arid abstractions.

      OK one more point: your identification of Tao with God is contraindicated by the text of the Tao te Ching itself:

      The Way is empty;
      Yet when you use it , you never need fill it again
      Like an abyss! It seems to be the ancestor the ten thousand things.

      Submerged! It seems perhaps to exist.
      We don՚t know whose child it is;
      It seems to have even preceded the Lord.

      • A.morphous, thanks for all the time and careful effort you have devoted to this exchange. I truly appreciate your thoughtful contribution. Our conversation has engendered some really cool insights for me; I’ve learned some good stuff about the implications of my own arguments. Viz., the notion of the Unknown Knower. That one will stick.

        I’ll respond to your last because – as usual – I find it fascinating and fun to do so. Please don’t feel obliged to rejoin – although I would welcome your rejoinder.

        … it’s your blog after all. That is, the rules of usage around here are based on older definitions of words …

        Yeah. To talk to each other usefully, it behooves us to understand and use the key terms of the conversation in the same way, or else we’ll just be talking past each other, or confusing each other.

        It’s not really the case that modern metaphysics dispenses with formal and final cause.

        Right. Like I said, modern metaphysics tried *and failed* to dispense with formal and final causation. Because they cannot get rid of formal and final causation altogether and still end up with a theory that can accommodate order and regularity, and thus enable science of any sort, modernists have no option but to smuggle final and formal causation back into their reductionist theories of causation, often without realizing they are doing so (the only option is chaos, which is the zero of order, and of science). Viz., a strange attractor in a configuration space is a final cause in a formal system of causes. The only reason it is “strange” is that the theorist has foreclosed to his conscious theoretical imagination any possibility that goodness and beauty – and, thus, a nisus inherent in actual events that inclines them toward the actualization of goodness and beauty, which imposes upon them a requirement of moral and aesthetic evaluation – might be implicit in the logical structure of formal systems (e.g.: logical validity is beautiful, and good). The attractor is strange because, under the reductionist metaphysic of the materialist, there is *absolutely no way to account for attractions to attractors.* The materialist has no way to understand why nature homeostatically seeks equilibria that he observes her everywhere homeostatically seeking.

        It’s true that [final and formal causes] are no longer considered fundamental, instead they are emergent properties of mechanical processes.

        The problem is that emergence is hand waving, like that of the magician on stage. The reality behind all the hand waving is that you can’t get something from nothing. You can’t emerge rabbits from hats that have no rabbits in them. To say that final and formal causes emerge from the operation of material and efficient causes is to say only that the final and formal causes must have been there all along, implicit ab initio in those material and efficient causes.

        If formal and final causes are at bottom simply not at all real – as strict reductionist materialism supposes is the case – then there is no question of their somehow or other emerging from what is in fact real. On the contrary. If they are not real, then *they are just not real,* and that’s all there is to it. They don’t really emerge at all. That’s the end of the story. What is not real cannot emerge from what is real. Because why? Because it *just isn’t real.* There is no rabbit anywhere.

        If there are no formal or final causes – if there is no formal or final aspect to things – then there are no such things as regularities, equilibria, constants, and so forth; or therefore science. But there are all those things. So there are formal and final causes.

        If final and formal causes are present in nature *at all,* then they must be present fundamentally, right along with material and efficient causes, and indeed integrally with them (for, as we have seen, they cannot have emerged from something utterly unlike themselves). And that’s just Aristotelian hylemorphism.

        [The classical definitions of metaphysical terms] are not “broader” or “more precise” (hard to be both of those at once!).

        Hard it is, indeed. That’s why it was so hard for Newton and Einstein to achieve what they did. Newtonian physics subsumes terrestrial and celestial phenomena under a single, much broader dynamical system of laws, and in the process greatly increases the theoretical precision of mechanics. Einsteinian physics subsumes the absolute frames of inertial reference of Galileo and Newton under a broader and more precise treatment of relative frames of inertial reference.

        [The classical definitions] are merely different, and more or less obsolete.

        They are fallen lately into desuetude, not because they are wrong, or therefore obsolete – what is true, or truly apt, cannot obsolesce – but rather, *because they are misunderstood* by those who have not troubled to understand them. I have never read an atheist who truly understood the theistic arguments. *All* the atheists I have read demolish straw men, that theists have never proposed. It’s really rather pathetic.

        But as a hip postmodern kind of guy, I have to be more humble and say that we just have two very *different* theories of motion and we can’t really judge one better than the other – they can and should co-exist, serving their different purposes.

        This actually makes a lot of sense. The modern usage of “motion” denotes something quite different – namely, change of location in some physical frame of extensive reference – than the ancient usage of the term, which denoted the quite general transition from potential to actual, *including* (as a special case) such transitions as changes of location within physical frames of extensive reference. The modern usage of the term is physical; the ancient usage of the term is metaphysical.

        “Motion” means something different to metaphysicians than it does to physicists. We are discussing metaphysical propositions, so I have been using the term the way metaphysicians do. And, NB: metaphysics is broader than physics *by definition.* As more abstract than physics, it is also capable of greater precision, in exactly the same way that logic and math are capable of greater precision than natural history. Physics, after all, is inherently tied to and concerned with discerning the lógos of this cosmos; whereas metaphysics is concerned to discern what must be true of lógoi in general, of any cosmos – and, therefore, of ours.

