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Topic: [HM] Creating Numbers
Replies: 5   Last Post: Jun 4, 2003 11:31 AM

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David Corfield

Posts: 10
Registered: 12/3/04
Re: [HM] Creating Numbers
Posted: Jun 2, 2003 8:25 AM
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John Conway's account of his "Fairy Tale Platonism" was very interesting.
"Fairy tale" because mathematics is not about things in the world,
"Platonism" because there is an objectivity involved.

A theme emerging in recent philosophy of mathematics is the idea
that we may have conflated two dimensions with our debates concerning
realism. On one dimension you wonder about the mode of existence of
mathematical entities (concrete like a chair, abstract like democracy,
fictional like Oliver Twist, etc.). The other dimension concerns what
it is that constrains the mathematician in her choice of research. Is
there something beyond logic, but short of mere fashionability which
dictates that some concepts just fly after they've been introduced
(e.g., quantum groups) while others never really make it?

In chapter 9 of my book (see website below), I discuss an argument
between mathematicians as to whether the groupoid concept is a "natural"
one. Points made for groupoids talk of the inevitability of their
appearance (occur in many fields independently), their simplicity
(especially in category theoretic language), the avoidance of arbitrary
choices when they are used. The opposition might admit their mild
convenience, but take groups to be the real McCoy when it comes to
symmetry measurement.

As for myself, I find debates aimed at this second dimension far more
interesting. To what extent are mathematicians constrained by their
psychology and the tastes of their community? Don't the surprising
interconnections achieved by some concepts (e.g., quantum groups to
knot theory) suggest a "getting things objectively right"? What should
we make of the often very distant empirical sources of mathematical

In the case of surreal numbers, once the generating mechanism of
gap-filling is in place presumably matters are fairly forced. But
before this, in the "space" of such mechanisms is there something that
singles out the gap-filling one as especially interesting?

David Corfield
Faculty of Philosophy
10 Merton St.
Oxford OX1 4JJ

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