In this post, I will follow closely the excellent summary of the nature of the debate on mathematical Platonism offered by Øystein Linnebo, though the book by James Brown on the philosophy of mathematics is also an excellent, if opinionated, source.
To begin with, just to clear the air of a possible misunderstanding, mathematical Platonism — despite the name — bears little correspondence to Plato’s theory of forms. The latter, it may be recalled, was based on the notion that the world as we perceive it is but a pale reflection, a shadow (as in the famous metaphor of the cave) of the real world of pure concepts, to which, however, it is related. So for Plato there are the chairs of our everyday experience and then there is the ideal of a Chair, there are good things in the world and the ideal of Good itself, and so on.
Mathematical Platonism, instead, is a much more metaphysically circumscribed notion about the ontology of a particular category of abstract objects, those of concern to mathematicians (like numbers, sets, and so on). To be precise, Linnebo defines mathematical Platonism as the conjunction of these three theses:
Existence: There are mathematical objects.
Abstractness: Mathematical objects are abstract.
Independence: Mathematical objects are independent of intelligent agents and their language, thought, and practices.
As we shall see, only the last thesis, Independence, is controversial, and whether one accepts all or only a subset of the above theses defines what sort of ontology one is willing to attribute to mathematical objects.
Let us then start with what is likely the least controversial thesis, that of Existence. Linnebo mentions that the famous logician Frege proposed the following argument in defense of the existence of mathematical objects:
Premise 1 (Truth): Most sentences accepted as mathematical theorems are true.
Premise 2: Let S be one such sentence.
Premise 3 (Classical Semantics): The singular terms of the language of mathematics — such as S — purport to refer to mathematical objects, and mathematical language’s first-order quantifiers purport to range over such objects.
Premise 4: By Classical Semantics, the Truth of S requires that its singular terms succeed in referring to mathematical objects.
Conclusion: Hence there must be mathematical objects, as asserted by Existence.
The second premise is a simple stipulation, so it cannot be challenged. Premise 1 could be challenged, but only at the cost of doing away with much mathematics and its well established applications to science, clearly not a viable route. (Linnebo presents a number of ways to defend the Truth premise, the most convincing of which is the so-called indispensability (of mathematics to science) argument proposed by Quine and Putnam.) Premise 4 is a straightforward derivation of Premises 1-3, so the only thing that could possibly be argued is the truth of Premise 3, but very few philosophers and logicians have seriously questioned classical semantics (see Linnebo’s article for a discussion of this point), therefore we have to agree that mathematical objects exist.
Now that we have Existence, what about the second thesis, Abstractness? Interestingly, this is by far the least controversial piece of the puzzle, as most philosophers think Abstractness so likely that there are few explicit defenses (or criticisms) of it. Linnebo summarizes the situation by simply stating that if mathematical objects were not abstract then mathematicians — like scientists — should be concerned about their location and other physical attributes. Since mathematicians, and — more importantly — the practice of mathematics itself, don’t concern themselves with such things, this is a good prima facie argument for the abstractness of mathematical objects.
Even if we accept both Existence and Abstractness we have not arrived at mathematical Platonism just yet. Rather, we can think of ourselves at this point as anti-nominalists, since nominalism is the philosophical position that there are no abstract objects. Anti-nominalism, it should be obvious, is logically weaker than full fledged mathematical Platonism (which, recall, requires all three theses: Existence, Abstractness and Independence).
Here Linnebo makes an intriguing observation: few philosophers deny the independence of mathematical objects from the existence of minds capable of thinking about them not much because there are a lot of arguments in favor of the thesis, but rather because it would not be at all clear what it would mean for the thesis to fail. Be that as it may, there are some arguments in favor of Independence, most famously those formulated by Kurt Gödel (he of the incompleteness theorems that famously undermined Russell and Whitehead’s quest for self-sufficient logical foundations of mathematics). Gödel proposed two arguments to establish Independence. Due to my limited understanding of mathematical theory, I will simply let Linnebo summarize them both:
First, “the legitimacy of impredicative definitions is best explained by the truth of some form of Platonism [according to Gödel], including something like [the] claim [of] Independence.” Second, “Much of the search for new axioms in set theory is today based on so-called ‘extrinsic justifications,’ where candidate axioms are assessed not just for their intrinsic plausibility but also for their capacity to explain and systematize more basic mathematical facts. Perhaps this methodology can somehow be used to motivate Independence.” Again, however, recall that the stronger argument in favor of Independence appears to be simply how hard it is to fathom the meaning of its failure.
There are, naturally, plenty of objections (and counter-objections) to the notion of mathematical Platonism. Arguably the most obvious one is the issue of epistemic access, which asks how exactly we can gain reliable mathematical knowledge (which we apparently do) if mathematical objects really are abstract and mind-independent. We know how we get epistemic access to mind-independent physical objects (planets, say), but what human sense could possibly be involved in mathematical knowledge? The epistemic access objection is based on a crucial demand for a causal explanation of the reliability of mathematical knowledge, but as it turns out some philosophers have proposed more minimal accounts of reliability that do not involve causality (not even Linnebo gets into this, but he does provide references to the relevant primary literature).
An interesting objection to mathematical Platonism is of a metaphysical nature, and it basically states that there is nothing to mathematical objects (say, numbers) outside relations to other such objects. In other words, there really aren’t “objects” at all, just relations. There are, naturally, counters to this argument too. The idea that natural numbers have only structural properties is apparently rejected by logicist and neo-logicist philosophers on the grounds that numbers are tied to the cardinals of the collections they number.
Moreover, structuralist philosophers reject the notion that there cannot be objects characterized only by structural properties. Which, as the attentive reader might have surmised, brings us right back to Ladyman and Ross, since their contention is that even what we think of as physical objects are nothing, at the bottom, but loci of relational properties (hence the title of their book, every thing must go). If that’s not a problem for physical objects it is hard to see why it would be for abstract ones.
To recap, there are strong positive arguments in favor of the Existence and Abstract theses, the acceptance of which at the very least commits one to anti-nominalism in the philosophy of mathematics. In order to be a full fledged mathematical Platonist one also has to accept (or at the least not reject) the further Independence thesis. We have seen that direct arguments in favor of this thesis do exist, but that the most convincing one for philosophers is actually the difficulty of making sense of the failure of the thesis.
Of course all of the above can and have been debated, but at the very least implies that mathematical Platonism is not at all a fanciful or irrational position (it is apparently accepted by most mathematicians, not just by philosophers of mathematics and logicians). Notice also that nothing that we have discussed is in any way incompatible with naturalism (as, again, Ladyman and Ross also stress). So until further notice consider me a naturalistic anti-nominalist with strong tendencies toward Platonism in mathematics.
First appeared on Rationally Speaking, 9/14/12