# Mathematical Proof

Edited by Jordan Bohall (University of Illinois, Urbana-Champaign)
 Summary Mathematical proof concerns itself with a demonstration that some theorem, lemma, corollary or claim is true. Proofs rely upon previously proven statements, logical inferences, and a specified syntax, which can usually trace back to underlying axioms and definitions. Many of the issues in this area concern the use of purely formal proof, informal proof, language, empirical methodologies, and everyday practice.
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 Introductions Horsten 2008
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1. Mathematical Fit: A Case Study†.Manya Raman-Sundström & Lars-Daniel Öhman - 2016 - Philosophia Mathematica 26 (2):184-210.
Mathematicians routinely pass judgements on mathematical proofs. A proof might be elegant, cumbersome, beautiful, or awkward. Perhaps the highest praise is that a proof is right, that is, that the proof fits the theorem in an optimal way. It is also common to judge that one proof fits better than another, or that a proof does not fit a theorem at all. This paper attempts to clarify the notion of mathematical fit. We suggest six criteria that distinguish proofs as being (...)

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2. Mathematical Justification without Proof.Silvia De Toffoli - forthcoming - In Giovanni Merlo, Giacomo Melis & Crispin Wright (eds.), Self-knowledge and Knowledge A Priori. Oxford University Press.
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 (...)

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3. Proofs for a price: Tomorrow’s ultra-rigorous mathematical culture.Silvia De Toffoli - 2024 - Bulletin (New Series) of the American Mathematical Society 61 (3):395–410.
Computational tools might tempt us to renounce complete cer- tainty. By forgoing of rigorous proof, we could get (very) probable results for a fraction of the cost. But is it really true that proofs (as we know and love them) can lead us to certainty? Maybe not. Proofs do not wear their correct- ness on their sleeve, and we are not infallible in checking them. This suggests that we need help to check our results. When our fellow mathematicians will be (...)

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4. Ratio.Michael Dummett & Philip Tartaglia (eds.) - 1963 - Duckworth.
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5. The Algorithmic-Device View of Informal Rigorous Mathematical Proof.Jody Azzouni - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2179-2260.
A new approach to informal rigorous mathematical proof is offered. To this end, algorithmic devices are characterized and their central role in mathematical proof delineated. It is then shown how all the puzzling aspects of mathematical proof, including its peculiar capacity to convince its practitioners, are explained by algorithmic devices. Diagrammatic reasoning is also characterized in terms of algorithmic devices, and the algorithmic device view of mathematical proof is compared to alternative construals of informal proof to show its superiority.

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6. The Social Epistemology of Mathematical Proof.Line Edslev Andersen - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 2069-2079.
If we want to understand why mathematical knowledge is extraordinarily reliable, we need to consider both the nature of mathematical arguments and mathematical practice as a social practice. Mathematical knowledge is extraordinarily reliable because arguments in mathematics take the form of deductive mathematical proofs. Deductive mathematical proofs are surveyable in the sense that they can be checked step by step by different experts, and a purported proof is only accepted as a proof by the mathematical community once a number of (...)

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7. Introduction to proofs and proof strategies.Shay Fuchs - 2023 - New York, NY: Cambridge University Press.
Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof as they transition to advanced mathematics. Using several strategies, students will develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow.

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8. The nuts and bolts of proofs: an introduction to mathematical proofs.Antonella Cupillari - 2023 - San Diego, CA: Academic Press, an imprint of Elsevier.
The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Fifth Edition provides basic logic of mathematical proofs and shows how mathematical proofs work. It offers techniques for both reading and writing proofs. The second chapter of the book discusses the techniques in proving if/then statements by contrapositive and proofing by contradiction. It also includes the negation statement, and/or. It examines various theorems, such as the if and only-if, or equivalence theorems, the existence theorems, and the uniqueness theorems. In (...)

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9. Taking the "oof!" out of proofs.Alexandr Draganov - 2024 - Boca Raton: CRC Press.
This book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one. Readers will see how various mathematical concepts are tied, will see mathematics is (...)

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10. The Epistemological Subject(s) of Mathematics.Silvia De Toffoli - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 1-27.
Paying attention to the inner workings of mathematicians has led to a proliferation of new themes in the philosophy of mathematics. Several of these have to do with epistemology. Philosophers of mathematical practice, however, have not (yet) systematically engaged with general (analytic) epistemology. To be sure, there are some exceptions, but they are few and far between. In this chapter, I offer an explanation of why this might be the case and show how the situation could be remedied. I contend (...)

