What Is It
Plato claimed that numbers exist in some mind-independent abstract heaven. Nominalists claim that there is no such heaven. Clearly, we can't see, hear, taste or feel numbers. But if there are no numbers what is mathematics all about? John and Ken count on a great discussion with Gideon Rosen from Princeton University, co-author of A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics.
What are mathematical objects? What kind of things are they? John claims that he has never seen a "three" or a "million". John mentions the notion of "ontology". The ontology of a particular thing refers to what it is made of, what its existence really entails.
Ken introduces Professor Gideon Rosen of Princeton University. John explains that school children are taught that numbers are just numerals. Plato on the other hand thought that numbers were non-physical, non-sensible things in a realm beyond time and space. Ken challenges the Platonist view. If numbers belong to this Platonic realm, they can not be causally active. If numbers are not causally active, how can be aware of them? Rosen explains that there are two answers to this question. One asserts that knowledge regarding numbers is acquired in a mysterious, para-psychological way. Rosen's view is that mathematical knowledge is required through doing mathematics.
Another view is that numbers are ideas in our minds. Rosen challenges this view. If "six" is just an idea, when I refer to "six", do I refer to the six in your mind or in my mind? There are infinitely many mathematical objects and most are not even thought of. Can there be "ideas" that no one has ever thought of?
Ken explains a third view: there are no numbers in the same way that there is no Sherlock Holmes. Mathematics is simply fiction with certain constraints. Rosen points out that according to this view, statements such as "there is an even number between 1 and 3" are all false, since there are no even numbers, which isn't that intuitive.
Yet another view is that numbers are just abstractions from physical magnitudes. John points out that different units lead to assignment of different numbers to the same measurement. Rosen argues that it is doubtful to think that numbers are part of the real world. There are infinitely many numbers but there isn't an infinite amount of any quantity in the real world.
Do mathematicians need to care about what numbers really are? Does the ontology of mathematics matter to the purposes of math and science? Rosen explains that mathematicians don't need to answer philosophical questions about mathematics. They postulate the existence of numbers, in the fictional or Platonist sense, as a theorem, just as "2+2=4" is a theorem.