Robert Inventor

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Infinity and very large numbers

back to: about me. This page is rather philosophical. If you want a page which is slightly more mathematical (though still mainly philosophy) you may like to start with:

The idea is that mathematical infinity arises through attempting to reach exceedingly large numbers with insufficient resources, where the numbers are more than astronomically large, quite unimaginably large. But just very very very large numbers, not truly infinite.

The idea isn't to downplay infinity particularly, rather the idea is that in maths we may think we have some grasp of infinity because we can put it into our equations etc - but that the real infinity is far vaster than anything we can imagine. So in some ways it is more realistic to treat the infinities of maths as just exceedingly large numbers. The idea is, if you make truly large numbers large enough then they are indistinguishable to us from true infinities as far as maths and practical purposes are concerned.

That is to say truly large numbers, far more than astronomically large, numbers like the number of electrons in the universe or the volume of the visible universe in cubic Plank units wouldn't even figure here at all. Even a number like 1000 ... 000 where the 1 is followed by ten thousand million zeroes (or in maths notation 10^(10^10) ) wouldn't count as particularly large for mathematicians in this field. I was interested in exploring the possiblity of treating even numbers "as small" as 10^(10^10) as infinite with very severe limitations on proof methods but normally the numbers involved would be far far larger even than this number.

The exact size doesn't matter, the main thing is just that it is finite though enormously large. In fact, I was thinking in terms of something so huge that no mathematician has yet defined anything so large in any way that lets it be compared in size with other numbers. A number so huge we can't define it precisely in any notation - but just guesture towards it by analogy - that we can define enormously large numbers - there must be other numbers beyond the horizon of anything that any mathematician has defined, there must be numbers even beyond anything any mathematician ever will define on this Earth. So - assume existence of such a very large number too large for us ever to define it precisely - beyond our horizons - and make a mathematics in which a number as large as that can be treated as infinite.

Then - as a sort of way of looking at the same idea in a microcosm, one could also try developing a mathematics that treats even just very large numbers like 10^(10^10) as infinite with very restricted proof methods.

Special Logic

It uses a special logic intermediate between the Yes / No type classical logic where all mathematical statements are considered to be ultimately decidable even if we don't know any way to settle the question - and the intuitionistic logic which admits the possibility of statements that haven't yet been proved to be true or false or even decidable (but may later on be shown to be one or the other).

Intuitionistic logic is very hard for a newbie to get used to, and unfortunately, compounding the problem, popular accounts of it are frequently way off the mark. It isn't fuzzy logic in any form at all (which has statements in an intermediate fuzzy state which are neither true nor false), as statements can only be true or false. It is a precise, two valued logic with no third truth value. But sometimes - until you find out more, you just have to give up the attempt to say anything about the truth value.

Statements of intuitionistic maths are not provisionally indeterminate either - "not yet true or false" - the truth isn't time dependent (this is another popular misconception). It is just that you can't say anything at all until you find a proof or disproof. Until you know more, you can't even say that they are indeterminate or not yet true or false, because you have no proof of that either (in fact you never can prove anything to be indeterminate in truth value in intuitionistic logic).

It is all about the statements you can make, so you simply don't talk about whether the statements are "really true" or "really false" or "really indeterminate" in intuitionistic maths, as that sort of way of talking doesn't make sense in intuitionist maths - all that is regarded as lingering vestiges of classical thinking requiring a pre-existing truth independent of any attempt to prove it or disprove it.

If everything is finite, then for intuitionist as well as the classical logic, everything is decidable (even if it takes a long time). So then both approaches say the same thing.

So anyway, in this special combined logic, the statements about the large numbers beyond the horizon of our resources are pre-existing true of false type statements (classical) and the truth of those statements do persist independent of attempts to prove or disprove them because they are regarded as really finite, just very very large, just as all finite statements are regarded in intuitionistic logic. But statements involving the infinities that arise when we try to comprehend truly large numbers are sometimes ones that we just have to give up and not try to assign a truth value to at all at our present state of knowledge (intuitionist, or constructivist maths) because they are about number series which are in progress, and unending.

This gives a way to bring together the two perspectives into a single mathematical system - the intuitionistic maths with its refusal to allow one to talk about truth or falsity of anything until it is proved or disproved or at least shown to be decidable in a finite number of steps - and the classical logic which allows things to be "really" true or false in a pre-existing type fashion, without any proof either way. Which is kind of satisfying philosophically, at least if you aren't an out and out intuitionist and have classical logic type sympathies as well. To me both seem to have an inkling of truth, yet both also seem to be incomplete, so melding them together, sort of like two sides of the same coin, feels very satisfying.

The neat thing about it mathematically is that infinitesimals arise so naturally, leading to a very easy way of deriving many theorems of calculus, much easier than the usual limits method or Robinson's original rather complicated "star transform" infinitesimals approach. Also the theorems need to be proved from scratch rather than just by star transforms of limits results, leading potentially perhaps even to new maths. For a constructivist the theorems are attractive as they are much stronger than the usual constructivist results, yet they are not quite classical. Also - well to my mind anyway - the proofs were rather appeallingly beautiful, the infinitesimals made them so much shorter and simpler to prove, the axioms fitted together well and the proofs were just nice to work with.

All the results I proved were very similar to the classical results - definitely stronger than intuitionist maths and very slightly weaker than classical results but for most practical purposes you wouldn't notice, and nothing a classical mathematician would consider to be new (just neater proofs of the same results) - except for one bit of work I did on a new form of Lebesque integration which seemed to have some potentiality just possibly for some novelty, though that was more or less the last thing I did and I didn't take it very far (Unfortunately I can't remember the details but have the notes somewhere).

