User:Garamond Lethe/sandbox/18 Unconventional Essays on the Nature of Mathematics

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18 Unconventional Essays on the Nature of Mathematics is an eclectic compilation of writing selected by Reuben Hersh based loosely around several issues in Philosophy of Mathematics. The authors range from mathematicians and philosophers to a computer scientist, a sociologist and a cognitive scientists. Topics range from traditional foundational questions to the practice of mathematics and the intersection and influence of mathematics, philosophy and science. The selection of the essays was based solely on those that happened to catch Hersh's eye. While there is no overarching theme or organization, the works express provocative, original ideas that, while not outside of the mainstream of philosophy of mathematics, are unapologetically minority opinions.

Background[edit]

Reuben Hersh received his Ph.D. in Mathematics at New York University and began writing about philosophical issues in mathematics in the 1970s, beginning with "Introducing Imre Lakatos" (1978)[citation needed] and "Some proposals for reviving the philosophy of mathematics" (1979)[citation needed]. In 1981 he co-authored (with Philip J. Davis) the award-winning The Mathematical Experience, which (in part) categorized existing philosophy of mathematics into three broad categories: platonism, formalism and constructivism.[citation needed]{{efn|More recent scholarship has created expanded taxonomies. For example, the Stanford Encyclopedia of Philosophy lists four major schools that were formed in the 19th century as a reaction against Platonism: Logicism, Intuitionism, Formalism and Predictivism.

In any discussion of the foundations of mathematics, three standard dogmas are presented: Platonism, formalism and constructivism.[1]: 356 

According to Monk, the mathematical world is populated with 65% Platonists, 30% formalists, and 5% constructivists. Our own impression is that the Cohen-Dieudonné picture is closer to the truth. The typical mathematician is both a Platonist and a formalist—a secret Platonist with a formalist mask that he puts on when the occasion calls for it. Constructivists are a rare breed, whose status in the mathematical world sometimes seems to be that of tolerated heretics surrounded by orthodox members of an established church.[1]: 360 

Most writers on the subject seem to agree that the typical working mathematician is a Platonist on weekdays and a formalist on Sundays. That is, when he is doing mathematics he is convinced that he is dealing with an objective reality whose properties he is attempting to determine. But then, when challenged to give a philosophical account of this reality, he finds it easiest to pretend that he does not believe in it after all.[1]: 359 

This books comes from the Internet. Browsing the Web, I stumbled on philosophers, cognitive scientists, sociologists, computer scientists, even mathematicians!—saying original, provocative things about mathematics. And many of these people had probably never heard of each other! So I ahve collected them here. This way, they can read each other's work. I also bring back a frew provocative oldies that deserve publicity.[2]: vii 

Critical Reception[edit]

1. Brief, enthusiastic review.[3]

2.

Proofs and Reutations was intended for philosophers of mathematics to be cognizant of the historical development of ideas. Yet, its popularizaton by Reuben Hersh (and Philip Davis)gradually led to the development of the so called"maverick" traditions" in the philosophy ofmathematics, culminating in the release of ReubenHersh's (2006) book 18 Unconventional Essays onthe Nature of Mathematics—a delightfulcollection of essays written by mathematicians,philosophers, sociologists, a cognitive scientistand a computer scientist. These essays are scattered "across time" in the fact that Herscollected various essays written over the last 60 years that support the "maverick" viewpoint. His book questions what constitutes a philosophy of mathematics and re-examines foundational questions without getting into Kantian, Quinean or Wittgensteinian linguistic quagmires. [4]

3. Full review.[5]

Mathematics is the invention and investigation of formal patterns, and good mathematics is the invention and investigation of deep and beautiful formal patterns. Let us call physics that portion of science that can be described, to a great extent, by a formal pattern, and call the rest of science biology. Then by definition mathematics is successful in physics.[5]

4. MAA Review[6]

5.

This is a very appealing collection of essays. Readers will not remain indifferent to any one of them, like it or hate it. I highly recommend this book to those wondering how math is carried out, both mathematicians and laymen. I also recommend it to educators interested in changing the dominant view of math and how to do math. Computing Reviews,[7]


6. MT&L,[8]

7. ZfDdM.[9]

The Essays[edit]

A Socratic Dialogue on Mathematics[10][edit]

The first of our eighteen articles is Alfréd Rényi's "..." (Dialogues on Mathematics, Holden-Day, 1967.) Rényi was a famous probabilist and nmber theorist, co-creator with Pal Erdos of the subject random graphs[sic], and for many years director of the Institute of Mathematics in Budapest. This is a most inviting, charming and thought-provoking tour de force and jeu d'esprit. It poses the basic problem, and answers it in a way that invites further questioning and deeper development.[2]: xi–xii 

"Introduction" to Filosofia e matematica[11][edit]

The next article, by the logician-philosopher Carlo Cellucci of Rome, is the introductory chapter to his book, Filosofia e matematica. He simply lists 13 standard assumptions about mathematics (what he calls "the dominant view") and demolishes all of them. A most impressive and stimulating performance.[2]: xii 

On Proof and Progress in Mathematics[12][edit]

