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Towards a Philosophy of Real Mathematics
David Corfield provides a variety of innovative approaches to research in the philosophy of mathematics. His study ranges from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics to the use of analogy; the prospects for a Bayesian confirmation theory; the notion of a mathematical research program; and the ways in which new concepts are justified. This highly original book will challenge philosophers as well as mathematicians to develop the broadest and most complete philosophical resources for research in their disciplines.
300 pages, Hardcover
First published April 23, 1999
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March 29, 2024
---Notes---
Chapter 1: intro
1.1 real math = what matters for mathematicians
1.2 Lakatos: should be able to study math as it evolves. unlike philosophy of physics, philosophy of math is too detached from mainstream math
1.3 major point that this book argues against: new math is of no philosophical interest (=: foundationalist filter)
category theory : 1. beginning of new foundational language; 2. math never stops evolving at its most foundational level.
1.4 ontological commitment is weird in the case of math - can't distinguish between well-recognized math concepts vs jibberih.
structural realism: science uncovers mathematical structures in the world <- No
inherent structurism: the world may be structured in the way we describes them <- promising
(!) math knowledge is historically situated, not timeless
1.5 connect developments in math with philosophical concerns
- describe how mathematicians work or have worked
- treat issues interior to branches of math, not only as a whole
-> no timelessness, no one-pass history
1.7 factors governing the way math proceeds:
- correctness within some existing calculus
- plausibility
- psychological factors
- sociological and institutional factors
- relation with other sciences
- inherent structures
Chapter 2: investigation of automated theorem proving can shed light on central aspects of math research activities
- computers' problem solving techniques are different from those of humans
- correctness vs explanatory proofs, conceptually illuminating
- go beyong results that are *humanly* provable
-> mathematicians may not be interested in computers' path of computation
-> mathematicians look for new *concepts*, *techniques* and *interpretations*
Chapter 3: automated conjecture formulation
construct a space of hypotheses by *weeding out* failures from a larger space of hypotheses devised by *overarching theoretical concerns*
Chapter 4: analogy in math
metaphors and similes used in informal math talks often reveal somethinga about their conceptualisation
Hilbert: in many situations, finding the "right" way to represent a domain would provide the key to cracking a more extensive one, thereby placing the riginal theory in its proper setting. "In perhaps most cases when we fail to answer a question, the failure is caused by unsolved or insufficiently solved simpler and easier problems. Thus all depends on finding the easier problem and soling it with tools that are as perfect as possible and with notions that are capable of generalization."
important dynamic for math development arises when between two branches there appears an unforseen partial structural analogy
—————begins comments—————
I’m largely in accord with Tim’s NDPR review. First off, Cornfield is one of the few philosophers of math who are adequately proficient in modern mathematics. He combines the best of both worlds - on one hand top-notch expository skills as well as broadness that even few mathematicians possess, and on the other hand fluency in topics and methods central to the development of modern mathematics that few philosophers are capable of. Secondly, this book is an informative guide to the historical and conceptual development of certain mainstream mathematics. I do not, however, regard Cornfield’s focus on mainstream mathematics as a shortcoming. What Tim refers to as “mainstream mathematics” is precisely areas that have been neglected (deliberately or due to incompetency) by the philosophy of mathematics. Of course that has to do with the sociological aspects (c.f. distinction between the expected proficiency in physics (resp. math) for beginning scholars in the philosophy of physics (resp. math). Thirdly, Cornfield fails to relate how modern mathematics may be of concern to philosophy. (To fully assess Tim’s accusation, however, it is necessary distinguish between two notions of “philosophy of mathematics”. One refers to the philosophical ideas internal to the theory of mathematics and to the development of mathematical ideas. This a priori has nothing to do with what philosophers are traditionally interested in, for example, metaphysics, epistemology, ontology etc. The other is of course what is interesting to philosophers. Such concern more often lie external to what may concern mathematicians. Tim, with his philosopher’s hat on, definitely is more interested in the latter. Not only is such criticism perhaps anticipated by the title of Cornfield’s book, but Cornfield also preempts full validy of such a point of view precisely suggesting a way to mediate between the two philosophies of mathematics at the end of Chapter 10.) Most papers in this book lack original insight, although they present in a very, very well manner material (eg. results, conjectures, thinkings of mathematicians) otherwise inaccessible to a lay person in a way fathomable to the greater philosophy community.
Therefore, this book is a great stepping stone for pushing the study of philosophy of mathematics towards a direction more adequately on par with modern mainstream mathematics. In a certain sense, I admire Cornfield’s book for the same reasons Cornfield admires Lakatos, but also see great limitations in his work likewise.
From the point of view of a mathematician, Cornfield’s work is enlightening in two aspects. It is almost exemplary in how one should present frontiers of mathematics in plain English. This is a quality that I’ve seen few living mathematicians possess, in number theory and algebraic geometry at least. To my limited knowledge, it almost seems as if the philosophical guidelines of mathematics have stopped evolving on a broader scale after the emergence of the Langlands Program, although it is definitely true that locally within the Program new ideas and methods have never stopped to develop. After Langlands Program, where will we go? Is mathematics also going towards some sort of grand unifying theory? After that’s done, will the global conceptual map of mathematics be complete? If so, what's next?
