FOM: rigidity

[Harvey Friedman]

>Dear FOMers,
>
>Friedman objects to infinitesimals on the grounds that one cannot
>give an example.


The word "object" here is an oversimplification of what I said.

>
>Note that it is impossible to give an example of a point in classical
>Euclidean geometry.  All the points are the same.  Geometry (as
>originally understood) is not rigid in the way that arithmetic is, and
>it is not self-evident that this is a disadvantage.  (Of course, this
>fact is disguised by the way we now implement geometry using R^2 or
>R^n).

This is precisely one of the main reasons that Euclidean geometry was 
viewed as being in need of a foundation in terms of rigid concepts. 
So one uses Euclidean space, or one uses suitable isomorphism types.

>
>To take an example from my own experience, if one works in NFU
>(Quine's NF as modified by Jensen to allow urelements) plus the axiom
>of choice one can prove that there are urelements, but one cannot
>produce one -- because they are indistinguishable.  It is important
>that there are urelements, but there is no particular important
>urelement.  The indiscernibility of the urelements is not a practical
>disadvantage for NFU as a theory of sets, at any rate.

These are among the many reasons why NF and NFU are not generally 
viewed as appropriate formalisms for the foundations of mathematics.

>
>Infinitesimals can be very useful without our having or even wanting
>an example of an infinitesimal.  Similarly, the fact that the reals
>can be well-ordered (which, to be used, requires occasional appeal to
>an "arbitrary" well-ordering of the reals) can be very useful without
>anyone being especially interested in producing an example (which is
>fortunate, since we can't produce one -- unless we are willing to
>assume V=L, of course, in which case we can give an example of an
>infinitesimal in a nonstandard model of the reals as well).
>

The issue of usefulness was discussed in a serious way in one of my 
previous postings.

We have an agreed upon observed reaction of the general mathematics 
community to the meaningfulness, importance of, fundamental nature 
of, etcetera, certain notions that are currently extramathematical 
(perhaps infinitesimals and perhaps "arbitrary objects"), and also of 
certain notions that are easily treated within set theory (e.g., 
arbitrary sets of reals, etcetera). I am giving various components of 
explanations as to why these attitudes prevail.

If you notice, I also prove theorems about these notions.

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