The basic maths isn't new to me - it is closely related to work by Vopenka and his group in Prague for his AST (Alternative Set Theory), and I took his work as my starting point - but the combination of the maths with the intermediate classical / intuitionistic logic was new, and some of the details of the formulation of the axioms and the techniques I used to develop the theorems of calculus were also new (there were only very sketchy hints available about how calculus might be developed in Vopenka's AST).


Philosophically, (for any specialist philosophers reading this) it involves Strict Finitism, and Vague predicates, and is related to the sorites paradox or the paradox of the heap -the paradox that if you remove a pebble from a heap, the result is still a heap - but if you were to keep that up for a long enough time eventually the heap would go away, but removing a single pebble can never turn a heap into a non heap - so at what point does the heap cease to be a heap?

If the heap is truly enormous, then the paradox goes away for all practical purposes, as you can never remove enough pebbles as you just don't have the time or resources to do that. So one gets the illusion of a heap that is infinite, and the results you prove are consistent so long as you operate within those limitations. This infinity is in a way illusory but the idea is that it is the nearest we can get to infnity that we can actually have any hope of really understanding from our own experience, and that it hints beyond the horizon to what real infinity is like. Sort of like treating the Earth as flat. I feel in some ways it is more valid to treat the world as flat, because it is so very large, that as soon as we try to think of it as round, we are bound to imagine something far too small, and it is impossible to imagine anything quite as large as our Earth, yet still curved - depending on your perspective - of course if you are an astronaut in orbit, it is more valid to treat the world as round, or if you want to emphasize its finiteness - but if you are a farmer or a builder building a house, it may a better approximation for you to treat it as flat.

Similarly in some contexts it is more valid to treat sufficiently large numbers as infinite rather than as finite, but even more so than for the flat earth. It is not impossible to think of the earth as round and at the same time, as large as it really is, merely very hard. You could imagine some being maybe with sufficient imagination doing it, imagining it in all its detail and vastness or indeed, perhaps one might really begin to get some idea if one were to set out to walk and swim around the world. But when it comes to infinity - it is impossible really for any finite being at all to comprehend infinity in the same way from a classical survey everything type perspective. So it is unclear that it is valid for us to use such a logic. It seems more "honest" or at least more "down to earth" to use an intuitionist / constructivist logic with some classical elements for the vastness beyond our horizons, at least in maths - well so it seems to me anyway.

Is there a maths of true infinity?

I don't know. But if there is, ZF (Zermelo Fraenkel set theory) - I feel that perhaps it expresses some kind of a truth but not a final truth.

Things like the axiom of choice, the continuum hypothesis, the Tarski paradox, and solutions to Russell's paradox - and all those orders of infinity in Cantor set theory - and specially, the classical distinction of sets and classes - it just doesn't seem like a final solution. Maybe it is expressing some kind of truth, but the truth seems in some way incomplete.

It's possible that taking the finitist maths with infinitesimals and applying classical logic to it instead of intuitionistic logic - and removing the idea that the very large number is truly finite in some sense - this can give another way of looking at set theory from a classical perspective with a different more finitist inspiration. In a way that is what Vopenka's AST does, a finitist inspired theory with infinites, infinitesimals, and using classical logic. It has a kind of elegant simplicity which may appeal to some.

But I feel doing that too isn't a final solution to understanding of infinity. Rather, it is an interesting alternative which can show e.g. that maths didn't have to follow the ZFC path but could have followed another route equally well.

Can a finite being can somehow transcend our finitist limitations? Perhaps they can in a non mathematical or intuitive way. Maybe even in maths too (e.g. Ramanujan?). If so - is it possible to bring back some kind of classical infinite maths for use with finite methods of reasoning - I don't know. I don't have any doubts about the validity of the maths as such, for practical purposes etc, as no-one seems to have found any obvious flaws in e.g. ZF set theory - but what we have seems limited and artificial from a finistist perspective or once you really take on board the validity of alternative approaches.

You can get similar things in finitist maths too - extra axioms, alternative ways of doing it and so on including finitist versions of the continuum hypothesis etc - finitist maths is no more unique than classical maths. But it is understandable that there is more than one way to set about things as you have much more limited objectives in your maths. Perhaps in some way the same still applies when you try to talk about true infinity, perhaps our objectives there should be more limited.

Anyway I prefer to leave true infinity as an unknown. To leave the sense of wonder about infinity. Accept the truths of ZF, AST etc but acknowledge that one isn't really sure what those truths mean (if one really did there would be no Russell's paradox). Focussing on an approach to maths that treats very large numbers as infinite, and showing that it can also work as well as the true infinity type theories perhaps can help with that.

Publication possibilities

Reactions to my research were very unfavourable, so I never tried to publish it, (and it put me off publishing my other research as well) though I think part of that was because I was rather poor at explaining the ideas involved in those days - many things that were clear in my own mind came over as very confused to others when they listened to what I said, and especially so when they read what I wrote. However, for me anyway, it answered the basic question that motivated the research, of course leading to many new questions - and lead to new areas for research which I enjoyed following up. So, it could possibly be of interest to others too.

This page is just an informal philosophical introduction to some of the ideas. The maths isn't available anywhere yet, nor is the detailed philosophical material. Here is another older page that gives a bit more detail of the maths - but not very much, just a few hints, no axioms and none of my proofs

I may return to it again some day. I have a lot of unpublished and perhaps unpublishable material (hundreds of pages though with some duplication), both mathematical and philosophical, which I could probably rewrite and make more publishable now as I am better than I was at communicating ideas, whether mathematical or otherwise. Probably also better now at sifting out the material that would be of interest to other mathematicians and philosophers of maths.

But it would take such a long time to do, as there is so much of it and it is so many years now since I did any work on the subject. Really I'd need to spend a year or two to get back into it, then read up the related research that's been done since then, to do a proper job of that. Whether that will ever happen I don't know.

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