Naturalism: A Search for Infinity Among the Numbers

It is obvious that all the things that pass away do indeed pass away. It is therefore silly to search among the things that pass away for something that does not. It is even sillier to conclude from the necessary failure of that search that there are no things that do not pass away.

That would be like searching for infinity among the numbers, and having failed to find it, concluding that there is no such thing as infinity.

47 thoughts on “Naturalism: A Search for Infinity Among the Numbers

  1. Pingback: Naturalism: A Search for Infinity Among the Numbers | @the_arv

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  3. There is no such ‘thing’ as ‘infinity’ among natural numbers (or anything else).
    ‘Infinity’ is merely a shorthand way of saying that there is no largest number.
    ie: to whatever is the largest number, 1 (or more) can always be added.
    In other words, the set of natural numbers is infinite.

      • 3 is not a thing either. Both 3 and infinity are mathematical concepts and have the kind of existence appropriate to their status; and mathematicians manipulate and reason with them in the course of their work.

        I think what you mean is that sets of physical objects (things) with a cardinality of infinity are not to be found in the physical world, which is certainly true and is one way in which 3 and infinity differ.

      • I think what you mean is that sets of physical objects (things) with a cardinality of infinity are not to be found in the physical world …

        That’s exactly right, thanks, a.morphous.

        3 is not a thing either.

        Certainly 3 is not a thing in the same way that my coffee cup is a thing. But that is not to say that it is not a thing at all. It is to say only that it is a different sort of thing.

      • Infinity, while not being a thing. is however, a fruitful mathematical concept. It is not merely a shorthand for “there is no largest number”. There are infinite number of rational numbers between any two integers. There are a yet infinite number of irrational numbers between any two rationals.
        And so on–the usage of the concept infinity is again unbounded in maths and physics.

      • Thanks, Bedarz. Just so. As Zeno made clear, every finite thing is infinitely deep. In that sense, infinity is indeed to be found among the numbers. Nevertheless it is never determinable as some specific number.

      • “There is no such ‘thing’ as ‘infinity’ among natural numbers (or anything else).”

        Note the inverted commas round ‘thing’ and ‘infinity’ and the bracketed (or anything else). That takes care of the Real Numbers, Rational Numbers, Irrational Numbers, Imaginary Numbers and everything else. Infinity is not a number for if it is, then so is Infinity + 1 or Infinity – 1

        ‘Infinite’ (= endless or boundless) is an adjective. ‘Infinity’ is that adjective used as a noun.

  4. Declarations that God does not exist because the observable universe cannot explain Him always struck me as Man planting a flag on the universe and declaring he has seen it all.

  5. Nothing is greater than God. In order for there to be something there must be a field of nothingness in which it exists. Similarly, finitude can only exist within a field of infinity.

    • Yes. In everything than which some greater can be conceived, that than which no greater can be conceived is logically implicit. Thus every finite thing presupposes that than which no greater can be conceived, or therefore be. This, in just the way that infinity is logically implicit in every number, so that every number logically presupposes infinity.

      That there are things entails that there is what is greater than things as such.

      • Well, strictly speaking, nothingness is logically impossible to implement. There can be no such state of affairs as nothingness; for, in the event that there was indeed nothing at all, it would be possible that there could be nothing at all, and in that event, there would be the possibility that there could be nothing at all. And a possibility is something. If there is the possibility that there could be nothing at all, then there is something: that possibility. But then the notion that it is possible that there could be nothing at all turns out to be autophagic. It turns out that it is logically impossible for there to be nothing at all. Ergo, it turns out that the opposite of nothingness is logically necessary.

        Also, we can’t coherently say that things exist in a field of nothingness. If there are any things at all, then there is no nothingness, period full stop. We must say rather that things exist in a field that is not itself a thing. That this field is not itself a thing does not mean it is not concrete.

        I almost responded to your first comment by saying that you were writing like Saint John of the Cross. I took your “nothingness” to be like his Darkness. I.e., I took you to be verging over – quite properly – from metaphysics into talk of the spiritual life. So, I took you to be speaking of nothingness metaphorically, rather than literally.

        In the final, strict analysis, the only way to speak of nothingness is to speak metaphorically.

    • Nothing is greater than God. — WinstonScrooge

      No… This is nonsensical. To be greater than God, some thing must be greater than God. “Nothing” is even less than “something.”

      • Exactly.

        I had taken Winston to be saying “no thing is greater than God.” But it seems that he was saying rather that “nothingness is greater than God.” Which, as you say, is logically incoherent.

      • In this case I was speaking metaphorically. But I disagree with your premise that nothing at all can be in or come from nothing. God created the universe from nothing.

