Bernays Platonism

This article appears to be the one in which the term "Platonism," as a name for a view about mathematics, was coined, in 1934.

Plato advocated, not taking abstract objects to exist, but universals, ("the red") which are more like properties. The medieval debates between the nominalists and the realists were not about abstract objects but about properties. At least, so I am told. Of course, I am also told that all the standard views about Plato are wrong.

At any rate, for these reasons, the term has become less fashionable of late, and people now tend to talk about realism about mathematical objects, mathematical realism (which is sometimes the view that mathematical statements can be true or false without a corresponding view about objects), and, usually, since the context is about mathematics, just realism.

Austin said that "real" is a trouser word: it is the contrast that wears the trousers. You don't know what real means without knowing first what the worry is:

Is it real cream? Or a hallucination. Is it real cream? Or coffee whitener? Is it real cream? Or cream from a genetically modified cow.

Is it (a tux) really black? Or is it blue and the light is just bad. Is it (her hair) really black? Or is it really gray? Here the question is not about the light, but about whether it is dyed.

Is it a real duck? Or a decoy? Is it a real decoy? Or a toy? Is it real toy? Or a display mockup?

My son told me he likes Zelda because it uses real mythical creatures.

"Real" doesn't mean a whole lot. That does not mean that the debates about mathematical realism, or Platonism, are useless, it just means that the first task in looking at any discussion of realism is to sort out what the problem under discussion really is.

Bernays talks about strengths of Platonism. In fact, he advocates "restricted Platonism," while claiming that "absolute or extreme Platonism" is refuted by the paradoxes. He also considers Platonism about only the natural numbers, Platonism about the natural numbers and the real numbers, …. Right away, we know that the simple idea of realism is in trouble: reality doesn't come in strengths: the natural numbers are more strongly real than the real numbers.

So what issues does Bernays actually associate with Platonism?

1. "The value of platonistically inspired mathematical conceptions is that they furnish models of abstract imagination." What is "abstract imagination" being contrasted with? "[Platonistically inspired mathematical conceptions] form representations which extrapolate from certain regions of experience and intuition." (p. 259)
   1. He contrasts Euclid's constructions with Hilbert's static facts: (Euclid) Given two points, one can produce a line that passes through both. (Hilbert) Any two points are contained by exactly one line. Hilbert's version is about things that already exist, independent of any actions or time.
   2. He doesn't say enough for this to be completely clear, but the kinds of things he mentions are, attributing to large, even infinite, mathematical system, the properties of small finite ones. Thus, given a finite collection, there is a collection of all the subcollections of it, combinatorially selected, not in virtue of any property, and so we suppose that the same is true of infinite collections. In taking infinite collections to have properties analogous to finite collections, we are, in a certain sense, taking them to be real and independent of us.
   3. The intuitionist balks at treating the infinite like the finite. Bernays points out that we could equally well balk at : We , including the intuitionists, take that number to have a representation in arabic numerals. It doesn't: the representation would be too big to fit into our universe. Even the intuitionist to some extent abstracts from "elementary evidence."
   4. Ed Nelson distinguishes between genetic numbers —ones you can count up to—and exponential numbers —like Bernays's example—and views it as a questionable hypothesis of the intuitionists that every exponential number is genetic.
2. The law of excluded middle. The law of the excluded middle is just the logical principle of the form . As Dummett has emphasised, the principle of bivalence is the principle about language that every appropriate sentence is either true or false. (By appropriate, I mean the sort of sentence that we ordinarily take to purport to be either true or false, ruling out questions, commands, and so forth.)
   1. In works of fiction, we obviously don't take every relevant sentence to be either true or false. It just isn't the case either that Santa's suit is made of cotton or not. It arguably is the case that "Either Santa's suit is made of cotton or it is not." However, "Santa's suit is made of cotton" does not seem to have a truth value.
   2. The point for our purposes is that when something exists, that gives us reason to endorse LEM and Bivalence about it. If something does not exist, that casts doubt on applicable instances of LEM and Bivalence. Thus, endorsing LEM and /or bivalence about some mathematical entities is a mark of realism about them. Dummett at one time, took that to be the definition of realism.
   3. This example allows the infinite set of all numbers, but does not allow infinitely many independent choices.
   4. Bernays's chief example is trichotomy: given any two real numbers, either the first is greater than the second, they are equal, or the second is greater than the first.
3. Predicativity. Can we pick out something among the class of all such things, or are we creating things, and hence without a stable class of all of them?
   1. Bernays's chief example is the classical reals versus Weyl's reals, which he assimilates to the arithmetic reals.
4. He doesn't emphasize one of things that today is taken as a mark of realism: mathematical statements are true or false.
   1. I think the reason is that he is comparing various ways of practicing mathematics, and mathematicians within every practice take their theorems to be true.
5. He gets off the train with the idea that the domain of discourse is itself a real mathematical object. That is, he rejects "absolute Platonism." Why? Suppose that "all" mathematical objects "exist." Then one of them is the collection of all mathematical objects. Call that collection C. Let be the collection of all collections in C that are not members of themselves. Is in C? Answer: it it is, we're up against Russell's paradox. Thus, it isn't in C. What I've proved, which is a perfectly ordinary theorem proved by a standard proof, is that given any collection, there is something that is not in it. This raises the question whether it makes sense to talk about "absolutely, unrestrictedly everything." See the book Absolute Generality, Uzquiano and ? eds.

The details of the various viewpoints I've presented are not the central issue. The central issue is that Bernays, unlike anyone else we've read, is happy to be eclectic: different points of view about logical and foundational issues motivate different mathematical systems. They have different things going for them, and comparison between them is not only philosophically but even mathematically illuminating. Everyone else we've read is trying to find and defend the one true mathematical faith.

-- ShaughanLavine - 09 Feb 2005 - 08 Feb 2007