Gottlob Frege: The machine in the ghost (2017)
prospectmagazine.co.uk
prospectmagazine.co.uk
> he is still very much ignored by the tradition he created
Really? When I studied philosophy[0], reading at least some Frege was required for Philosophy of Logic and Philosophy of Language. Frege wasn't treated as obscure or unimportant; quite the reverse. Have things changed?
[0] University of London, early 80's. Philosophical tastes vary rather widely from one institution to another.
So he is partly famous for not being famous, a paradox he would hopefully appreciate.
Why? I think you're referring to Principia, which is a predecessor to Coq and Lean. While Coq and Lean don't use Principia's formalism, they do try and tackle the same problem. What did Principia fail at?
[edit]
Or are you referring to the programme of using formal logic to understand language and reliable everyday reasoning?
GIT is more a statement about the limitations of the proof-driven approach to mathematics than it is about any formalisation of mathematics like Principia's. It shows that whatever assumptions proofs are based on cannot be shown consistent except by assuming something even stronger. Principia is affected to exactly the same degree as all proof-driven mathematics, and no more.
I'm not sure whether there has ever been a satisfactory answer to the 2nd GIT among the mathematical establishment. I guess mathematicians just assume that everything is consistent, and go about proving theorems as normal. It seems that almost no mathematician understands - let alone cares about - these foundational concerns.
Principia is the predecessor to modern software like Lean, Coq and Agda. Nothing got destroyed.
Is that wrong? If I've got that wrong, I have a big attitude adjustment coming down the track towards me.
I'm usually not sure if I understand it (reverse: Im usually sure that I don't understand it). Also: I'm not a philosopher - I'm an old man that did a philosophy BA 40 years ago. I can't unpack this stuff.
I see a consonance between Gödel's theorems, Heisenberg's principle, Turing's work on computability, and some of the stuff in Frege and Wittgenstein. Maybe even Cantor. I wish I could to say this clearly-enough for someone to knock down. But I can't do the maths; I'm entirely dependent on the explanations of others.
Let's assume that mathematics is a job that follows the following rules: You use two languages: Predicate Logic and Natural Deduction. You use Predicate Logic to write claims like "There are infinitely many prime numbers", and you use Natural Deduction to construct proofs of such claims. Both languages are formal. And you can even use a computer to verify that you're using them correctly.
To prove things, you need to assume that some things are true. These "assumptions" are called axioms, and are expressed using Predicate Logic. Philosophically speaking, the rules of Natural Deduction can be thought of as axioms for how to correctly construct proofs, but they're usually called "inference rules" and not axioms for minor technical reasons.
The methodology which we described arises inevitably from a proof-driven approach to mathematics, because mathematicians need a way to agree which methods of argumentation they think are correct, and also to prevent people from making things up. Admittedly, this is only true for those mathematicians who have looked into these foundational matters.
Some axioms can lead to an accident where you can prove a proposition and its negation simultaneously. (An example in set-theoretic foundations is Unrestricted Comprehension.) By a simple result called the principle of explosion, in such a situation you can prove every proposition and its negation. So such axiom sets are useless. We call them inconsistent.
A question arises: Given a set of axioms which is sufficient to serve as a foundation of mathematics - like ZFC set theory for instance, or less obviously Peano arithmetic - is it possible to construct a proof of their own consistency from those same axioms alone? The answer is no, and this is Goedel's 2nd Incompleteness Theorem.
You can - in a way - prove that Peano arithmetic (for instance) is consistent, but you need to assume axioms which are strictly more powerful than Peano arithmetic. And to show the consistency of those axioms, you need to assume yet stronger ones. This is quite a disturbing situation.
Ultimately, the theorem is a statement about the methodology for doing mathematics which we described. But there is no escaping it if mathematics is about stating claims and proving them.
Aside from Wittgenstein, Frege is the “analytic” philosopher I most admire. There’s a certain creative genius that shines through his work that I feel is absent from the work of many other analytic philosophers. His short Foundations of Mathematics text quite literally changed my life by making math interesting to me for the first time. I was lucky enough to read a copy of the Begriffs chrift since my school’s library had it, it’s a work of beauty!
You would think people such as Boole, Peirce, Łukaiewicz, Tarski, Brouwer, Heyting, Gentzen, Prawitz, Girard, and Martin-Löf are mere footnotes in the history of logic. Yet their work is more relevant and used by working mathematicians and computer scientists.
Each in turn:
Boole invented propositional logic. Peirce supposedly invented a lot, but got overlooked. Łukaiewicz, I'm not sure. Tarski employed quantifier elimination to prove the decidability of various theories like Euclidean geometry. Brouwer invented intuitionistic logic, and a(n arguably) viable approach to doing analysis and topology intuitionistically. Heyting provided semantics to intuitionistic logic. Gentzen created the proof calculus Natural Deduction which resembles informal proofs (as well as the closely related proof calculus Sequent Calculus). Prawitz I don't know. Girard, I don't know. Martin-Löf derived Dependently Typed Lambda Calculus using the Curry-Howard correspondence.
I guess all this stuff gets overlooked because Russell's goal was very ambitious: He wanted to connect formal logic with everyday language, and how to perform sound everyday reasoning. In that sense, his ambition was similar to the Stoics. The above stuff mostly doesn't address this use-case.
I would describe Frege as the philosopher I had never heard of before showing up at a philosophy department whom I then heard the most about.
His insights were "forgotten" for a time, but have been central to discussions for several decades.
One can regret the opinions in Frege's diaries without it affecting what one thinks about the work.
If we can build up a picture of someone from anecdotes about their life we can use this information to inform our understanding of why their work might have taken one form rather than another.
For example, Schopenhauer criticised the notoriously obscure schemata of Kant's Critique of Pure Reason by saying that it was a result of Kant's psychological need for architectonic symmetry.
Great love for Wittgenstein, Russell, Frege, the Vienna Circle—their philosophical and logical writings—and I do feel alienated when I encounter their popular treatments, because it makes my ego feel special having read their most important works, but I bet the Wittgenstein-Russell fame was good for lots of people getting exposed to the path: I heard about them in a Time Magazine special edition retrospective on artists and thinkers of the 20th Century 22 years ago.