Abstract
The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the target of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitive capacities demanded to the practitioners; and (3) the specific forms of representation and notation shared and selected by the practitioners. Moreover, it is claimed that a broadening of the notion of ‘permissible action’ as introduced by Larvor (2012) with respect to mathematical arguments, allows for a consideration of all these three elements simultaneously. Second, a case from topology—the proof of Alexander’s theorem—is presented to illustrate a concrete analysis of a mathematical practice and to exemplify the proposed method. It is discussed that the attention to the three elements of the practice identified above brings to light philosophically relevant features in the practice of topology: the need for a revision in the definition of criteria of validity, the interest in tracking the operations that are performed on various notations, and the constant and fruitful feedback from one representation to another. Finally, some suggestions for further research are given in the conclusion.
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Notes
- 1.
Lakatos (1976) was one of the first to allow for the simultaneous presence of these opposite elements in mathematics into the philosophical discussion.
- 2.
- 3.
The term “maverick” is taken from the Introduction of Aspray and Kitcher (1988).
- 4.
Larvor mentions Alexander’s theorem as an example of informal argument (Larvor 2012, p. 727), referring to Jones’ presentation (Jones 1998, pp. 209–213). We will expand on that and present the case in detail relating it to our general framework. Elsewhere, we have defended an analogous approach to diagrammatic reasoning in mathematics by offering other case studies such as knot theory (De Toffoli and Giardino 2014) and low-dimensional topology (De Toffoli and Giardino 2015).
- 5.
The cycle of conferences that brought to this collection of essays was precisely devoted to pinpoint such a notion.
- 6.
See (Lawrence this volume) in this volume for a description of such a stereotype.
- 7.
This communities do not have necessarily to share the same location: contemporary technology allows for communities to form even if the experts are geographically apart.
- 8.
Empirical studies would provide evidence for four of these ‘core’ systems and hint at a fifth one: these systems work to represent (i) inanimate objects and their mechanical interactions, (ii) agents and their goal-directed actions, (iii) sets and their numerical relationships of ordering, addition, and subtraction, (iv) places in the spatial layout and their geometrical relationships, and possibly (v) members of one's own social group in relation to members of other groups thus guiding social interactions (see (Kinzler and Spelke 2007) for reference).
- 9.
We align with the literature by using the term ‘set’, but we specify that it should be intended in an informal sense. In our opinion, ‘collection’ would be a more appropriate term, but cognitive scientists do not seem to differentiate between the two. We thank José Ferreiros for having pointed out this terminological problem to us.
- 10.
See for reference (Hutchins 2001).
- 11.
The convention of indicating crossings by double points was used by early knot theorists, see for example (Alexander 1928).
- 12.
However, the second one has the advantage that when drawing a knot diagram, we can start with the associated planar graph and only later decide which strand goes under and which over.
- 13.
The figure is taken from Wikimedia Commons, the free media repository.
- 14.
Other grouping laws belonging to the primary process are the following: vicinity, same attribute (like color, shape, size or orientation), alignment, symmetry, parallelism, convexity, closure, constant width, amodal completion, T-junctions, X-junctions, Y-junctions. See for reference (Kanizsa 1986).
- 15.
Choosing among different possible notations is a very deep and complex matter in the practice of mathematics. In knot theory, many different notations are needed and there are no ‘more natural’ ones. See for reference (Brown 1999) as a starting point and our previous study on knot diagrams (De Toffoli and Giardino 2014).
- 16.
- 17.
See (De Toffoli and Giardino 2014) for a philosophical discussion on knot theory and knot diagrams.
- 18.
See (Cromwell 2004, p. 52).
- 19.
See (De Toffoli and Giardino 2014).
- 20.
For this result, Jones was awarded the Field medal in 1990.
- 21.
It is possible to close the braids in other ways so that we can obtain different knots, but this is not relevant here.
- 22.
In order to extend the operation to braids we would need to verify that by composing different diagrams of the same braid, we obtain the same braid (which is a straightforward result, which is omitted here).
- 23.
A video would be very effective to show this isotopy. In the discussion, we will assess the informative value of videos for mathematics and for topology.
- 24.
See (Lickorish 1997, Chap. 2) for reference.
- 25.
- 26.
It is a deep result that for knot theory working on the category of smooth curves is equivalent to working in the PL category of piece-wise linear segments.
- 27.
This example is taken from (Jones 1998, p. 211) with some modifications.
- 28.
See (Murasugi and Kurpita 1999, Chap. 9).
- 29.
There are actually exceptions, but we have no time to discuss them here.
- 30.
References
Adams, C. C. (1994). The knot book. Providence: American Mathematical Society.
