Choices of Axiomatic and Number Systems / Sets and Alternatives

[Jarvis323]

I know that the number systems we use are typically constructed from axiomatic set theory, and overall our choices along the way seam to have been largely informed by practical consideration (e.g. to resolve ambiguities, or do away with limitations).

Today I randomly started to think deeper about number systems and philosophy. Here are the thoughts/questions I am pondering.

Why do we not typically (or primarily) use hyperreals, hyperintegers, hypernaturals, etc. Why do we often just say infinity=infinity, and in what instances such a choice might fail us, or limit us? For example, explicit numbers have structure, they have or lack embedded patterns. Why choose to strip such potential properties from infinitely large or infinitesimal numbers? Shouldn't there be some cases where this might be useful? Could it be possible that such 'objects' exist in the physical universe, or that properties of the universe could require such tools to properly describe?

Why should numbers be represented by linear/directional orderings of symbols in the first place. Why must they have a first symbol, but not necessarily a last? Why not allow them to expand infinitely at either the beginning, middle, or end (or do we in some cases)? For example, a number that can only be described by an infinite number of symbols, yet has a start and end? Or why the concept of first and last should even apply to numbers in the most pure sense?

Are we humanizing mathematics (or making it feasible for humans to comprehend) to a degree that is useful, but unrealistic? Should we assume the mathematical universe in its most pure form resembles the mathematical universe we have constructed, and should we assume that the mathematical universe we have constructed accurately represents reality? Is such a form of mathematics that is designed to be human interpretable likely to be powerful enough to describe it? Could we develop a more "natural" and powerful axiomatic system that can better tackle difficult theoretical physics problems?

For example, (and this is not a serious suggestion just a hint at what I'm getting at) rather than using set theory to define axioms, and then building sets of numbers the way we do, maybe we could do something like construct infinite dimensional geometric objects in a similar way, except such that the objects we use like numbers, each have some inherent unique structure and complexity explicitly attatched to them.

There must be many possible alternative mathematics systems that would seam completely alien to us. Maybe there can be two choices, one which is intuitive and practical at the start but builds into something much more complex, and another which begins complex and unintuitive but ultimately allows for a simpler and more intuitive description of higher level concepts. I.e., maybe axiomatic systems / systems of mathematics can be considered to have complexity scaling properties so to speak, or even asymptotic complexity scales that fundamentally limit human participation at certain levels.

Hope this question isn't too far from the mainstream, I am not trying to claim anything and hope to not encourage too much speculation or crackpottery. Some of these questions might be silly. But at least some of these questions must have been discussed seriously in some academic circles, I just don't know how to easily find it and see the big picture.

 

[nomadreid]

First, a number of your questions are "why don't we do X?" when we do X. Examples: if you study Model Theory, you will definitely not go around saying infinity=infinity; what you can do with the existence of measurable cardinals is much different from what you can do with just assuming inaccessibles exist. Secondly, linear order is not assumed: yes, it appears (albeit there is no proof) that the (heretofore defined) large cardinal axioms are linearly ordered by consistency strength, but there are lots of other aspects of the infinite cardinals which are not linearly ordered. But you don't have to go to infinities: partial orders that are not linear orders pop up all over the place; a good example is in the lattices used to define truth values. Then, as far as weird exceptions, they are not ignored: this is why the expressions such as "almost all", "significant", "under the standard model", and so forth exist. It is true that we focus on the theories that have the best chance of being useful -- after all, there is funding to think about, as well as the corollary of competition for publishing approvals -- but looking through mathematics journals, one realizes that the exceptions are out there -- mathematics for mathematics' sake. My favourite passage concerning this is from Terry Pratchett's "Small Gods": what he says aboutthe philosophers can also be said about mathematicians in fields of pure mathematics:
"What's a philosopher?" said Brutha.
"Someone who's bright enough to find a job with no heavy lifting," said a voice in his head.
"An infidel seeking the just fate he shall surely receive,' said Vorbis. "An inventor of fallacies. This cursed city attracts them like a dung heap attracts flies."
"Actually, it's the climate," said the voice of the tortoise. "Think about it. If you're inclined to leap out of your bath and run down the street every time you think you've got a bright idea, you don't want to do it somewhere cold. If you do do it somewhere cold, you die out. That's natural selection, that is. Ephebe's known for its philosophers. It's better than street theater."
"What, a lot of old men running around the streets with no clothes on?" said Brutha, under his breath, as they were marched onward.
"More or less. If you spend your whole time thinking about the universe, you tend to forget the less important bits of it. Like your pants. And ninety-nine out of a hundred ideas they come up with are totally useless."
"Why doesn't anyone lock them away safely, then? They don't sound much use to me," said Brutha.
"Because the hundredth idea," said Om, "is generally a humdinger."
"What?"
"Look up at the highest tower on the rock."
Brutha looked up. At the top of the tower, secured by metal bands, was a big disc that glittered in the morning light.
"What is it?" he whispered.
"The reason why Omnia hasn't got much of a fleet any more," said Om. "That's why it's always worth having a few philosophers around the place. One minute it's all Is Truth Beauty and Is Beauty Truth, and Does a Falling Tree in the Forest Make a Sound if There's No one There to Hear It, and then just when you think they're going to start dribbling one of 'em says, Incidentally, putting a thirty-foot parabolic reflector on a high place to shoot the rays of the sun at an enemy's ships would be a very interesting demonstration of optical principles," he added. "Always coming up with amazing new ideas, the philosophers. The one before that was some intricate device that demonstrated the principles of leverage by incidentally hurling balls of burning sulphur two miles. Then before that, I think, there was some kind of an underwater thing that shot sharpened logs into the bottom of ships."
(with gratitude to http://www.cocoa.uk.com/v1/text/smallgods.html)

 

[fresh_42]

First, a number of your questions are "why don't we do X?" when we do X. Examples: if you study Model Theory, ...

... and I like to mention the variety of logical systems beside the cardinals: https://en.wikipedia.org/wiki/Formal_system and the links therein, but even more https://ncatlab.org/nlab/show/Science+of+Logic.

So there is a lot going on outside the mainstream, e.g. three valued logical systems.