This month, we sit down with Patricia Blanchette to discuss the work of Gottlob Frege, one of the most influential philosophers of the 20th century. Click here to listen to our conversation.

We saw in our episode on the philosophy of mathematics how difficult it was to say what numbers are. What is the number three, and how do I come to know things about it? (Like that it’s odd, that it’s prime, that it’s the lowest number that’s both odd and prime, that it’s a factor of 135, and so on for all the things a mathematician might teach you about it.) Frege thought we could make some headway on these questions if we could show that arithmetic was really just complicated logic. And one way of demonstrating that arithmetic is just complicated logic is by showing that you can translate any statement about arithmetic into a statement about logic without changing its meaning.

Logic, you ask? What do numbers have to do with logic? Well, logicians study what are called *valid* inferences: inferences in which the premises, if true, *guarantee* the conclusion no matter what. For example, if Jane is riding a bus, it follows that someone is riding a bus–no matter what. *If* it’s true that Jane is riding the bus, then it *has to *be true that someone is.

What does this have to do with numbers? Well, think back to what we said about the number zero during our interview with Agustin Rayo. ‘The number of dinosaurs is zero’ really just means ‘there are no dinosaurs’–or, more stiltedly, ‘it is not the case that something is a dinosaur.’ Logicism, the idea that arithmetic is really just disguised logic, is based on the idea that any statement about numbers can be translated into a ‘something’ statement, in more or less the way we did for statements involving the number zero.

Tune in to hear Patricia Blanchette explain how the whole thing works!

*Matt Teichman*

Excellent podcast. My only slight contention is that she wasn’t quite right that Logicism (as I think she put it) failed because the truths of arithmetic were unmanageable.

What Gödel’s Incompleteness Theorems showed was that if a formal system is expressive enough to serve as a foundation for arithmetic, it has to be incomplete or inconsistent. The real problem that Russell’s Paradox caused was that it trivialized the naive set theory by way of the Principle of Explosion: P & ~P -> Q (from a contradiction, anything & everything follows, aka “trivialism”).

Of course, this brings up an interesting avenue that some logicians have been & are currently exploring: What if we keep the naive set theory, we keep Russell’s Paradox, but we change our logic from Frege’s Classical Logic? This has been explored by logicians like Brady & Weber by switching from Classical Logic to a Paraconsistent Logic. And the interesting thing is that it seems like we *can* reduce mathematics to Paraconsistent Logic, since those systems of logic can tolerate contradictions (like Russell’s Paradox) without the theory falling to trivialism. And this is right in Gödel’s Incompleteness theorems; they allow for a foundation of mathematics to be inconsistent. The problem was that no one (outside of a couple of Russian logicians) knew about Paraconsistent Logic in those times, so they thought a contradiction was indisputably the death of a theory.

/long comment 🙂

My only slight contention with MindForgedManacle is that he (I assume from the Twitter handle) isn’t quite right (actually not right at all) to say that Prof Blanchette wasn’t quite right to say that Logicism failed because the truths of arithmetic were (interestingly) unmanageable (by which she only meant they are not consistently axiomatizable). For one thing, Dr Blanchette was referring to Logicism in the axiomatic style of Frege — she leaves open the possibility that the truths of mathematics are (unaxiomatizable) truths of logic. Moreover, she was talking about the failure of Logicism within the classical logical paradigm in which Frege was working. What she said was entirely correct. Likewise, MFM also isn’t quite right to say that what “the REAL problem” caused by Russell’s Paradox is that it trivialized naive set theory via Explosion. There is no such thing the REAL problem caused by the paradox. We know that naive set theory and classical logic are inconsistent. One can deal with this either by revising the principles of naive set theory (or replacing them entirely) or by revising the underlying logic, hopefully, in either case, with some reasonably satisfying philosophical justification. But no such “solution” (so far, anyway) is entirely free of technical and philosophical difficulties.

Well ACTUALLY, mindforgedmanacle….

MindForgedManacle, a few corrections:

First, to say that “no one (outside of a couple of Russian logicians) knew about Paraconsistent Logic in those times” is oddly eurocentric and, to be frank, offensively dismissive of those Russian (and, slightly later, Polish and South American) logicians, who were — and are — both creative and brilliant. Just because a few arrogant logicians working in primarily English-speaking circles were unaware of work being done on other continents doesn’t mean that “no one… knew.” What you mean, I think, is that the Vienna Circle folk were unaware of the work being done elsewhere. Their bad, of course. But Prof Blanchette isn’t making any normative claims about what logicists should or should not have explored, and so your argument that logicism failed because the logicists failed to consider something other than logicism (i.e., an axiomatic system in a non-classical logic) is a bizarre red herring.

Second, the remark that “we *can* reduce mathematics to Paraconsistent Logic, since those systems of logic can tolerate contradictions (like Russell’s Paradox)” is similarly laden with non-universal assumptions — in this case, about what is meant by “mathematics.” I’ll assume, for the sake of argument, that when you say mathematics you mean axiomatized systems of basic arithmetic, e.g., Peano Arithmetic, or perhaps Robinson’s arithmetic (a finitely axiomatized fragment of Peano Arithmetic). But then what you say is false, because Peano arithmetic presumes an underlying logic (embedded in what used to be called the equality relation axioms) — which of course means that PA *cannot* be reduced to, for example, the paraconsistent system LP, because LP lacks some of the basic rules of inference (such as modus ponens) that are required by the underlying logic of PA. Sure, we can change the logic of PA. But that is simply to modify the arithmetic or mathematics that we’re trying to reduce so as to make it reducible — not to actually reduce PA.

Finally, it’s misleading to suggest that paraconsistent theories somehow permit a way around the incompleteness result by embracing the inconsistent fork of the dichotomy. Stewart Shapiro’s 2002 “Incompleteness and Inconsistency” is a good start for those who want to grasp the fact that paraconsistent logics don’t provide a direct path to completeness — or, for that matter, even to the promise of a comfortably controlled form of inconsistency. Shapiro’s results show that if one adds Priest’s dialethic semantics to PA to produce a recursively axiomatized system PA* that contains it’s own truth predicate, there are purely arithmetic (Π0 — !) sentences that are both provable and refutable in PA* — and, even worse, there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA*. In other words: paraconsistent mathematical theories come in many different flavors, and will not always be complete. Depending on what the theory takes to be true, and the strength of the deductive system, there might well be unprovable truths.

Matt — many thanks for posting this excellent discussion with Prof Blanchette!