        If therefore you critique my usage of “motion” because it does not agree with the way that physicists use the term, well then, you have fallen prey to yet another category error; as if you were to critique economic projection because it is not exactly like psychological projection.

        NB however that I don’t have a different theory of physical motion than Einstein. The motions of Einsteinian physics are all changes from a potential state of affairs to an actual state of affairs, and are therefore easily accommodated under the categories of classical metaphysics of motion (albeit not, to be sure, of Aristotelian *physics* of merely physical motion).

        On the contrary, you can’t even have a First Thing without time, because the concept of “First Thing” is also temporal at its essence. As is “derivative.”

        A First Thing is an ultimate axiom, without which ratiocination cannot occur. Axioms in logical calculi are not prior in time to their derivate theorems, but rather are prior logically. The theorems that derive from those axioms are logically implicit in them. It is not as though the theorems don’t exist until someone takes the time to derive them from the axioms. On the contrary: they can’t be derived by anybody unless they are implicit in the axioms to begin with – i.e., eternally.

        I’ll grant that you are probably using “First Thing” metaphorically. You don’t really mean “before everything else in time,” but something like “logically or metaphysically prior.”

        Exactly. Except it isn’t a metaphor.

        If you are deriving a from b you already in a temporal realm. In eternity, nothing is derived, because everything just is. No metaphysical ordering is possible, just as no action is possible.

        When you engage in an act of derivation, *you* are already in a temporal realm. But a and b in themselves are not, and nor are their logical or mathematical relations, such as that of the derivation of b from a, or that of the integration of b in a. The derivation of b from a could not proceed at all (whether temporally, or eternally) if it was not eternally true that b derives from a. Analogously, you can’t count from 1 to 10 unless 10 is out there to begin with. 10 is not generated by your counting.

        The only thing that needs to be tweaked in order to improve the analogy star : planet :: God : creation is to stipulate that in the analogy, the star defines an absolute frame of gravitational reference.

        That makes the entire analogy pointless. It becomes a disanalogy. There are no unmoved movers in physics. And the historical trend is to make that even more the case (e.g., the advancement of Einsteinian over Newtonian theory).

        *Of course* there are no unmoved movers in physics: physics *just is* the study of things in motion. This is *obvious.* So physics *simply cannot comprehend motionlessness.*

        But we are not talking physics, here, OK? We are talking *metaphysics.* To treat of God under the terms of physics is just obtuse. You might as well treat of him under the terms of dentistry.

        The only way to draw an analogy between a physical system and a metaphysical system so as to make the latter more comprehensible under the more familiar terms of the former is to map the former to the latter (messy and difficult though that might be), and not vice versa. I’m not asking you to treat a physical star as if it defined an absolute physical frame of inertial reference. That’s a silly notion. I’m asking you to imagine a *metaphysical* star, and think of it as defining an absolute *metaphysical* frame of reference.

        Honestly, a.morphous, it sometimes seems to me that your analogy engine is busted. It’s almost as if I were to say, “Think of an elephant,” and you were to reply, “Elephants can’t fit in our skulls, so it is impossible to think of them.” To which I might respond, “Oh, really? How did you come to think that of elephants, if you can’t think of them to begin with?”

        If God is responsible for all movement, then he’s basically the same thing as the laws of nature. The laws of nature may be worthy of awe or even worship, but they don’t have *agency.* The term “God” is just putting a human mask on something profoundly not-human.

        If the laws of nature do not, qua laws, have agency – which obviously they do not, I agree – then they cannot be said to be responsible for anything. Only agents can be responsible. So it must not be quite right to say that God is just the laws of nature on account of the fact that he is responsible for all movement. After all, if God is responsible for all movement, and so just is the laws of nature, why then the laws of nature, being responsible for all movement, do definitely have agency. They don’t have agency. So God, who does have agency, is not tantamount to the laws of nature.

        But then, I did not say that God is responsible for all movement. To say that would be to say that we have no responsibility for our own movements. God forbid! Obviously we do have responsibility for our own movements.

        I said rather, only, that God moves us.

        Perhaps you mean to suggest that the God I am describing sounds an awfully lot like the laws of nature, but if he is, it is odd to ascribe personal attributes to him, when after all the laws of nature are not personal.

        To that I would say simply that the God I am describing might sound to you awfully like the laws of nature, and nothing more, but he is in fact much more than a mere set of formal relations. Indeed, he is among other things the set of all coherent formal relations as exemplified in an eternal actual substantive being. Recall that under Aristotelian hylemorphism, the forms do not exist apart from their instantiation in concrete actualities. The basic idea is that ideas can’t have themselves: forms must be forms of some actual thing or other, or else they are … not forms of anything; which is to say, not forms, period full stop.

        Considered abstractly, and in themselves, the Aristotelian forms have the same sort of merely notional existence as his Prime Matter, that has as yet taken no form. There is no such actual thing as Prime Matter. All matter has taken some form or other; to be formed as something or other in particular is the only way to be material. Likewise, to be concresced actually as something or other in particular is the only way to be formal.

        If we can’t know what [love, etc.] mean to “him,” how do we know they mean anything at all, applied to him? That is basically my point. “Love” or “knowledge” may derive from God’s effect on us, but that’s only because you’ve identified God with all causality, so everything is thus derived, so you haven’t really said very much.

