Abstract
This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case study on Riemann’s work on complex functions, I propose that mathematicians deal with systems of representations and that truth—or what we can prove—depends on available representations in some context where the problem can be solved.
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Notes
Field’s (1980) Science without Numbers is the standard example of rewriting science. The second option of rewriting mathematics is undertaken by Kitcher, introducing ideal agents performing actions on physical objects (modified by Hoffman (2004)), and in Chihara’s (2004) structuralist, constructibility theory.
See the recent book by Leng (2010) for an updated discussion on these arguments.
Riemann characterises a function in the following way “Denote by \(z\) a variable that can take successively all possible real values. When there is a unique value of the variable \(w\) corresponding to each \(z\), we say that \(w\) is a function of \(z\)”. (p. 1) (In original: Denkt man sich unter \(z\) eine veränderliche Grösse, welche nach und nach alle möglichen reellen Werthe annehmen kann, so wird, wenn jedem ihrer Werthe ein einziger Werth der unbestimmten Grösse \(w\) entspricht, \(w\) eine Function von \(z\) gennant (p. 1).
Functions that are holomorphic except for isolated points, where they have singularities, are still possible to represent by power series. In such cases these series have (possibly an infinite number of) terms with negative powers. Such functions are called meromorphic functions.
Die Auffassung einer solchen Veränderlichkeit, welche sich auf ein zusammenhängendes Gebiet von zwei Dimensionen erstreckt, wird wesentlich erleichtert durch einer Anknüpfung an räumliche Anschauung (p. 3).
Conformal mappings play an important role for solving problems in physics. (See http://en.wikipedia.org/wiki/Conformal_map) Conformal maps preserve harmonic functions, i.e., functions that fulfil differential equations like \(\frac{\partial ^2f}{\partial x_1^2} +\frac{\partial ^2f}{\partial x_2^2}=0\). Examples include electromagnetic and gravitational fields that are defined by such equations. It may be that the solution of such an equation is difficult to handle, but that the problem can become tractable by transforming it with a conformal map to a different space.
A more formal definition of a branch point is the following. Denote by \(\Vert w(z)\Vert \) a multivalued function, indicating that for each value of \(z\) there may be more than one value of \(w(z)\). Take as an example the argument function, \(\Vert arg(z) \Vert \). The argument of a complex number \(z=x+iy\) denotes the angle that the line from the origin to the point \(z (x,y)\) forms with the positive real axis (measured anti-clockwise). Note that the choice of angle is not unique, the argument of \(z\) will be \(\Vert arg(z)\Vert = \{\theta + 2k\pi , k\in \mathbb Z \}\), for some choice of angle \(\theta \). \(a\) is a branch point for \(\Vert w(z)\Vert \) if, for all (sufficiently small) \(r>0\), it is not possible to choose a single valued function \(f(z)\in \Vert w(z)\Vert \) so that \(f\) is a continuous function on the circle with centre in \(a\) and radius \(r\).
Es soll kurz angedeutet werden, was durch unsere Untersuchung für die Theories solcher Functionen gewonnen ist. Die bisheringen Methoden, diese Functionen zu behandeln, legten stets als Definition einen Ausdruck der Function zu Grunde, wodurch ihr Werth für jeden Werth ihres Argumentes gegeben wurde; durch unsere Untersuchung ist gezeigt, dass, in Folge des allgemeinen Charakters einer Funktion einer veränderlichen complexen Grösse, in einer Definition dieser Art ein Theil der Bestimmungsstücke eine Folge der Übrigen ist, und zwar ist der Umfang der Bestimmungsstücke auf die zur Bestimmung nothwendigen zurückgefärt worden. Dies vereinfacht die Behandlung derselben wesentlich (Werke pp. 35–36, italics in original).
Talking about mathematical practice in terms of representations is by no means new. Ken Manders has for many years stressed the importance of artifact-use in mathematics. More recently Grosholz (2007) has pointed to the ambiguous use of representations in mathematical reasoning.
Another example is given in (Carter 2010).
Laugwitz (2008), pp. 133–139, gives a nice presentation of this work.
\(\epsilon \cdot e^{\phi i}\) is defined as \(\epsilon \cdot (\cos (\phi ) + i \sin (\phi )).\)
I will not discuss the status of concepts, although I will point to work by Marcus Giaquinto that to me indicates we are able to generate certain geometrical concepts via perceptual evidence. See for example, Giaquinto (2008).
