Mathematics (3)
Acceleration-free Motion Models on Two-element Number Planes
„Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.”
(Jordan Ellenberg, How not to be wrong)
I started to clarify the concepts of differential and integral calculus on two-element numbers, with the exception of complex numbers, of course, since its analysis is a well-developed area of mathematics. The article on the calculus was getting too long, so I will summarise its background, and more specifically the important connections between the two-element numbers – which form the basis for their analysis – in this paper. Some of the relations presented here have already been mentioned in my previous articles, but now I have listed the mathematical foundations in the context of differential and integral calculus, in preparation for it, adding new ones and their interpretation.
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Limits and Continuity on Two-element Numbers
“„One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory.””
(Einstein , The meaning of relativity)
The problem with mathematical continuity is similar to the problem with infinity; continuity in reality is very different from the concept defined in mathematics. Just as we do not experience a quantitative infinite that actually exists, we cannot experience continuity in the sense that mathematics defines it, and the two are causally related.
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Is the existence of transcendental numbers an actual existence?
Contents
- Introduction
- The potential and the actual existence
- The intensive and the extensive infinite
- The new concept of qualitative infinity
- Duality of extensive and intensive infinity in qualitative infinity
- The actual existence of transcendental numbers
- The actual existence of transcendental numbers in positional numeral systems
- The actual existence and geometry
- Continuity and infinity
- An artificial intelligence (AI) answer to the question of the actual existence of transcendental numbers
- The idealization of "arbitrarily large finite" with potential infinity
- Summary
Appendix A- Elementary properties of two-element numbers
Appendix B - Two-element numbers and homogeneous coordinates
Appendix C - Axiomatisation of set theory with the new qualitative infinite
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