Heterogeneity in Mathematics: Proofs, Narratives, Rhetorics, and Style

1. Proofs, narratives, rhetorics, style and other mathematical elements Jean Paul Van Bendegem Vrije Universiteit Brussel Centrum voor Logica en Wetenschapsfilosofie Universiteit Gent

2. Starting hypothesis Mathematics is a heterogeneous activity A first analysis produces (at least) five different types: Type I: The search process to find a solution to a mathematical problem typically falls apart in different stages that require different strategies, including reformulations of the problem, Type II: The proximity of mathematical problems need not be related to a proximity of the corresponding proofs or similar problems can require different proofs and distinct problems can be solved by similar proofs,

3. Type III: A mathematical problem can change drastically when transposed from one mathematical background to another, up to the point of its disappearance or becoming uninteresting, Type IV: Mathematical theories that form the background for a (set of) problems are to be considered different because of the different proof strategies and related concepts that are being used, Type V: Mathematical explanation, whatever it might be, depends (though perhaps not solely) on proof and it thus ‘inherits’ its heterogeneity.

4. x³ + y³ + z³ = n, for 29  n  33 • x³ + y³ + z³ = 29 has at least two simple solutions: • x = 3, y = z = 1 and x = 4, y = –3, z = –2, • x³ + y³ + z³ = 30 has a smallest solution: x = –283059965, y = –2218888517, z = 2220422932, • x³ + y³ + z³ = 31: no solutions, • x³ + y³ + z³ = 32: no solutions, • x³ + y³ + z³ = 33: as it happens this problem is still open and, apparently, no one seems to have an idea how to handle it.

5. How were these results found? The cases n = 29 and 30 are the outcome of computer searches The cases n = 31 and 32: elementary mathematics m 0 1 2 3 4 5 6 7 8 m²0 1 4 9 16 25 36 49 64 m³ 0 1 8 27 64 125 216 343 512 m³ 0 1 -1 0 1 -1 0 1 -1 (mod9) Case n = 33 stillunsolved (though a solution for n = 52 is known)

6. Additional curious features: For the case n = 1 (Mahler 1936): (9t4)³ + (3t − 9t4)3 + (1 − 9t3)3 = 1 For the case n = 2 (Werebrusov 1908): (1 + 6t³)³ + (1 − 6t³)³ + (−6t²)³ = 2 ‘Oddest’ feature: turn theequationaroundand a new problememerges: n = x³ + y³ + z³ A problem about decomposition of natural numbers in powers

7. Thesis put forward Aspects of mathematical practice such as Beauty Style Rhetoric inherit this heterogeneity Two consequences Each of these aspects supports the others Mathematical foundations serve to reduce the heterogeneity

8. Mathematicians care about beauty • Poincaré, Hardy, Hadamard, Atiyah, Papert, Rota andmanyothers • However: notthatmanysystematic approaches • Most oftenreferredto: Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty” (1997)

9. Importance of making distinctions • Beauty betweenmathematicsand art • Beauty betweenmathematiciansand non-mathematicians • Beauty withinmathematics: • Theorems • Proofs • Steps in a proof • Theories • Definitions

10. Importance of making distinctions • Beauty betweenmathematicsand art • Beauty betweenmathematiciansand non-mathematicians • Beauty withinmathematics: • Theorems • Proofs • Steps in a proof • Theories • Definitions

11. Importance of making distinctions • Beauty betweenmathematicsand art • Beauty betweenmathematiciansand non-mathematicians • Beauty withinmathematics: • Theorems • Proofs • Steps in a proof • Theories • Definitions

12. David Wells in The Mathematical Intelligencer, 1988-1990, produces a list of 24 theoremsrankedby beauty • ei= -1 (7,7) • V + F – E = 2 (7,5) • The number of primes is infinite (7,5) • There are 5 regularsolids (7,0) • 1 + 1/2² + 1/3² + ... 1/n² + ... = ²/6 (7,0) • The fixed-point theorem (6,8) • 2 is irrational(6,7) •  is transcendental (6,5) • The four-colour theorem (6,2) • Any prime of the form 4n+1 is uniquely the sum of 2 squares (6,0)

