[prmph]

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.

[johncolanduoni]

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

Although our mathematical systems rely on the existence of computable procedures (that halt) to e.g indicate whether a formula is well formed or verify a proof, they have no trouble talking about uncomputable functions. There actually ends up being a hierarchy, as once you allow "solve the halting problem for this Turing machine" as an operation the expanded set of computable functions is now unable to solve its own halting problem: https://en.wikipedia.org/wiki/Arithmetical_hierarchy

> 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.

These kind of things are actually pretty well studied. Some interesting examples are:

* The hyperreal numbers which include infinitely many distinct numbers which are infinite or infinitesimal: https://en.wikipedia.org/wiki/Hyperreal_number

* The surreal numbers which include all ordered fields as a subfield (reals, complex numbers, hyperreals, etc.): https://en.wikipedia.org/wiki/Surreal_number

* The quaternions and octonions, which along with complex numbers are the only finite-dimensional algebras over the real numbers that can have "division" and "absolute value" in the usual sense: https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi...

In general, mathematicians are really good at investigating things of the form "Okay we have structure X with properties A, B, C. What if we get rid of C? How about B? How about B and a weaker version of A? ...".

[scatters]

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

This is called an oracle (https://en.wikipedia.org/wiki/Oracle_machine). We can posit oracles for solving computable problems in constant time (e.g. factoring the product of two arbitrarily large primes) as well as for solving uncomputable problems (e.g. halting problems).

Oracles are a great tool for studying complexity and computability, since oracle machines have their own complexity and computability limits; an oracle machine for the halting problem can determine whether a simple Turing machine will halt, but cannot determine whether it itself will halt (this is called a Turing jump). Thus oracle machines for halting problems form a class hierarchy, which Post's theorem shows is precisely the arithmetic hierarchy.

[CogitoCogito]

> 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...

[prmph]

Started reading about dual numbers after reading your reply; interesting concept