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11. Understanding mathematical proof.John Taylor - 2014 - Boca Raton: Taylor & Francis. Edited by Rowan Garnier.
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 techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own. The (...)

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12. The story of proof: logic and the history of mathematics.John Stillwell - 2022 - Princeton, New Jersey: Princeton University Press.
How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the (...)

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13. The meaning of proofs: mathematics as storytelling.Gabriele Lolli - 2022 - Cambridge, Massachusetts: The MIT Press. Edited by Bonnie McClellan-Broussard & Matilde Marcolli.
This book introduces readers to the narrative structure of mathematical proofs and why mathematicians communicate that way, drawing examples from classic literature and employing metaphors and imagery.

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14. An introduction to proof via inquiry-based learning.Dana C. Ernst - 2022 - Providence, Rhode Island: MAA Press, an imprint of the American Mathematical Society.
An Introduction to Proof via Inquiry-Based Learning is a textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry, the book features a highly structured set of leading questions and explorations. The reader is expected to construct their own understanding by engaging with the material. The content ranges over topics traditionally included in transitions courses: logic, set theory including cardinality, the topology of the real line, a bit of number theory, and more. The (...)

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15. Introduction to mathematics: number, space, and structure.Scott A. Taylor - 2023 - Providence, Rhode Island: American Mathematical Society.
This textbook is designed for an Introduction to Proofs course organized around the themes of number and space. Concepts are illustrated using both geometric and number examples, while frequent analogies and applications help build intuition and context in the humanities, arts, and sciences. Sophisticated mathematical ideas are introduced early and then revisited several times in a spiral structure, allowing students to progressively develop rigorous thinking. Throughout, the presentation is enlivened with whimsical illustrations, apt quotations, and glimpses of mathematical history and (...)

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16. Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).
Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...)

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17. Mathematical Hygiene.Andrew Arana & Heather Burnett - 2023 - Synthese 202 (4):1-28.
This paper aims to bring together the study of normative judgments in mathematics as studied by the philosophy of mathematics and verbal hygiene as studied by sociolinguistics. Verbal hygiene (Cameron 1995) refers to the set of normative ideas that language users have about which linguistic practices should be preferred, and the ways in which they go about encouraging or forcing others to adopt their preference. We introduce the notion of mathematical hygiene, which we define in a parallel way as the (...)

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18. Beweistheorie.K. Schütte - 1960 - Berlin,: Springer.
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19. Aspetti della dimostrazione per assurdo.Roberto Senes - 1969 - Trieste,: Tip. Villaggio del fanciullo.
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20. Proof requires a dialogue between agents to clarify obscure inference steps, fill gaps, or reveal implicit assumptions in a purported proof. Hence, argumentation is an integral component of the discovery process for mathematical proofs. This work presents how argumentation theories can be applied to describe specific informal features in the development of proof-events. The concept of proof-event was coined by Goguen who described mathematical proof as a public social event that takes place in space and time. This new meta-methodological concept (...)

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21. Purity and Explanation: Essentially Linked?Andrew Arana - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 25-39.
In his 1978 paper “Mathematical Explanation”, Mark Steiner attempts to modernize the Aristotelian idea that to explain a mathematical statement is to deduce it from the essence of entities figuring in the statement, by replacing talk of essences with talk of “characterizing properties”. The language Steiner uses is reminiscent of language used for proofs deemed “pure”, such as Selberg and Erdős’ elementary proofs of the prime number theorem avoiding the complex analysis of earlier proofs. Hilbert characterized pure proofs as those (...)

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23. Syllogistic Logic and Mathematical Proof.Paolo Mancosu & Massimo Mugnai - 2023 - Oxford, GB: Oxford University Press. Edited by Massimo Mugnai.
Does syllogistic logic have the resources to capture mathematical proof? This volume provides the first unified account of the history of attempts to answer this question, the reasoning behind the different positions taken, and their far-reaching implications. Aristotle had claimed that scientific knowledge, which includes mathematics, is provided by syllogisms of a special sort: 'scientific' ('demonstrative') syllogisms. In ancient Greece and in the Middle Ages, the claim that Euclid's theorems could be recast syllogistically was accepted without further scrutiny. Nevertheless, as (...)

