I wanted to share an interesting passage I came across when doing research for an upcoming blog post that mentions Kant's philosophy of arithmetic1. I'll start by sharing a bit of background context.
The core question at stake is whether you can defend arithmetical judgements like 2 + 2 = 4 purely with an appeal to basic logical laws, or whether some other faculty is required. In Kant's case specifically, the additional faculty in question is imagination.
My current understanding of Kant's view is that to judge that 2 + 2 = 4 you must firstly construct in imagination 2 objects and then 2 more objects beside them. Secondly you count the total as 4. Lastly you observe that nothing in this process depended on the particular objects involved, but only on the general procedure 2 for being able to generate two things in the imagination. This means that this judgement holds necessarily and universally for any case of two and two things. 3 I think he also perhaps thought you could do this by working with these imaginative procedures in other ways - which is how it is still possible to do arithmetic on unimaginably large 4 numbers.
With this context out of the way, here is the passage5 I came across:
I was personally struck by how subtle and elegant the objection made to Leibniz's approach here is.
Associativity is the basic algebraic property that (a + (b + c)) = ((a + b) + c). It is generally so obvious that we never even bother to teach it to children learning math. We also never in practice write brackets like these, because once you prove associativity, you can prove that any placement of the brackets doesn't make a difference to the final answer.
I claim the reason that the assumption of associativity is so easy to miss is precisely because the operation of our imagination being modelled by addition is associative. When you imagine (2 things next to (1 thing next to another 1 thing)) that is just the same as imagining ((2 things next to 1 thing) next to another 1 thing).
The approach amounts to saying that even if you symbolically define numbers in terms of units, and addition as a binary operation which combines these units, you still don't get the axiom of associativity for free, it has to come from somewhere else.
This led me to ask three more questions:
- Who was Johann Schultz, and what else did he have to say about Kantian theories of arithmetic?
- How did the 20th century logicists (mathematicians and philosophers with the view that mathematics should only be placed on logical foundations, without appeal to other faculties) respond to this?
- If we want to entertain the idea that associativity is not a logical requirement for arithmetic: can we find mathematical structures with all the same logical structure as arithmetic but for which associativity doesn't hold?
I haven't gotten very far answering (1). Much of Schultz' work and the commentary on it is in German, and hasn't been translated into English. However there is a great open access paper that translates some relevant sections of his work and compares it to 19th century axiomatisations of arithmetic.
Most of the answer to the second question is also outlined in that paper. Reading it reminded me that you can prove associativity by mathematical induction. This turns out to be what Frege relies on to establish associativity in his book The Foundations of Arithmetic.
What remains is for Frege to answer whether mathematical induction is something you can found on purely logical assumptions or whether it itself requires appeal to an imaginative faculty. My understanding is that if you follow Frege's approach, the crux turns out to be whether you think that second-order logic, in particular quantifying over properties and relations, is an acceptable case of a pure logical operation that doesn’t rely on imaginative faculties6.
The third question about whether non-associative arithmetics exist is similar to the question of whether non-Euclidean geometries exist. My initial knee-jerk guess was that magmas might fit the bill; but as structures go these ones on their own are too simple to do anything that really counts as arithmetic - for example they don't come with a concept of subtraction, division or multiplication. Some further searching online led me to discover that a few structures do support all four basic arithmetic operations but are not associative - with the canonical example being octonion algebras
-
In general, I am trying to spell out what a Kantian philosophy of computer science might look like. ↩
-
I think Kant would call this general procedure the "schema" for the concept of the number 2, however I can only find evidence of him talking about the schema of the concept of "number" in general, as opposed to of any particular number. (B180) ↩
-
Note this particular example of imagination is not a question about whether we learn addition empirically (that is, by extrapolating from experiences of the world), nor is this a question about whether you need any particular sense like sight to do addition. You can just as easily read this post by imagining numbers as tones in a sequence of sounds, or as taps on your shoulder and the same points will hold. That said, if arithmetic requires this kind of imaginative faculty, it does place constraints on the kinds of beings that can do mathematics: they must at the very least possess something like what Kant calls "forms of intuition" (ways of representing objects in space and time) similar to ours. ↩
-
In this case I think for most people this already has broken down for numbers greater than 10. ↩
-
This is from Charles Parsons' 1964 Paper: "Kant’s Philosophy of Arithmetic" ↩
-
I was going to say here that this approach inevitably leads to the famous Russell’s paradox, and while this was historically true it looks like recent literature has shown that Frege could have essentially avoided the paradox by instead accepting as foundational a basic axiom known as Hume’s principle. Presumably Frege chose not to do this as he was not satisfied that this axiom was substantively more logically well-founded than something like associativity. ↩