Ferreirós (2007) comments that this went against the trend of the nineteenth century, where analysis was gradually being freed from its geometrical foundation.
References
Baker, R., Christenson, C., & Orde, H. (2004). Collected Papers. Bernhard Riemann. Translation from the (1892nd ed.). Heber City: Kendrick Press.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. New York: Oxford University Press.
Birkhoff, G. (1973). A source book in classical analysis. Cambridge: Harvard University Press.
Burgess, J. P. (2004). Mathematics and Bleak house. Philosophia Mathematica, 12(3), 18–36.
Burgess, J. P., & Rosen, G. (1997). A subject with no object: Strategies for nominalist interpretations of mathematics. Oxford: Oxford University Press.
Carter, J. (2004). Ontology and mathematical practice. Philosophia Mathematica, 12(3), 244–267.
Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1), 1–14.
Carter, J. forthcoming. Mathematics dealing with ‘hypothetical states of things’.
Chihara, C. (2004). A structural account of mathematics. Oxford: Oxford University Press.
Ferreirós, J. (2007). Labyrinth of thought. A history of set theory and its role in modern mathematics. Basel: Birkhauser.
Field, H. (1980). Science without numbers. Princeton: Princeton University Press.
Giaquinto, M. (2008). The philosophy of mathematical practice. In P. Mancosu (Ed.), Philosophy of mathematical practice. Oxford: Oxford University Press.
Gray, J. (2008). Plato’s ghost. Princeton: Princeton University Press.
Grosholz, E. (2000). The partial unification of domains, hybrids, and the growth of mathematical knowledge. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 81–92). Dordrecht: Kluwer Academic Publishers.
Grosholz, E. (2007). Representation and productive ambiguity in mathematics and the sciences. Oxford: Oxford University Press.
Hartshorne, C., & Weiss, P. (1965). Collected papers of Charles Sanders peirce vol. I and II. Cambrigde: The Belknap Press of Harvard University Press.
Hoffman, S. (2004). Kitcher ideal agents, and fictionalism. Philosophia Mathematica, 12(1), 3–17.
Katz, J. (1998). Realistic rationalism. Cambrigde, MA: MIT Press.
Kjeldsen, T. H., & Carter, J. (2012). The growth of mathematical knowledge—introduction of convex bodies. Studies in History and Philosophy of Science Part A, 43, 359–365.
Laugwitz, D. (2008). Bernhard Riemann 1826–1866. Birkhäuser: Turning points in the conception of mathematics. Translated by A. Shenitzer. Boston.
Leng, M. (2005). A revolutionary fictionalism: A call to arms!. Philosophia Mathematica, 13(3), 277–293.
Leng, M. (2010). Mathematics and reality. New York: Oxford University Press.
Lützen, J. (2010). The algebra of geometric impossibility: Descartes and Montucla on the impossibility of the duplication of the cube and the trisection of the angle. Centaurus, 52(1), 4–37.
Manders, K. (1999). Euclid or Descartes?. Representation and responsiveness: Unpublished draft.
Oliveri, G. (2007). A realist philosophy of mathematics, Milton Keynes: Texts in, Philosophy 6.
Scholz, E. (1982). Riemanns frühe Notizen zum Mannigfaltigkeitsbegriff und zu den Grundlagen der Geometrie. Archive for History of Exact Sciences, 27, 213–232.
Weber, H. (1892). Bernhard Riemann’s Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass. Lepzig: Teubner.
Weyl, H. (1913). Die Idee der Riemannschen Fläche. Leipzig und Berlin: B.G. Teubner.
Acknowledgments
I wish to thank the two anonymous referees for their insightful comments on earlier versions of this paper. In addition I would like to thank participants of the conference ‘International Conference on the History of Modern Mathematics 1800–1960’ held in Xi’an August 2010, where I first presented this material. In particular Colin Mc Larty, Norbert Schappacher and Karine Chemla.
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Carter, J. Handling mathematical objects: representations and context. Synthese 190, 3983–3999 (2013). https://doi.org/10.1007/s11229-012-0241-5
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DOI: https://doi.org/10.1007/s11229-012-0241-5