What are Junk Theorems?
Sometimes mathematicians prove theorems that they don’t want – even worse: theorems that don’t make sense. Junk theorems are a byproduct of the foundations of mathematics. How do they come about? And are they a problem?
A foundation of mathematics uses axioms to describe certain entities as a basis for all of mathematics: Such entities could be sets, categories, or types. Their axiomatisations may give rise to well-known systems such as Zermelo-Fraenkel Set Theory ZF, the elementary theory of abstract categories ETCC, or Homotopy Type Theory HoTT.
These systems then fulfil their foundational role like this: All mathematical objects, such as numbers, groups and spaces, are built up from the basic entities of these systems. Junk theorems arise when this process of ‘building maths up’ has unintended consequences.
Let’s have a look at an example. Set theorists usually define the natural numbers in the following way. The number 0 is defined to be the empty set.
Next, they define a successor operation, that successively gives rise to all natural numbers as we know them. If we have defined the natural number n, then we define the number n+1 to be the set-theoretic union of the number n and the set that contains the number n. Just like this:
From this definition, we get a representation of the natural numbers as sets in set theory: The number 0 is defined to be the empty set. The number 1 is defined as the set-theoretic union of the number 0 and the set containing the number 0:
If we keep applying the successor operation, we get that the number 2 is the set {0,1}. The number 3 is {0,1,2}, and so on.
It turns out that this representation of natural numbers in set theory allows to prove all their usual properties. That is, set theory as a foundational system succeeds in giving a foundation for the natural numbers. They behave like they should: addition is symmetric (x + y = y + x), addition and multiplication are distributive (x(y+z) = xy + xz), and so on. (Of course, you need to define addition and multiplication in the right way first!)
However, there are also unintended consequences. The number 4 is an element of the number 100: 4 ∈ 100. And the set-theoretic intersection of 3 and 4 is 3: 3 ∩ 4 = {0,1,2} ∩ {0,1,2,3} = {0,1,2} = 3.
In fact, you can easily prove a theorem stating that:
Let n and m be natural numbers. Then n < m if and only if n ∩ m = n.
But this looks a bit like rubbish, doesn’t it? That’s why such theorems are sometimes called junk theorems: they do not prove anything sensible that we would expect to be true of natural numbers. On the face of it, it surely looks wrong to calculate the intersection of two natural numbers.
Are junk theorems problematic?
Now, you might wonder: If our foundational system has unintended consequences, such as the junk theorem above, then can it be the right foundation? Shouldn’t a good foundation of mathematics avoid such weird theorems?
The philosopher Imre Lakatos argues that:
… if we insist that a formal theory should be the formalisation of some informal theory, then a formal theory may be said to be ‘refuted’ if one of its theorem is negated by the corresponding theorem of the informal theory. (A Renaissance of Empiricism in the Recent Philosophy of Mathematics, p. 214)
Let’s try to understand this quote by applying it to our example concerning natural numbers.
We start from a certain informal theory about natural numbers. That is the theory that we use when we argue about natural numbers, and write down proofs about them (but maybe without explicitly referring to a particular axiom system). The goal of the foundational theory — set theory in our case — is to provide a formal foundation for the informal theory.
Lakatos argues that such a foundational theory does something wrong if it allows to prove theorems that are the negation of a theorem that the informal theory proves. In such a situation, the informal theory refutes the foundational theory.
Does such a case of refutation happen in our example? Did we show modern set theory wrong? After all, our informal theory certainly doesn’t prove that 3 ∩ 4 = 3.
I don’t think so. Foundational theories cannot be refuted on the basis of junk theorems. The reason is this: Junk theorems only arise because of some additional structure that the foundational theory provides but the informal theory doesn’t. Set theory allows to talk about ‘intersections,’ ‘unions’ and ‘elements.’ Arithmetic, however, speaks only about ‘plus’ and ‘times.’
Whatever informal theory of natural numbers you have in mind, it certainly does not say anything meaningful about ‘set-theoretic intersections of numbers.’ The junk theorem only arises because the set-theoretic machinery allows us to see details of formalisation that form no part of an informal theory.
Junk theorems might feel a bit like what happens if you look under the hood of codings used in programming languages.
The ASCII code for the character ‘.’ (dot) is the number 46, and the code of ‘<’ (less-than) is 60. So, in a programming language that uses this coding you get that: ‘.’ + ‘<’ = ‘j’ because 46 + 60 = 106, and that’s the code of the letter ‘j.’
But certainly: the programming language was not designed, in the first place, to perform arithmetical calculations on letters and interpunction signs.
Nonetheless, junk theorems can be problematic as Joel Hamkins points out in his recent book:
The objectionable aspect of junk theorems are not cases where the foundational theory S proves additional theorems beyond T in the language of T, but rather where it proves S-features of the interpreted T-objects. (Lectures on the philosophy of mathematics, p. 19)
Let’s unfold this a bit: Essentially, Hamkins is saying that a junk theorem about the natural numbers is problematic only in those cases where it contradicts our pre-foundational knowledge of how the natural numbers should behave. In particular, the junk theorem ‘3 ∩ 4 = 3’ is not problematic because the set-theoretic intersection ‘∩’ does not form part of the language of arithmetic. It’s not something we talk about when we do basic arithmetic.
We’d be in trouble, however, if our interpretation of natural numbers in set theory allowed to prove a junk theorem like ‘1 + 2 = 4’ because this is a claim that we care about when we do arithmetic: it’s false! Luckily, set theory doesn’t prove such non-sense.
So what are junk theorems? Junk theorems are unwanted byproducts of formalising mathematics in a foundational theory. Are they a problem? Mostly not, they are problematic only under special circumstances when they demonstrate that the formalisation is wrong.
Can you find more examples of junk theorems? This phenomenon isn’t limited to set-theoretic foundations.
