Abstract
This article suggests a novel way to advance a current debate in the philosophy of mathematics. The debate concerns the role of diagrams and visual reasoning in proofs—which I take to concern the criteria of legitimate representation of mathematical thought. Drawing on the so-called ‘maverick’ approach to philosophy of mathematics, I turn to mathematical practice itself to adjudicate in this debate, and in particular to category theory, because there (a) diagrams obviously play a major role, and (b) category theory itself addresses questions of representation and information preservation over mappings. We obtain a mathematical answer to a philosophical question: a good mathematical representation can be characterized as a category theoretic natural transformation. Assuming that this is not some reductio against the maverick approach to these issues, this in turn moots some of the disagreement in the philosophical debate and provides better questions with which to go on.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
An initial worry about this whole approach may be put as follows (and thanks to a referee for doing so). If an answer is mathematical then that means that the question turns out to be mathematical as well—for wouldn’t it be very surprising if questions in metaphysics or epistemology had mathematical answers? It is the burden of this paper and similarly motivated projects to show how mathematical philosophy can be fruitful, noting for now that the idea has been around at least since Descartes and Leibniz. As a research program, the mathematical approach would be interesting and informative even if it is ultimately unsuccessful.
- 2.
This is to use ‘modern’ in the technical sense of Jeremy Grey: “Modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated—indeed, anxious—rather than a naive relationship with the day-to-day world, which is the de facto view of a coherent group of people, such as a professional or discipline based group that has a high sense of the seriousness and value of what it is trying to achieve” [17, p. 1].
- 3.
Gauss’ definition of intrinsic curvature repeats this result, since the curvature \(\kappa(t) = \frac{1}{r}\) of the osculating circle is identically 0 when r goes to infinity [21, p. 3].
- 4.
Stillwell (not historically unproblematically) observes that “the Greeks used curves to study algebra rather than the other way around” [28, p. 64].
- 5.
- 6.
As long as Algebra and Geometry were separated, their progress was slow and their use limited; but once these sciences were united, they lent each other mutual support and advanced rapidly together towards perfection. (Lagrange 1795)
- 7.
A diagram like this one commutes when g∘h=f, i.e. g(h(x))=f(x).
- 8.
Einstein wrote to Levi-Civita, “I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.” Quoted in Goodstein, J.R.: The Italian mathematicians of relativity. Centaurus 26(3), 241–261 (2007).
- 9.
Dirac: “With a network of curvilinear coordinates the g μν , given as functions of the coordinates, fix all the elements of distance; so they fix the metric. They determine both the coordinate system and the curvature of space” [10, p. 9].
- 10.
If the speed of light is taken to be unity, the metric on Minkowski space is
$$\bigl(dx^{0}\bigr)^{2} - \bigl(dx^{1} \bigr)^{2} - \bigl(dx^{2}\bigr)^{2} - \bigl(dx^{3}\bigr)^{2} $$Through some notational conventions for moving indices up and down, this reduces to dx μ dx μ, with Einstein’s summation convention understood. A coordinate system with coefficients g μν is introduced. The invariant distance between a point x μ and a nearby point x μ+dx μ is
$$g_{\mu \nu} dx^{\mu}dx^{\nu} $$again with summations understood. If we define a Christoffel symbol
$$\varGamma_{\mu \nu \sigma} := 1/2(g_{\mu \nu , \sigma} + g_{\mu \sigma , \nu} - g_{\nu \sigma , \mu}) $$then we can define the Riemann-Christoffel curvature tensor
$$R_{\nu \rho \sigma}^{\beta}:= \varGamma_{\nu \sigma , \rho}^{\beta} - \varGamma_{\nu \rho , \sigma}^{\beta} + \varGamma_{\nu \sigma}^{\alpha}\varGamma_{\alpha \rho}^{\beta} - \varGamma_{\nu \rho}^{\alpha}\varGamma_{\alpha \sigma}^{\beta} $$Lowering suffixes, the tensor R μνρσ has 256 components. One can prove that space is flat if and only if R μνρσ vanishes. The Ricci tensor is defined: \(R_{\nu \rho \mu}^{\mu} = R_{\nu \rho}\).
- 11.
For a rigorous but accessible presentation, see [27].
- 12.
A vector space may have many different bases, but there are always the same number of basis vectors in each; a vector space is uniquely determined by its dimension, up to isomorphism.
- 13.
