FOM: the ongoing discussion of "arbitrary object"

[Martin Davis]

This thread was started by Charles Silver's enthusiasm for a book by Kit 
Fine on this subject.

I am puzzled. Everyone on this list understands that there is no problem in 
formalizing the reasoning in proofs in which the locutions in question 
occur. So one could let matters rest regarding the mathematical usage as a 
figure of speech.

But sometimes something is gained by not letting matters rest in such 
situations. Reasoning about infinitesimals was relegated to a figure of 
speech by the Cauchy-Weierstrass epsilon-delta reasoning. But then Abraham 
Robinson taught us that something can be learned from a new and deeper 
analysis of the notion of infinitesimal.

I am quite prepared to believe that Kit Fine has done something analogous 
with "arbitrary object". What puzzles me is that with all of the ongoing 
discussion, no one has taken the trouble to write a paragraph or two 
summarizing the nature of Fine's contribution (as for example I could 
readily do for non-standard analysis). I for one would read such a 
contribution with considerable interest.

Martin


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com