Practical Platonism

 

We have  again and again  glanced  at the question of Realism in mathematics, from a philosophical perspective.

Here is a case in which the Platonist perspective is actually integral to daily practice, rather than merely something to be speculated about postprandially, over brandy and cigars:

The complication with impredicative definitions  is that we lose track of how new sets are introduced.   To justify them, we have to assume that sets are more or less already there, so that the definition merely serves to describe certain properties of pre-existing things, rather than to bring the set defined  into being.

-- Hao Wang, From Mathematics to Philosophy  (1974), p. 78

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Bertrand Russell writes, (Introduction to Mathematical Philosophy, chapter 13 – p. 137 of the Dover edition):

Classes are logical fictions; and a statement which appears to be about a class  will only be significant if it is capable of translation into a form in which no mention is made of the class.

Now, if anyone has a right to such nominalism about classes (a.k.a. sets), it is Bertrand Russell, who gave up the best and most difficult years of his life, laboring on Principia Mathematica, which attempted to found the whole of mathematics upon no more than logic and set theory – truly the case of angels dancing on the mosh-pit of a pin. The man has earned his stripes.  But if classes are logical fictions, please inform me what is not.

            Consider the humble hamburger.  The following are the available entrees for Friday, March 41^st, at the Gödel Middle School:

            hamburger; hot dog; mystery meat.

We do not say “the singleton set consisting of the hamburger”, although really that might be more accurate (an actual *hamburger* is hot, and made of meat; how could *it* figure on a menu? and would it have ketchup if it did?).  But consider this.  The following are the meals available at said Middle School, for the price of two lunch coupons:

{hamburger, french fries, pickle}; {hotdog, potato chips; cole slaw}; {mystery meat; mystery fries; unidentified side-dish}.

            You can’t strip the set-parentheses out of that:  you’d get a jumble, the leftovers from some legendary food-fight.  Three classes, each consisting of three members, are staring us in the face.  And the food-lady is getting impatient.  Anyone have a Loewenheim-MacDonald’s model, in which the classes somehow disappear?

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Back to Wang:

A Platonic world of ideas, unlike material things in space and time which form the basis of the physical siences, seems to have very little explanatory power in mathematics.

-- Hao Wang, From Mathematics to Philosophy  (1974), p. 49

My heart sank as I read this.  Yet on the very next page  he contradicts that diffuse and unsupported sentiment,  in quite a concrete way:

Extensions of the usual set of arithmetic axioms  seem to be just as natural, e.g. the addition of transfinite induction to the first epsilon number.  This tends to indicate that there is something absolute in the concept of number, and that we only gradually approximate it through mental experimentations.  Or, at least, we have no full control over our intentions and mental constructions, which, once in existence, tend to live a life of their own.

-- Hao Wang, From Mathematics to Philosophy  (1974), p. 50