William Thurston's friendly, down-to-earth article, "..." provides a rare, invaluable glimpse for outsiders at some aspects of mathematical cration at the highest level. Its frank, unpretentious look at what really is done, what really happens at that level is told in a style and language accessible to anyone. It was published in the Bulletin of the AMS, one of the responses to the Jaffe-Quinn proposal mentioned above.[2]: xii 

The Informal Logic of Mathematical Proof[13][edit]

The U.S.-based English philosopher Andrew Aberdein's article, "...." draws on "informal logic," a subject that was revived by Stephen Toulmin. "Informal logic" is closely allied to "rhetoric". An old article by Phil Davis and myself called "Rhetoric and Mathematics" may be relevant to Aberdein's article. (It appeared in The Rhetoric of the Human Sciences, edited by John S. Nelson, Allan Megill and Donald N. McCloskey, University of Wisconsin Press, 1987 and also as a chapter in our book, Descartes' Dream.)[2]: xii 

Philosophical Problems of Mathematics in the Light of Evolutionary Epistoemology[14][edit]

The article by the Israeli-French mathematician Yehuda Rav is "...". He shows that the human ability and inclination to mathematize can be understood as the result of natural selection. It is necesarry and advantageous for our survival as a species. It was first published in the journal Philosophica, and reprinted in the anthology Math WOrlds: Philosophical and social Studies of Mathematics and Mathematics Education...[2]: xii 

Towards a Semiotics of Mathematics[15][edit]

The English-American mathematician-turned cognitive scientist, Brian Rottman, provides a surprising insight into mathematics in the language of semiotics. His article, "..." clarifies what you do when you write mathematics. Three diffrent personae participate: first of all, ther eis the disembodied pure thinker, the impersonal voice who calls himself "we." Secondly, ther eis also an imaginary automaton, who in imagination ("in principle") carries out any calculations or algorithms that "we" mention. ANd yes, there is also an actual live flesh-and-blood human being, who is sitting in your chair. This article first appeared in Semiotica 72-1/2(1988).[2]: xii 

Computers and the Sociology of Mathematical Proof[16][edit]

Donald Mackenzie's article, "..." gives a detailed history of the computer scientist's search for program correctness, and thereby shines a searchlight on the notion of mathematical certainty. It was presented at a conference at the University of Roskilde, Denmark, in 1998, whose proceedings were published as New Tends in the History and Philosophy of Mathematics, University Press of Southern Denmark, 2004.[2]: xiii 

From G. H. H. and Littlewood to XML and Maple[2]: 147–159 [edit]

Do Real Numbers Really Move? Language, Thought and Gesture: The Embodied Cognitive Foundations of Mathematics[17][edit]

Does Mathematics Need a Philosophy?[2]: 182–200 [edit]

How and Why Mathematics is Unique as a Social Practice[18][edit]

The Pernicious Influence of Mathematics upon Philosophy[19][edit]

The Pernicious Influence of Mathematics on Science[20][edit]

What Is Philosophy of Mathematics Looking For?[21][edit]

Concepts and the Mangle of Practice Constructing Quaternions[22][edit]

Mathematics as Objective Knowledge and as Human Practice[23][edit]

The Locus of Mathematical Reality: An Anthropological Footnote[24][edit]

Inner Vision, Outer Truth[2]: 320–326 [edit]

References[edit]