        I didn’t identify God with all causality. A.morphous, you simply must be more careful about this sort of thing. All I said was that God moves us, and that the way that God moves us can tell us something about him.

        The question of your first sentence though is apt: If we can’t know what [love, etc.] mean to him, how do we know they mean anything at all, applied to him? The answer is to be found in the Scholastic maxim, ex nihilo nihil fit: from nothing, nothing is made. A total lack of x cannot give rise to a single jot of x. You can’t transmit momentum that you have not got. The Conservation Laws and the laws of thermodynamics all express this principle; so do GIGO, and TANSTAAFL. It is the refutation of emergence: you can’t pull a rabbit out of a hat that has no rabbits in it.

        So, you can’t pull love out of a state of affairs in which there is no love. If you see some love around in the cosmos, it must come originally from something extracosmic that is aptly characterized as lovely. Ditto for goodness, agency, being, beauty, order, and so forth: none of them can come to pass from an utter absence of such things. Indeed, because the effect must be proportionate to the cause, the extracosmic being, love, etc., that cause the being, love, etc., of this cosmos must be at least commensurate thereto, if not greater. The cosmos can inherit its love, order, etc., only from something prior that is at least as lovely, orderly, and so forth as is the cosmos in toto.

        Whatever love means to the creator of all cosmic love, it must mean something at least as lovely as love is to us.

        Consider then that there is no reason to suspect that there is a limit on the number of cosmoi. Then the love and order and beauty manifest in the whole array of cosmoi must all derive originally from a source at least as lovely and orderly and beautiful as all of them put together: i.e., a source that is *infinitely* lovely, orderly, and beautiful.

        While it seems pointless to be opposed to the eternal in and of itself, because by definition it is the matrix in which we exist, it is quite pointed to be opposed to various attempts to enlist the eternal on one side or another in human affairs.

        The mathematical truths of game theory are eternal. All biology runs on game theory, including natural selection. The eternal Lógos is opposed to lethal mutations, whether of genes or memes. He destroys them, relentlessly (albeit, as sometimes it seems, altogether too slowly). Thus murder is proscribed, not just by happenstance, but by eternal logic. The instability of evil is the morality of the cosmos. Evil *just doesn’t work.* And this is written in the eternal logic of game theory. Evil is *illogical.* It is therefore insane. It tends, not to the end of the game, but to its dissolution.

        Consider contraception in all its forms, including abortion and infanticide, masturbation and homosexuality, the whole gamut. *Contraception forecloses both adaptation and maladaptation.* It frustrates *the entire procedure of mutation and natural selection.* It stops evolution; it stops life. No babies, then no opportunities either of human success or failure. Literally everything that is human, both good and bad, supervenes upon more babies. Stop babies, and you stop man as such.

        So with all evil.

        Eternity, then, even if only in its eternal game theoretical department, does indeed inform human affairs pervasively. And there is no reason to think that the information of human affairs by eternity is limited to its game theoretical department. On the contrary: if as you say eternity is by definition the matrix of our existence, then *every bit of human affairs is pervasively conditioned by eternity.* Viz., the spooky applicability of mathematics in general to the behavior of the cosmos.

        If eternity, which for humans can only be an abstraction, must have an enemy, it is the concreteness of the present, of the now, of the body, of the specific. All of which are irreducibly temporal. We are embedded in time in such a way that the present moment is distinct from all others, and all our knowledge, even the most abstract concepts, is tied to this embedding. Time is at the core of our Being, and the Now is the cure for being trapped in these arid abstractions.

        Why must eternity have an enemy? How is eternity at war with the concreteness of the present Now of the human moment? It is not. If as you say eternity is the matrix in which we exist, then our temporal existence and eternity cannot conflict. Their agreement must rather be perfect. What disagrees with the matrix of existence disagrees with its own existence.

        It seems to me that in this paragraph you have descended into the Romantic from the Analytic.

        If eternity is the matrix of our existence, then each Now is first a moment in eternity – which is itself a Now.

        OK one more point: your identification of Tao with God is contraindicated by the text of the Tao te Ching itself:

        The Way is empty; Yet when you use it, you never need fill it again. Like an abyss! It seems to be the ancestor the ten thousand things. … Submerged! It seems perhaps to exist. We don’t know whose child it is; It seems to have even preceded the Lord.

        On Christian metaphysics, the Suprapersonal Godhead precedes the three Persons of the Trinity – not in time or sequence, mind you, but in logic – and the Father likewise precedes his only begotten Son, who is the Lord. Then the Father and the Son likewise precede the Holy Spirit. But all four – the Godhead, the Father, the Son, and the Holy Spirit – are One. So all four are the Tao. The Son of the Tao is the Tao; the Father of the Son is the Tao. And the Holy Spirit, who is the Way of Heaven, is the Tao.

      • A.morphous, thanks for all the time and careful effort you have devoted to this exchange. I truly appreciate your thoughtful contribution.

        You are very welcome. For my part, I՚m enjoying the chance to argue and feel like I՚m getting something out of it, although I couldn՚t tell you exactly what.

        I՚ll say one thing for you: like the Infinite, you seem inexhaustible. I feel I am at a disadvantage, having only finite resources of time and patience.

