Abstract
If there is a ‘platonic world’ \(\mathcal {M}\) of mathematical facts, what does \(\mathcal {M}\) contain precisely? I observe that if \(\mathcal {M}\) is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent of us. Both alternatives challenge mathematical platonism. I suggest that the universality of our mathematics may be a prejudice and illustrate contingent aspects of classical geometry, arithmetic and linear algebra, making the case that what we call “mathematics” is always contingent.
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Notes
For an overview, see for instance (Linnebo 2013).
Contemporary mathematicians that have articulated this view in writing include Roger Penrose (Penrose 2005) and Alain Connes (Connes).
Hard to deny the existence of mathematical objects if to exist is defined “to be the value of a variable”(Quine 1953)!
Here I use “axiom” as in the classic definition of a group: A group is a set together with an operation that satisfy four ‘group axioms’ (Herstein 1975). One may object that a mathematical object (a group, for instance) is something else from a set of axioms it satisfies. This objection is not relevant for the following, because whenever different extensions of a set of axioms are possible, these define objects that mathematicians consider distinct. Therefore the infinite multiplicity of the set of inequivalent consistent axioms pointed out below cannot be regarded as redundancy in description.
Hegel utilized this Yiddish saying to ridicule Schelling’s notion of Absolute, meaning that –like mathematical platonism– this included too much and was too undifferentiated, to be of any relevance (Hegel 1977).
Cosmological measurements indicate that spacetime is curved, but have so far failed to detected a large scale cosmological curvature of space. This of course does not imply that the universe is flat (Ellis 2005), for the same reason for which the failure to detect curvature on the fields of Egypt did not imply that that the Earth was flat. It only shows that the universe is big. Current measurements indicate that the radius of the Universe should be at least ten time larger than the portion of the Universe we see (Hinshaw and WMAP Collaboration 2009). A ratio, by the way, quite similar to the Egyptian case.
Dante mentions Euclid in the Commedia, and refers vaguely to his geometry, but gives no signs of having studied it (Rak 1970).
Kant was notoriously wrong in stating that the Euclidean geometry of physical space is true a priori (Kant 2008). Reality is that this is neither true a priori, nor true a posteriori: it is false. But even Wittgenstein appears to dangerously assume a unique possible set of laws of geometry for anything spatial: “We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry” [Tractatus, Proposition 3.0321] (Wittgenstein 2007). Really?
This might be why ancient humans attributed human-like mental life to animals, trees and stones: they were perhaps utilizing mental circuits developed to deal with one another –within the primate group– extending them to deal also with animals, trees and stones.
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Thanks to Hal Haggard, Andrea Tchertkoff and Filippo Cesi for a careful reading of the manuscript and comments.
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Rovelli, C. Michelangelo’s stone: an argument against platonism in mathematics. Euro Jnl Phil Sci 7, 285–297 (2017). https://doi.org/10.1007/s13194-016-0159-8
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DOI: https://doi.org/10.1007/s13194-016-0159-8