It's the first day of Trends in Logic VII, aka Trends in the Philosophy of Mathematics. So far, we're past an opening, an opening leture by Ryszard Wójcicki, and a splendid conference dinner.

Ryszard Wójcicki, an excellent "hardcore" logician known for his work on consequence operations and Polish-style meta-theory of propositional calculi, has recently decided to think about more philosophical issues. He was talking about Two sources of mathematical truth. The main gist was that the key "source" of mathematical truth was "conceptual realities" (the other source being empirical domains). Alas, I didn't quite get what being a source of truth is, how conceptual realities are supposed to be different from mathematical structures, what their ontological status is, and why they're supposed to exist. Having said that, it was interesting to hear a real "hardcore" researcher say what he thinks about the philosophical status of his own field.

My general impression is that if a "hardcore" scientist of any specific sort suddenly starts to philosophize, it's bound to be slightly weird stuff from the philosopher's perspective (it's not as bad as a philosopher trying to do science, though). What slightly surprised me was that this also holds for logicians. On the other hand, I do think that one of the problems that analytic philosophy in Poland is facing is that there are many excellent logicians doing highly technical stuff but having no philosophical interests or well developed intuitions, and there are many philosophers with highly developed intuitions, but with almost no grasp of logic or attention to arguments and details whatsoever.

Ryszard Wójcicki, an excellent "hardcore" logician known for his work on consequence operations and Polish-style meta-theory of propositional calculi, has recently decided to think about more philosophical issues. He was talking about Two sources of mathematical truth. The main gist was that the key "source" of mathematical truth was "conceptual realities" (the other source being empirical domains). Alas, I didn't quite get what being a source of truth is, how conceptual realities are supposed to be different from mathematical structures, what their ontological status is, and why they're supposed to exist. Having said that, it was interesting to hear a real "hardcore" researcher say what he thinks about the philosophical status of his own field.

My general impression is that if a "hardcore" scientist of any specific sort suddenly starts to philosophize, it's bound to be slightly weird stuff from the philosopher's perspective (it's not as bad as a philosopher trying to do science, though). What slightly surprised me was that this also holds for logicians. On the other hand, I do think that one of the problems that analytic philosophy in Poland is facing is that there are many excellent logicians doing highly technical stuff but having no philosophical interests or well developed intuitions, and there are many philosophers with highly developed intuitions, but with almost no grasp of logic or attention to arguments and details whatsoever.

## Comments

Grzegorczyk, as a logician, is (I think) "hardcore" in this sense. This doesn't exclude him from being interested in philosophy. But when you look at his stuff on ethics, it's quite different both in style and justification standards (I'm slightly inclined to think Grzegorczyk is another case of a hardcore logician turning to philosophy).

As for Wójcicki being a philosophical logician. Again, the issue is a tad terminological, but what I meant is that Wójcicki is mostly known for his stuff on meta-theory of propositional consequence operations, and if you look at his Brazil lectures, for instance, they're pretty much devoted to proving theorems and defining new mathematically interesting concepts, not to a philosophical discussion of these things. Compare it, for instance, to the stuff that Kit Fine, Timothy Williamson, or Krister Segeberg are doing. And really, if you listen to some more philosophical things that Wójcicki is saying, despite the fact that he's a great logician, he seems to lack some of the conceptual finesse displayed by many analytic philosophers of mathematics.

This discussion is weird. Modern Logic is recognized as the branch of science, which arose from the philosophical motivations in the foundations of mathematics. Nowadays, it naturally splits into two subbranches: pure logic and applied logic. Logic for philosophical purposes is what is called "philosophical logic". Philosophical logic is the proper part of applied logic. Logic in Computer Science is also the proper part of applied logic.

Wojcicki in itself is not "hardcore " logician. Living hardcore logicians are Shelah and Vaananen for example.

I advise any philosopher to not engage in any interaction with above mentioned guys. Nervous breakdown guaranteed.

You refer to Segerberg. This is the interesting example of pure logician with some possible philosophical import. Segerberg is the best known probably from his contributions to dynamic logic ( completeness proofs, Segerberg's induction axiom), where dynamic logic itself can be used to model certain philosophical concepts ( i. e philosophy of action).

Kit Fine he is metaphysician who did some work in logic. I regard metaphysics as non-sensical. The attempts to formalize some metaphysical claims at least show that they are not inconsistent. This what Fine's work is good for to be honest ...

Well, I'm suspicious toward the mainstream philosophy of logic and mathematics. Especially if it dissociated with any mathematical practice and proper education in the field. Obviously it takes time and effort to master the field and then eventually draw any reasonable philosophical conclusions. I consider myself as the student of logic.

The standard practice these days is to base philosophical theoretizing on some kind of intuitions. What is an absurd.

Logic serves purposes and his own set of criteria what can be regarded as pure logic ( the expressive power, computational behavior etc. )

What about computational complexity, what about logic and language? How would you classify that? What about logic and games? This is the emerging paradigm in logic, which will dominate the field for the couple of decades.

Philosopher does not need to use only logic as his tool of clarification or formalization. Probability and game - theory are natural candidates. Sometimes better suited than logic.

"Well, I'm suspicious toward the mainstream philosophy of logic and mathematics. Especially if it dissociated with any mathematical practice and proper education in the field."

There is no mainstream philosophy of logic and mathematics which satisifies this condition, I think.

"Obviously it takes time and effort to master the field and then eventually draw any reasonable philosophical conclusions. I consider myself as the student of logic."

I'm quite convinced that most of good philosophers working on philosophy of mathematics have a rather good grasp of theories they're talking about and have quite decent background in mathematics - this is especially so since most of philosophical issues pertaining to mathematics arise even when we consider fairly well-understood theories (various set theories, ZFC among them, PA or Q when it comes to arithmetic, analysis, ...).

"The standard practice these days is to base philosophical theoretizing on some kind of intuitions. What is an absurd."

This is too vague to respond. And what tells you that using your intuitions is absurd? Your intuition, perhaps? And are you saying that mathematicians don't use intuition at all?

"What about computational complexity, what about logic and language? How would you classify that? What about logic and games? This is the emerging paradigm in logic, which will dominate the field for the couple of decades."

Uhm, this depends what you do with them. If you work on formal results in complexity theory or game theory, you're doing "hardcore" logic/math. If you think about what philosophical consequences or interpretationts certain results may have, you're doing philosophical logic. Not sure if you don't overestimate game theory, though. My bet would be on category theory.:)