three aspects of mathematical explanation: bertrand’s postulate alan baker department of...
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Three Aspects of Mathematical Explanation:
Bertrand’s Postulate
Alan Baker
Department of Philosophy
Swarthmore College
“Mathematical Aims Beyond Justification"
“Mathematical Aims Beyond Justification"
Focus 1: Explanation
“Mathematical Aims Beyond Justification"
Focus 1: Explanation
Focus 2: Proof
M S
M MEM MES
S SEM SES
SEM
Skow, B. (forthcoming) “Are There Genuine Physical Explanations of Mathematical Phenomena?”, British Journal for the Philosophy of Science.
SES
(i) Theoretical virtues (for scientific theories) vs. ‘explanatory virtues’ (for mathematical proofs).
(ii) Theory choice (between alternative scientific theories) vs. ‘proof choice’ (between alternative mathematical proofs).
SES
(i) Theoretical virtues (for scientific theories) vs. ‘explanatory virtues’ (for mathematical proofs).
A candidate ‘explanatory anti-virtue’: proof by cases
SES
(i) Theoretical virtues (for scientific theories) vs. ‘explanatory virtues’ (for mathematical proofs).
A candidate ‘explanatory anti-virtue’: proof by cases
“A method of mathematical proof in which the statement to be proved is split into a finite number of cases, and each case is checked to see if the statement in question holds.”
Bertrand’s Postulate: There is always a prime between n and 2n
Bertrand’s Postulate: There is always a prime between n and 2n
Conjectured: 1845 (Bertrand)First proof: 1850 (Chebyshev)First elementary proof: 1932 (Erdös)
pr < 2n10 252 22 . 32 . 7 5
12 924 22 . 3 . 7 . 11 6
30 24 . 32 . 5 . 17 . 19 . 23 . 29 15
pr < 2n10 252 22 . 32 . 7 5
12 924 22 . 3 . 7 . 11 6
30 24 . 32 . 5 . 17 . 19 . 23 . 29 15
“very small” < √2n “small” < 2n/3
gap 2n/3 < p < n BP n < p < 2n
For n large enough, the binomial coefficient must have another prime factor, which can only lie between n and 2n.
Analytic proof holds for all n > 4,000.
For n large enough, the binomial coefficient must have another prime factor, which can only lie between n and 2n.
Analytic proof holds for all n > 4,000.
“Landau’s trick”:
2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001
A sequence of primes, each smaller than twice the previous one.
MEM
Proofs from the Book
El Bachraoui, M (2006) “Primes in the Interval [2n, 3n],” Int. J. Contemp. Math. Sci., 1 (13), 617 – 621.
Loo, A. (2011) “On the Primes in the Interval [3n, 4n], Int. J. Contemp. Math. Sci., 6(38), 1871 – 1882.
Balliet, K. (2015) “On the Prime Numbers in the Interval [4n, 5n], arXiv:1511.04571v1.
Shevelev, V. (2013) “On Intervals [kn, (k+1)n] Containing a Prime for All n > 1,” Journal of Integer Sequences, 16.
[2n, 3n]: holds for all n
lower bound of analytic proof = 945
[3n, 4n]: holds for all n
lower bound of analytic proof = e12 ≈ 162,755
[4n, 5n]: holds for all n > 2
lower bound of analytic proof = 6,817
“The difference between mathematical and physical explanations of physical phenomena is now amenable to analysis. In the former, as in the latter, physical and mathematical truths operate. But only in mathematical explanation is [the following] the case: when we remove the physics, we remain with a mathematical explanation -- of a mathematical truth!”
(Steiner [1978a, p. 19])
Steiner’s Hypothesis
M (M* S*)
(M M*) S* e
where there is some appropriate (structural?) correspondence between the mathematical theorem, M*, and the physical phenomenon, S*.
The Honeycomb Theorem
(1) It is advantageous for bees to minimize the amount of wax per unit area of their honeycomb cells. [biological law]
(2) Regular hexagons have the least perimeter per unit area of any partition of the plane into equal areas. [geometrical theorem]
--------------------------------------------------------------(3) Hence, bees have evolved to build hexagonal
cells in their honeycombs.
Regular hexagons have the least perimeter per unit area of any partition of the plane into equal areas.
= The Honeycomb Theorem
(Thomas Hales, 1999)
Regular hexagons have the least perimeter per unit area of any partition of the plane into equal areas.
= The Honeycomb Theorem
(Thomas Hales, 1999)
Baker, A. (2012) “Science-Driven Mathematical Explanation,” Mind, 121, 243-267
Why are the cicada life-cycle periods prime?
Why are the cicada life-cycle periods prime?
Given a synchronized, periodic life-cycle, is there some evolutionary advantage to having a period that is prime?
Avoiding Predators
“For example, a prey with a 12-year cycle will meet – every time it appears – properly synchronized predators appearing every 1, 2, 3, 4, 6 or 12 years, whereas a mutant with a 13-year period has the advantage of being subject to fewer predators.”
(Goles, Schulz & Markus [2001])
(1) It is advantageous to have a life-cycle period which minimizes intersection with other periods. [biological law]
(2) Prime periods minimize intersection frequency. [number-theoretic theorem]
--------------------------------------------------------------(3) Hence, organisms with periodic life-cycles
are likely to evolve periods that are prime.
Lemma 1: the lowest common multiple of m and n is maximal if and only if m and n are coprime.
Lemma 2: a number, m, is coprime with each number n 2m, n m if and only if m is prime.
(3) Organisms with periodic life-cycles are likely to evolve periods that are prime.
(4) Periodical cicadas are restricted to periods between 12 and 18 years.
[ecological constraint]
(5) Hence, periodical cicadas are likely to evolve 13-year or 17-year periods.
(3) Organisms with periodic life-cycles are likely to evolve periods that are prime, if there are primes in their ecological range.
(3’) If the ecological range is of the form [2n, 3n] then there is at least one prime.
[El Bachraoui’s Theorem]
(4) The ecological range of periodical cicadas is [12, 18] [ecological constraint]
(5) Hence, periodical cicadas are likely to evolve prime periods.
[2n, 3n]: holds for all n
lower bound of analytic proof = 945
[3n, 4n]: holds for all n
lower bound of analytic proof = e12 ≈ 162,755
[4n, 5n]: holds for all n > 2
lower bound of analytic proof = 6,817
M S
M MEM MES
S SEM SES