      • Indeed he did. But he didn’t create the universe from a state of affairs in which there was absolutely nothing. He created the universe from a state of affairs in which there was God. He created the created order from a complete absence of anything that was not God.

      • Not quite. As to the second question, God’s presence eliminated absolute nothingness. If even one thing exists, there is no absolute nothingness. There is however relative nothingness. Say that there’s a hammer and nothing else. In that case, there is no absolute nothingness, because of the hammer. But there is in that case a nothingness of nails.

        As to the first, no. I’m saying only that God did not create using stuff that was already there. He created stuff that was not himself out of an utter absence of such stuff.

        God does not need stuff in order to create stuff. He doesn’t need any stuff of the sort that is not God, nor does he need any stuff of the sort that is God.

      • Which assertion? You’ll have to get more specific. I’m not prepared to disgorge citations in support of *all* the assertions I have made. That would amount to the writing of a long and technical book, which I don’t quite have time for this afternoon. But then, I feel sure that you are not asking me to do that, but rather have some particular assertions in mind. Some of the things I write are a matter of mere logic (e.g., that absolute nothingness is logically impossible). Others are derived from various philosophers or theologians, or Doctors or Fathers, or Scripture.

        I am betting that you are asking what authority supports my assertion that God creates not out of any stuff that is not himself, nor out of any stuff that is himself. Much has been written on this topic, but paragraph 296 of the Catechism of the Catholic Church should suffice:

        We believe that God needs no pre-existent thing or any help in order to create, nor is creation any sort of necessary emanation from the divine substance. God creates freely “out of nothing”: If God had drawn the world from pre-existent matter, what would be so extraordinary in that? A human artisan makes from a given material whatever he wants, while God shows his power by starting from nothing to make all he wants.

        That said, I recommend also a short article by philosopher Bill Vallicella on a reconciliation of creatio ex nihilo with ex nihilo nihil fit. Vallicella’s nifty solution fits neatly with the doctrine of the Church set forth in the Catechism.

      • If God created the world out of nothing, that means nothing or “nothingness” precedes creation. It is the womb from which creation springs forth in other words. It is in this sense that nothing is greater than God.

      • As infinite, God is ubiquitous. There is nowhere “outside” God for there to be nothingness. God does not reach outside himself to create, because there is no outside of God.

        Winston, I get the impression that you are stuck – as I once was – thinking of God as a sort of glob floating in a vacuum, so that the vacuum is the environment of God, and thus prior to him. In this vision, God speaks and created beings – angels, worlds, etc. – pop into existence in that same vacuum, orbiting God as it were.

        It’s a natural way for humans to think. But it isn’t like that. God fills the whole vacuum. He is the environment of the vacuum.

        Because God fills the whole vacuum, there is no part of it in which there is simply nothing at all.

      • It is indeed tricky, to be sure.

        That God is ubiquitous does not mean that there is no room anywhere for anything other than God. It means only that whatever happens, he is in it immanent, so that nowhere can there possibly be nothing at all. Everywhere, necessarily, no matter what else might or might not be there, there is God. So nowhere is there complete nothingness, properly so called. But, that there is something at locus x does not mean that there may be nothing else at x. E.g., at the locus of my body right now are also the United States of America, the offices of my investment firm, neutrinos, an NPR broadcast, on and on.

        Try thinking of it this way. It isn’t meant to describe a temporal sequence, but a logical order.

        1. There is only God. God is infinite. Nowhere is there nothing at all, because God is everywhere. This is a permanent state of affairs, a necessity. No matter what happens, it must remain the case that God is everywhere, and therefore immanent in whatever happens.
        2. God creates beings that are not God. They can’t be somehow made of God, for in that case they would simply be bits of God, rather than beings in their own right. They are not bits of God. On the contrary, they are creatures: disparate, definite, finite, contingent, and so forth.
        3. These creatures are pervasively environed by God. In him they live, move, and have their being. They are entirely dependent upon him for their existence. So they partake or participate God; he touches and informs and so forms them in every aspect of their being, so it is impossible for them to do otherwise. But they are not God. And they are not made of God.

          Think by analogy of how the United States informs and so forms me here in my office this evening. I am not the US, and the US is not me. I am not made of the US.

        4. Prior (logically, not temporally) to God’s creation of the creatures, there is nothing that is not God. Posterior (logically, not temporally) to that creation, there are things that are not God. The creatures appear out of a total absence of anything that is not God; for, apart from the creatures, there is nothing that is not God.

        Make sense? The nothing out of which God creates creatures is an absence of what is not God.

