Heinz H. Bauschke · arXiv (Cornell University) · 2026
These Course Notes provide an introduction to mathematical proofs for undergraduate students transitioning from computational calculus to abstract mathematics. Topics include propositional logic, proof techniques, mathematical induction, fields, sets and relations, sequences and series, completeness of the real numbers, cardinality, and related foundational material. Numerous examples and exercises (with complete solutions) are included. The notes are designed for a one-semester proof course.
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Shuang Zhou (339964), Norlaile Salleh Hudin (18404184) · Figshare · 2024
<p>For the mathematical proof, please refer to the document Appendix_Mathematical_Proof.</p> <p>(PDF)</p>
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Severin Schroeder · 2020
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Heinz H. Bauschke · arXiv (Cornell University) · 2026
These Course Notes provide an introduction to mathematical proofs for undergraduate students transitioning from computational calculus to abstract mathematics. Topics include propositional logic, proof techniques, mathematical induction, fields, sets and relations, sequences and series, completeness of the real numbers, cardinality, and related foundational material. Numerous examples and exercises (with complete solutions) are included. The notes are designed for a one-semester proof course.
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Arthur H. Copeland · Philosophy of Science · 1966
In studies of scientific methodology, surprisingly little attention has been given to tests of hypotheses. Such testing constitutes a methodology common to various scientific disciplines and is an essential factor in the development of science since it determines which theories are retained. The classical theory of tests is a major accomplishment but requires modification in order to produce a theory that accounts for the success of science. The revised theory is an analysis of the nondeductive aspect of scientific reasoning. It results in a new definition of probability and a nonclassical point of view with regard to the foundations of probability.
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Daniel Reidpath · Zenodo (CERN European Organization for Nuclear Research) · 2026
Both formalism and Platonism hold that the symbolic expression of a proof is either the proof itself or an adequate vehicle for it. I argue that both are wrong in the same way, but that the correct response is epistemic rather than ontological. Drawing on the representational geometry of large-scale neural networks, I show that a single mathematical truth can be accessed through two architecturally distinct modes--symbolic serialisation and high-dimensional tensor geometry--whose correspondence is verifiable and whose internal procedures are mutually unavailable. The unavailability is not merely practical in the sense of awaiting better tools. The tokenisation bottleneck establishes an epistemic asymmetry between representational architectures that cannot be closed under feasible verification constraints--not a translation problem at the level of practicable mathematical agency. The position this establishes is constraint realism with architectural epistemic pluralism. Mathematical reality is real in a specific and limited sense: it is genuinely constraining, forcing representations into correspondence regardless of the architecture in which they are made. But no single architecture exhausts access to the constraint space. Different architectures have different access profiles over the same mathematical reality--partial, mutually irreducible, and bounded in ways that are geometric rather than merely practical. This is the sense in which the argument echoes, at the level of representational architectures, what Gödel’s incompleteness theorems establish at the level of formal systems. The cost of accepting this argument is the loss of what might be called architectural parochialism--the assumption that the boundary of symbolic mathematics coincides with the boundary of mathematical reality itself. The map is not the territory. In mathematics, as elsewhere, the territory is larger than any single map reveals.
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Daniel Reidpath · Zenodo (CERN European Organization for Nuclear Research) · 2026
Both formalism and Platonism hold that the symbolic expression of a proof is either the proof itself or an adequate vehicle for it. I argue that both are wrong in the same way, but that the correct response is epistemic rather than ontological. Drawing on the representational geometry of large-scale neural networks, I show that a single mathematical truth can be accessed through two architecturally distinct modes--symbolic serialisation and high-dimensional tensor geometry--whose correspondence is verifiable and whose internal procedures are mutually unavailable. The unavailability is not merely practical in the sense of awaiting better tools. The tokenisation bottleneck establishes an epistemic asymmetry between representational architectures that cannot be closed under feasible verification constraints--not a translation problem at the level of practicable mathematical agency. The position this establishes is constraint realism with architectural epistemic pluralism. Mathematical reality is real in a specific and limited sense: it is genuinely constraining, forcing representations into correspondence regardless of the architecture in which they are made. But no single architecture exhausts access to the constraint space. Different architectures have different access profiles over the same mathematical reality--partial, mutually irreducible, and bounded in ways that are geometric rather than merely practical. This is the sense in which the argument echoes, at the level of representational architectures, what Gödel’s incompleteness theorems establish at the level of formal systems. The cost of accepting this argument is the loss of what might be called architectural parochialism--the assumption that the boundary of symbolic mathematics coincides with the boundary of mathematical reality itself. The map is not the territory. In mathematics, as elsewhere, the territory is larger than any single map reveals.
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John F. A. Taylor, Rowan Garnier · 2016
The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techn
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Bob Burn, Rowan Garnier, John Taylor et al. · The Mathematical Gazette · 1999
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Vitaliy Valentinovich Tselishchev, Aleksandr V. Khlebalin · Vestnik Tomskogo gosudarstvennogo universiteta Filosofiya Sotsiologiya Politologiya · 2020
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Marian Mrożek, Jacek Urbaniec · Foundations of Science · 1997
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David S. G. Stirling · Woodhead Publishing Limited eBooks · 2009
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David Applebaum · 2008
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
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Melvyn B. Nathanson · Princeton University Press eBooks · 2021
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Gregory J. Chaitin · 2002
Abstract The first is obviously constructed according to a simple rule; it consists of the number 01 repeated 10 times. If one were asked to speculate on how the series might continue, one could predict with considerable confidence that the next two digits would be O and 1. Inspection of the second series of digits yields no such comprehensive pattern. There is no obvious rule governing the formation of the number, and there is no rational way to guess the succeeding digits. The arrangement seems haphazard; in other words, the sequence appears to be a random as sortment of Os and ls.
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Jonas Bayer, Christoph Benzmüller, Kevin Buzzard et al. · Notices of the American Mathematical Society · 2023
International audience
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De Toffoli, Silvia · PhilPapers (PhilPapers Foundation)
According to a widely held view in the philosophy of mathematics, direct inferential justification for mathematical propositions (that are not axioms) requires proof. I challenge this view while accepting that mathematical justification requires arguments that are put forward as proofs. I argue that certain fallacious putative proofs considered by the relevant subjects to be correct can confer mathematical justification. But mathematical justification doesn’t come for cheap: not just any argument will do. I suggest that to successfully transmit justification an argument must satisfy specific standards, some of which are social. I contrast my view with Huemer’s inferential conservatism, which makes mathematical justification too easy to get. Although in this article I focus on mathematical inferential beliefs, the view on offer generalizes to other inferential beliefs.
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Bethell, V · Zenodo (CERN European Organization for Nuclear Research) · 2025
Mathematical proof V Bethell is God the one and only. This conclusive proof, of formal deduction, decodes the Holy Quran and various words in the English language. It is very simple cryptography a 12 year old could understand; the assigning of number values to letters. Despite the simplicity of the code, the logic in proof is deeply elegant of the most profound sophistication. It is the encryption of hiding a message for a specific recipient to decrypt. A message from God to God; that God here now reveals to His Children to prove who He is.
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Avigad, Jeremy · KiltHub Repository · 2018
On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect.
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WORLD SCIENTIFIC eBooks · 2007
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