How real are real numbers? (2004)

An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders?

I have this concept of "materialization" and wonder if there is a formal mathematical term for it. Complex numbers are actual, in the sense that they can be used in calculations that finally would give us a number we can make sense of (materialization), even if we cannot actually imagine a complex quantity.

In the same way, what if we invent a new class of real numbers that are quantized (such that there exists a smallest quantity x)?

What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?

These mathematical objects do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.

My philosophical take: the halting problem illustratres how free will can exist in a deterministic universe

> "pi" and "e" - among the uncountably infinite transendentals - are probably reasonable responses to this article in it's entirety

pi and e are computable; we even know a bunch of programs which will compute them (to arbitrary precision; Google for "spigot algorithms"). Computable numbers are countable, so we can identify them with the natural numbers, and there's no need for "reals". For example, we might think about the tapes of a binary universal turing machine: we can enumerate them (empty; 0; 1; 00; 01; etc.), and hence the tapes/programs are isomorphic to the natural numbers; the computable numbers are the results of running those programs (this is easier to imagine with a separate "monotone" tape for writing "output").

What about the rest of the "uncountably infinite transcendentals"? I would claim that such an argument is too hand-wavey in the form you've presented it. Can you write down a uniquely determined transcendental number which is uncomputable?

Such numbers can be defined; Chaitin's "omega" is an example. However, it's difficult to decide whether such numbers "exist" or not: we can assume that the physical world "exists", by definition; things which can't exist physically, but can be explored mathematically, like pi and e, can be said to exist, although it's a leap of faith; uncomputable numbers can't really be explored mathematically, or in any way that we know. Can we still say they "exist"?

Consider that the digits of omega: whatever formal system you use to study it, the digits you "discover" are nothing more than a scrambled representation of your formal system's axioms; since the axioms are assumed rather than proven, the "discovery" is actually a circular argument.

> Whitehead & Russell's Principia Mathematica which (and please correct me) used logic to construct the integers

It's easy to construct the integers "using logic". For example, Peano constructed the naturals like this (more or less):

    The symbol '0' is a natural
    The symbol 'S', followed by a natural, is a natural

We can count through Peano's naturals as '0' (0), 'S0' (1), 'SS0' (2), 'SSS0' (3), etc.

We can likewise construct the integers like this:

    A natural, followed by the symbol ':', followed by a natural, is an integer

We can interpret the first natural as positive, the second as negative, and the resulting integer as their sum. Hence '0:0' is 0 (0 + 0), and so is 'SSS0:SSS0' (+3 + -3); 'SSS0:0' is 3, '0:SSS0' is -3, and so on.

It's not too hard to construct something logically. The problem is what assumptions you're willing to make. Whitehead and Russell wanted to prove everything using only the assumptions of set theory; that's why it took them hundreds of pages to prove seemingly trivial theorems about integers.

Goedel's incompleteness theorem(s) showed that no set of assumptions is complete and consistent; hence proving some result using set theory is no better than proving it with some other system. In particular, if it's easier to construct a theory of integers (like the one above) than it is to derive one from set theory (like Whitehead and Russell), we might as well take the easy path :)

I've taken a minor personal interest in some of Ramanujan's more quickly converging sequences for pi. Giving myself a crash course in computable analysis/computable reals. Wish there was more out there about them (particularly limitations vis a vis traditionally defined reals).

> An interesting idea that follows from this: what other kinds of "numbers" might we come up with if we relax our logical blinders?

It depends on exactly what you mean by this, but I would argue that we do this absolutely everywhere. For example, matrices have similar properties to numbers. You can add/subtract them, you can multiply them, you can (sometimes) divide them. Depending upon how you restrict your set of matrices, ab might equal ba or it might be the case that ab makes sense while ba does not (i.e. not only is it not commutative, but it doesn't even make sense to multiply the other way).

> In the same way, what if we invent a new class of real numbers that are quantized (such that there exists a smallest quantity x)?

Here's an example of something similar:

https://en.wikipedia.org/wiki/Dual_number

Basically you're adding in a number smaller than everything else that squares to 0.

> What if we propose the existence an "imaginary" algorithm (call it omega, if you like) that can decide the halting problem?

I'm no logician, but I believe that they would refer to this sort of thing as "model theory". I.e. you're basically taking proofs (sequences of logical statements) and studying different logical systems in which these statements make sense. For example, you could extend your theory/model to take as an axiom that there exists an algorithm to solve the halting problem (though this would be stupid since you can already prove within most theorems that the halting problem is impossible...i.e. you would have a self-contradictory axiomatic system).

> These mathematical object do not have to "exist" for them to be useful. I think there is a whole lot of new and interesting math that would be unlocked by dispensing with our logical blinders.

I 100% agree. Math is a tool that we invent to better understand the world and our thoughts (as well as do more). However, you'll probably have to track down a true logician if you were to go down that rabbit hole...