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I was surprised to learn that Quine is a mathematical realist (See this interview for example). I always assumed that his "Two Dogmas of Empiricism" and specifically his dissolution of the Analytic/Synthetic distinction was a strong challenge to mathematical realism. My reasoning was the following:

  • (a) Quine shows that what seemed like a perfect example of analytic proposition "all Bachelors are unmarried" is actually based on empirical data inferred from the way the word Bachelor is used. There are possible worlds where "Married bachelor" is a valid statement.
  • (b) Based on the same reasoning "Odd numbers aren't divisible by 2" can also be considered an empirical statement, and there are possible worlds where "There are Odd numbers divisible by 2" is a valid statement.

Quine later in the paper argues that even the laws of logic are themselves subject to potential change based on empirical data. The way I understand it, this, along with the reasoning in (b), would seem to indicate that math is arbitrary. Based on different empirical data and suitably chosen axioms of logic, just about any mathematical system can be constructed, where statements like "prime that has an integer square root" or "1+1=3" are completely valid.

Towards the end of the paper, he famously states:

"Physical objects are conceptually imported into the situation as convenient intermediaries not by definition in terms of experience, but simply as irreducible posits comparable, epistemologically, to the gods of Homer . . ."

My questions:

  • If even the law of excluded middle is subject to revision based on empirical data, then isn't "2+2=4" subject to revision as well?
  • Is my analogy between (a) and (b) correct? Or even if (a), (b) is still incorrect?
  • Isn't his stance as a mathematical realist contradicted by the conclusion of the paper? If even physical objects are posits epistemically on par with Homeric gods, then don't mathematical objects have the same epistemic challenge as well? How can they be real?
  • @Conifold Your thoughts? – Alexander S King May 11 '16 at 21:08
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    It's a good question and I don't think I know enough about Quine to give a good answer, but here's a thought: just as the fact that the meaning of bachelor is subject to revision doesn't necessarily imply that there are no people who are bachelors, so too the fact the math is subject to revision doesn't necessarily imply that there are no math objects. – Eliran May 11 '16 at 21:20
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    @EliranH see my edit: Your though is compelling, but Quine's comparing of physical objects to Homeric gods strikes me applying to mathematical objects as well. – Alexander S King May 11 '16 at 21:34
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The short answer is that Quine is not a mathematical realist as intended in the question (on my reading of it). Why does he call himself a realist? Because he practices what he preaches. Indeterminacy of translation, and hence meaning, implies that words only mean as relata in a scheme, not as individual references to raw reality or mental content. Holism of verification implies that conceptual scheme is only testable as a whole, although he moderated this to "chunks" with a "critical mass" in Two Dogmas in Retrospect (his dismissal of analyticity is somewhat moderated there as well, so it should be read in conjunction with the original Two Dogmas for a full picture):

"Looking back on it, one thing I regret is my needlessly strong statement of holism... "no statement is immune to revision". This is true enough in a legalistic sort of way, but it diverts attention from what is more to the point: the varying degrees of proximity to observation..."

So "2+2=4" is subject to revision only in a legalistic sort of way, in practice it is highly likely to stick around under the maxim of "minimal mutilation". But we could make it false by changing the use of symbols 2, +, = and 4, in the absence of intrinsic meanings this is hardly surprising.

What of existence? This is clarified in On What There Is:

"To be assumed as an entity is, purely and simply, to be reckoned as the value of a variable", “to be is to be the value of a variable” in a scheme. "We look to bound variables in connection with ontology not in order to know what there is, but in order to know what a given remark or doctrine, ours or someone else's, says there is".

So when Quine says that mathematical and physical objects "really exist" what it amounts to is that they can not be eliminated from our current scientific scheme by paraphrase, like "the current king of France" can be. In his own words from Theories and Things:

"I see no way of meeting the needs of scientific theory... without admitting universals irreducibly into our ontology... Nominalism... is evidently inadequate to a modern scientific system of the world".

This came to be known as the "indispensability argument" against nominalism, and it is in this way that Quine is a realist about universals. And this realism has no need for the analytic/synthetic distinction.

This is clearly not the colloquial meaning of "existence" or "realism", although Quine would claim that he is using the words in the same role relative to the scientific scheme as they are used colloquially relative to the naive everyday "scheme". And if a paraphrase, known as "nominalistic reconstruction", of universals were to be found, well, then presumably they won't exist no more. Burgess, a traditional realist, gives an illuminating explanation with a bit of history in Why I am Not a Nominalist:

"Some antinominalists have argued that the conflict between nominalism and science is so strong that nothing like modern science as we know it could survive if the nominalist ban on mathematical abstractions were accepted. Such a position has been reluctantly maintained by the ex-nominalist Quine ever since the failure of his joint attempt with Goodman at nominalistic reconstruction. Such a position was also maintained, under Quine's influence, by Hilary Putnam, during his phase of enthusiastic realism... In short, Quine and Putnam have maintained that mathematical objects are scientifically indispensable... Quine and Putnam have been false friends of numbers in making the case, for their acceptance seems to depend on a claim of indispensability".

Well, where Quine and Goodman failed in 1940s Field and Chihara largely succeeded in 1980s. Predicative mathematics has been nominalistically reconstructed, and it does seem to suffice at least for most of science. So presumably Quine would go back to nominalism now, we know Putnam did.

Finally, it is tangential to Quine, but traditional mathematical realists (Platonist or Aristotelian, see For a mathematical realist, is there a distinction between real mathematical objects and constructed mathematical objects?) have no need for the analytic/synthetic distinction either. According to them mathematical knowledge derives from ideal perception just as physical knowledge derives from sense perception, so it is in a way "empirical". Of course, they have the formal/material distinction depending on the source instead, hence Husserlian division into formal/material sciences in Logical Investigations. It is empiricists like Russell or Carnap, who consider the five senses to be the sole source of knowledge (no innate ideas, rational intuition or mental a priori), but wish to carve out a privileged epistemological status for mathematics, that need analyticity to get it done.

  • What about the process of mathematical proof? I mean, we use analyticity step by step at every stage of a proof... what does Quine think about this process? – Ameet Sharma Aug 10 '17 at 2:02
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    @AmeetSharma Quine rejects foundationalist model of justification: the truth of mathematical claims does not rely on self-evidence of axioms ("foundation") and analyticity of proofs, mathematics like science is justified holistically a posteriori. Proofs follow inference rules ("steps") just as science follows its methodology, they are subject to the same "tribunal of experience" as the content they deliver. And they are subject to it only in "chunks" with content, which means that the distinction between analytic and synthetic, formal and factual, is at best pragmatic and blurry. – Conifold Aug 10 '17 at 5:40

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