Monday, September 06, 2010

Did Euler’s ‘2’ refer to {Ø, {Ø}}?

The foundation of mainstream mathematics is standard set theory, within which ‘2’ usually refers to {Ø, {Ø}} (when we are considering the natural numbers, see comments below).
......Alexander Paseau (2009: ‘Reducing Arithmetic to Set Theory,’ in Otávio Bueno & Øystein Linnebo, New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 35–55) thinks that those reducing arithmetic to set theory in such a way—perhaps they want their ontology to include only sets, not also non-set-theoretic numbers—may also take most mathematicians, past and present, to have been referring to the standard set {Ø, {Ø}} with their ‘2’s. He (2009: p. 42) made the following analogy: ‘When the ancient Greeks spoke about the sun, they spoke, unknowingly, about a hydrogen-helium star that generates its energy by nuclear fusion.
......By contrast, anyone taking ‘say, a carrot to be the referent of “2” in Euler’s mouth’ should, he (2009: p. 43) thinks, ‘be an error theorist about Euler’s claims involving “2”,’ and so risk taking too many—according to Hartry Field (2001: Truth and the Absence of Fact, Clarendon Press, p. 214)—of Euler’s words to be untrue. According to Paseau (2009: p. 43), our ‘less eccentric reductionists need not interpret Euler’s arithmetical claims error-theoretically and may respect his intended truth-values.’
......They could, he thinks, take that view even if ‘2’ referring to the standard set {Ø, {Ø}} was not so much a discovery about 2 as a technical convention. As Paul Benacerraf (1965: ‘What Numbers Could Not Be,’ Philosophical Review, 74, pp. 47–73) famously observed, another possible referent is {{Ø}}. So for another analogy, suppose some chromatographers took ‘orange’ to refer to wavelengths of light within some definite range, in order to avoid vagueness and because such a stipulation was sufficient for their scientific needs. They would surely need further reasons to take us to be referring to such wavelengths with our uses of ‘orange’. And similarly, if our less eccentric reductionists have not so much discovered the referent of ‘2’ as accepted a useful technicality, then it seems to me that they should not be taking our ‘2’s—nor Euler’s—to be referring to the standard set {Ø, {Ø}}.


Firionel said...

Hooray for that post!

I have been trying to point out to people that not only the concepts of more complex mathematical objects (like, say, a surface) change over time, but also the ones concerning the most primitive ones. That problem is vastly underappreciated in the philosophy of science, and especially mathematics.

What we need is a kind of inter-temporal cultural relativism.

Xamuel said...

But wait a second, "2" is a lot more ambiguous than that.

The natural number 2 is {Ø,{Ø}}, fair enough.

The integer 2, depending how you define your integers, may or may not be something like the ordered pair (2,0) by which we of course mean {{{Ø,{Ø}}},{{Ø,{Ø}},Ø}}, or may even be some sort of equivalence class with infinitely many elements ((2,0), (3,1), and so on).

Then there's the rational number 2, which is really the equivalence class of the pair (2,1) relative to the equivalence relation (a,b)~(c,d)<->ad=bc. Writing this out directly is a bit tedious...

Then you've got the real number 2. Dedekind cut? Equivalence class of Cauchy sequences? Something else entirely? Very hard to write out directly, in any case.

Finally, there's the complex number 2, which is really the coset 2+(x²+1) of R[x]/(x²+1) (where the latter "2" is the integer). Writing this out directly counts as one of the seven tasks of Hercules ;)

Xamuel said...

Err, correction, the "latter" 2 in the last paragraph is actually the *polynomial* 2 of R[x], which of course introduces the whole new question of how you construct R[x]! ;)

enigMan said...

Thanks Firionel and Xamuel...

I don't know what you mean by 'inter-temporal', Firionel. Also, I'm arguing (as in today's post, which continues this one) that the concept of the natural number 2 hasn't changed, and that it shouldn't change. So I may disagree with you about what we need.

You're right about '2', Xamuel, I should've mentioned that I was talking about the natural number 2. It might help if we used '+2' to refer to the integer 2, '+2/1' to refer to the rational 2, '+2.000...' to refer to the real 2, and '+2.000... + +0.000...i' to refer to the complex 2. Perhaps I could argue that the use of '2' in such cases is really just a shorthand. But in any case, the basic meaning of '2' is the natural number.

enigMan said...

(I've added a PS to the first line of this post, in light of your comments Xamuel, which may unfortunately make them look less apposite:)