Mark Burgin’s Contribution to the Foundation of Mathematics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I read the paper with great pleasure and interest. There are many aspects related to the foundation of mathematics and why abstract concepts like continuity, and infinity can be helpful in the analysis of natural phenomena or can lead to various sorts of inconsistency. It was very fascinating to see how Mark Burgin approached these issues from different angles. For example, the application of non-Diophantine finite arithmetic to Feynman's divergent integrals was very impressive. One part especially affected me: if there is no actual infinity in our real world then how our mind (brain), as a part of the real world, can produce infinite theories like ZFC, prove, e.g., the law of prime numbers distribution x/ln(x) for any! x when nobody can compute 10^10^10^...^10s prime number, because the number of elementary particles is less than this number - and this law is verified with a high precision until the computational devices can manipulate with the corresponding numbers. I'm not even talking about the Riemann hypothesis in which infinity plays a significantly crucial role. Is it maybe an infinite machine inside our heads?  However, the assumptions about the existence of an infinite machine inside our brain suffer from paradoxes: from the classical Thomson lamp paradox to paradoxes of predictability, see recent paper "Programming Infinite Machines" https://link.springer.com/article/10.1007/s10670-019-00190-7. (The last paradox, I believe, can probably help in the future solve the Navier-Stokes problem, because all the velocities inside the infinite machine demonstrating paradoxical behavior are finite and can be produced in, probably, a fluid mechanics framework because this framework is rich enough. But, of course, this is only a hypothesis.) I highly recommend this article about Mark Burgin's approaches for publication.

Author Response

I am grateful to Reviewer 1 for his/her comments, and I agree with them. Indeed, ZFC, the law of prime numbers and the Riemann hypothesis involve infinities, but we can verify only statements involving finite numbers. ZFC is the basis of mainstream mathematics but as is clear even from the beginning of Sec. 2, Mark Burgin believed that even entire natural series will not be used in the future fundamental mathematics which will describe fundamental physics. Also, as I note in my response to Reviewer 2, a widely discussed approach in the literature is that the future fundamental physics will be based on finite mathematics.

Reviewer 2 Report

Comments and Suggestions for Authors

see attached file

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The author may want to check the text once more for correct language.  E.g. on p. 3, last paragraph, what's an "existed experience"?

Author Response

see details in the attachment

Author Response File: Author Response.pdf