Talk:Gottlob Frege

From Academic Kids

How is the work of Gottlob Frege related to that of George Boole? --Hirzel 03:06 15 Jun 2003 (UTC)

You could say that Frege's work was later, was research rather than exposition, was deeper as an enquiry into language. In fact there is no serious intellectual connection, though obviously there might be some comparison in the use of algbra-style notations. User:Charles Matthews

As I know, Boole mostly worked on the theory of classes in logic, Frege was involved rather in the theories of first-order and second-order languages. It's a more general theory, because class theory is a logical theory, the theories of logical languages are metalogical theories (in a particular, relative meaning of metalogics). These statements are a bit inexact, but I can't explain it detail, cause English is not my original language, however it is true the relation between Boole and Frege is a quite neglected area in the Frege-research. I have no sources about it and I think even if some exists so they are unpopular (f.e. not foundable in internet or in public libraries). Gubbubu 20:37, 6 Aug 2004 (UTC)

Comments, questions and quibbles 25-aug-2004

...such as the use of quantification...

• Common logician's error here: in normal english phrases like "two horses", two is a quantifier: no need for variables, bound or otherwise, to have quantification. Frege's contribution was the way he expressed quantification by means of variables; he did not invent quantification itself.

Frege was the first to devise an axiomatization of propositional logic and of predicate logic.

• Frege did not create a separate theory of propositional logic, as the above suggests, and actually it is an achievement of Frege's to combine propositional connectives and quantifiers in one calculus.

• In fact, the formalisation of predicate logic took almost another 60 years to complete, with the publication of Hilbert and Ackermann's book in 1928.

Ludwig Wittgenstein and Edmund Husserl were among the other philosophical notables strongly influenced by Frege.

• With Wittgenstein this is undeniable (although I think the sentence goes better elsewhere), but just how important an influence was Frege on Husserl. Husserl first agrees with Frege's objections, and then goes on to change his mind about it. That Husserl had an important correspondence is clear, but strong influence I think is going too far.

Nice article, though: I was surprised by how many things I learned reading it. ---- Charles Stewart 22:30, 25 Aug 2004 (UTC)

Failure of logicist programme

The article says:

Frege was the first major proponent of logicism -- the view that mathematics is reducible to logic. ... Russell discovered the paradox which bears his name, and that the axioms of the Grundgesetze led to this contradiction; he wrote to Frege, who acknowledged the contradiction in an appendix to volume two of the Grundgesetze, noting what he perceived to be the faulty axiom. Frege never did manage to amend his axioms to his satisfaction, however; and after Frege's death, Kurt Gödel's incompleteness theorems showed that Frege's logicist program was impossible.

This strongly suggests that logicism was a failure. But I don't think that is correct; it seems to me that Frege's program was a success. Mathematicians still view set theory and logic, as set forth by Frege, to be the proper foundations for mathematics. It's only in recent years that an alternative, in the form of category theory, has appeared.

The Russell paradox was satisfactorily resolved by Russell and Whitehead's theory of types and later by Zermelo's work on the axiom of foundation. It is not a serious hindrance today.

The remark about Gödel's incompleteness theorems is a non-sequitur. The incompleteness theorems show that there can be no formal axiomatization of all of mathematics. This no more invalidates the logicist programme than it invalidates the idea of doing mathematics at all. Mathematics can be founded on logic, and frequently is; the fact that the Gödel theorems say that there will be true theorems that are not provable does not negate the usefulness or soundness of the foundation.

For these reasons, I have rewritten this paragraph of the article and removed the reference to Gödel entirely. -- Dominus 14:31, 14 Jan 2005 (UTC)