      • Do you think that when the new heavens and the new earth commences? That the entire universe would be undergoing continuous improvement. Ascending to ever greater heights of beauty and a myriad of other perfections?

      • Is separation from God ever truly possible in your understanding, Kristor? We’re told that this does happen to the damned in Hell.

      • Nothing can separate us from the love of God. Romans 8:38-39. How could it? There is no way to get away from Ubiquity. The damned in Hell are not separate from God, for he is immanent in them there, as he is immanent in all things whatever. He is closer to them than they are to themselves. They are not ever possibly separate from him, for he is the source and foundation of their being wherever they may be.

        All that said, all beings that are not God are *disparate* from him; as in, distinct from him. But, they are not *distant* from him. You can’t be distant from everywhere.

        The damned are not *separate* from God, then, but *alienate.* They have turned away from him, and inward. They are mewed in to the prisons of themselves, prisons of their own making. To adduce the metaphor of Plato’s cave, the damned are those who sit in it facing the shadows cast by the fire behind them, *and who have ripped out their own eyes.*

        Nothing can penetrate their self-imposed solipsism.

        There is more. The damned cannot even apprehend *themselves* rightly. If they could, they’d remember that God is the source of their being, informs their every moment, and loves them dearly. Everyone not already incorrigibly damned knows this perfectly well, no matter how depraved he may still be by his sins, and no matter how his inward vision is thus warped. This is why we feel guilty, and imperfect, and sorrowful about our failures, and nostalgic for Paradise: we know our natures, so we know perfectly well what would constitute our perfection, and so we know how far we have from it fallen, and disappointed our proper appointments. For, God is at the root of every being, right there ready to be heard, a still small voice, when all the distractions are stripped away (this is why severe ascesis is so universally efficacious and beneficial, even for pagans and atheist Buddhists).

        All this, the damned cannot any longer see. The ugly consequence: they rather *like* their damnation. They think it natural, right, and as good as good can really be. They think notions of anything better are simply insane delusions.

        As CS Lewis rightly points out, the gates of their prisons are locked on the inside. Only the damned can possibly unlock those gates, for those locks are entirely of their own making. This does not mean that they can ever be unlocked. If you’ve locked a gate and destroyed the only key, the jig is up.

        So then: the damned are not separate from God, but alienate; so that – God being the source of all beings, and the medium of their interactions – their phenomenal experience is of radical ontological loneliness and separation from all other beings. I cannot imagine a more horrible torment for a being that by its nature is made to interact causally in harmony and justice with others – i.e., in love – as we creatures all are.

        But, again: the damned have lost all memory of love. They think their permanent exile right, and good, and the best that can be expected. They suffer torments. But they do not remember the sunlight. Any other ways that everlasting life might be seem to them delusional. So the torments of Hell seem to them just the way everlasting life is, and not too bad withal – especially since they have themselves chosen it as preferable to all others.

        Full disclosure, in light of Winston Scrooge’s legitimate inquiry about the authority standing behind my assertions: the stuff about separation versus alienation of the damned is pretty standard Christian doctrine, while the stuff about the overall contentment of the damned with their phenomenal lot is my own logical inference. Perhaps it is supported in the Fathers or the Doctors; I don’t know yet. What I do know is that – so far – I have eventually discovered that all such logical inferences of my own are right there in the Magisterium, somewhere or other; and, never have I found such inferences contradicted by the Magisterium. Had I known about the presence of such inferential notions in that corpus of wisdom hard won, I’d have saved a lot of intellectual labor. So it goes. As it is, I find it peculiarly satisfying – and reassuring – to discover that an arduous logical leap I had taken on my own is in fact quite ancient.

      • [The damned’s] phenomenal experience is of radical ontological loneliness and separation from all other beings. — Kristor

        This is a succinct summation of a “radical autonomy” only “enjoyed” in Hell while hopelessly desired for in an oppressively mudane world.

      • Exactly, yes, thank you, Thordaddy. Radical autonomy *just is* is radical annihilation of the self. The self is perfected only insofar as it is in communion with others, and therefore causally efficacious – as in its given nature it is intended to be. And this communion cannot be obtained under autonomy. It can be obtained only under conditions of heteronomy: of influence of others upon each self.

        Heteronomy is necessary to the formation of a causal system – of a world. No causal interaction, no world. Autonomy then amputates the world from the self. So doing, it annihilates the self, properly so called.

        Radical autonomy then is auto-deletion. Insofar as it succeeds in attaining its selfish objective, it depraves and degrades and deletes the self.

        What’s the right neologism for auto-deletion? Autodely?