13. David Wells in The Mathematical Intelligencer, 1988-1990, produces a list of 24 theoremsrankedby beauty • ei= -1 (7,7) • V + F – E = 2 (7,5) • The number of primes is infinite (7,5) • There are 5 regularsolids (7,0) • 1 + 1/2² + 1/3² + ... 1/n² + ... = ²/6 (7,0) • The fixed-point theorem (6,8) • 2 is irrational(6,7) •  is transcendental (6,5) • The four-colour theorem (6,2) • Any prime of the form 4n+1 is uniquely the sum of 2 squares (6,0)

14. Whatiftheoremandproof are related? George David Birkhoff (1884-1944) “AestheticMeasure” (1933) • Not taken veryseriously • Startedformal-mathematicalaesthetics • Linkedto information theory (Shannon) • Alsoappliedtoethics

15. M = O / C M = measure of beauty O = order C = complexity Applied to geometrical figures, tilings and ornaments and to music Note: explicit argument to “derive” the formula starting from M = f(O, C) to M = f(O/C) and, by simplicity, to O/C

16. Example: the beauty of a square M = (V + E + R + HV – F)/C V = vertical symmetry = 1 E = 1 if V is 1 (corresponds to equilibrium) R = rotational symmetry/2 = 4/2 = 2 (turns) HV = 2 iff the polygon fits in a horizontal-vertical grid F = 0 (if the figure is sufficiently regular) C = 4 = number of distinct straight lines = number of sides of the polygon Hence M = (1 + 1 + 2 + 2 – 0)/4 = 6/4 = 1.5

19. How to apply to theorems and proofs? Most often heard: “Most beautiful is a simple proof for a deep result” Suggest an interpretation: O = measure of what a “deep” result is C = simplicity of the proof M = the beauty of a proof = O / C

20. Some direct consequences • A shorterproofforthesametheoremincreasesits beauty • A similarprooffor a deeperresult is more beautiful Problem: Sinceonlythe ratio is considered, anunimportantresultwith a really short proofcouldbejust as beautiful

21. Some more detail: What is a deep result? Core concept: connectedness Example: If many theorems have been proved of the form “If A, then B”, then A is deeply connected But how does it relate to proof?

22. Theorem: √2 is irrational Proof: suppose √2 = a/b such that gcd(a,b) = 1 a2 = 2b2 hence a = 2c 2c2 = b2 hence b = 2d thus gcd(a,b) ≠ 1 Theorem: √2 is irrational Proof: suppose √2 = a/b Hence a2 = 2b2 Number of prime factors in a2 is even, in 2b2 odd, which is impossible

23. Anotherproof: • Ratheranoddity • Rather “surprising” a b

24. This measure of beauty corresponds to the beauty of the efficient, economic problem-solver Hence it expresses a basic feature of the activity of a mathematician Thus serves to identify exemplars Hence it is essential and not a mere “side-effect”

25. Some further consequences 1. Demonstrates the importance of proof methods • The reductio proof • Proof by infinite descent • Proof by cases • Mathematical induction • Career induction • (Partial) computer proof • “Experimental” proof

26. 2. Shows the importance of research programmes in mathematics to identify the well-connected problems and the favoured proof methods 3. Explains why outside mathematics other aesthetic standards occur 4. Explains why beauty can change over time 5. In principle extendable to theories and definitions

27. As mentioned this cannot be the whole story Another approach to understand the diverse forms of beauty is to make the connection with style However style turns out to be equally heterogeneous: Applied to a person Applied to a connected group of persons Applied to a cultural setting (time, place)

28. Style related to different phases in the mathematical process • Style of discovery • Style of translation in written text • Style of the finalized text, a proof François LE LIONNAIS (1901-1984): “Les grands courants de la penséemathématique“ (1948) Distinction between romanticist and classicist attitude to mathematics, where romanticist corresponds to (1) classicist corresponds to (2)

29. Ad (1): Problem-solving involves • “discovering” new problem domains • covering new grounds • “uncharted” territory Better term: complexity-seeking style Classic examples: • Introduction of imaginary numbers • Cantor’s theory of infinities • Riemann’s hypothesis

31. Ad (2): Notes, roughly noted ideas, letters generative style (?)

32. Ad (3): complexity-reducing style simplicity-seeking style unification-seeking style Best known and best studied at the present Bourbaki style is a perfect example Set a format for producing mathematical texts (related to foundational studies and their role in mathematical practice)

33. Enter rhetorics Again a complex story: Inside or outside of mathematics? Withinmathematics: Belongstotheformal or theinformal? Belongstoparticularelements or tothestructure? Applicabletoproofs or toothermathematicalelements?

