Thought and Language in Mathematics - A paradigmatic example

In this talk I will discuss the relation between two theorems and their proofs: the older one is due to Pierre Samuel (1948), the newer one to Peter Freyd (1964). While Samuel’s result is hardly known at all, Freyd’s is (one of) the most important theorems of elementary category theory — known as Freyd’s Adjoint Functor Theorem. Both results provide constructions which can be applied to quite a number of non-trivial situations in different areas of mathematics as, e.g., the existence of free topological groups or the Stone-Cech-compactification. Thus, ˇ the obvious question is: How are these related? The somewhat surprising answer to this question is: Though Freyd’s result is more general then Samuel’s, their proofs are essentially the same — up to the language of mathematics available to the authors. Thus, they show by example what Wittgenstein expressed by writing “The limits of my language are the limits of my world“. Because of this I will start my talk (maybe taking owls to Athens) by briefly recapitulating the development of the language of mathematics during the first half of the 20th century. For the category theorists: An alternative title of this talk, thus, could have been (explaining why I present it this year): Does the GAFT turn 60 this year, or turned it 75 already last year?