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24. Logical and Semantic Puritiy.Andrew Arana - 2008 - In Gerhard Preyer (ed.), Philosophy of Mathematics: Set Theory, Measuring Theories, and Nominalism. Frankfort, Germany: Ontos. pp. 40-52.

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25. Beweisen im Mathematik-Unterricht: didakt, Anwendungen d. Lehre vom log. Schliessen.Peter Zahn - 1979 - Darmstadt: Wissenschaftliche Buchgesellschaft, [Abt. Verl.].
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26. Lehren des Beweisens im Mathematikunterricht.Walter Witzel - 1981 - Freiburg (Breisgau): Hochschulverlag.

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27. Mathematical Proving as Multi-Agent Spatio-Temporal Activity.Ioannis M. Vandoulakis & Petros Stefaneas - 2016 - In Ioannis M. Vandoulakis & Petros Stefaneas (eds.), Modelling, Logical and Philosophical Aspects of Foundations of Science. Lambert Academic Publishing. pp. 183-200.
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28. Objectivity and Rigor in Classical Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2022 - Noesis 38:195-212.
The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...)

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29. Rigor and the Context-Dependence of Diagrams: The Case of Euler Diagrams.David Waszek - 2004 - In A. Blackwell, K. Marriott & A. Shimojima (eds.), Diagrammatic Representation and Inference. Springer. pp. 382-389.
Euler famously used diagrams to illustrate syllogisms in his Lettres à une princesse d’Allemagne [1]. His diagrams are usually seen as suffering from a fatal “ambiguity problem” [11]: as soon as they involve intersecting circles, which are required for the representation of existential statements, it becomes unclear what exactly may be read off from them, and as Hammer & Shin conclusively showed, any set of reading conventions can lead to erroneous conclusions. I claim that Euler diagrams can, however, be used (...)

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30. Introduction to mathematical proof: a transition to advanced mathematics.Charles E. Roberts - 2015 - Boca Raton: CRC Press, Taylor & Francis Group.
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31. Building proofs: a practical guide.Suely Oliveira - 2015 - New Jersey: World Scientific. Edited by David Stewart.
This book introduces students to the art and craft of writing proofs, beginning with the basics of writing proofs and logic, and continuing on with more in-depth issues and examples of creating proofs in different parts of mathematics, as well as introducing proofs-of-correctness for algorithms. The creation of proofs is covered for theorems in both discrete and continuous mathematics, and in difficulty ranging from elementary to beginning graduate level. Just beyond the standard introductory courses on calculus, theorems and proofs become (...)

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32. Proof theory: sequent calculi and related formalisms.Katalin Bimbó - 2015 - Boca Raton: CRC Press, Taylor & Francis Group.
Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems are, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations completely as an abundant source of meta-logical results. The goal of this book is to provide a fairly comprehensive view of sequent calculi -- including a wide (...)

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33. An introduction to proof through real analysis.Daniel J. Madden - 2017 - Hoboken, NJ: Wiley. Edited by Jason A. Aubrey.
An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis A mathematical proof is an inferential argument for a mathematical statement. Since the time of the ancient Greek mathematicians, the proof has been a cornerstone of the science of mathematics. The goal of this book is to help students learn to follow and understand the function and structure of mathematical proof and to produce proofs of their own. An Introduction to Proof through Real Analysis is based on (...)

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34. A bridge to higer mathematics.Valentin Deaconu - 2017 - Boca Raton: CRC Press, Taylor & Francis Group. Edited by Donald C. Pfaff.
This is an introduction to proofs book for the course offering a transition to more advanced mathematics. It contains logic, sets, functions, relations, the construction of rational, real and complex numbers and their properties. It also has a chapter on cardinality and a chapter on counting techniques. The book explains various proof techniques and has many examples which help with the transition to more advanced classes like real analysis, groups, rings and fields or topology.

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35. Reverse mathematics: proofs from the inside out.John Stillwell - 2018 - Princeton: Princeton University Press.
This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse (...)

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36. A transition to proof: an introduction to advanced mathematics.Neil R. Nicholson - 2018 - Boca Raton: CRC Press, Taylor & Francis Group.
A Transition to Proof: An Introduction to Advanced Mathematics describes writing proofs as a creative process. There is a lot that goes into creating a mathematical proof before writing it. Ample discussion of how to figure out the "nuts and bolts'" of the proof takes place: thought processes, scratch work and ways to attack problems. Readers will learn not just how to write mathematics but also how to do mathematics. They will then learn to communicate mathematics effectively. The text emphasizes (...)

