Mathematician Saunders Mac Lane, 95, Dies

Saunders Mac Lane, one of the country's leading mathematicians and a professor at the University of Chicago for nearly four decades, has died. He was 95.

He died April 14 in San Francisco after a long illness, according to a statement released Thursday by the university.

In a landmark paper he co-authored with Samuel Eilenberg in 1945, Mac Lane developed new ways of thinking about mathematics _ introducing what are known as "categories," "functors" and "natural transformations," the statement said.

"A very great deal of mathematics since then would quite literally have been unthinkable without that language," University of Chicago math professor Peter May said in the statement.

Mac Lane received the nation's highest award for scientific achievement, the National Medal of Sciences, in 1989.

Mac Lane, who was born in Norwich, Conn., earned a bachelor's degree from Yale University and a master's from the University of Chicago. He received a Ph.D. in 1934 from the Mathematics Institute in Gottingen, Germany.

Mac Lane began teaching math at the University of Chicago in 1947, retiring in 1982.

He wrote or co-authored six books, including "A Survey of Modern Algebra," which became a leading text in the field.

Mac Lane also wrote extensively on the history of mathematics, said Peter Johnstone, another University of Chicago math professor.

He said Mac Lane "left a unique body of material for future historians of 20th century mathematics, written by someone who ... knew what it was like to be working at the cutting edge of mathematical research.

"Natural Transformations" are the closest thing to GOD in Mathematics.]

I'm sorry. I only got through two years of college calculus and I have NO IDEA WHAT THIS IS.

http://encyclopedia.laborlawtalk.com/Natural_transformation

I tried to read through some of the explanations but I couldn't even understand the language.

"Seven years of college down the drain."

:^)

The best summary of Category Theory I can find is in: Stanford Encyclopedia of Philosophy: Category Theory. Category Theory is an alternative to Set Theory for the foundations of mathematics.

Mathematical foundations are very much like starting off to build whatever you can out of Legos, where you constrain yourself to the most basic and essential minimum kinds of Legos. One starts off with a few axioms which specify that certain Sets or Categories exist. This is just like saying "Let's suppose we have all the 4-bump legos we want; then let's see what we can build."

Read the Zermelo-Frankel Set Theory Axioms You will see things like

There's an empty set.
If x and y are sets, then the pair {x, y} is also a set.
If x is a set of sets, then there is another set which is the union of all the sets in x.

This is just mathematicians saying "suppose we start with these - now what can we build?" It turns out they can build all the familiar mechanisms of mathematics - arithmetic, algebra, calculus, geometry, and so forth. A French group, writing under the name Nicolas Bourbaki, produced perhaps the most famous exposition of mathematics, starting with just basic Set Theory.

In Category Theory, we start out with the basic idea of mappings, or morphisms. We suppose that we have the identity morphism (maps objects to themselves) and that for any two morphisms, we have the composition (if one morphism F maps a to b, and another G maps b to c, then we suppose we also have another morphism H that maps a directly to c. Then, as with Set Theory, we build the rest of mathematics on this foundation.

This is perhaps a better description of Category theory http://en.wikipedia.org/wiki/Category_theory

John Baez tries to squeeze all physics into categories. I do not think that it is useful, but his "This Week's Finds in Mathematical Physics" http://math.ucr.edu/home/baez/TWF.html are worth the time it takes to read them.

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