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.

After registration, you may ask for a translation of the full text by email.

On Freedom and Free Will again

I've already written an article about freedom and free will, but I've been listening to so many conversations on the internet about this topic lately that I have to write about it again. Of these conversations, I will only mention the name of László Bernáth, because at last I have a contemporary Hungarian philosopher with whom I agree on more things than I disagree. However, I will not discuss philosophical approaches in my writing, but I will outline my own world view on the basis that I would like to grasp the whole of existence in my thinking, including the functioning of free will.

After registration, you may ask for a translation of the full text by email.

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.

After registration, you may ask for a translation of the full text by email.

Coordinate Systems and Spacetime

“It does not do harm to the mystery to know a little about it. For far more marvelous is the truth than any artists of the past imagined!”
(Richard Feynman)¹

Although the article contains many and deep mathematical connections, it also includes many interpretations and explanations, which may make the old/new concepts of number and infinity in mathematics understandable to those who are not involved in mathematics, and reveal the mystical beauty of mathematical connections.

_____________________________________

¹ Richard Feynman, The Feynman Lectures on Physics Vol. I Ch. 3 / Footnote 3-4 Astronomy

After registration, you may ask for a translation of the full text by email.

From my Diary, 23.01.2024.

I listened to the latest conversation¹ of the podcast Mindenségit! with brain researcher Balázs Hangya while doing housework today.

I would like to highlight one topic in the discussion, the one that science has no real theories on yet, so I will write a few words about consciousness.

_____________________________

¹ Balázs Hangya: Brain research, brain function, learning, consciousness, emotions / Mindenségit!
https://www.youtube.com/watch?v=e7U04VjmfHE 

After registration, you may ask for a translation of the full text by email.

Is the existence of transcendental numbers an actual existence?

Contents

1. Introduction
   1. The potential and the actual existence
   2. The intensive and the extensive infinite
2. The new concept of qualitative infinity
3. Duality of extensive and intensive infinity in qualitative infinity
4. The actual existence of transcendental numbers
   1. The actual existence of transcendental numbers in positional numeral systems
   2. The actual existence and geometry
   3. Continuity and infinity
   4. An artificial intelligence (AI) answer to the question of the actual existence of transcendental numbers
5. The idealization of "arbitrarily large finite" with potential infinity
6. 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

After registration, you may ask for a translation of the full text by email.

„Life without holidays is a long way without an inn.”
/Democritus/

Today, 19 October 2023, I celebrate the 12th birthday of my website.

As a joke, I asked Bing to write a little birthday poem. I quote only one part from the poem, leaving out the parts about me:

„May this birthday be a special one,
Filled with joy and cheer.
May your site continue to grow,
And bring happiness far and near.”
/Bing/

Comment on a CERN experiment

Although the new test results suggest that the scientists' conclusions are at odds with my mathematical model¹ I have for antigravity between clay and antimatter, I was pleased to read the reports in Nature² on 27 September 2023 - and in Scientific American³ online on - of a CERN experiment using antihydrogen atoms to test the behaviour of antimatter in the Earth's gravitational field. I was happy to see the tests because, as beautiful as a theory is, experience has the final word.

____________________________

¹ See the article titled „Chatting with Bing about the direction of matter-antimatter gravity”;
Chatting with Bing about the direction of matter-antimatter gravity (infinitemath.hu)

² See Antimatter falls down, not up: CERN experiment confirms theory (nature.com)

³ See What Happens if You Drop Antimatter? New Gravitational Test Sees First Fall - Scientific American

After registration, you may ask for a translation of the full text by email.

Reflections on the Informative and the Deceptive Similarity

In physics - and in the natural sciences in general - similarity models are methods that exploit the similarity of certain properties of objects or processes, and can be useful to better understand a less well known object or process, assuming that other unknown properties can be inferred from the similarity. Of course, the similarity itself may be wrong, but even if it is true, we may not be able to infer other similarities well. Nevertheless, the recognition of similarities is one of the most important scientific methods of gaining knowledge, alongside the search for related symmetries.

I'm not thinking about deceptive similarities now, I'm more interested in informative similarities, and among them those that are clearly visible in a given case, but which we overlook or, if we do notice them, think are irrelevant, perhaps deceptive.

After registration, you may ask for a translation of the full text by email.

Mathematical Magic, Magical Mathematics

"Mathematics is like a beautiful garden. It is full of beautiful flowers, which, if we take care of them, we can witness wonderful magic."
(Rózsa Péter)

Ever since Euler, we have admired the relationship named after him:

e^iπ=-1

It seems almost mysterious how we can get an integer from the operation between two transcendental numbers: e and π, and the imaginary number i. It has long been known – to anyone fascinated by mathematics – that the above formula is the exponential form of -1 in the complex numbers. Paul J. Nahin has devoted a whole book¹ to this fabulous formula of Euler.

_______________________________

¹ Dr. Euler’s Fabulous Formula – Cures Mathematical Ills, Paul J. Nahin, Princeton University Press, 2006.

After registration, you may ask for a translation of the full text by email.