Category: Philosophical theories

Mathematical Platonism

“Mathematical Platonism” designates a school in philosophy of Mathematics. According to that school mathematical objects are abstract entities existing by themselves independently of the mathematician that investigates them.

Interpretations with respect to “Platonism” and to what it consists in vary from downright realistic to nominalistic and transcendental. Mathematical Platonism, however, cannot but be making part of Realism. It differentiates itself from schools of philosophy of Mathematics that consider mathematical objects/theorems as constructions of our consciousness (Intuitionism), or as the result of a ‘mere game between symbols’ (Formalism), or believe that a mathematical formula can be true, if and only if it is provable (Antirealism).

According to Mathematical Platonism: (i) the objects of mathematics exist ‘out there’, independently of us, just as physical objects, with the difference that they are abstract; (ii) the theorems of mathematics represent the objective relations that govern the same objects. This combined claim results in a series of positions and methodological practices that we will now briefly go through.

The term “Mathematical Platonism” has appeared during the first half of the 20th century, but it retrospectively claims many philosophers and schools. The most emblematic among the schools of Philosophy of Mathematics that are platonic is Logicism. Its main representatives are Bertrand Russell and, according to some interpretation at least, Gottlob Frege. According to this school, natural numbers (i.e. the fundamental mathematical objects) are some specific kind of sets. (Number n is the set of all sets that are constituted of n objects.) These sets are abstract objects of our world that exist independently of our consciousness. Logicism considers that mathematical theorems represent the objective relations by which these objects are governed and that the mathematician is simply called to discover them. On the other hand, non platonic/realistic schools uphold that mathematical objects are constructions of our consciousness (Intuitionism), or that they result by transformations of some initial set of axioms that are here considered as sequences of meaningless symbols (Formalism). The mathematician is called to discover which sequences of symbols result from the axioms, upon some already given transformation rules. Intuitionism, as well as Formalism (at least the one of Hilbert), have Kantian roots and, thereby, render consciousness a parameter in mathematical practice.

The most characteristic, perhaps, proponent of Mathematical Platonism during the previous century was Kurt Gödel. In two landmark papers on philosophy of Mathematics, he sketches the main principles of Mathematical Platonism. Despite the fact that his Platonism has not been completely free of intuitionistic elements, these latter concern his mathematical epistemology rather than his ontology. In substance, Gödel’s Mathematical Platonism has kept the technique of Formalism intact, while, at the same time, does not consider mathematical formulas to be meaningless. Gödel believes that the mathematician is under the constant responsibility of shaping the formal framework she is working in so that it best depicts what her intuition tells her about the mathematical universe. For example: Gödel believes that the Continuum Hypothesis is true. However, none among the current set theories is able to prove it, and, moreover, it has also been shown that, if these latter are consistent, they will never prove it, even if it is true. Unlike formalists, for which the realization of its undecidability by the same set theories is enough, Gödel thinks that our intuition should see us through a revision/change of the axioms of current set theories in a way that allows the Continuum Hypothesis to be proved.

One of the most interesting criticisms against Mathematical Platonism is the so-called “problem of Benacerraf”. Mathematical objects, according to Mathematical Platonism, are both abstract and self-subsistent. The problem that Benacerraf has raised is –in variations– the following: If the above two claims about mathematical objects are true and if, moreover, (i) the knowing subject cannot reach any knowledge of any object, unless it is involved in some causal interaction with it, and (ii) the mathematician, through her work, reaches some kind of knowledge of mathematical objects, then it follows that something abstract, i.e. the mathematical object, is in causal interaction with a spatiotemporal entity, i.e. the mathematician. Now, since it is far from clear how something that is abstract can be in causal interaction with anything whatsoever, the assumption that we can, in principle, have knowledge of the Platonic mathematical objects becomes equally far from obvious. It is of some interest here that Plato himself elaborates in the Sophist (247d-249d, esp. 248c-249a) a similar argument about the (immaterial) ideas and how it is possible for them to be involved in any causal interaction with us. For, if we claim that it is possible to obtain knowledge of the ideas, and every knowledge presupposes some causal interaction between the knower and the known, how can we know the ideas, if they are immaterial?

Author: Doukas Kapantais
  • Benacerraf, P. "Mathematical Truth." The Journal of Philosophy 70 (1973)
  • Dummett, M. "Realism." Synthese 52/1 (1982)
  • Hilbert, D. "Über das Unendliche [Αγγλική μετάφραση στο, Van Heijenoort, J., From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931 (Cambridge Mass. 1967)].." Mathematische Annalen 95 (1926)
  • Frege, G. Die Grundlagen der Arithmetic. Breslau, 1884.
  • Gödel, Κ. American Mathematical Monthly. 1947.
  • Gödel, Κ. "Russell’s mathematical logic." Schilpp, P. ed. The Philosophy of Bertrand Russell. New York, 1951.
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