Proofs, narratives, rhetorics, style and other mathematical elements

Jean Paul Van BendegemVrije Universiteit Brussel

Centrum voor Logica en WetenschapsfilosofieUniversiteit Gent

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,

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.

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.

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 8m² 0 1 4 9 16 25 36 49 64m³ 0 1 8 27 64 125 216 343 512m³ 0 1 -1 0 1 -1 0 1 -1(mod9)

Case n = 33 still unsolved (though a solution for n = 52 is known)

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 the equation around and a new problem emerges:

n = x³ + y³ + z³

A problem about decomposition of natural numbers in powers

Thesis put forward

Aspects of mathematical practice such as Beauty Style Rhetoricinherit this heterogeneity

Two consequences Each of these aspects supports the others Mathematical foundations serve to reduce the

heterogeneity

Mathematicians care about beauty

Poincaré, Hardy, Hadamard, Atiyah, Papert, Rota and many others

However: not that many systematic approaches

Most often referred to: Gian-Carlo Rota, “The Phenomenology of Mathematical Beauty” (1997)

Importance of making distinctions

Beauty between mathematics and art Beauty between mathematicians and

non-mathematicians

Beauty within mathematics: Theorems Proofs Steps in a proof Theories Definitions

Importance of making distinctions

Beauty between mathematics and art Beauty between mathematicians and non-

mathematicians

Beauty within mathematics: Theorems Proofs Steps in a proof Theories Definitions

Importance of making distinctions

Beauty between mathematics and art Beauty between mathematicians and non-

mathematicians

Beauty within mathematics: Theorems Proofs Steps in a proof Theories Definitions

David Wells in The Mathematical Intelligencer, 1988-1990, produces a list of 24 theorems ranked by beauty

1. ei = -1 (7,7)2. V + F – E = 2 (7,5)3. The number of primes is infinite (7,5)4. There are 5 regular solids (7,0)5. 1 + 1/2² + 1/3² + ... 1/n² + ... = ²/6 (7,0)6. The fixed-point theorem (6,8)7. 2 is irrational (6,7)8. is transcendental (6,5)9. The four-colour theorem (6,2)10. Any prime of the form 4n+1 is uniquely the

sum of 2 squares (6,0)

David Wells in The Mathematical Intelligencer, 1988-1990, produces a list of 24 theorems ranked by beauty

1. ei = -1 (7,7)2. V + F – E = 2 (7,5)3. The number of primes is infinite (7,5)4. There are 5 regular solids (7,0)5. 1 + 1/2² + 1/3² + ... 1/n² + ... = ²/6 (7,0)6. The fixed-point theorem (6,8)7. 2 is irrational (6,7)8. is transcendental (6,5)9. The four-colour theorem (6,2)10. Any prime of the form 4n+1 is uniquely the

sum of 2 squares (6,0)

What if theorem and proof are related?

George David Birkhoff (1884-1944)“Aesthetic Measure” (1933)

Not taken very seriously Started formal-mathematical aesthetics Linked to information theory (Shannon) Also applied to ethics

M = O / C

M = measure of beautyO = orderC = 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

Example: the beauty of a square

M = (V + E + R + HV – F)/C

V = vertical symmetry = 1E = 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 gridF = 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

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 isC = simplicity of the proof

M = the beauty of a proof = O / C

Some direct consequences

A shorter proof for the same theorem increases its beauty

A similar proof for a deeper result is more beautiful

Problem:Since only the ratio is considered, an

unimportant result with a really short proof could be just as beautiful

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?

Theorem: √2 is irrational

Proof: suppose √2 = a/b such that gcd(a,b) = 1

a2 = 2b2

hence a = 2c2c2 = b2

hence b = 2dthus 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

Another proof: Rather an oddity Rather “surprising”

ab

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”

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

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

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)

Style related to different phases in the mathematical process

(1)Style of discovery(2)Style of translation in written text(3)Style of the finalized text, a proof

François LE LIONNAIS (1901-1984): “Les grands courants de la pensée mathématique “ (1948)

Distinction between romanticist and classicist attitude to mathematics, where

romanticist corresponds to (1)classicist corresponds to (2)

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

Ad (2): Notes, roughly noted ideas, lettersgenerative style (?)

Ad (3): complexity-reducing stylesimplicity-seeking styleunification-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)

Enter rhetorics

Again a complex story:

Inside or outside of mathematics?Within mathematics: Belongs to the formal or the informal? Belongs to particular elements or to the

structure? Applicable to proofs or to other

mathematical elements?

“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.

Structure = narrative?

“The ludic is a treatise of the ideal type […]: a work based on obtaining results in surprising, intricate ways, where the author brings out his own voice in rich, modulated ways, and where the textual surface is often made deliberately opage by, say, long passages of calculation.” (p. 108)

Reviel Netz: Ludic Proof. Greek Mathematics and the Alexandrian Aesthetic. Cambridge: CUP, 2009.

(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.

Introduces the broader field of semiotics

Semiotic analysis by Brian Rotman Semiotic analysis by Paul Ernest Not restricted to the educational setting

Important forerunner: the Significs

The Signific Movement

Members: Gerrit Mannoury, L.E.J Brouwer, Frederik van Eeden, van Ginniken, …

First quarter of 20th century Linked to the Wiener Kreis Core thoughts:

Language is a “living” thing Primarily a social process Should be studied as such Distinction between speaker and hearer Also applicable to mathematics

Where does this leave us?

In a semiotic framework all mentioned aspects can be brought together: beauty, style, rhetoric, …

In such a framework these aspects depend on one another

Each of these elements inherits the heterogeneity of mathematical practices

Suggestion: Better to talk about ‘varieties of X’ rather than simply X

And explanation?

Thesis: the same holds for explanation

Support: There are at least two basic proposals (Steiner

& Kitcher) that do not seem to exclude one another

The on-going discussion shows that not everyone is satisfied with these two accounts

Inside or outside mathematics? If inside, related to particular practices or not?

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) …

Final thought: Does not the heterogeneity imply that

every possible theory of mathematical practices will be necessarily incomplete?

Mathematicians are familiar with “surprises”

PSLQ algorithm (Helaman Ferguson)

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