### Logic as Invariant-Theory

#### Posted by David Corfield

Greetings from Oberwolfach, where I’m attending a workshop on *Explicit Versus Tacit Knowledge in Mathematics*. ‘Tacit knowledge’ is a term we owe to Michael Polanyi, though we haven’t heard all that much about his views this week. (In fact, the attentive Café visitor may have seen more of Polanyi here over the years, here, here and here.)

But there are always plenty of reasons to hope to gain from a workshop, and I was particularly interested by a talk by Christophe Eckes on the debt Hermann Weyl owed to Felix Klein and his group-theoretic approach. I was already alerted to the theme earlier in the week, having found out about a paper An Extension of Klein’s Erlanger Program: Logic as Invariant-Theory in which F. I. Mautner had attempted to carry over Weyl’s treatment of the Erlanger Program from Classical Groups to logic. Essentially we’re seeing the action of the symmetric group, $S_n$, on, $X$ a set of $n$ elements and on powers of $X$. Invariants of the full symmetric group are logical constructions. We even get to see a tensor notation in whose terms propositions may be expressed.

That put me in mind of Todd’s posts on ‘Concrete Groups and Axiomatic Theories’ I and II. I’ll need to look a little closer, but I think Mautner gets the point that there’s a theory corresponding to each subgroup of $S_n$. It seems he didn’t see the Galois correspondence with complete theories.

Looking back at those posts by Todd, just see what we have never got to hear:

a whole slew of interesting developments, in which we view Jim’s orbi-simplex idea as a geometric description of a general axiomatic theory, which in turn is related to the idea of viewing Tits buildings as “quantized” axiomatic theories, and also perhaps to the theory of classifying toposes and their “Galois theory”.

I see Todd mentions

Alfred Tarski was also interested in applying Klein’s Erlanger Programm to logic, in his “What are Logical Notions?”,

but Mautner got there first.

## Re: Logic as Invariant-Theory

Hi David,

I haven’t had a chance to look at Mautner’s article yet. But could you give here an illustrative example of this statement:

? Thanks!