      • ”All this, the damned cannot any longer see. The ugly consequence: they rather *like* their damnation. They think it natural, right, and as good as good can really be. They think notions of anything better are simply insane delusions. ”

        Its akin to people who find stalinist brutalist grey architecture with its hellish designs as beautiful. And Piss Christ is sublime art.

      • Kristor, thanks for the detailed response as always.

        the stuff about separation versus alienation of the damned is pretty standard Christian doctrine, while the stuff about the overall contentment of the damned with their phenomenal lot is my own logical inference.

        Do you have a Magisterial source regarding the separation versus alienation distinction? I ask because CCC 1035 reads “The chief punishment of hell is eternal separation from God” (in Latin, “aeterna separatione a Deo”). I find it odd that the Catechism would use this language if, as you say, “the damned are not separate from God, but alienate.”

      • You are welcome, cpt. I’m going to have to dig around for a cite to alienation versus separation, and won’t have time to do so until this evening. If even then: the grandchildren are coming over!

        As for the terms of CCC 1035, they are often used imprecisely. The Church speaks often of our inheriting eternal life, e.g., when obviously only God can be eternal. Properly speaking, our lives are not eternal, but everlasting.

        The alienation of the damned amounts to separation in its phenomenal effects. A thing can be present to you immediately, yet you can be completely ignorant of it. So the experience of the damned then is indeed an experience of separation from God. I meant to put that in my comment above, but forgot.

      • CPT: in his Letter for the Doctrine of the Faith to All Bishops – Recentiores Episcoparum Synodi – of 1979, Saint John Paul II writes that, “[the Church] believes that … the sinner [will] be deprived of the sight of God.” Denziger 4657. Such is the poena damni common to all the sinners in Hell. This pain of loss “consists in the loss of the beatific vision and in so complete a separation of all the powers of the soul from God that it cannot find in Him even the least peace and rest. It is accompanied by the loss of all supernatural gifts, e.g. the loss of faith. … And yet the pain of loss is but the natural consequence of that aversion from God which lies in the nature of every mortal sin.” Catholic Encyclopedia, Hell.

        The separation of the sinner from God in Hell then is not so much geographic – the sinner is not separated from God by a geometric extensive distance of some sort – as it is spiritual. And the spirit that is afflicted – the spirit that is affected, that is changed – by damnation is certainly not the immutable spirit of God, but rather the spirit of the sinner. His aversion from God – this turn being the essence of sin – literally turns away all the powers *of his own soul* from readiness to the apprehension of the ever ready influx of Divine Grace. It is a turn inward, and away from reality. This turn prevents enjoyment of any of the spiritual gifts, and in particular the gift of faith – so that the damned are *atheists.*

        The inward turn of the damned also, by the way, prevents their enjoyment of any other positive, real pleasures, all of which involve apprehension of reality. The turn away from God, being a turn away from Reality as such and in its essence, is ipso facto also a turn away from all other lesser reals, other than the self.

        The fundamental phenomenal character of beatitude is the apprehension of God. The fundamental phenomenal character of damnation is inapprehension of God – i.e., of the Good, and so of all lesser goods.

        Damnation then is solipsism perfected. A perfect solipsist has no data from anywhere other than himself that might disturb his stability in his isolate condition. This is why damnation is permanent, and incorrigible. Nothing from outside himself can get to the sinner, that might distract him from his sin.

  6. No modern has much of a fight going on with the Infinite or Ultimate or Absolute. OTOH, almost everyone has some kind of dastardly grudge against “white supremacy” signaling his cloaked resistance to objective Supremacy.

  7. Free will ensures us that that race of man (or men) who genuinely desire Perfection will separate (at least consciously) from the rest of “humanity.” Practically-speaking, this is theosis. And “theosis” is the “why” a “totalitarian integration” cannot be.

    • Must disagree. *All* peoples tend to separate themselves from others. If only those who genuinely desire Perfection did so, they would be the only societies that waged wars or established borders. But all societies war and establish borders.

      Moreover, theosis already has a practical definition. The term has nothing to do with social relations.

      • *All* peoples tend to separate themselves from others. — Kristor

        Which is just to say that *all* people to some degree or another desire Perfection, ie., conscious separation from a degenerate other. Which then only makes more explicit the nefarious nature of the anti-white Supremacist and his demand for “totalitarian integration,” ie., diversity.

      • OK. Although I would say rather that our adversaries are anti-white, than anti-white-supremacist. They’ve ripped off their masks in the last two weeks, and gone from shouting about diversity – and, thus, the relative social derogation of whites – to shouting that white men must be killed and fed to swine.

  8. People who enjoy teasing about various senses of “nothing” and how “nothing” can precede “something” will appreciate Dorothy Sayers’ discussion of the matter in The Mind of the Maker.


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