Abstract
There are at least two versions of modern structuralism and each has its proper difficulties: if one adopts the in re version, the crucial feature is that the background ontology is not understood in structural terms; if one adopts the ante rem version, the crucial feature is that the talk about structures is exposed to a kind of third man objection.
The main thesis of this paper is that Poincaré ’s conventionalism and Lautman ’s structuralism must be ranked among these sources of structuralism that try to escape the mentioned difficulties.
Poincaré uses a psycho-physiological approach in order to justify his conventionalism in geometry, which is an improvement of an attenuated version of ante rem structuralism , and Lautman proposes a metaphysical dialectic in order to justify his anti-foundationalist position, which brings ante rem and in re structuralism together. Poincaré’s approach fails for technical reasons whereas Lautman’s approach fails for its aporetic conceptual vagueness.
My present concern is to incorporate the French historical inheritance in the systematic discussion of mathematical structuralism .
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Notes
- 1.
Cf. for the following (Heinzmann/Stump 2013).
- 2.
Because Poincaré distinguished very well between analytic and synthetic sentences, the analogy to Quine must be restricted to conventions.
- 3.
Representation of an object in the sensible space means nothing else than the deliberate and conscious reproduction of muscular sensations thought necessary to reach the object: “When it is said, […] that we “localise” an object in a point of space, what does it mean? It simply means that we represent to ourselves these movements that must take place to reach that object […] When I say that we represent to ourselves there movements, I only mean that we represent to ourselves the muscular sensations which accompany them” (Poincaré 1902, 57; our emphasis).
- 4.
Reprinted in (Lautman 2006).
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Heinzmann, G. (2014). Does the French Connection (Poincaré, Lautman) Provide Some Insights Facing the Thesis That Meta-mathematics Is an Exception to the Slogan That Mathematics Concerns Structures?. In: de Paz, M., DiSalle, R. (eds) Poincaré, Philosopher of Science. The Western Ontario Series in Philosophy of Science, vol 79. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8780-2_7
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