        …modern metaphysics tried *and failed* to dispense with formal and final causation. Because they cannot get rid of formal and final causation altogether and still end up with a theory that can accommodate order and regularity, and thus enable science of any sort, modernists have no option but to smuggle final and formal causation back into their reductionist theories of causation, often without realizing they are doing so

        I՚m not sure what you think you accusing modernists of. If they prefer to talk about attractors rather than final causes, but mean the same thing…so what? How have they failed?

        The only reason it is “strange” is that the theorist has foreclosed to his conscious theoretical imagination any possibility that goodness and beauty – and, thus, a nisus inherent in actual events that inclines them toward the actualization of goodness and beauty, which imposes upon them requirement of moral and aesthetic evaluation – might be implicit in the logical structure of formal systems (e.g.: logical validity is beautiful, and good). The attractor is strange because, under the reductionist metaphysic of the materialist, there is *absolutely no way to account for attractions to attractors.*

        “Strange” has a particular technical meaning that has nothing to do with strangeness in the everyday sense. https://en.wikipedia.org/wiki/Attractor#Strange_attractor

        The problem is that emergence is hand waving, like that of the magician on stage. The reality behind all the hand waving is that you can’t get something from nothing.

        Why not? You say this as some kind of axiom, but where else would we get something from, if not nothing?

        If formal and final causes are at bottom simply not at all real – as strict reductionist materialism supposes is the case – then there is no question of their somehow or other emerging from what is in fact real. On the contrary. If they are not real, then *they are just not real,* and that’s all there is to it.

        “real” is one of the most confusing and poorly-defined words there is. I՚d advise against using it.

        You seem to be arguing that final causes must be metaphysically fundamental or they can՚t exist at all (are not real). I would disagree – this is why natural selection and cybernetics were such big deals, because they showed how purpose and design could have a perfectly explicable material or naturalistic basis.

        “Motion” means something different to metaphysicians than it does to physicists. We are discussing metaphysical propositions, so I have been using the term the way metaphysicians do.

        I don՚t really think that՚s the case. You can have a metaphysics of motion, but there is no special kind of motion peculiar to metaphysics (I may be wrong, it՚s not really my area of expertise). It sounds like what you mean is the Aristotilean theory of motion, with its associated metaphysics. That՚s fine, but the modernist theory of motion has its own associated metaphysics.

        So what you are really saying is that you are using Aristotelian terminology, while I am using modern scientific terminology, to describe the same thing, that is, motion or change. But I won՚t accept that that either is more or less metaphysical than the other.

        Honestly, a.morphous, it sometimes seems to me that your analogy engine is busted. It’s almost as if I were to say, “Think of an elephant,” and you were to reply, “Elephants can’t fit in our skulls, so it is impossible to think of them.” To which I might respond, “Oh, really? How did you come to think that of elephants, if you can’t think of them to begin with?”

        There is a difference between understanding an analogy and accepting it as valid.

        The question of your first sentence though is apt: If we can’t know what [love, etc.] mean to him, how do we know they mean anything at all, applied to him? The answer is to be found in the Scholastic maxim, ex nihilo nihil fit: from nothing, nothing is made. A total lack of x cannot give rise to a single jot of x. … You can’t transmit momentum that you have not got. The Conservation Laws and the laws of thermodynamics all express this principle; so do GIGO, and TANSTAAFL. It is the refutation of emergence: you can’t pull a rabbit out of a hat that has no rabbits in it.

        Except it isn՚t a refutation, it՚s just a bad opinion.

        The mathematical truths of game theory are eternal. All biology runs on game theory, including natural selection. The eternal Lógos is opposed to lethal mutations, whether of genes or memes. He destroys them, relentlessly (albeit, as sometimes it seems, altogether too slowly).

        Thus murder is proscribed, not just by happenstance, but by eternal logic. The instability of evil is the morality of the cosmos. Evil *just doesn’t work.* And this is written in the eternal logic of game theory. Evil is *illogical.* It is therefore insane. It tends, not to the end of the game, but to its dissolution.

        Oh good lord is that a bad argument. Game theory may be eternally true, and may accurately describe some aspects of biology. It is not the case that “all biology runs on game theory”. And things that don՚t work do not need to be proscribed. For this and many other reasons, using evolutionary arguments for moral laws is an extremely dubious move.

        The instability of evil is the morality of the cosmos. Evil *just doesn’t work.*

        Well OK then, we can all relax.

        I find this an odd posture for a rightwing activist to take. If goodness is going to inevitably triumph due to the eternal truths of game theory, then why bother struggling against evil or working for good?

        Consider contraception in all its forms, including abortion and infanticide, masturbation and homosexuality, the whole gamut. *Contraception forecloses both adaptation and maladaptation.* It frustrates *the entire procedure of mutation and natural selection.* It stops evolution; it stops life. No babies, then no opportunities either of human success or failure. Literally everything that is human, both good and bad, supervenes upon more babies. Stop babies, and you stop man as such.

        That is a completely idiotic argument. Sorry to be rude, but I can՚t otherwise indicate how stupid I feel this is. Stupid or dishonest, and I feel you are trying to be honest.

        Contraception and the rest “forecloses adaptation” only if it is *universal*. Given that it is not, it is merely one of many mechanisms by which biological systems control themselves via the radical negative feedback of death. And it is no more a departure from the Good than is cellular apoptosis, or the fact that a large fraction of human zygotes spontaneously abort, or that natural selection involves a great deal of death (that is what “selection” means).

