Wednesday, May 29, 2024

Mathematical Platonism As a Problem for Physicalism

Comerica Park, Detroit

(At dinner tonight my friends J and D and me discussed mathematical Platonism and the ontological status of abstract objects. I'm re-posting this for them.)

University of Toronto philosopher James Robert Brown argues for Mathematical Platonism in Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Brown uses Plato's theory of Ideas and applies it to math, arguing that non-physical mathematical ideas have ontological status; i.e., mathematical ideas exist. 

See Chapter Two - "Platonism." This was the chapter that "disturbed" Massimo Pigliucci. (See here.)

Brown is not the only mathematical Platonist. A number of mathematicians are, to include Frege, Godel, et. al.  Pigliucci writes:

"If one ‘goes Platonic’ with math , one has to face several important philosophical consequences, perhaps the major one being that the notion of physicalism goes out the window. Physicalism is the position that the only things that exist are those that have physical extension [ie, take up space] – and last time I checked, the idea of circle, or Fermat’s theorem, did not have physical extension. It is true that physicalism is now a sophisticated doctrine that includes not just material objects and energy, but also, for instance, physical forces and information. But it isn’t immediately obvious to me that mathematical objects neatly fall into even an extended physicalist ontology. And that definitely gives me pause to ponder."

Brown cites the connection between Platonism and semantic theory. The logic of mathematical Platonism runs like this.  He writes:

"Let us suppose the sentence 'Mary loves ice cream' is true. What makes it so? In answering such a question we'd say 'Mary' refers to the person Mary, 'ice cream' to the substance, and 'love' refers to a particular relation which holds between Mary and ice cream. It follows rather trivially from this that Mary exists. If she didn't, then 'Mary loves ice cream' couldn't be true, any more than 'Phlogiston is released on burning' could be true when phlogiston does not exist.

The same semantic considerations imply Platonism. Consider the following true sentences: '7+5 = 12', and '7 >12'. Both require the number 7 to exist, otherwise the sentences would be false. In standard semantics the objects denoted by singular terms in true sentences ('Mary', '7') exist. Consequently, mathematical objects do exist." (Brown, 13)

So, the number '7', and 'pi', and you-number-it, exist. But where? Surely, not in physical reality. I just hit the number 7 key on my keyboard. The number 7 key exists physically. But I won't be hitting the number 7 anytime in the future.

Brown states: "Mathematical objects are outside space and time." (Ib.) They are non-physical, abstract objects with ontological status.

For more see Pater van Inwagen and William Lane Craig, Do Numbers Exist?