Abstract
Sylvestre Huet: To start this chapter on the nature of mathematics and of its relationship with reality, I would like to quote you two extracts from interviews.
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Notes
- 1.
A French daily, translator’s note.
- 2.
E.P. Wigner, Communications on Pure and Applied Mathematics, XIII (1960), 1–14.
- 3.
See for example Michel Detlefsen (2004), Formalism, in: Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of mathematics and Logic, Oxford, Oxford University Press, pp. 236–317.
- 4.
This thesis states that it is impossible to test empirically a mathematical or physical hypothesis in isolation, but a bundle of physical and mathematical hypotheses can be tested.
- 5.
What are the mathematical entities truly vital for a given scientific theory? If it could be shown that these are numbers, the Platonic answer could reduce to a constructivist position! Which principles concerning these entities are necessary for the required mathematics?.
- 6.
Giuseppe Peano (1858–1932), Italian mathematician and linguist. A pioneer in the formalist approach to mathematics, he contributed to the development of an axiomatization of arithmetic.
- 7.
By using the term “invented”, I obviously do not mean that the mathematician invents material objects, but that the mathematical conceptualization of reality is not a description, but preserves a certain degree of freedom. To illustrate what I want to say, it is convenient to take the traditional example of geometry before 1905 (space-time); I can either take Euclidean geometry or one of the non-Euclidean geometries to measure street angles; none of the descriptions obtained is more true than the others and for this reason none is a description in the sense of the expression of a biunivocal relation, but is an “invention”. Distinct theories are, therefore, empirically equivalent or, as formulated by American philosopher Willard Ban Orman Quine: theories are sub-determined by experience.
- 8.
The osculating circle of a curve at a point is the circle best approximating the curve passing through the point, keeping its direction and curvature.
- 9.
Niccolo Tartaglia (around 1500–1557) was an Italian mathematician, who gave away his technique for finding the roots of polynomial equations of the third degree to his fellow countryman Girolamo Cardano (1501–1571), Cardan in French, translated Euclid into Italian, and among other domains worked in ballistics.
- 10.
Isaac Newton (1642–1727), English mathematician, physicist, philosopher and astronomer. This emblematic scientific figure is especially known for having founded classical mechanics, elaborated the theory of universal gravitation and, along with Leibniz, developed infinitesimal calculus.
- 11.
See note p. 52.
- 12.
A matrix is a square or rectangular array of numbers, such as
$$\begin{aligned}\begin{pmatrix}0 &{} 1 \\ -1 &{} 0 \\ \end{pmatrix}\,\hbox {or}\, \begin{pmatrix}2 &{} 3 &{} 4 \\ 9 &{} 8 &{} 7 \\ \end{pmatrix}.\end{aligned}$$Two matrices of the same size (\(2\times 2\) for example) can be added by adding the corresponding entries together:
$$\begin{aligned}\begin{pmatrix}7 &{} 5 \\ 4 &{} 3 \\ \end{pmatrix}+\begin{pmatrix}2 &{} 8 \\ 9 &{} 11 \\ \end{pmatrix}=\begin{pmatrix}7+2 &{} 5+8 \\ 4+9 &{} 3+11 \\ \end{pmatrix}\end{aligned}$$There is also a multiplication rule. Let’s just say that:
$$\begin{aligned}\begin{pmatrix}0 &{} 1 \\ -1 &{} 0 \\ \end{pmatrix}\times \begin{pmatrix}0 &{} 1 \\ -1 &{} 0 \\ \end{pmatrix}=\begin{pmatrix}-1 &{} 0 \\ 0 &{} -1 \\ \end{pmatrix}\,\hbox {so that the matrix}\, \begin{pmatrix}a &{} b \\ -b &{} a \\ \end{pmatrix}\end{aligned}$$is a faithful representation of the complex number \(a+bi\). As a result, calculation rules for complex numbers are justified by matrix calculus. In geometry, a rotation on the plane (or space) is represented by a \(2\times 2\) (or \(3\times 3\)) matrix and matrix multiplication corresponds to the composition of rotations.
- 13.
Werner Heisenberg (1901–1976), German physicist, who received the Nobel prize in physics in 1932, famous for his uncertainty principle.
- 14.
Max Born (1882–1970) German theoretical physicist, who received the physics Nobel prize in 1954 for his remarkable work in quantum theory.
- 15.