        So yes, if you aborted *every* fetus that would be “stopping babies” and a bad thing, from an evolutionary perspective as well as a moral perspective. But aborting one fetus stops one baby, not all babies. That may not make it morally palatable to you, but it does not have the kind of existential consequences you are talking about.

        Your understanding of natural selection seems extremely backwards. The central lesson of natural selection is that creativity and purposefulness can arise from a random process of variation and a non-random process of selection. We do indeed get something from nothing this way, but it՚s not for free, it՚s an enormous cost in competition, struggle, death, and waste. And dead babies, so many dead babies from natural or human causes, but evolution only cares about the ones who survive and reproduce.

        Why must eternity have an enemy?

        I don՚t know that it must. However, you seem to be setting yourself up as a spokesperson for the eternal, and given my opposition to your point of view, that makes me an enemy of sorts. But I don՚t think of myself as an enemy of eternity itself, just to its misuse and misrepresentation.

      • I’ll say one thing for you: like the Infinite, you seem inexhaustible. I feel I am at a disadvantage, having only finite resources of time and patience.

        It helps that we are taking at least a few days to rest up between each of our comments. We’ve been at this discussion since January! I actually think that’s kind of neat.

        I’m not sure what you think you [are] accusing modernists of. If they prefer to talk about attractors rather than final causes, but mean the same thing … so what? How have they failed?

        I’m taking “modernist” to include “materialist.” Materialism reduces all effects to material and efficient factors, putatively eliminating the putatively unnecessary explanatory entities of formal and final factors from its account of reality. It’s a radical application of Ockham’s Razor. On strict materialism, then, the formal order and final regularity so evident and so pervasive in our experience can’t be explained or understood. The difficulty of course is that our experience is in fact pervasively characterized by order and regularity. They can’t be gainsayed, other than by gainsaying experience per se. The stricter sort of materialists resort to that very move, asserting that subjective experience is illusory, so that their assertions are illusory, and that there are no such things as illusions. The looser, hipper (and saner) materialists smuggle formality and finality back into their quondam materialism via final attractors in a formal configuration space that is nowhere to be found in physical reality, which is therefore immaterial (at least, with respect to the matter of our own cosmos), and that just happens to describe physical reality to an uncanny degree of verisimilitude, for no reason whatever, move along now, nothing to see here.

        It is in this sense that the materialist Ockhamian project of cutting formal and final causation out of the picture can be said to have failed. If attractors actually attract, materialism is false.

        “Strange” has a particular technical meaning that has nothing to do with strangeness in the everyday sense.

        It was an attempt at an artful figure of speech. I should just have said that, “The attractor is perplexing to the materialist because, under his reductionist metaphysic, there is *absolutely no way to account for attractions to attractors.* The materialist can understand *that* nature homeostatically seeks equilibria that he observes her everywhere homeostatically seeking, but he has no way to understand *why* she does this.”

        You say [that we can’t get something from nothing] as some kind of axiom, but where else would we get something from, if not nothing?

        From something. Duh. Go ahead: show me how you can pull a rabbit out of an empty hat. Show me an instance of that, anywhere. NB: virtual particles of the quantum foam don’t count, because a vacuum is not nothing.

        If formal and final causes are at bottom simply not at all real – as strict reductionist materialism supposes is the case – then there is no question of their somehow or other emerging from what is in fact real. On the contrary. If they are not real, then *they are just not real,* and that’s all there is to it.

        “Real” is one of the most confusing and poorly-defined words there is. I’d advise against using it.

        Oh, give me a break. You know perfectly well what I meant. Why not respond to that? Can you?

        Try this reformulation:

        If at bottom there is no such thing as formal and final causes – as strict reductionist materialism supposes to be the case – then there is no question of their somehow or other emerging from anything. On the contrary. If there is no such thing, then *there is no such thing,* and that’s all there is to it.

        You seem to be arguing that final causes must be metaphysically fundamental or they can’t exist at all (are not real).

        Not quite. I do say that:

        If final and formal causes are present in nature *at all,* then they must be present fundamentally, right along with material and efficient causes, and indeed integrally with them (for, as we have seen, they cannot have emerged from something utterly unlike themselves).

        So, I do think that final causes are metaphysically indispensable, ergo fundamental; but that’s not what I’m arguing in the passage in question. I’m arguing that they must be real, or you wouldn’t find them anywhere. If there is no such thing as rabbits, you won’t ever find rabbits emerging from hats, or anywhere else for that matter. If on the other hand rabbits are real, then you’ll find them in all sorts of places, and not just emerging from hats; the rabbits won’t need hats as a subvening substrate. Alternatively, you might find that hats all have rabbits in them, and that all rabbits are in hats. That’s what the assertion of emergence amounts to. And that’s just Aristotelean hylemorphism that dares not speak its name – that, indeed, is buried so deep in its closet that it does not know about its closet, or what hides therein.

        I except David Chalmers and the other panpsychists from that criticism. They have come out of the closet (Whitehead – the archon of panpsychism – is of course metaphysically a Platonist, and an Aristotelian, and what is more an Augustinian). Gay : homosexual :: Panpsychism : hylemorphism.

