# Mark Burgin’s Contribution to the Foundation of Mathematics

## Abstract

**:**

## 1. Problems in the Foundation of Classical Mathematics

- (a)
- If those numbers are greater than L, then the result equals L. Say that $L=40$, then the experiment will confirm that $10+20=30$, while if $L=25$, then we have that $10+20=25$.
- (b)
- If those numbers are ≥L, then the result equals the standard result but modulo L. Say that $L=40$, then the experiment will confirm that $10+20=30$; if $L=25$, then we have that $10+20=5$.

## 2. Mark Burgin’s Approach to the Problem of the Foundation of Mathematics

**Definition 1.**

**Z**but operations with them are defined in a different way.

**Z**:

**Z**in the formal limit $L\to \infty $. However, as noted above, from the point of view of verificationism, the value of L should be finite. For illustration, Mark considers examples of operations in the arithmetic ${\mathbf{A}}_{L}$, where $L={10}^{100}$. If ⊕, ⊖ and ⊗ are used to denote addition, subtraction, and multiplication in this arithmetic, respectively, then:

**Proposition 1.**

**Theorem 1.**

- (a)
- Addition and multiplication are commutative in the arithmetic ${\mathbf{A}}_{L}$;
- (b)
- Addition in the arithmetic ${\mathbf{A}}_{L}$ is not always associative;
- (c)
- Multiplication in the arithmetic ${\mathbf{A}}_{L}$ is always associative;
- (d)
- Multiplication in the arithmetic ${\mathbf{A}}_{L}$ is not always distributive with respect to addition;
- (e)
- The results of addition, subtraction, and multiplication in the arithmetic ${\mathbf{A}}_{L}$ cannot be greater than L and less than $-L$.

## 3. Elimination of Divergences in Quantum Electrodynamics

## 4. Discussion

- The initial states in A and B are the same;
- All particles in the finite states of A and B have the same momenta and spins but the final state of B contains an additional (soft) photon with very small energy $E=\hslash \omega $.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wigner, E. The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math.
**1960**, 13, 291–306. [Google Scholar] [CrossRef] - Misak, C.J. Verificationism: Its History and Prospects; Routledge: New York, NY, USA, 1995. [Google Scholar]
- Ayer, A.J. Language, Truth and Logic. In Classics of Philosophy; Oxford University Press: New York, NY, USA; Oxford, UK, 1998; pp. 1219–1225. [Google Scholar]
- William, G. Lycan’s Philosophy of Language: A Contemporary Introduction; Routledge: New York, NY, USA, 2000. [Google Scholar]
- Grayling, A.C. Ideas That Matter; Basic Books: New York, NY, USA, 2012. [Google Scholar]
- Popper, K. Stanford Encyclopedia of Philosophy. Available online: https://plato.stanford.edu/entries/popper/ (accessed on 26 December 2023).
- Burgin, M.S. Non-Diophantine Arithmetics or Is It Possible That 2 + 2 Is Not Equal to 4? Ukrainian Academy of Information Sciences: Kyiv, Ukraine, 1997. [Google Scholar]
- Weinberg, S. The Quantum Theory of Fields; Cambridge University Press: Cambridge, UK, 1999; Volume I. [Google Scholar]
- Weinberg, S. Living with Infinities. arXiv
**2009**, arXiv:0903.0568. [Google Scholar] - Burgin, M.; Czachor, M. Non-Diophantine Arithmetics in Mathematics, Physics and Psychology; World Scientific: New York, NY, USA; London, UK; Singapore, 2020. [Google Scholar]
- Burgin, M.S. Non-Classical Models of Natural Numbers. Adv. Math. Sci.
**1977**, 32, 209–210. (In Russian) [Google Scholar] - Rashevsky, P.K. On the Dogma of the Natural Series. Adv. Math. Sci.
**1973**, 28, 243–246. (In Russian) [Google Scholar] - Burgin, M. Introduction to Projective Arithmetics. arXiv
**2010**, arXiv:1010.3287. [Google Scholar] - Burgin, M. Introduction to Non-Diophantine Number. Theory Appl. Math. Comput. Sci.
**2018**, 8, 91–134. [Google Scholar] - Burgin, M. On Weak Projectivity in Arithmetic. Eur. J. Pure Appl. Math.
**2019**, 12, 1787–1810. [Google Scholar] [CrossRef] - Burgin, M.; Lev, F. An Approach to Building Quantum Field Theory Based on Non-Diophantine Arithmetics. Found. Sci.
**2023**, 1–26. [Google Scholar] [CrossRef] - Flynn, M.J.; Oberman, S.S. Advanced Computer Arithmetic Design; Wiley: New York, NY, USA, 2001. [Google Scholar]
- Parhami, B. Computer Arithmetic: Algorithms and Hardware Designs; Oxford University Press: New York, NY, USA, 2010. [Google Scholar]
- Bogolubov, N.N.; Shirkov, N.N. Introduction to the Theory of Quantized Fields; Interscience Publishers: Geneva, Switzerland, 1960. [Google Scholar]
- Lev, F. Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory. With Application to Gravity and Particle Theory; Springer: Cham, Switzerland, 2020; ISBN 978-3-030-61101-9. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lev, F.M.
Mark Burgin’s Contribution to the Foundation of Mathematics. *Philosophies* **2024**, *9*, 8.
https://doi.org/10.3390/philosophies9010008

**AMA Style**

Lev FM.
Mark Burgin’s Contribution to the Foundation of Mathematics. *Philosophies*. 2024; 9(1):8.
https://doi.org/10.3390/philosophies9010008

**Chicago/Turabian Style**

Lev, Felix M.
2024. "Mark Burgin’s Contribution to the Foundation of Mathematics" *Philosophies* 9, no. 1: 8.
https://doi.org/10.3390/philosophies9010008