  1. ^ a b c Davis, P. J.; Hersh, R.; Marchisotto, E. A. (2012), The Mathematical Experience: Study Edition, Springer, doi:10.1007/978-0-8176-8295-8_7, ISBN 978-0-8176-8294-1
  2. ^ a b c d e f g h i j k Hersh, Reuben, ed. (2006), 18 Unconventional Essays on the Nature of Mathematics (PDF), Springer Science+Business Media, ISBN 978-0387-25717-4
  3. ^ Häggström, Olle (2007), Sriraman, Bharath (ed.), "Objective Truth Versus Human Understanding in Mathematics and Chess", The Montana Mathematics Enthusiast, 4 (2): 140–153, ISBN 9787774569278, ISSN 1551-3440 {{citation}}: Unknown parameter |month= ignored (help)
  4. ^ Sriraman, Bharath; English, Lyn (2010), "Surveying Theories and Philosophies of Mathematical Education", in Sriraman, Bharath; English, Lyn (eds.), Theories of Math Education: Seeking New Frontiers, Springer-Verlag, p. 11, ISBN 978-3-642-00742-2
  5. ^ a b Nelson, Edward (2007), "Review: 18 Unconventional Essays on the Nature of Mathematics" (PDF), American Mathematical Monthly, Mathematical Association of America {{citation}}: External link in |journal= (help); Unknown parameter |month= ignored (help)
  6. ^ Satzer, William J. (2006), Review: 18 Unconventional Essays on the Nature of Mathematics, Mathematical Association of America Reviews {{citation}}: Unknown parameter |month= ignored (help)
  7. ^ Zenil, Hector (2008), "Review: 18 Unconventional Essays on the Nature of Mathematics", Computing Reviews(subscription required) {{citation}}: External link in |journal= (help); Unknown parameter |month= ignored (help)
  8. ^ Sriraman, Bharath (2007), "Beyond Traditional Conceptions of the Philosophy of Mathematics", Mathematical Thinking and Learning, 9 (2): 173–178, doi:10.1080/10986060709336814, S2CID 121148422
  9. ^ Sriraman, Bharath (2006), "Bridging the communities of mathematics and mathematics education: Is reconceptualizing the philosophy of mathematics the answer?", Zentralblatt für Didaktik der Mathematik, 38 (4): 361–365, doi:10.1007/BF02652796
  10. ^ Rényi, Alfréd (1967), Dialogues on Mathematics, Holden-Day, OCLC 712329, reprinted in Hersh pages 1–16.
  11. ^ Cellucci, Carlo (2002), "Introduction", Filosofia e matematica, Laterza, ISBN 9788842067665, translation of a revised version of this introduction printed in Hersh, pages 17–36.
  12. ^ Thurston, William (1994), "Proof and Progress in Mathematics" (PDF), Bulletin of the American Mathematical Society, 30 (2): 161–177, arXiv:math/9404236, Bibcode:1994math......4236T, doi:10.1090/S0273-0979-1994-00502-6 {{citation}}: Unknown parameter |month= ignored (help), reprinted in Hersh pages 37–55
  13. ^ Aberdein, Andrew (2007), "The Informal Logic of Mathematical Proof", Perspectives On Mathematical Practices, Logic, Epistemology, and the Unity of Science, vol. 5, pp. 135–151, arXiv:math/0306298, doi:10.1007/1-4020-5034-8_8, ISBN 978-1-4020-5033-6, S2CID 773550, reprinted in Hersh pages 56–70.
  14. ^ Rav, Yehuda (1992), "Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology", Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education, State University of New York Press, pp. 80–112, ISBN 978-0-7914-1330-2, OCLC 25282846, reprinted in Hersh pages 71–96
  15. ^ Rotman, Brian (1988), "Towards a Semiotics of Mathematics", Semiotica, 72 (1––2): 1–36, doi:10.1515/semi.1988.72.1-2.1, ISSN 0037-1998, S2CID 201699632, reprinted in Hersh pages 97–127
  16. ^ MacKenzie, Donald (1998), "Computers and the Sociology of Mathematical Proof", Proceedings of the 3rd BCS–FACS Conference on Northern Formal Methods: 13–{{citation}}: CS1 maint: extra punctuation (link), reprinted in Hersh pages 128–146
  17. ^ Núñez, Rafael (2004), "Do Real Numbers Really Move? Language, Thought and Gesture: The Embodied Cognitive Foundations of Mathematics", Embodied Artificial Intelligence, Lecture Notes in Computer Science, vol. 3139, Springer, pp. 54–73, doi:10.1007/978-3-540-27833-7_4, ISBN 978-3-540-22484-6, reprinted in Hersh pages 160–181
  18. ^ Azzouni, Jody (2007), "How and Why Mathematics is Unique as a Social Practice", Perspectives On Mathematical Practices, Logic, Epistemology, and the Unity of Science, vol. 5, pp. 3–24, arXiv:math/0306298, doi:10.1007/1-4020-5034-8_8, ISBN 978-1-4020-5033-6, S2CID 773550, reprinted in Hersh pages 201–219.
  19. ^ Rota, Gian-Carlo (1997), Palombi, Fabrizio (ed.), The Pernicious Influence of Mathematics upon Philosophy, Birkhäuser, pp. 89–103, ISBN 978-0-8176-3866-5, reprinted in Hersh pages 220&ndash230
  20. ^ Schwartz, Jack (1966), "The Pernicious Influence of Mathematics on Science", Studies in Logic and the Foundations of Mathematics, 44: 356–360, doi:10.1016/S0049-237X(09)70603-2, ISBN 9780804700962, reprinted in Hersh pages 231–235
  21. ^ Ávila del Palacio, Alfonso C. (1997). "What Is Philosophy of Mathematics Looking For?". Revista de la Universidad Autonoma de Chihuahua. {{cite book}}: |journal= ignored (help), revised and translated version appears in Hersh, pages 236&ndash&249
  22. ^ Pickering, Andrew (1995), "Concepts: Constructing Quarternions", The Mange of Practice: Time, Agency and Science, University of Chicago Press, pp. 113–156, ISBN 978-0-2266-6803-1, reprinted as "Concepts and the Mangle of Practice Constructing Quaternions" in Hersh, pages 250–288
  23. ^ Glas, Eduard (2007), "Mathematics as Objective Knowledge and as Human Practice", Perspectives On Mathematical Practices, Logic, Epistemology, and the Unity of Science, vol. 5, pp. 25–42, arXiv:math/0306298, doi:10.1007/1-4020-5034-8_8, ISBN 978-1-4020-5033-6, S2CID 773550, reprinted in Hersh pages 289–303.
  24. ^ White, Leslie A. (1947), "The Locus of Mathematical Reality: An Anthropological Footnote", Philosophy of Science, 14 (4): 289–303, doi:10.1086/286957 {{citation}}: Unknown parameter |month= ignored (help), reprinted in Herst, pages 304–319
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