        … this is why natural selection and cybernetics were such big deals, because they showed how purpose and design could have a perfectly explicable material or naturalistic basis.

        That’s another way of saying that natural selection and cybernetics showed how matter and nature have a formal and final basis, so that they are inherently characterized by design and purpose.

        The bottom line is that the order of things makes no sense unless we posit that things have forms, and that the regularity of events makes no sense unless we posit that things have final causes. In fact, it goes deeper: to say that things are ordered *just is* to say that they have forms, and to say that events are regular *just is* to say that they are finally caused.

        “Motion” means something different to metaphysicians than it does to physicists. We are discussing metaphysical propositions, so I have been using the term the way metaphysicians do.

        I don’t really think that’s the case. You can have a metaphysics of motion, but there is no special kind of motion peculiar to metaphysics (I may be wrong, it’s not really my area of expertise). It sounds like what you mean is the Aristotelean theory of motion, with its associated metaphysics. That’s fine, but the modernist theory of motion has its own associated metaphysics.

        Close, but again, not quite. There is a special kind of motion peculiar to metaphysics – the motion from potential to actual – of which the motions of physics, of displacement from one extensive location to another, are types. So, for example, a change in the energy of a particle might not involve physical motion from one extensive locus of its own frame of inertial reference to another, but it does involve a motion from one state to another. A change in energy (or any other physical property) without a change in locus is an example of a motion in the metaphysical sense, but not in the physical sense.

        There is a difference between understanding an analogy and accepting it as valid.

        Sure. But you have not *shown* that the analogy is invalid. You’ve simply asserted that it is: “That’s a disanalogy.”

        Since you seem to be having such trouble imagining a frame of inertial reference specified by reference to a single celestial body, allow me to try to refine your brilliant analogy of a sun and its planet in a slightly different way. Let’s use Sol and Earth. From Sol’s frame of inertial reference, Sol itself is not moving, while the rest of the universe is moving around it – particularly its planet Earth.

        Meanwhile from Earth’s frame of inertial reference, Earth is motionless and the rest of the universe is moving around it – most importantly its star Sol. To itself, each of the pair is motionless, while the other is moving. To itself, each of the pair is an unmoved mover. Yet both bodies know and affect each other in a dynamical system.

        Taking time as basic is analogous to Ptolemaic astronomy, which takes the perspective of Earth as basic. Taking eternity as basic is analogous to Copernican astronomy, which takes the perspective of Sol as basic. The latter is more accurate.

        But the perspective of eternity no more rules out the perspective of time, NB, than Copernicanism rules out Terran motion. On the contrary: eternity can accommodate time, whereas not vice versa.

        [Ex nihilo nihil fit: from nothing, nothing is made] is the refutation of emergence: you can’t pull a rabbit out of a hat that has no rabbits in it.

        Except it isn’t a refutation, it’s just a bad opinion.

        If you can produce an absolutely empty hat that has a rabbit in it – i.e., if you can perform a contradiction of the Law of Noncontradiction – I’d sure like to see you do it.

        Game theory may be eternally true, and may accurately describe some aspects of biology. It is not the case that “all biology runs on game theory.”

        I should have expressed myself more carefully: game theory is eternally true, and accurately describes some aspects of all biological phenomena. I did not mean to say – and did not in fact say – that all biology runs *only* on game theory. Biology runs on other sorts of maths, as well. As I wrote:

        And there is no reason to think that the information of human affairs by eternity is limited to its game theoretical department. On the contrary: if as you say eternity is by definition the matrix of our existence, then *every bit of human affairs is pervasively conditioned by eternity.* Viz., the spooky applicability of mathematics in general to the behavior of the cosmos.

        My general point stands: the truths of mathematics are eternally true, and all reality runs on mathematics; so eternity pervasively conditions temporal reality. That being the case, it cannot be true that temporality and eternity are anywise in conflict.

        And things that don’t work do not need to be proscribed.

        So you are saying that murder works OK? That its proscription is therefore inapt? You are saying that of two societies otherwise exactly the same, and in which there might be murders, the one that proscribes murder won’t do better than the one that doesn’t?

        … using evolutionary arguments for moral laws is an extremely dubious move.

        Agreed. I have often rejected that move. As a sort of consequentialism, it’s anathema to Reaction.

        Also, as consequentialism, it’s fundamentally amoral: for, it suggests that whatever works is “good,” and whatever doesn’t work is “bad.” When, obviously, you can’t tell what “works” unless you first know what is good. And when, obviously, amoral procedures cannot give rise to morality, properly so called. If the world is basically immoral, then the world is immoral through and through, period full stop.

        I suppose you must have overlooked the fact that this response of yours was to a paragraph in which I adduced as the source of moral laws, not the happenstantial course of untrammeled evolution, forsooth, but rather the eternal truths of math.

        Evil does indeed have bad consequences, but it is not evil because it has bad consequences. To say so is to say only that evil is evil. It runs the other way: some sorts of acts have bad consequences – e.g., bad evolutionary consequences – because those acts are evil inherently, in and of themselves, and by their very nature. Their evil is prior to their enaction, or therefore to any of their consequences, because it is written in eternity before all worlds or their evolutionary histories.

        The instability of evil is the morality of the cosmos. Evil *just doesn’t work.*

        Well OK then, we can all relax.

