Tarski's automorphsisms
November 14, 2024I first came across Albert Lautman (nLab) after reading Fernando Zalamea’s remarks on his importance to structural mathematics in Synthetic philosophy of contemporary mathematics.
While reading the n-category cafe article on Albert Lautman, I encountered some ideas which have been coming up in my work on my masters thesis and seem to originate with Alfred Tarski.
From Tarski’s 1966 lecture “What are logical notions?” (published posthumously in 1986).
What would an answer to the question of the title look like?
people speak of catching the proper, true meaning of a notion, something independent of actual usage, and independent of any normative proposals, something like the platonic idea behind the notion. This last approach is so foreign and strange to me that I shall simply ignore it for I cannot say anything intelligent on such matters (Tarski 1986, p. 145).
Tarski’s account was to capture a certain common use of the concept of logical notion.
in answering the question ‘What are logical notions?’ what I shall do is make a suggestion or proposal about a possible use of the term ‘logical notion’. This suggestion seems to me to be in agreement, if not with a prevailing usage of the term ‘logical notion’, at least with one usage which actually is encountered in practice (Tarski 1986, p. 145).
What are notions?
I use the term “notion” in a rather loose and general sense, to mean, roughly speaking, objects of all possible types in some hierarchy of types like that in Principia mathematica. Thus notions include individuals (…), classes of individuals, relations of individuals, classes of classes of individuals, and so on (Tarski 1986, p. 147).
Logical notions are those invariant under automorphisms of the universe of discourse.
consider the class of all one-one transformations of the space, or universe of discourse, or “world”, onto itself. What will be the science which deals with the notions invariant under this widest class of transformations? Here we will have very few notions, all of a very general character. I suggest that they are the logical notions, that we call a notion “logical” if it is invariant under all possible one-one transformations of the world onto itself (Tarski 1986, p. 149).
further discussion Concrete groups and axiomatic theories II
see also Boolean algebras with operators part I, Alfred Tarski