Today, 19 October 2022, I celebrate the 11th birthday of my website. 

Or the Intensive and the Extensive Infinite

„In order to know the nature of the numbers according to which „geometrical distance” to be defined, for example, it would be necessary to know what happens both at indefinitely tiny and indefinitely large distance. Even today, these questions are without clearcut resolution.” (Roger Penrose, The Road to Reality)

Abstract

Today's mathematics is based on the continuum hypothesis (CH) about infinitesimals. Although Cantor formulated CH for extensive infinity, it can also be reformulated for intensive infinity. Thus, we have the foundation of a richer set theory, including classical mathematical analysis, in which the two-element numbers can now model the potential infinities corresponding to our experience, and the actual forms of infinities are captured not by their quantitative but by their special qualitative properties, and are modelled both "small" and "large".

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Briefly on László Mérő's lecture

Abstract

As László Mérő puts it, in the "world of miracles" there is no such thing as probability.  Well, this is a serious mistake. Hard-working hands and hard-working minds are needed to elaborate the already existing but as yet undeveloped mathematics of the world of miracles, which is the world of hyperbolic probabilities.

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Warning, Spoiler!

Quote from the article

I am busy with a large-scale project. In my new article titled "On Intensive and Extensive Infinity", I'm summarising what I know about infinity in small and large. Modelling infinity with imaginary numbers reveals some very interesting things. For example, it turns out, why Abraham Robinson's non-standard analysis did not bring anything new. The reason is that it is not fundamentally different from standard analysis. It is also based on a kind of infinitesimal concept, only in classical analysis this is hidden by the definitions with "epsilon-delta".

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A Very Personal Dream

„There are more things in heaven and earth, Horatio,
than are dreamt of in your philosophy.”
(William Shakespeare, Hamlet)

I have only one consolation as I watch the warfare and hatreds of our human world, which many of us feel is unbelievable in the 21st century. This consolation is the idea of a parallel between the cosmic big bang and the current chaos on earth.

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Gaps in Solutions of Polynomial Equations

„Mathematics is not settled. Even concerning the basic objects of study, like numbers and geometric figures, our ignorance is much greater then our knowledge” (Jordan Ellenberg)¹

The consequences of the modification of the mathematical notion of infinity are manifold, and I have already mentioned several of them, though I have not explained them all in detail. An important property of this notion of infinity is that it is closely related to Cantor's continuum hypothesis (CH) and with it to the axiomatic foundation of set theory, since with a quantified version of CH and its alternatives² we can "produce" three different set theories, similar to the way we arrive at different geometries in geometry depending on the definition of the sums of angles of triangles. A quantified version of CH can be used to unify probability calculus, and three different versions can be axiomatized.³ If all this is not enough, it is necessary to mention that the most elementary relations describing our real spaces can be described by the numerical models of these CH-variations, the two-element numbers, since the multiplication by the parabolic unit vectors models the Galilean transformation, and the multiplication by the hyperbolic unit vector models the Lorentz transformation.⁴

But I have not written about the need for changes to elementary algebra, which also have far-reaching consequences. One of these is the subject of this article.

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¹ Jordan Ellenberg, How Not to Be Wrong: The Power of Mathematical Thinking

² See for example article titled „Linking Hilbert’s 1st and 6th Problems”:
https://www.infinitemath.hu/en/archieve/others/373-linking-hilbert-s-1st-and-6th-problems

³ See article titled „Towards a Universal Probability Theory, Part III (Final part)”;
https://www.infinitemath.hu/en/archieve/mathematics/375-towards-a-universal-probability-theory-part-iii-final-part

⁴ See article titled „The Galilean Transformation and the Parabolic Numbers”;
https://www.infinitemath.hu/en/mathematics/413-the-galilean-transformation-and-the-parabolic-numbers 

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Musings

„From time immemorial, the infinite has stirred men's emotions more than any other question. Hardly any other idea has stimulated the mind so fruitfully. Yet, no other concept needs clarification more than it does.”
(David Hilbert)¹

Contents

1. Introduction
2. Two-element numbers and the qualitative nature of infinity
3. Two-element numbers and the concepts of counting and measurement
4. Conclusions

Appendix – Elementary properties of two-element numbers

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¹ David Hilbert, „On the infinite”,
https://math.dartmouth.edu/~matc/Readers/HowManyAngels/Philosophy/Philosophy.html 

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A subjective look back and forward

Abstract

I see two important elements as essential to transform our approach. On the one hand, the reinterpretation of the concept of infinity brings a paradigm shift that affects the whole of mathematics, and on the other hand, the recognition of the physical nature of information and its classification as a type of energy also brings a paradigm shift in physics, which also affects the whole of physics. These paradigm shifts are still in their infancy, as they have yet to be fully developed and rolled out, but they already offer bright prospects.

Contents

1. Introduction
2. Paradigm shift in mathematics
3. Paradigm shift in physics
4. Quantum mechanics and the hidden variables
5. Conclusion

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Contents

1. Introduction
2. Symmetry of motion
3. Measurement and counting
4. Elementary motions and spacetime
5. Summary

Appendix

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