The Myths of Mathematical Platonism

Page last updated 25 Sep 2020

The standard Platonist claim is that mathematical entries ‘exist’ in some non-physical sense, completely independently of any language or definition. See also Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.

The origin of the myths of Platonism might be attributed to the usage in mathematics of expressions such as “there exists an x that satisfies the condition …”, where the word ‘exists’ is used in the same way as when it is used to refer to the physical existence of physical objects. Many Platonists claim that one is thereby inferring the actual non-physical ‘existence’ of something by the use of the word ‘exists’. (Footnote: See, for example, the introduction to the book: A Subject with No Object: Strategies for nominalistic interpretation of mathematics, Burgess, John P. & Gideon Rosen, Clarendon Press, 1997, ISBN: 9780198250128 A Subject with No Object: Details.) But there is no inherent need to use a term such as ‘exists’. One could equally well use an expression such as, “it is possible for an x to satisfy the condition …”, and where the domain of x is simply a specified set of symbol sequences of the language in question.

Clearly, since the postulated Platonist entities cannot have any physical form, the only way to argue for the ‘existence’ of such entities is by language. This is quite different to the case of a reference in language to a physical entity, where the existence of a physical entity can be established by physical means.

And so immediately we can perceive a problem with the Platonist claim, and the problem is this. Words in language do not have any inherent correspondence to physical entities, and any such correspondence has to be established by the utilization of ostensive means that are not part of the language itself.

For example, consider the sentence, “There is a chair in the room.” The word ‘chair’ does not have any inherent connection to the concept of a chair. That connection has to be established, and such a connection might be established by pointing to a chair while saying the word chair. Of course, the objection might be raised that one could use other words to describe a chair. While this is true, it merely moves the necessity of a demonstrative ostensive connection of words to physical objects elsewhere, and at some point in such recursive definitions a demonstrative connection must be made, and which requires the use of something other than words - a physical demonstration. Of course, once such an initial correspondence is established between words and physical objects, one can then describe new physical objects by the use of words. But the requirement for an initial establishment of a correspondence of words to physical objects, by means other than only the use of words, must be recognized as being absolutely essential. (Footnote: It might be claimed, for example, that an alien intelligence might be able to analyze a human language and deduce which physical things correspond to words of that language. But that ability itself depends on the alien having already acquired a knowledge of the intended correspondence of his own alien language to physical objects.)

It is only once that this correspondence of words to physical objects has been initiated, one can then use words to describe things as having various physical attributes, but which do not actually exist - for example, one can describe unicorns and their physical attributes in intricate detail, but that does not indicate that unicorns exist.

Since this is the case, we can readily see that the Platonist claim that certain words (and other sequences of symbols) imply the actual ‘existence’ of non-physical entities has no logical basis whatsoever. Since a description in words has no inherent correspondence to any physical thing, and since words/sentences may describe things that do not physically exist, there is no logical reason to infer that, simply because there is a description in words (Footnote: Note that here we use the term ‘description in words’ as a convenient term that includes any description by some sequence of symbols, such as mathematical definitions.) of a non-physical thing, that thing must ‘exist’ in some non-physical sense.

There is no finer example of a logical non sequitur.

Furthermore, it need hardly be pointed out that no-one has ever made a verified correspondence of a non-physical entity to a word or expression of language.

Of course, in the case of physical objects, once the initial correspondence of words to physical objects is established, one can also refer, using words, to properties of physical objects. And one can also refer to properties that collections of objects have (such as numbers), even if there is no individual object that can have that property. And those properties can themselves have properties, and so on. But regardless of the amount of recursive definition, all such properties ultimately derive from an initial correspondence between words and physical objects. Nowhere in such definitions is there any logical implication whatsoever of the independent non-physical ‘existence’ of any such properties.

It might be argued that, for example, the number five does not refer to any individual thing, but it can be used to refer to a collection of things, such as a collection of properties, and that, as such, it is independent of any particular physical object. But it is also the case that, for example, the word ‘chair’ is independent of any particular physical object, but we would not normally postulate that there ‘exists’ an independent non-physical thing that is associated with the word ‘chair’. (Footnote: One might note that this is what Plato originally postulated, that there ‘exist’ perfect forms in some sort of ethereal realm, and that these perfect forms hold the essence of all the actual physical things that derive from the perfect form. So Plato’s view was that there ‘exists’ a perfect non-physical chair that instills the essence of ‘chairness’ into all physical chairs.) Instead, we would recognize that the word ‘chair’ is a term that we use to indicate the general concept of a member of the set of both actual physical chairs and hypothetical imagined chairs. In the same way, it is logical to recognize that the word ‘five’ is a term that we can use to refer to any collection of five things, or any collection of five hypothetical imagined things, or any collection of five describable properties, and that it indicates the general concept of a collection of five physical things or a collection of five properties, and similar collections.

Of course, it can be the case that one can refer to relationships between properties, and one does not need to specifically reference the physical objects that originally gave rise to those properties. As such all one needs is an alphabet, and a set of rules of grammar as to what symbol sequences (words and sentences) are allowable, and a set of rules of logic as to what symbol sequences can be derived from other sequences of symbols. Which is, of course, what we have in mathematics - and where there is no logical implication of the independent non-physical ‘existence’ of things that correspond to any of those symbol sequences.

See also Numbers, chairs and unicorns, and for more on Platonism see Platonism, Mark Balaguer and Platonism, Platonism’s Logical Blunder, and the posts Moderate Platonism and Descartes’ Platonism.