Why Is There Philosophy of Mathematics At All?

✍ Scribed by Ian Hacking

• Publisher: Cambridge University Press
• Year: 2014
• Tongue: English
• Leaves: 308
• Category: Library

✦ Synopsis

This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent (and astonishing) facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright
Dedication
Epigraph
Contents
Foreword
1. A Cartesian Introduction
1. Proofs, applications, and other mathematical activities
2. On jargon
3. Descartes
A. Application
4. Arithmetic applied to geometry
5. Descartes' Geometry
6. An astonishing identity
7. Unreasonable effectiveness
8. The application of geometry to arithmetic
9. The application of mathematics to mathematics
10. The same stuff?
11. Over-determined?
12. Unity behind diversity
13. On mentioning honours - the Fields Medals
14. Analogy - and Andre Weil 1940
15. The Langlands programme
16. Application, analogy, correspondence
B .Proof
17. Two visions of proof
18. A convention
19. Eternal truths
20. Mere eternity as against necessity
21. Leibnizian proof
22. Voevodsky' s extreme
23. Cartesian proof
24. Descartes and Wittgenstein on proof
25. The experience of cartesian proof: caveat emptor
26. Grothendieck's cartesian vision: making it all obvious
27. Proofs and refutations
28. On squaring squares and not cubing cubes
29. From dissecting squares to electrical networks
30. Intuition
31. Descartes against foundations?
32. The two ideals of proof
33. Computer programmes: who checks whom?
2. What Makes Mathematics Mathematics?
1. We take it for granted
2. Arsenic
3. Some dictionaries
4. What the dictionaries suggest
5. A Japanese conversation
6. A sullen anti-mathematical protest
7. A miscellany
8. An institutional answer
9. A neuro-historical answer
10. The Peirces, father and son
11. A programmatic answer: logicism
12. A second programmatic answer: Bourbaki
13. Only Wittgenstein seems to have been troubled
14. Aside on method - on using Wittgenstein
15. A semantic answer
16. More miscellany
17. Proof
18. Experimental mathematics
19. Thurston's answer to the question 'what makes?'
20. On advance
21. Hilbert and the Millennium
22. Symmetry
23. The Butterfly Model
24. Could 'mathematics' be a 'fluke of history'?
25. The Latin Model
26. Inevitable or contingent?
27. Play
28. Mathematical games, ludic proof
3. Why is There Philosophy of Mathematics?
1. A perennial topic
2. What is the philosophy of mathematics anyway?
3. Kant: in or out?
4. Ancient and Enlightenment
A. An answer from the ancients: proof and exploration
5. The perennial philosophical obsession ...
6. The perennial philosophical obsession ... is totally anomalous
7. Food for thought (Matiere a penser)
8. The Monster
9. Exhaustive classification
10. Moonshine
11. The longest proof by hand
12. The experience of out-thereness
13. Parables
14. Glitter
15. The neurobiological retort
16. My own attitude
17. Naturalism
18. Plato!
B. An answer from the Enlightenment: application
19. Kant shouts
20. The jargon
21. Necessity
22. Russell trashes necessity
23. Necessity no longer in the portfolio
24. Aside on Wittgenstein
25. Kant's question
26. Russell's version
27. Russell dissolves the mystery
28. Frege: number a second-order concept
29. Kant's conundrum becomes a twentieth-century dilemma: (a) Vienna
30. Kant's conundrum becomes a twentieth-century dilemma: (b) Quine
31. Ayer, Quine, and Kant
32. Logicizing philosophy of mathematics
33. A nifty one-sentence summary (Putnam redux)
34. John Stuart Mill on the need for a sound philosophy of mathematics
4. Proofs
1. The contingency of the philosophy of mathematics
A. Little contingencies
2. On inevitability and 'success'
3. Latin Model: infinity
4. Butterfly Model: complex numbers
5. Changing the setting
B. Proof
6. The discovery of proof
7. Kant's tale
8. The other legend: Pythagoras
9. Unlocking the secrets of the universe
10. Plato, theoretical physicist
11. Harmonics works
12. Why there was uptake of demonstrative proof
13. Plato, kidnapper
14. Another suspect? Eleatic philosophy
15. Logic (and rhetoric)
16. Geometry and logic: esoteric and exoteric
17. Civilization without proof
18. Class bias
19. Did the ideal of proof impede the growth of knowledge?
20. What gold standard?
21. Proof demoted
22. A style of scientific reasoning
5. Applications
1. Past and present
A. THE EMERGENCE OF A DISTINCTION
2. Plato on the difference between philosophical and practical mathematics
3. Pure and mixed
4. Newton
5. Probability - swinging from branch to branch
6. Rein and angewandt
7. Pure Kant
8. Pure Gauss
9. The German nineteenth century, told in aphorisms
10. Applied polytechniciens
11. Military history
12. William Rowan Hamilton
13. Cambridge pure mathematics
14. Hardy, Russell, and Whitehead
15. Wittgenstein and von Mises
16. SIAM
B. A VERY WOBBLY DISTINCTION
17. Kinds of application
18. Robust but not sharp
19. Philosophy and the Apps
20. Symmetry
21. The representational-deductive picture
22. Articulation
23. Moving from domain to domain
24. Rigidity
25. Maxwell and Buckminster Fuller
26. The maths of rigidity
27. Aerodynamics
28. Rivalry
29. The British institutional setting
30. The German institutional setting
31. Mechanics
32. Geometry, 'pure' and 'applied'
33. A general moral
34. Another style of scientific reasoning
6. In Plato's Name
1. Hauntology
2. Platonism
3. Webster's
4. Born that way
5. Sources
6. Semantic ascent
7. Organization
A. ALAIN CONNES, PLATONIST
8. Off-duty and off-the-cuff
9. Connes' archaic mathematical reality
10. Aside on incompleteness and platonism
11. Two attitudes, structuralist and Platonist
12. What numbers could not be
13. Pythagorean Connes
B. TIMOTHY GOWERS, ANTI-PLATONIST
14. A very public mathematician
15. Does mathematics need a philosophy? No
16. On becoming an anti-Platonist
17. Does mathematics need a philosophy? Yes
18. Ontological commitment
19. Truth
20. Observable and abstract numbers
21. Gowers versus Connes
22. The 'standard' semantical account
23. The famous maxim
24. Chomsky's doubts
25. On referring
7. Counter-platonisms
1. Two more platonisms - and their opponents
A. TOTALIZING PLATONISM AS OPPOSEDTO INTUITIONISM
2. Paul Bernays (1888-1977)
3. The setting
4. Totalities
5. Other totalities
6. Arithmetical and geometrical totalities
7. Then and now: different philosophical concerns
8. Two more mathematicians, Kronecker and Dedekind
9. Some things Dedekind said
10. What was Kronecker protesting?
11. The structuralisms of mathematicians and philosophers distinguished
B. TODAY'S PLATONISM/NOMINALISM
12. Disclaimer
13. A brief history of nominalism now
14. The nominalist programme
15. Why deny?
16. Russellian roots
17. Ontological commitment
18. Commitment
19. The indispensability argument
20. Presupposition
21. Contemporary platonism in mathematics
22. Intuition
23. What's the point of platonism?
24. Peirce: The only kind of thinking that has everadvanced human culture
25. Where do I stand on today's platonism/ nominalism?
26. The last word
Disclosures
References
Index
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