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.

30 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.

  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 crushing 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.

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