F. William Lawvere

To guide the learning, development, and use of mathematical sciences, the laws of possible rational passage from one assertion to another need to be made explicit.

As a young mathematician I learned about those laws, and more, from the professional ­logicians Tarski and Scott (whom I observed to be also mathematicians at heart).

But assertions are phrased in terms of concepts, therefore it is also urgent to make explicit the laws of possible rational passage from one concept to another. In order to meet and discuss with professional philosophers of science who are investigating that question, I participated in 1964, 1975, and 2011 in the congresses of the LMPS series.

I was encouraged to find that some are seriously struggling (like me) to throw off the philosophical straight jackets that we ­inherited from Balfour, Peano, Russell, and others.

The most extensive approximation to an understanding of the laws of passage from one concept to another I found in the ­theory of categories, developed by Eilenberg, Mac Lane, Kan, and ­Grothendieck.

Unfortunately, like other recent advances, this theory is also an attractive source of mystifying buzz words, which can discourage its detailed learning as a mathematical subject.  

It is truly gratifying, therefore, to find that there are professional logicians and professional philosophers of science who are indeed seriously learning and applying the methodology of ­category theory.

F. William Lawvere - University at Buffalo