- #1

- 309

- 179

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.

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.

Last edited: