### What Did the Philosophers Ever Do For Us?

About a year ago, I gave a list of achievements of analytic metaphysics.

A couple of friends said to me, "Look - that's not metaphysics! It's mathematics!". Ok. Fair enough - but then you can't have it both ways. Here we go with a list of the parts of modern mathematics invented by philosophers, in the period roughly 1879-1922:
1. The theory of relations.
2. The theory of quantification.
3. The analysis of what a variable is.
4. Truth tables/compositionality.
5. The formulation of formation and inference rules.
6. The recognition of the intimate relation of $A$ and "$A$ is true".
7. Higher-order logic.
8. The theory of types.
9. The theory of identity.
10. The concept of an abstraction principle.
11. The definition of cardinality.
12. The definition of ancestral (transitive closure).
13. The derivation of Peano's axioms from an abstraction principle (Frege's Theorem)
14. The recognition of some need for levels/types/orders.
(Admittedly, mostly logic and foundations, and mainly Frege and Russell. There's overlap, of course, with Dedekind, Cantor, Peano, Zermelo; and somewhat later, Hilbert, von Neumann and so on. I have also ignored the Poles. Not on purpose. Mainly because I know much less about Lesniewski's work.)

To be a bit more serious, maybe we should think of these as all simultaneously parts of mathematics, philosophy and logic.

I neglected to mention The Romans.

1. Apart from 15 (obviously) and perhaps 6 (less obviously) this is a list of the achievements of Frege, whose post at Jena was in *mathematics*. So it's a list of contributions from a *mathematician*.

2. Robert, agreed, if one defines it as the post held. Note that Einstein was an office worker from 1902-1909!
(I need to check for the exact ref, but you find (6) in "Ueber Sinn und Bedeutung" and "Der Gedanke").

Still, Frege says somewhere that the response from colleagues to his work was that this was metaphysics. Just looked it up: "Metaphysica sunt, non leguntur". And I agree with that!

Jeff

1. Let me just get this straight:

Frege, a mathematician, makes major contributions to mathematics which are also of major significance for philosophy (in which he is also interested).

So Frege is really a philosopher.

So this is an example of a philosopher making major contributions to mathematics.

Have I got that right? Well, compare:

Darwin, a biologist, makes major contributions to biology which are also of major significance for theology (in which he is also interested).

So Darwin is really a theologian.

So this is an example of a theologian making major contributions to biology.

2. Hi Robert, this seems to be a black & white fallacy. An intellectual is usually defined in terms of their *scholarly output*, not by a label on the building in which they work. For example, by your criterion, Albert Einstein in 1905 was an office worker (his building was the Swiss Patent Office). By my criteria, Einstein was a physicist (and was also a philosopher, heavily influenced by Hume and Mach). Saul Kripke has never been employed in a mathematics department. Does this logically imply that he is not a mathematician?

Next, note that Frege's output was in philosophy, logic and the foundations of mathematics. So, e.g., his classic 1892 article "Ueber Sinn und Bedeutung" was published in Zeitschrift für Philosophie und philosophische Kritik. The first sixty-nine sections of Die Grundlagen (1884) are (very clear) analytic philosophy. He states the "Philosophical Motivations" in Sc. 3.

On the other hand, Charles Darwin had no University position; by your criteria, he was perhaps a sailor! Darwin is, primarily, a naturalist, but had interests in theology. I do not believe he ever published in theology though.

By your "office label" criterion, Hobbes, Descartes, Locke, Spinoza and Hume were not philosophers. And Darwin was a sailor, and Einstein not a physicist and Kripke not a mathematician. Etc.

Cheers,

Jeff

3. If you don't stop at 1922, you can have recursion theory and large cardinals. Before 1922, you could perhaps count Special Relativity and Weyl's work on field theories too, for something more applied.

4. Yes, thanks - I thought of maybe including Special Relativity. Hermann Weyl not so sure, but the general point is right. I was thinking of 1922 (Tractatus) as a kind of stopping point; after then, the detailed developments were made by people like Hilbert, von Neumann, Godel, etc., ... Still, running to 1930, we could add Ramsey theory, decision theory, ... Going beyond that, we have the apparatus of modal logic (syntactic and model-theoretic), from C.I. Lewis, through Carnap and Barcan, to Kripke and beyond.

Jeff

5. Which work of a philosopher are you alluding to in your nr. 3?
- Frege's remarks on Latin letters' (Bs §1, Gg §17, Grundlagen der Geometrie)?
- Chapter VIII of the Principles of Mathematics (The variable is perhaps the most distinctively mathematical of all notions; it is certainly also one of the most difficult to understand.')?
- Or perhaps Wittgenstein's obscure remarks in TLP 3.31's (`Die Festsetzung der Werte ist die Variable' etc.)?

6. Thanks, Ansten

No, I am referring to Gottlob Frege, and I mean his various explanations of Latin, Greek and in particular German letters in Begriffsschrift (1879), §11-12. Here, Frege thinks of variables as analogous to arguments in a function. This motivates the approach set out in Begriffsschrift, along with the deductive system. A nice explanation can be found here

http://dec59.ruk.cuni.cz/~kolmanv/Begriffsschrift.pdf

Frege return to the topic in later works: e.g., "Function and Concept", online here:

http://fitelson.org/proseminar/frege_fac.pdf

If anything, Russell is something of a step backwards, as both Godel and Quine were to point out much later.

Not sure if there is a special reason to include Wittgenstein here, except to remark on his attempt to ensure that variables never co-denote. Kai Wehmeier has written on this topic,

The culmination of this is the truth/satisfaction definition that Tarski gives in the early 1930s for quantified formulas, assigning variables to terms in sequences.

Cheers,

Jeff

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