Creating something from nothing and formal number theory

A different type of post today. Not a rant by any stretch. Something positively funny, for a change.

I’ll assume I’ll be excused for the the catchy title, designed to lure you into reading this post. However, the actual content isn’t that far from it. For the rest of this post, I’ll just use my memory. I don’t – alas – have my books here with me and I trust not much of information that can be got on the internet. So bear with me.

Something that has always caught my imagination when I studied Mathematical Logic, some thirty years ago now, was the magic behind the formal definition of Natural Numbers (N), 0, 1,2,3 and so on for the uninitiated, or the Positive Integers, for the sophisticated (Z+).
In both the functional (functional theory, anyone of them, Kleene’s recursive functions, Lambda Calculus and what have you) and the extensional definition (formalised set theory, with my mind going to the Zermelo-Fraenkel-Skolem axiomatisation – ZFS), N is defined as the class of objects containing 0 and – in the former flavour (functional) – f(n+1) where n is already in N (your typical mathematical induction definition), or for the extensive approach, zero is “{}”, the set containing no elements whatsoever, with n+1 = {{…}}, where the “…” means n-1 many curly brackets. And nobody is ranting or disputing that here. That’s the foundation of formal number theory in the two existing flavour.

But, over the years, I’ve sometimes gone back with my mind to what has always seemed to me a bit of trickery (something a bit formalistic that is, (David Hilbert), consistent and yet devoid of any actual human meaning) and wondered how in hell it could be that you get something out of nothing. I mean, only the gods are said to be able to do that, right? Then, today, while stirring my home made Béchamel sauce and – admittedly – with a moderate amount of Heineken flowing in my blood, I was actually thinking about that again. Then it struck me. Socrates, goddammit!

So, say you know nothing. That’s the zero.

Then you realise that you know nothing. That’s 1. Socrates’ level. You know that you know nothing.

Then you go on and you know that you know that know nothing. That’s Descartes level, whereby I realise that I’m thinking (since I know that I have thought, that of knowing that I know nothing) and – therefore – that I exist (at least according to Descartes). Now Descartes didn’t say that we’re necessarily existing as physical objects, to my knowledge. We could be like Agent Smith, a program that has become self aware. yet that’s still existence (who said that to exist is to be able to appear as the value of a variable, to instantiate a variable? I think it was Quine who said that, but I might wrong).

And then you go on and on. And now it makes human sense. It’s no longer a formal trickery!

But, wait! Alas! now I realise my fallacy: we did not start from nothing. We started from knowing nothing. That’s already 1 for the functional approach (f(0)).

Back to trickery status then and me trying and figuring this out from a human perspective!