34. “The myth of totally rigorous, totally formalized mathematics is indeed a myth. Mathematics in real life is a form of social interaction where “proof” is a complex of the formal and the informal, of calculations and casual comments, of convincing argument and appeals to the imagination and the intuition.” (p. 68) Philip J. Davis & Reuben Hersh: “Rhetoric and Mathematics”. In: John S. Nelson, Allan Megill & Donald N. McCloskey (eds.): The Rhetoric of the Human Sciences: Language and Argument in Scholarship and Public Affairs. Madison: The University of Winconsin Press, 1987, pp. 53-68.

35. Structure = narrative? “The ludic is a treatise of theideal type […]: a workbased on obtainingresults in surprising, intricateways, wheretheauthorbrings out his ownvoice in rich, modulatedways, andwherethetextualsurface is often made deliberatelyopageby, say, long passages of calculation.” (p. 108) RevielNetz: Ludic Proof. Greek Mathematics and the Alexandrian Aesthetic. Cambridge: CUP, 2009.

36. (a) basis: suppose that n = 0, then (a1) 1 + 2 + 3 + … + n = 0 and (a2) n.(n+1)/2 = 0 and (a3) so they are equal. (b) induction step: suppose the statement holds for n, so: 1 + 2 + 3 + … + n = n.(n+1)/2 (c) add to both sides n+1: 1 + 2 + 3 + … + n + (n+1) = n.(n+1)/2 + (n+1) (d) the right-hand side can be transformed into: (d1) n.(n+1)/2 + (n+1) = (n+1).(n/2 + 1) (d2) n.(n+1)/2 + (n+1) = (n+1).(n + 2)/2 (d3) n.(n+1)/2 + (n+1) = (n+1).((n+1)+1)/2 (e) put this together, and one finds: 1 + 2 + 3 + … + n + (n+1) = (n+1).((n+1)+1)/2 which is precisely the statement to be proven for n+1. (f) by mathematical induction, for all n, 1 + 2 + 3 + … + n = n.(n+1)/2.

38. Introducesthebroader field of semiotics • Semiotic analysis by Brian Rotman • Semiotic analysis by Paul Ernest • Notrestrictedtotheeducational setting • Important forerunner: theSignifics

39. The SignificMovement • Members: Gerrit Mannoury, L.E.J Brouwer, Frederik van Eeden, van Ginniken, … • First quarter of 20th century • Linkedtothe Wiener Kreis • Corethoughts: • Language is a “living” thing • Primarily a socialprocess • Shouldbestudied as such • Distinctionbetween speaker andhearer • Alsoapplicabletomathematics

40. Where does thisleaveus? In a semioticframeworkallmentionedaspectscanbebroughttogether: beauty, style, rhetoric, … In such a framework these aspectsdepend on oneanother Each of these elementsinheritstheheterogeneity of mathematicalpractices Suggestion: Betterto talk about ‘varieties of X’ ratherthansimply X

41. And explanation? Thesis: the same holds for explanation Support: There are at leasttwo basic proposals (Steiner & Kitcher) that do notseemtoexcludeoneanother The on-goingdiscussion shows thatnoteveryone is satisfiedwith these two accounts Inside or outsidemathematics? Ifinside, relatedtoparticularpractices or not?

42. Foundational studies Homogeneity as compensation Along different dimensions: A foundational theory (ZFC, category theory, …) An idealized mathematician (Brouwer’s creative subject) An idealized community of mathematicians (logical modelling of groups of epistemic agents) A standardized form of proof (related to computer proofs) …

43. Finalthought: Does nottheheterogeneityimplythateverypossibletheory of mathematicalpracticeswillbenecessarilyincomplete? Mathematicians are familiarwith“surprises” PSLQ algorithm (Helaman Ferguson)