        So you are saying that you would be pretty relaxed about my murdering your children, amputating your legs, and taking your woman and all your stuff? Because, after all, the morality of the cosmos would sooner or later redound to me, as bad karma, so, what’s the problem?

        Bad karma would indeed redound to me for such acts, to be sure. But karma is mediated by *acts.* It is mediated, in particular, by *our* acts. We – and all other creatures – are agents willy nilly of the moral law of the cosmos. It behooves us then to recognize our actual status as such, and then to own our share of the responsibility for the moral order of the cosmos, and then to take that responsibility – i.e., precisely *not* to relax about evil, but rather to be ever on guard against it, and to make war upon it with all our might. Only insofar as creatures are trying themselves to be good, to do good, and to confound evil, can the morality of the universe be effectuated in and by their acts.

        If goodness is going to inevitably triumph due to the eternal truths of game theory, then why bother struggling against evil or working for good?

        Because it is good to do so; much better than not.

        I’m kind of shocked that I have to explain this.

        You are an agent of and in the game. If you don’t play by the rules, and as well as you can, and as a good sport, why then you’ll ruin the game for everyone, including yourself. Goodness will win out in the long run in which we all are dead because the game is over; that’s for sure. That will happen no matter how we play. The point of the game for any of us players then is to be on the winning side, which is the side that is better, more beautiful, and nicer for everyone withal.

        So yes, if you aborted *every* fetus that would be “stopping babies” and a bad thing, from an evolutionary perspective as well as a moral perspective. But aborting one fetus stops one baby, not all babies.

        I think I did not express myself clearly enough. If you kill one baby, you stop all the historical developments that might have eventuated from his life. You thereby forestall exploration of a huge set of lines of inquiry of the biological solution space, which he and his descendants might otherwise have undertaken. His whole line of descent then is excised from the game thenceforth.

        It is in this sense that a single abortion has the effect of destroying the game at the margin, and impoverishing it. It doesn’t destroy the game altogether, to be sure. It destroys the game only a bit. But the point is this: an act that would indeed destroy the game completely if it were universally enacted – such as murder, contraception, theft, and so forth – is an inherently evil act that should be proscribed.

        Question: If aborting one baby is not bad from an evolutionary perspective, as you suggest, then what is it that prompts you to say that aborting all babies would be bad from that same perspective? Species die, after all; what’s the *problem*?

        On strict materialism, there is no such thing really as a problem, no such thing as good or bad, no such thing as intentions or their frustrations. There’s just stuff happening in an incredibly orderly and regular way for no reason.

        The central lesson of natural selection is that creativity and purposefulness can arise from a random process of variation and a non-random process of selection. We do indeed get something from nothing this way …

        The non-randomness of natural selection is a manifestation of the morality of the cosmos. It is *not* nothing. It is built into the something of the material past, and the formal future, of the cosmos. The only way that creativity and purposefulness can be anywhere evident in the cosmos is if they are built into it from the get go. They can’t emerge if they aren’t already there, waiting to emerge.

        Why must eternity have an enemy?

        I don’t know that it must. However, you seem to be setting yourself up as a spokesperson for the eternal, and given my opposition to your point of view, that makes me an enemy of sorts. But I don’t think of myself as an enemy of eternity itself, just to its misuse and misrepresentation.

        You had said that if eternity has an enemy, it is the temporal Now. You had also said that eternity is the matrix of time – i.e., of every Now. My point in asking why eternity must have an enemy was not to suggest that you were setting yourself up as such, but rather only to point out that the eternal matrix of the temporal Now cannot be at enmity with that Now; for, if it were, then, since the eternal matrix of the Now is logically prior thereto, the Now would lose that contest, with the result that there could be no such thing as a Now.

        There is such a thing as a temporal Now. Thus eternity cannot have an enemy in the Now. It must then be possible to reconcile the two notions. I have tried in this exchange to show how to perform that reconciliation. Pro tip: it is not to be done by taking time as basic; for, eternity is, as we have agreed, the matrix of time, and not vice versa. It is on the contrary to be done, if at all, *only* by taking eternity as basic, and time as derivative. That’s what I’ve been doing. That’s what you have not been doing.

        A.morphous, this has continued fun for me, but we have wandered wonderfully far afield from what I take to be your basic difficulty with the argument of the post, which it seems to me you expressed in this passage:

        [Eternity and its mathematical] objects, for all their beauty, are static. Minds, at least the ordinary human ones we know about, are dynamic, they interact with their environment, learn, are born and eventually die, and these qualities are essential to their nature. An unchanging mind is not a mind.

        To that, I responded:

        Are you quite sure that the only sorts of minds are those that change? Are you quite sure that change is that basic? I mean, sure, change is basic to the sorts of lives that men lead, for we are embodied, and as such, temporal. But are you sure it is basic to every sort of life whatever?

        To which, you replied:

        I don’t think I am sure in the sense that you are asking, but the notion of an unchanging mind seems obviously oxymoronic to me, given my concept of what a mind is.

        If minds are indeed by definition changing, as your concept of mind insists, then “changeless mind” is indeed oxymoronic. But that’s so only on your concept of mind. I admit that your concept of mind is natural to us all, just as Ptolemaic astronomy is natural to us – and that it is every bit as difficult to transcend it, and as weird, as it was for our ancestors to transcend Ptolemaic astronomy. But the question at issue is whether your concept of mind exhausts the category; i.e., whether your concept of mind per se is absolutely correct.

