FOM: reply to the "list 2" crowd
pratt at cs.Stanford.EDU
Fri Jan 23 14:36:45 EST 1998
>This may just be obtuseness on my part, but I don't understand how
>the notion of a "continuum" is used here as a foundational concept. I
>admit that I've been "corrupted" by set theory, and it may be that I have
>a slightly different notion of "continuum" in my head.
Indeed. The difficulty is almost certainly that your notion starts
from the points of the continuum and equips them with topology.
Instead picture the continuum as an atomic entity whose properties
One difficulty with this point of view is our natural tendency to picture
atomic entities as pointlike, which of course contradicts our intuition
about space. One is tempted then to picture the properties as somehow
stretching this point out into a line. Instead, think of a point in
the frequency spectrum, and consider the spatial meaning of that point:
it is a wave spread out in time or space. A morphism has spatial extent
from the beginning, before the detailed properties of space have emerged.
The most basic property is connectivity, the underlying graph structure
of a category. Next comes subdivisibility, via composition.
>I think the picture you present above is of ZF tumbling out of
>first-order logic on its own and later being interpreted (or '*read*') by
>us in a certain way. I don't think this is quite correct. It seems to me
>that the axioms are constructed by us in the first place in order to
>capture the conception of interest.
I see these as consistent. ZF was indeed not brought down from a mountain
engraved on tablets, it is as you say a formalization of our intuitions
about collections. My point is that, at least in our austere Platonic
world of pure mathematics, the role of this formalization is not as
a supplement to our intuition but as a formal replacement for them.
If ZF were only a supplement we would presumably be allowed to refer
to our intuitions in formal proofs. While we certainly do so in our
informal practice of mathematics, this is not allowed formal proofs.
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