There is a related category theoretic notion of representation. A representation is a natural homomorphism (in the technical sense of natural, above) between S and a set theoretic object. One can prove that a functor \(G: \mathcal{D} \longrightarrow \mathcal{C}\) has a left adjoint iff each hom(X,G−) is representable, for every X∈C.
- 14.
An adjunction between two categories can also be an equivalence relation. An equivalence on functors \(F: \mathcal{C} \longrightarrow \mathcal{D}\), \(G: \mathcal{D} \longrightarrow \mathcal{C}\) is such that
$$\mathit{hom}(Z, X) \cong \mathit{hom}\bigl(F(Z), F(X)\bigr) $$This is “isomorphism up to isomorphism” [2, p. 148]. Writing Z=G(Y), it follows by using some natural isomorphisms that hom(G(Y),X)≅hom(F(G(Y)),F(X))≅hom(Y,F(X)). Therefore any equivalence is an adjunction [2, p. 181].
- 15.
The figures-and-formulae functor is not the most general schema for the sort of representations we mean to capture, since it requires the existence of a unit element 1 X for any X. It seems clear that there can be interesting instances of X with no such element. So this schema can be taken as a partial result, that holds over domains with a sufficient amount of structure. Similarly, not every mathematical object is a group or a ring, but still group and ring theory are quite ubiquitous, and have a lot to tell us.
- 16.
Thanks to: Philip Catton, Cathy Legg, Clemency Montelle, Peter Smith, and Koji Tanaka.
References
Avigad, J., Dean, E., Mumma, J.: A formal system for Euclid’s elements. Rev. Symb. Log. 2(4), 700–768 (2010)
Awodey, S.: Category Theory. Oxford Logic Guides (2006)
Azzouni, J.: Tracking Reason: Proof, Consequence, and Truth. Oxford University Press, New York (2006)
Barwise, J., Etchemendy, J.: Visual information and valid reasoning. In: Allwein, G., Barwise, J. (eds.) Logical Reasoning with Diagrams, pp. 3–25. Oxford University Press, New York (1996)
Birkhoff, G., Maclane, S.: Algebra. Am. Math. Soc., Providence (1999) (First edition 1967)
Brown, J.: Philosophy of Mathematics, 2nd edn. Routledge, London (2008)
Cohn, P.M.: An Introduction to Ring Theory. Springer, Berlin (2000)
Corfield, D.: Towards a Philosophy of Real Mathematics. Cambridge University Press, Cambridge (2003)
Coxeter, H.M.: Introduction to Geometry, 2nd edn. Wiley, New York (1969)
Dirac, P.M.: The General Theory of Relativity. Wiley, New York (1975)
Ferreirós, J., Grey, J.J. (eds.): The Architecture of Modern Mathematics. Oxford University Press, Oxford (2006)
Francis, G.K.: A Topological Picture Book. Springer, Berlin (2007)
Frankel, T.: The Geometry of Physics: An Introduction, 2nd edn. Cambridge University Press, Cambridge (2004)
Giaquinto, M.: Visual Thinking in Mathematics. Oxford University Press, Oxford (2007)
Gilles, D. (ed.): Revolutions in Mathematics. Clarendon, Oxford (1992)
Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960)
Grey, J.: Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton University Press, Princeton (2008)
Grosholz, E.R.: Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford University Press, Oxford (2007)
Hartshorne, R.: Geometry: Euclid and Beyond. Springer, Berlin (2000)
Jänich, K.: Vector Analysis (trans.: Kay, L.). Springer, New York (2001)
Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature. Springer, New York (1997)
MacLane, S.: Mathematics, Form and Function. Springer, Berlin (1986)
Mancosu, P. (ed.): The Philosophy of Mathematical Practice. Oxford University Press, New York (2008)
Mancosu, P., Jorgensen, K.F., Pedersen, S.A. (eds.): Visualization, Explanation and Reasoning Styles in Mathematics. Synthese Library, vol. 327. Springer, Dordrecht (2005)
Netz, R.: The Shaping of Deduction in Greek Mathematics. Cambridge University Press, Cambridge (1999)
Shin, S.-J.: The Logical Status of Diagrams. Cambridge University Press, Cambridge (1994)
Stewart, I.: Galois Theory, 3rd edn. Chapman & Hall, London (2004)
Stillwell, J.: Mathematics and Its History. Springer, New York (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Weber, Z. (2013). Figures, Formulae, and Functors. In: Moktefi, A., Shin, SJ. (eds) Visual Reasoning with Diagrams. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0600-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0600-8_9
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0599-5
Online ISBN: 978-3-0348-0600-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)