Thursday, August 21, 2014

Mathematical Platonism as a Contemporary Example of Plato's Theory of Ideas

Comerica Park, Detroit

I'm prepping for the Intro to Western Philosophy class I'm teaching this fall at MCCC. The first section will be on ancient Greek philosophy - Pre-Socratic, Plato, and Aristotle. I'll be giving contemporary examples that illustrate the relevance of what the ancient Greek thinkers were dealing with.

The current example I'm using for the relevance of Plato's Theory of Ideas is mathematical Platonism. Today I am especially looking at University of Toronto philosopher James Robert Brown's Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. Chapter Two is entitled "Platonism."

This was the chapter that "disturbed" Massimo Pigliucci. (See here.) Pigliucci writes:

"If one ‘goes Platonic’ with math [note: a number of mathematicians are mathematical Platonists], 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."

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

"Let us suppose the sentence 'Mary loves ice cream' is true. What makes it so? In answering such  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 semantical considerations imply Platonism. Consider the following true sentences: '7+5 = 12', and '7 >12'. Both of these 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.

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