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37. Fundamentals of mathematical proof.Charles A. Matthews - 2018 - [place of publication not identified]: [Publisher Not Identified].
This mathematics textbook covers the fundamental ideas used in writing proofs. Proof techniques covered include direct proofs, proofs by contrapositive, proofs by contradiction, proofs in set theory, proofs of existentially or universally quantified predicates, proofs by cases, and mathematical induction. Inductive and deductive reasoning are explored. A straightforward approach is taken throughout. Plenty of examples are included and lots of exercises are provided after each brief exposition on the topics at hand. The text begins with a study of symbolic logic, (...)

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38. Transition to analysis with proof.Steven G. Krantz - 2018 - Boca Raton: CRC Press/Taylor & Francis Group.
Transition to Real Analysis with Proof provides undergraduate students with an introduction to analysis including an introduction to proof. The text combines the topics covered in a transition course to lead into a first course on analysis. This combined approach allows instructors to teach a single course where two were offered. The text opens with an introduction to basic logic and set theory, setting students up to succeed in the study of analysis. Each section is followed by graduated exercises that (...)

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39. Mathematical proofs: a transition to advanced mathematics.Gary Chartrand - 2018 - Boston: Pearson. Edited by Albert D. Polimeni & Ping Zhang.
For courses in Transition to Advanced Mathematics or Introduction to Proof. Meticulously crafted, student-friendly text that helps build mathematical maturity Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. Written in a student-friendly manner, it provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as optional excursions into fields such as number (...)

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40. Mathematical maturity via discrete mathematics.Vadim Ponomarenko - 2019 - Mineola, NY: Dover Publications.
Geared toward undergraduate majors in math, computer science, and computer engineering, this text employs discrete mathematics to introduce basic knowledge of proof techniques. Exercises with hints. 2019 edition.

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41. Proof complexity.Jan Krajíček - 2019 - New York, NY: Cambridge University Press.
Proof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions, and it favors more generalizable statements. Lower bounds for lengths of proofs, often regarded as the key issue in (...)

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42. 99 Variations on a Proof.Philip Ording - 2018 - Princeton: Princeton University Press.
An exploration of mathematical style through 99 different proofs of the same theorem This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice (...)

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43. An introduction to mathematical proofs.Nicholas A. Loehr - 2020 - Boca Raton: CRC Press, Taylor & Francis Group.
This book contains an introduction to mathematical proofs, including fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The book is divided into approximately fifty brief lectures. Each lecture corresponds rather closely to a single class meeting.

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44. Proof and the art of mathematics.Joel David Hamkins - 2020 - Cambridge, Massachusetts: The MIT Press.
A textbook for students who are learning how to write a mathematical proof, a validation of the truth of a mathematical statement.

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45. The science of learning mathematical proofs: an introductory course.Elana Reiser - 2021 - New Jersey: World Scientific.
College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through difficult (...)

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46. Proofs 101: an introduction to formal mathematics.Joseph Kirtland - 2020 - Boca Raton: CRC Press, Taylor & Francis Group.
Proofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and Linear Algebra. It prepares students for the proofs they will need to analyse and write, the axiomatic nature of mathematics, and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of (...)

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47. Proof and the art of mathematics: examples and extensions.Joel David Hamkins - 2021 - Cambridge, Massachusetts: The MIT Press.
An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs. This book offers an introduction to the art and craft of proof-writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to (...)

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48. Idéaux de preuve : explication et pureté.Andrew Arana - 2022 - In Andrew Arana & Marco Panza (eds.), Précis de philosophie de la logique et des mathématiques, Volume 2, philosophie des mathématiques. Paris: Editions de la Sorbonne. pp. 387-425.
Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

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49. On Different Ways of Being Equal.Bruno Bentzen - 2020 - Erkenntnis 87 (4):1809-1830.
The aim of this paper is to present a constructive solution to Frege's puzzle (largely limited to the mathematical context) based on type theory. Two ways in which an equality statement may be said to have cognitive significance are distinguished. One concerns the mode of presentation of the equality, the other its mode of proof. Frege's distinction between sense and reference, which emphasizes the former aspect, cannot adequately explain the cognitive significance of equality statements unless a clear identity criterion for (...)