For those who don't know, τ is just 2π. The defenders of τ argue (as seen in the above links) that using it instead of π makes many things much clearer and simpler/ more elegant.

Let's assume for present purposes that the τ-proponents turn out, in the end, to be right. I want to ask a further question: what would this then say about Quine's denial of the analytic-synthetic distinction? Quine's denial, virtually all agree, is the claim that all rational belief change is pragmatic, i.e. there is no principled difference between questions of evidence/justification on the one hand, and questions of efficiency and expedience on the other (= between external and internal questions, i.e. between practical questions of which language-form to adopt, and questions of whether a particular empirical claim is supported by the available evidence).

So here's my question: if Quine is right, then is our old friend

*C*=2π

*r*simply wrong (and

*C*=τ

*r*right)? If not, how can a Quinean wiggle out of that consequence? And if so (i.e.

*C*=2π

*r*really is wrong), does the Quinean have any way of softening the sting of this apparently absurd consequence?

## 6 comments:

What do you mean by "wrong"?

Presumably the Tau and Pi proponents both agree that C=2πr and that C=τr.

The expressions are equivalent. So it is not that either claim is "wrong" in the sense of "false" or "not to be asserted", but that one is better suited for some practical purposes (like introductory trigonometry).

Hi Lewis --

Thanks for stopping by! Yes, what you say matches my intuitions/ way of thinking exactly.

The question I'm wondering about is whether that intuition that we agree on creates a modus tollens against Quine's "thoroughgoing pragmatism" (last section of "Two Dogmas"), which erases the distinction between epistemic and pragmatic justification. If τ really is pragmatically superior to π, then it seems like a Quinean will have to say that

C=2πris in the same justificatory bucket as geo-centric astronomy and phlogiston chemistry.In short, because of the point we agree on, I think the Quinean view is incorrect. But I would like to hear any arguments to the contrary.

I am not yet seeing the challenge for Quine's "thoroughgoing pragmatism". Let's take a slightly different example:

One might say:

E: A number n is even iff n is divisible by 2 without remainder.

One could also less helpfully say:

E': A number n is even iff (n+1) has a remainder of 1 when divided by 2.

Is there some reason Quine couldn't accept both of these as true? I don't see any. Even though E is more useful than E' for most mathematical purposes, Quine can surely acknowledge that E' follows from E (plus some other mathematical premises). And it is surely useful to allow for the assertion of various consequences of useful claims.

Note that, with phlogiston or geo-centrism, the dubious claims are in some sense at odds with the useful claims (oxygen, heliocentrism).

The case with tau is more like E/E' than like oxygen/phlogiston. Put another way, 4 is also expressable as (2^2), or as (2*2), or as (2+2). Should Quine have to pick exactly one of the following as correct?

3+1 = 4

3+1 = (2^2)

3+1 = (2*2)

3+1 = (2+2)

I am assuming you don't think Quine would have trouble allowing all four of the above. But if so, I am not clear on what problem arises for Tau and Pi.

Thanks Lewis, that helps me. (I don't really have an 'inner Quine' that I can channel very well.) I like your idea that accepting all the logical consequences of everything we accept is just one more piece of accepting the most pragmatically-epistemically best-justified system.

I'm not 100% I can articulate why, but the 'even' example felt more persuasive for me... I think because that case more clearly matches the tau vs. pi case: there will probably be no (or very few) situations where your alternative characterization of 'even' is better than the usual one -- which is what the tau-proponents claim about pi -- but there are clearly certain mathematical circumstances where we need to think of 4 as 2 squared, and other times as 3+1, and other times as 2!, etc.

Addendum: at the end of my last comment, "where we need to think of..." should be understood as "where it is simplest/ most efficient/ most useful to think of ..."

I think the fundamental issue is that the pragmatism should kick in between conflicting claims, not between complementary claims competing for pedagogical priority.

The Pi and Tau formulations don't conflict with each other, even though they might compete for a privileged role in our mathematical practices.

I don't have an inner Quine, per se, but I usually consult my inner Carnap on these issues.

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