One, Two, Three, Four
This week I have mostly been thinking about the philosopher Gottlob Frege, which is not a thing I’d necessarily recommend. Frege did his main body of work a little over a century ago, and is generally credited with having invented, at least, the second order predicate calculus. He also, and it is this I want to write about, tried very hard to put numbers on a sound footing. Not particularly elaborate numbers either; Frege is interested in the numbers that answer the question “how many”. For example, “How many lions are there in Trafalgar Square?” Four. There are four lions in Trafalgar Square.
You might very well think that it is blindingly obvious where the numbers come from, and in this you would have a lot in common with Frege’s contemporaries. But Frege was convinced that they were wrong, or at least unsound, and his short book Grundlagen der Arithmetik (1884, in English The Foundations of Arithmetic) sets out his attempt to do better. Rather by accident, I have recently read this entire book cover to cover a couple of times. I’ve also read a number of lengthy critical essays on the subject. Do I understand it? Hardly; I have never pretended to be a philosopher. Does it worry me? Yes, it does.
Canadian singer-songwriter Feist is best known for her song “One, Two, Three, Four”, which rose to popularity after being used as an advertising jingle. But it’s not that version of the song I’m interested in, but this one, which she did a little later, for Sesame Street.
In this song, Feist counts monsters, penguins and chickens, and relates them to a large orange model 4, which is in no way a definition of number. She points to the representation of “4”, and then to the penguins, and says “I see four here, and I see four there.” But does she? And do we? What are we doing when we make comparisons of this kind?
The first half of Frege’s book is largely a grumpy takedown of other people working in the field. Frege divides these into mathematicians and philosophers, and skewers each in turn. Some just by concept, but many by name. That includes, unsurprisingly, people who resort to the physical representation of number as numerals. When he exhausts them all, he starts to have his own go at the problem.
Remember I said earlier he’s credited with the invention of the second order predicate calculus? Well, I’m not going to explain what that is. But part of Frege’s construction is ‘concepts’, which are, roughly, functions that map to either ‘True’ or ‘False’ — what we would now call boolean values.(Philosophers tend to agree that Frege was not familiar with this portion of Boole’s work). So a concept might be “lions in Trafalgar square” — the four lions around the base of Nelson’s column map to True, but everything else maps to False.
He then has a first go at defining numbers straightforwardly from this definition, but rejects his first attempt for not being sufficiently rigorous. Subsequent commentators (you might call them Fregean apologists) have argued that this model works pretty well in practice, especially when taken with what is now known as “Hume’s Principle” — that two things have the same number if there is a one-to-one correspondence between them.
Instead, in one of those ways that probably seemed like a good idea at the time, Frege goes on to define the idea of ‘extension’. This is not the clearest of ideas, so he does it using an analogy rather than directly, by defining the direction of a line a to be the extension of the concept “lines parallel to line a”, and the shape of a triangle T to be the extension of the concept “triangles similar to T”. Just as you’re letting that soak in, he gets round to defining number. In what must be one of the most opaque sentences ever written, he says “The Number which belongs to the concept F is the extension of the concept ‘equal to the concept F.’ ” For anyone whose head is now spinning, he offers a helpful footnote; “I assume that it is known what the extension of a concept is”.
Well, it obviously isn’t. But to our jaded modern eyes, extensions look a bit like sets, and his definition would apply the number four to the set of all things that look four-like, like monsters walking cross the floor, penguins on the door, and chickens coming from the shore. He then goes on to show that from his definition, you can establish Hume’s principle, and from that, all of the basics of arithmetic.
So now we see what Feist is up to when she’s counting things; she’s observing that the extension of the concept “chickens coming from the shore” is the one that we habitually call ‘four’. She’s also helpfully identifying the sequential position of that extension as occurring immediately after “three” and immediately before “five” in Frege’s conception of the natural numbers, and remarking on our habitual association of this extension with the symbol “4”. Doesn’t that enhance your insight when watching the video?
Back in the history of mathematical philosophy, Frege’s work was under-appreciated by his fellow logicians at the time. He’d ignored, or was possibly unaware of, the work of George Boole, in a very similar space, and he introduced his own, relatively impenetrable, logical notation. But sometime later, Sesame Street favourite Bertrand Russell wrote Frege a super famous letter that said, roughly, gosh, Frege, what a very clever chap you are. Tell me, what do you think about the extension of the concept “thing that doesn’t contain itself”? Does it contain itself, or not? At which point Frege, and all his foundations, disappeared with a loud bang and a puff of purple smoke. In fact, Frege correctly predicts Gödel’s work in his reply, saying “It is all the more serious since […] not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic, seem to vanish.” And so it proved; it turns out that no arithmetic system can be both complete and consistent.
Despite this obvious setback, everything turned out all right in the end for Frege. I mean, ‘in the end’ means posthumously in this case, but still. All of modern mathematical philosophy and a fair chunk of the logic stem from his attempts to put arithmetic on a sounder footing. One, two, three, four.