Urbain Le Verrier (1811–1877), French astronomer and mathematician specialized in celestial mechanics, discoverer of Neptune and founder of modern French meteorology.
- 16.
Entropy is a quantity characterizing the disorder of a system.
- 17.
Ludwig Boltzmann (1844–1906), Austrian physicist, founder of statistical mechanics.
- 18.
James Clerk Maxwell (1831–1879), Scottish physicist and mathematician, famous for his equations unifying electricity and magnetism, as well as for his work on the kinetic theory of gases.
- 19.
A complex number: it is known that the square of an arbitrary number (integer or decimal) is positive, so \(-1\) cannot be a square. However, if we wish all second-degree equations, without exception, to have two roots, then admittedly, the equation \(x^2+1\) has two roots, i and \(-i\), using conventional notation. If we combine the new number i with the old ones, and want to perform all the usual operations (addition, subtraction, multiplication, division), all these numbers can be reduced to the normal form \(a+bi\), which defines complex numbers (as opposed to usual numbers such as a and b said to be “real”). Gauss, Argand, and Cauchy’s great discovery around 1800 was to give a geometric representation of these numbers: \(a+bi\) corresponds to a point on the plane with coordinates (a; b). As Euler found out, trigonometry can be very much simplified using complex numbers.
- 20.
Jean-Robert Argand (1768–1822), Swiss mathematician, who worked in Paris and was known for having introduced the planar representation of complex numbers in 1806.
- 21.
Reflections on the General Cause of Winds, 1745.
- 22.
Such as when drawing the map of some region of the Earth, angles are respected, but not lengths...
- 23.
Moritz Pasch (1843–1930), German mathematician, who wrote one of the first books axiomatizing geometry (1882).
- 24.
David Hilbert, Foundations of Geometry (1899), ed. Jacques Gabay, 1997.
- 25.
Posterior Analytics, Book I, Part 7.
- 26.
Michael Detlefsen, Andrew Arana (2011), “Purity of Methods”, Philosophers imprint 11 (2), 1–20, p. 5.
- 27.
Karl Weierstrass (1815–1897), German mathematician considered to be the founder of modern analysis.
- 28.
Henri Poincaré, The Value of Science, p. 23.
- 29.
Lex Davidovitch Landau (1908–1968), Russian theoretical physicist, who received the Nobel prize in physics in 1962 “for his pioneering theories on the condensed state of matter, in particular liquid helium”.
- 30.
Benoît Mandelbrot (1924–2010), French-American mathematician, famous for having developed a new class of mathematical objects: fractal objects (or fractals).
- 31.
Yuri Linnik (1915–1972), Russian mathematician whose work was mainly in number theory, probability theory and statistics.
- 32.
Paul Erdös (1913–1996), Hungarian mathematician who largely contributed to the development of number theory and combinatorics.
- 33.
Exhibition Mathematics, a sudden change of scenery (Cartier Foundation) welcomed 80 000 visitors between October 21, 2011 and March 18, 2012.
- 34.
The Value of Science, pp. 77–78.
- 35.
Science and Hypothesis, p. 17.
- 36.
See Henri Poincaré, Analyse des travaux scientifiques de Henri Poincaré faite par lui-même, Acta Mathematica, 38, 1921, pp. 1–135, p. 101.
- 37.
See his biography in Erdös, the man who loved only numbers, Berlin, 2000.
- 38.
Israel Gelfand (1913–2009), Russian mathematician, student of Andrei Kolmogorov, whose work is related to every mathematical field. He was, however, mostly known for his contributions to functional analysis and their impact in quantum mechanics.
- 39.
Caroline Julien, Esthétique et mathématiques. Une exploration goodmanienne, Presses universitaires de Rennes, 2008.
- 40.
Maxwell imagined a demon opening a door for fast molecules and closing it for slow ones. Precise experiments simulate such a demon.
- 41.
Isabelle Stengers, born in 1949, Belgian philosopher and historian of science.
- 42.
Ilya Prigogine, born in 1949, Belgian philosopher and historian of science of Russian origin, who received the Nobel Prize in chemistry in 1977 for his contribution to the thermodynamics of irreversible processes and to the theory of dissipative structures. In particular, he showed that when matter moves away from its equilibrium state, self-organization occurs—a phenomenon that can be observed in physics, biology and climatology.
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Cartier, P., Dhombres, J., Heinzmann, G., Villani, C. (2016). Mathematics and Reality. In: Freedom in Mathematics. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2788-5_2
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