A while ago in the intellectual communities on Facebook there was a discussion in which someone came up with the idea that God necessarily needs to exist because only God could read (infinite) irrational numbers like pi. Another one countered that Gödel´s incompleteness theorem would allow that (i.e. that stuff exists that cannot be proven or verified). Somehow both assertions seem inadequate, but make you think about the nature of numbers (and mathematics) which is actually a mysterious and haunting subject. However, upon reflection, numbers simply express how quantifiable properties relate to each other. Out of an r you can construct a circle with pi, an ever more perfect circle with the more digits of pi you know, the perfect circle, constructed with infinite precision, cannot be constructed in a finite universe. Likewise, if you have a basket with two apples and want to have one with three, you can do that, with infinite precision, by adding one apple to the basket. Numbers, in themselves, are neither platonic nor are they real, they are virtualities / virtual entities.
I have thought about the continuum hypothesis, the orders of the infinite, the incompleteness theorem, whether the universe is a mathematical system or a logical syllogism a while ago. Some say that by applying logics they can see it all, and maybe that is true, nevertheless with logics you can construct pretty much anything of your liking (apart from that a logically correct conclusion need not be based on a correct assumption). Lots of stuff, for instance proofs of God, have been constructed with logic – but all of them can also be refuted by using logics (see, comprehensively, John Mackie´s The Miracle of Theism if you´re interested). Usually the philosophers and theologians coming up with their proofs of God were thinking that they did not prove the existence of God by using logic but, literally, that they were proving the necessary existence of God out of logic, although to every neutral observer it was apparent that there was something wrong, wobbly, uncanny in their proofs, although it is not necessarily easy to exactly tell what the problem is. Often it may require an entirely new heuristics, and for instance it took centuries to exactly tell what is wrong with Zeno´s paradoxa. Metaphysical questions may be undecidable, not least because they´re paradoxical in nature.
And then, the incompleteness theorem… Despite its apparent gravity and the mysteriousness it seems to imply the incompleteness theorem hardly affects professional mathematician´s business. I have never read Gödel´s original paper and maybe would not understand it since maths, among other things, is not my speciality, however as far as I can see it is about the „paradox“ of the barber who shaves anyone but himself or the Cretans lying. Despite there is no logical solution to that paradoxes they will somehow be solved in practice without too many trouble (or if we applied „fuzzy logics“ we could formalize stuff or so, idk…). Maybe a kind of solution to it, respectively shedding some light on the mystery the incompleteness theorem seems to imply, comes in a way Cantor „solved“ the mystery of infinite sets – when he made the „paradoxes“ they carry their defining element. There is also this stuff: hyperinfinite sets. They can be constructed, but their existence cannot be proven, and under Occam´s Razor they may seem a nuisance (because they seem to add more orders of the infinite that seems to be needed). Given the incompleteness theorem, the mysterious hyperinfinite sets may either exist or not. However, certain mathematical objects, like knots, can be better conceptualised under the assumption that hyperinfinite sets do exist, be their existence only theoretical (under the assumption of hyperinfinite sets something is possible to construct about the understanding of knots, as an „indirect“ proof that would lead to the possibility of a more direct proof that could eventually do without the assumption of hyperinfinite sets). Apparently, the virtual again. When you think about numbers (and mathematical objects), especially about odd numbers, complex numbers – or negative numbers, or zero, or infinite sets that have puzzled humans for so long, you may become aware that they´re virtual entities.
WIthout the concept of the virtual we´re actually pretty fucked up if we tried to understand the nature of numbers, I guess. It can be argued that numbers are, e.g., platonic, and there are some indications to it, likewise there are other indications that taking them as platonic entitities does not actually apply. With Virtual Reality the notion of the virtual has become somehow more mainstream. Before that it has been prominent within the philosophy of Gilles Deleuze. In thinking and conceptualising about the virtual Deleuze draw on fellow philosopher Henri Bergson – and on Marcel Proust, i.e. neither a philosopher nor a scientist nor a mathematician but a literary genius who, concerning the virtual, was ruminating about how to grasp the qualities of memory. I cannot remember who it was but it was some eminent mathematician who noted that even the most abstract and aloof maths sooner or later is bound to somehow become applicable when trying to get a grasp on something in reality. It is all a gigantic network, hahahahaha.
In his book Infinity and the Mind Rudy Rucker described how it was when he had a personal encounter with Gödel. Despite popular beliefs that he was bizarre the elderly Gödel had, as it seemed to Rucker, the statue of a very wise man who seemed to have thought about everything in life, thoroughly and concise; something that people would also remark about the elderly Wittgenstein. Rucker noted that Gödel had the habit that when completing a sentence or statement he would often exalt his voice and break into a ringing laughter, in an obvious gesture of adding some irony and leaving room for calling into question the things he just stated with such rigid logic and that seemed to be so perfectly concise – bravo, that´s the spirit! Wittgenstein was also so eminent at logics that he used logic for accelerating perplexedness. When the elderly Wittgenstein displayed the profoundly wise man to others the effect was ambiguous, as Wittgenstein on the one hand seemed to have thought about everything, including the more mundane things in life, but would enter a discussion about everything with great intensity, devotion and sternness (including conversations about the more mundane things in life), so that people usually on the one hand felt enriched and that they had received valuable advice but that they sort of had been overrun by a tank on the other hand (conversations with Emily Dickinson seemed to have been of a similar quality). – A while ago I have noticed that Kripke is considered as one of the definitely most important philosophers of the last 200 years. Kripke explained Wittgenstein to a more general population after Wittgenstein´s death. Kripke is an analytical philosopher and so far I have not read much about him. I read however that most of his (more recent) works are lectures and he himself does not seem to care so much about them being published, because his mind is obviously working too fast for caring about such mundane things – bravo, that´s the spirit! Kripke however is silent about many other things a philosopher would be expected to be vocal about. I have read that, in personal encounters, Kripke appears like a very intelligent person, yet something somehow is missing, a certain human element. – I said this about Kripke because as an association it came to my mind, it also somehow fits into this note and it is, apart from that, informative, and I like to inform people about all kind of stuff because I like to get informed about all kind of stuff myself.
This note about numbers may be dilettante, I am not a professional mathematician, I have not thought a lot about it, and I am occupied with doing other things at the moment. But I don´t see an error with conceptualising numbers as virtual entities. So far for now.