------
p. 31 “Instead of seeing mathematical entities and constructions merely as ultimately composed of set theoretic dust, we should take into account structural considerations, rather as the student of anatomy gets little by viewing the human skeleton merely as a deposit of calcium.”
哈哈哈哈哈哈哈哈哈真棒
Chapter 1: intro
1.1 real math = what matters for mathematicians
1.2 Lakatos: should be able to study math as it evolves. unlike philosophy of physics, philosophy of math is too detached from mainstream math
1.3 major point that this book argues against: new math is of no philosophical interest (=: foundationalist filter)
category theory : 1. beginning of new foundational language; 2. math never stops evolving at its most foundational level.
1.4 ontological commitment is weird in the case of math - can't distinguish between well-recognized math concepts vs jibberih.
structural realism: science uncovers mathematical structures in the world <- No
inherent structurism: the world may be structured in the way we describes them <- promising
(!) math knowledge is historically situated, not timeless
1.5 connect developments in math with philosophical concerns
- describe how mathematicians work or have worked
- treat issues interior to branches of math, not only as a whole
-> no timelessness, no one-pass history
1.7 factors governing the way math proceeds:
- correctness within some existing calculus
- plausibility
- psychological factors
- sociological and institutional factors
- relation with other sciences
- inherent structures
Chapter 2: investigation of automated theorem proving can shed light on central aspects of math research activities
- computers' problem solving techniques are different from those of humans
- correctness vs explanatory proofs, conceptually illuminating
- go beyong results that are *humanly* provable
-> mathematicians may not be interested in computers' path of computation
-> mathematicians look for new *concepts*, *techniques* and *interpretations*
Chapter 3: automated conjecture formulation
construct a space of hypotheses by *weeding out* failures from a larger space of hypotheses devised by *overarching theoretical concerns*
Chapter 4: analogy in math
metaphors and similes used in informal math talks often reveal somethinga about their conceptualisation
Hilbert: in many situations, finding the "right" way to represent a domain would provide the key to cracking a more extensive one, thereby placing the riginal theory in its proper setting. "In perhaps most cases when we fail to answer a question, the failure is caused by unsolved or insufficiently solved simpler and easier problems. Thus all depends on finding the easier problem and soling it with tools that are as perfect as possible and with notions that are capable of generalization."
important dynamic for math development arises when between two branches there appears an unforseen partial structural analogy
—————begins comments—————
I’m largely in accord with Tim’s NDPR review. First off, Cornfield is one of the few philosophers of math who are adequately proficient in modern mathematics. He combines the best of both worlds - on one hand top-notch expository skills as well as broadness that even few mathematicians possess, and on the other hand fluency in topics and methods central to the development of modern mathematics that few philosophers are capable of. Secondly, this book is an informative guide to the historical and conceptual development of certain mainstream mathematics. I do not, however, regard Cornfield’s focus on mainstream mathematics as a shortcoming. What Tim refers to as “mainstream mathematics” is precisely areas that have been neglected (deliberately or due to incompetency) by the philosophy of mathematics. Of course that has to do with the sociological aspects (c.f. distinction between the expected proficiency in physics (resp. math) for beginning scholars in the philosophy of physics (resp. math). Thirdly, Cornfield fails to relate how modern mathematics may be of concern to philosophy. (To fully assess Tim’s accusation, however, it is necessary distinguish between two notions of “philosophy of mathematics”. One refers to the philosophical ideas internal to the theory of mathematics and to the development of mathematical ideas. This a priori has nothing to do with what philosophers are traditionally interested in, for example, metaphysics, epistemology, ontology etc. The other is of course what is interesting to philosophers. Such concern more often lie external to what may concern mathematicians. Tim, with his philosopher’s hat on, definitely is more interested in the latter. Not only is such criticism perhaps anticipated by the title of Cornfield’s book, but Cornfield also preempts full validy of such a point of view precisely suggesting a way to mediate between the two philosophies of mathematics at the end of Chapter 10.) Most papers in this book lack original insight, although they present in a very, very well manner material (eg. results, conjectures, thinkings of mathematicians) otherwise inaccessible to a lay person in a way fathomable to the greater philosophy community.
Therefore, this book is a great stepping stone for pushing the study of philosophy of mathematics towards a direction more adequately on par with modern mainstream mathematics. In a certain sense, I admire Cornfield’s book for the same reasons Cornfield admires Lakatos, but also see great limitations in his work likewise.
From the point of view of a mathematician, Cornfield’s work is enlightening in two aspects. It is almost exemplary in how one should present frontiers of mathematics in plain English. This is a quality that I’ve seen few living mathematicians possess, in number theory and algebraic geometry at least. To my limited knowledge, it almost seems as if the philosophical guidelines of mathematics have stopped evolving on a broader scale after the emergence of the Langlands Program, although it is definitely true that locally within the Program new ideas and methods have never stopped to develop. After Langlands Program, where will we go? Is mathematics also going towards some sort of grand unifying theory? After that’s done, will the global conceptual map of mathematics be complete? If so, what's next?
------
p. 31 “Instead of seeing mathematical entities and constructions merely as ultimately composed of set theoretic dust, we should take into account structural considerations, rather as the student of anatomy gets little by viewing the human skeleton merely as a deposit of calcium.”
哈哈哈哈哈哈哈哈哈真棒
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