        You have not *shown* that it is. All you have done is *declare* that it is, and then proceed to assert your incomprehension of the possibility that it is not. And all your incomprehension stems from your inability – or perhaps your unwillingness – to transcend your concept of mind, and to entertain a hypothesis radically new and strange (to you), and different.

        I have furnished arguments and demonstrations and analogies that show how it is intelligible to construe the Eternal One as mindful, dynamic, scient, active, lively, and so forth. To these, you have responded, not with refutations, but with quibbles about this or that point, mostly of rhetoric. I have then answered your quibbles, and you have then quibbled about those answers. That cycle has reiterated. That’s how we wandered so far afield.

        And again: it has been lots of fun.

        But nowhere in our conversation have you *shown* that an eternal mind is impossible. All you’ve done is insist that an eternal mind wouldn’t be temporal like ours, and so can’t be a temporal mind like ours. Which is tautological, and which begs the question.

        I have been asking you to consider that there might be a mind that is not of the ordinary human sort that we know about from being such minds, but which rather transcends them utterly. All you have been doing is saying, “No; can’t do that; won’t.” Do you see how jejune that is?

  9. you are interpreting terms – and reality – under the modern, nominalist, post-Cartesian metaphysic that has dispensed with the ancient formal and final sorts of causation,

    Uh yeah, because I live in a modern society. This isn՚t a Renaissance Faire, I՚m not cosplaying at thinking, I՚m doing it for real.

    This is not to say that classical concepts are not worthy of study, or potentially useful in some way. But it is very confusing to have an entirely separate meaning for terms like “motion” that have nothing to do with the more modern and everyday idea of motion. You should find something else to call it.

    But maybe this *is* like a Renaissance Faire – it՚s your blog after all. That is, the rules of usage around here are based on older definitions of words and maybe I՚m just being obnoxious and obtuse by insisting on their modern everyday meaning. I’ll try to stop, that doesn’t seem very productive.

    BTW It՚s not really the case that modern metaphysics dispenses with formal and final cause. It՚s true that those are no longer considered fundamental, instead they are emergent properties of mechanical processes. This is what the intellectual revolutions of evolution by natural selection, and later cybernetics were all about.

    Thus when I use terms such as “archetype,” “action,” or “motion,” I mean to refer to their broader more precise classical meanings – which include their modern meanings as special cases or analogues.

    They are not “broader” or “more precise” (hard to be both of those at once!). They are merely different, and more or less obsolete. If I was a bit more nerdy than I actually am, I՚d say that your definition of motion is just plain wrong, after Newton and Einstein and others we have a much better theory of motion than Aristotle had.

    But as a hip postmodern kind of guy, I have to be more humble and say that we just have two very *different* theories of motion and we can՚t really judge one better than the other – they can and should co-exist, serving their different purposes.

    One such error is to think of time as basic, so that all things must be temporal; when, of course, time *cannot* be basic, since every temporal event derives from other events; so that, the entire temporal extent being derivative, time *cannot* be a First Thing… The only way to understand time is as a derivate of eternity; and the only way to understand the derivation of time from eternity is to understand dynamism, action and motion as basically atemporal,

    On the contrary, you can՚t even have a First Thing without time, because the concept of “First Thing” is also temporal at its essence. As is “derivative”.

    I՚ll grant that you are probably using “First Thing” metaphorically. You don՚t really mean “before everything else in time”, but something like “logically or metaphysically prior”. But that too has a temporal aspect at its core. If you are deriving a from b you already in a temporal realm. In eternity, nothing is derived, because everything just is. No metaphysical ordering is possible, just as no action is possible.

    The only thing that needs to be tweaked in order to improve the analogy star : planet :: God : creation is to stipulate that in the analogy, the star defines an absolute frame of gravitational reference.

    That makes the entire analogy pointless. It becomes a disanalogy. There are no unmoved movers in physics. and the historical trend is to make that even more the case (eg the advancement of Einsteinein over Newtonian theory).

    God cannot be moved; he cannot be known. Yet he moves; he moves *us;* and because we are by him moved, and in his gravitational field live and have our being, from his motions upon us we can know something about him, insofar as he is manifest in us his creatures.

    If God is responsible for all movement, then he՚s basically the same thing as the laws of nature. The laws of nature may be worthy of awe or even worship, but they don՚t have *agency*. The term “God” is just putting a human mask on something profoundly not-human.

    What does the configuration of the planet, and of the dynamical system in which it orbits, tell us about the Utterly Strange Attractor that lies at the center of the system, and that defines it qua system? That’s where we start using terms like love, knowledge, will, mercy, judgement, and so forth. These are terms that, as applied to God, derive from his effects upon us. Because he is the Unknown Knower, we can’t know what they mean to him.

    If we can՚t know what they mean to “him”, how do we know they mean anything at all, applied to him? That is basically my point. “Love” or “knowledge” may derive from God՚s effect on us, but that՚s only because you՚ve identified God with all causality, so everything is thus derived, so you haven՚t really said very much.

  10. Pingback: XXXIX – The Other Half of Reality – Times-Dispatch of Vichy Earth

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