Alexander, J. W. (1928). Topological invariants of knots and links. Transactions of the American Mathematical Society, 30(2), 275–306.
Alexander, J. W. (1923). A lemma on systems of knotted curves. Proceedings of the National Academy of Science of the United States of America, 9(3), 93–95.
Aspray, W., & Kitcher, P. (1988). History and philosophy of modern mathematics. Minneapolis: University of Minnesota Press.
Brown, J. R. (1999). Philosophy of mathematics. New York: Routledge.
Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1), 1–14.
Chemla, K. (2005). The interplay between proof and algorithm in 3rd century china: the operation as prescription of computation and the operation as argument. In P. Mancosu, K. F. Jørgensen, & S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 123–145). Berlin: Springer-Verlag.
Corfield, D. (2003). Towards a philosophy of real mathematics. Cambridge: Cambridge University Press.
Colyvan, M. (2012). An introduction to the philosophy of mathematics. Cambridge: Cambridge University Press.
Cromwell, P. R. (2004). Knots and links. Cambridge: Cambridge University Press.
Dalvit, E. (2011). Braids. Ph.D. Thesis, University of Trento.
Dalvit, E. (2012). Braids. A movie. http://matematita.science.unitn.it/braids/. Accessed in 11 Nov 2013.
De Toffoli, S. & Giardino, V. (2015). An inquiry into the practice of proving in low-dimensional topology. Boston Studies in the Philosophy and History of Science, 308, 315–336.
De Toffoli, S., & Giardino, V. (2014). Forms and roles of diagrams in knot theory. Erkenntnis, 79(3), 829–842.
Ferreiros, J. (2015). Mathematical knowledge and the interplay of practices. Princeton: Princeton University Press.
Giaquinto, M. (2007). Visual thinking in mathematics. Oxford: Oxford University Press.
Høyrup, J. (2005). Tertium non datur: On reasoning styles in early mathematics. In P. Mancosu, K. F. Jørgensen, & S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 91–121). Berlin: Springer-Verlag.
Hutchins, E. (2001). Distributed cognition. In The International Encyclopaedia of the Social and Behavioral Sciences, 2068–2072.
Jones, V. F. R. (1998) A credo of sorts. In H. G. Dales, G. Oliveri, (Eds.), Truth in mathematics (pp. 291–310). Oxford: Oxford University Press.
Jones, V. F. R. (1985). A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society (N.S.), 12(1), 103–111.
Kanizsa, G. (1986). Grammatica del Vedere: Saggi su Percezione e Gestalt. Bologna: Il Mulino.
Kitcher, P. (1984). The nature of mathematical knowledge. Oxford: Oxford University Press.
Kinzler, K. D., & Spelke, E. S. (2007). Core systems in human cognition. Progress in Brain Research, 164, 257–264.
Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
Larvor, L. (2012). How to think about informal proofs. Synthese, 187(2), 715–730.
Lawrence, S. (this volume). What are we like.
Lickorish, R. (1997). An introduction to knot theory (Graduate Texts in Mathematics). New York: Springer.
Mancosu, P. (Ed.). (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.
Mancosu, P., Jørgensen, K. F., & Pedersen, S. A. (Eds.). (2005). Visualization, explanation and reasoning styles in mathematics. Berlin: Springer-Verlag.
Manders, K. (1999). Euclid or descartes: Representation and responsiveness, unpublished.
Murasugi, K., Kurpita, B. I. (1999). A study of braids, volume 484 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers.
Starikova, I. (2012). From practice to new concepts: Geometric properties of groups. Philosophia Scientiae, 16(1), 129–151.
Starikova, I. (2010). Why do mathematicians need different ways of presenting mathematical objects? The Case of Cayley Graphs. Topoi, 29, 41–51.
Sullivan, J. M. (1999). The “Optiverse” and other sphere eversions. arXiv:math/9905020v2.
Sullivan, J. M., Francis, G, Levy, S. (1998). The optiverse. Narrated videotape (7 min). In H. K. Hege, & K. Polthier, (Eds.), VideoMath Festival at ICM’98. Springer.
Thurston, W. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2).
Acknowledgments
We want to thank in particular Brendan Larvor for the organization of the cycle of conferences on Mathematical Cultures. A previous version of this paper was presented at the first meeting in 2010. We thank the participants for useful feedback. Thanks to John M. Sullivan for his comments. We also thank Ester Dalvit for having given us permission to reproduce part of her work and for her useful feedback.
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De Toffoli, S., Giardino, V. (2016). Envisioning Transformations—The Practice of Topology. In: Larvor, B. (eds) Mathematical Cultures. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28582-5_3
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