Mark Burgin’s Contribution to the Foundation of Mathematics

Lev, Felix M.

doi:10.3390/philosophies9010008

Philosophies

2023

DOI: 10.3390/philosophies9010008

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Mark Burgin’s Contribution to the Foundation of Mathematics

Felix M. Lev

Abstract: In this paper, I attempt to describe Mark Burgin’s results in non-Diophantine mathematics, which are important for the foundation of mathematics and its application in quantum field theory. In particular, the elimination of divergences in quantum electrodynamics is described.

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2024

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Refining Mark Burgin’s Case against the Church–Turing Thesis

Daylight

2024

Philosophies

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The outputs of a Turing machine are not revealed for inputs on which the machine fails to halt. Why is an observer not allowed to see the generated output symbols as the machine operates? Building on the pioneering work of Mark Burgin, we introduce an extension of the Turing machine model with a visible output tape. As a subtle refinement to Burgin’s theory, we stipulate that the outputted symbols cannot be overwritten: at step i, the content of the output tape is a prefix of the content at step j, where i<j. Our Refined Burgin Machines (RBMs) compute more functions than Turing machines, but fewer than Burgin’s simple inductive Turing machines. We argue that RBMs more closely align with both human and electronic computers than Turing machines do. Consequently, RBMs challenge the dominance of Turing machines in computer science and beyond.

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Refining Mark Burgin’s Case against the Church–Turing Thesis

Daylight

2024

Philosophies

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Mark Burgin’s Contribution to the Foundation of Mathematics

Abstract: In this paper, I attempt to describe Mark Burgin’s results in non-Diophantine mathematics, which are important for the foundation of mathematics and its application in quantum field theory. In particular, the elimination of divergences in quantum electrodynamics is described.

Cited by 1 publication

References 10 publications

Refining Mark Burgin’s Case against the Church–Turing Thesis

Refining Mark Burgin’s Case against the Church–Turing Thesis

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