Tenth Arche Abstraction Workshop (22-23 October 2004)

Timetable

Friday

4.00-5.30	Paul McCallion (Glasgow)	'Abstraction and the Complex Numbers'
5.30-6.00	Tea and Coffee
6.00-7.30	Bob Hale (Glasgow)	'Kit Fine's Limits'
8.00-10.00	Dinner at Pizza Express

Saturday

9.00-10.30	Robbie Williams (St Andrews)	'Complex Numbers and Reference'
10.30-11.00	Tea and Coffee
11.00-12.30	Peter Simons (Leeds)	'Why are Complex Numbers Applicable?'
12.30-2.30	Lunch (own arrangements)
2.30-4.00	Peter Sullivan (Stirling)	'Dummett's Case for Constructivist Logicism'
4.00-4.30	Tea and Coffee
4.30-6.00	Daniel Isaacson (Oxford)	'How Determinate is the Concept of Real Number?'
6.00-8.00	Free time
8.00-10.00	Dinner at Cafe India

Paul McCallion (Glasgow)
Abstractive accounts of a mathematical theory may aim to be hermeneutic (capturing the content of the targeted theory) or merely re-constructive (capturing the content of a replacement theory). Are complex number terms (e.g. ‘i’) singular? If so, abstraction will easily yield a re-constructive account of the complex numbers, but there are difficulties for any account which aims to be hermeneutic. However, there are some reasons to doubt that complex number terms are singular. An alternative account of the content of complex number sentences is offered which treats them as generalities. Attempts are made to combine this with various structuralist treatments. Of these, the eliminative structuralist account is found to fare best. An account of complex number theory as a theory of double generality – over structures, and over objects – is offered.

Kit Fine’s Limits

Bob Hale (Glasgow)
Although Kit Fine allows that a significant theory of abstraction can be developed in which some fundamental mathematical theories may be reconstructed, the overriding message of his book, as its title suggests, is largely negative. In particular, Fine is quite out of sympathy with the idea that abstraction might play any significant or useful rôle in providing a philosophical foundation for mathematics—even for those parts of which can be obtained on abstractionist principles. I suspect that, given certain assumptions underlying and shaping Fine’s approach to the subject, this negative assessment is more or less inevitable. In this paper, I want to explore what I believe to be some of the more important of those assumptions, and suggest some reasons why we might not feel under irresistable pressure to subscribe to them.

Complex Numbers and Reference

Robbie Williams (St Andrews)
The neo-Fregean, buying into a 'minimalist' account of reference, appears committed to a referential interpretation of complex number theory. Brandom and others have argued that the conjugation automorphism of the complex numbers challenges Fregean accounts of reference to such entities. I explore whether this challenge can be answered by holding that reference, in such cases, is indeterminate.

Why are Complex Numbers Applicable?

Peter Simons (Leeds)
The complex numbers C arose to serve a need within pure mathematics, namely that of finding solutions for equations (initially cubics) that otherwise lacked them. Their interest and beauty are undeniable, but by contrast with real numbers, whose applications are obvious and legion, complex numbers saw no serious use outside pure mathematics until the rise of quantum theory. Physicists' explanation for their applicability and usefulness there is that they just turn out to be needed to make the physics work, and that's that. For anyone sympathetic, as I am, to Frege's view that an adequate account of mathematical systems should embody an account of how their application is possible, this non-explanation is totally unhelpful. The physicists may ultimately be right, but we should not give up on an explanation without trying harder first. Since I do not think the standard geometrical and ordered pair representations of C are central to their nature or applications, I shall be looking for an account of how they are applicable which stresses (non-geometrical) periodicity, such as is manifested in the phase factor in quantum equations. Whether this is the right way to go, and what it tells us about the inner nature of complex numbers, is what I shall be trying to get clear on in my talk.

Dummett's Case for Contructivist Logicism

Peter Sullivan (Stirling)
Dummett's strongest case for constructivism does not depend on any general meaning-theoretic argument, applicable as much in history or astronomy as in mathematics, but on a specific contention about mathematical concepts and theories: in a mathematical theory, truth cannot outrun what is fixed as true by our conception of the objects it concerns. The contention is sound. It does not immediately rule out classicism; it does show how committed classicism has to be.

How determinate is the Concept of Real Number?

Daniel Isaacson (Oxford)
Determinacy of mathematical concepts may, from a structuralist perspective, be established by categoricity. One powerful means by which this can be done is via the quasi-categoricity of second-order Zermelo-Fraenkel set theory. A drawback to this argument is that the central result from mathematical logic on which it is based is obscure to mathematicians (as well as philosophers). In this talk I want to focus instead on a result that is specific to the real numbers and widely known and accepted among mathematicians, namely that the real numbers are a topologically complete ordered field and any two topologically complete ordered fields are isomorphic to each other. Also, the isomorphism between any two such structures is unique. I shall compare this situation with that of the complex numbers as an algebraic extension of the reals.

-- CarrieJenkins - 23 Aug 2004

OneOffEvent
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Contact:	CarrieJenkins
Startdate:	22 October 2004
Enddate:	23 October 2004
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