I’ve just realised that, as well as being the number of our house, and the number of varieties of Heinz sauce, it is also central to the example at the heart of Kripke’s argument for scepticism about meaning and reference.* This last being what I spent my post-doc at University College Dublin investigating.
There are infinitely many numbers, but obviously no one has calculated with all of them. So there’s bound to be a sum involving numbers no one has ever actually used before. For concreteness, says Kripke, let’s suppose the sum is 57+68 = ?
and no one has calculated as high as 57 before. Most people, we assume would answer `125′.
Next, he says, imagine a bizarre sceptic challenges you: “how can you be so sure that answering `125′ doesn’t represent a change in your previous usage? Perhaps as you used the terms in the past you should now answer `5′.”
The idea is that perhaps `+’ stood for a function that agrees with addition over precisely the range of calculations we did in the past, but differs regarding the answer required in the case at issue. It’s called a `bent’ rule because it is aligned with the rule of addition for all previously considered cases but kinks off, or diverges, from what addition requires for the case at hand. If `+’ stood for a bent rule, then using `+’ as we used it in the past would now require us to answer not `125′ but `5′.
Now, it’s clearly an absurd scenario. It just defies credibility to think that we should answer `5′ if we’re to answer in accordance with how we used `+’ in the past. Why would we be following a bent rule? But as Kripke says, if we’re not following a bent rule, there must be something about us that rules it out. Something about our past behaviour, or mental lives, or social and physical environment (e.g. a linguistic community), or our dispositions to answer related questions in the future, something must rule the bent possibility out.
But does anything rule it out? By hypothesis no one thought about numbers as high as 57 before. They weren’t in our minds’ eyes when we learned how to calculate, so how could what we learned have dictated one sort of response over others? Well, Kripke admits, we did learn a procedure for addition (e.g. count out x many marbles. count out y many marbles. put the piles together and count out the combined pile), but the terms in that procedure could themselves have been bent, so that `count’ really stands for the practice of enumerating objects one after another until you get to 57 objects after which the tally is always `5′. We may have learned a procedure for addition, but if that procedure involved bent rules, then a bent solution would be demanded for the question `57+68=?’.
Kripke goes through a number of possible candidates for a fact that might rule out bent interpretations, but he rejects them all. For instance, other speakers in our the linguistic community can’t come to the rescue, because if there’s no fact of the matter about what an individual means by a word, there can be no fact of the matter about what an aggregation of individuals means.
The one promising candidate for a fact about an individual that would show they weren’t following a bent rule is the fact that most people are disposed to answer `125′ not `5′. This is where many philosophers think the sceptic’s challenge can be answered. But Kripke makes a very strong prima facie case that dispositions will not answer the sceptic. The first thing he notes is that it is not in fact true that given any addition problem, most people are disposed to answer in accordance with the addition function. A lot of people make mistakes. It is very hard to manually add up a column of figures correctly. If you use a calculator, you are literally using a bent rule, because when the numbers get high enough, the calculator responds with `E’ (or `ERROR’ or some other output). The calculator follows addition some of the way and then diverges when the numbers are too large. Some numbers are infinitely big. You would literally die before finishing hearing them. Humans don’t have dispositions to answer in accordance with addition (as opposed to alternative bent functions).
One strategy to overcome this, which I think fails, is to consider our dispositions as complex products of dispositions to answer in accordance with addition and dispositions to deviate from addition in certain conditions. For example, when tired we have a disposition to misread a column of figures and misalign numbers so as to put the calculation off. If these dispositions to error can be factored out of our actual dispositions, then we can site our disposition to add as the fact about us that shows we weren’t following the bent rule.
The trouble is that the factoring out has to be more than a merely nominal one. After all, our actual dispositions can in principle be factored into a disposition to follow a bent rule and auxiliary dispositions to err etc, as well. We need to go further and say that whereas we only nominally have a disposition to follow the bent rule, we really have a disposition to follow a true one. I don’t believe we can do this. If there were some way to independently characterise the sort of dispositions bending our behaviour away from a true disposition to add, then we could say we really had that disposition to add. But the characterisation of auxiliary dispositions is always itself susceptible to sceptical reinterpretation and nominal factorisation into different bent dispositions. So the independent characterisation isn’t to be had.
The sceptical conclusion of the preceding train of thought is that there is no fact that shows I mean addition by `+’ (and should answer 125). The argument generalises to all words, so the final conclusion is that there is no fact of the matter as to what we mean by any word.
I just think that’s a blast. It pleases me that I have a little reminder of this argument in my house number. I thought I’d share the argument with you to see if it stimulates any thoughts.
*Saul Kripke is an American Philosopher and widely regarded as one of the best philosopher’s of the 20th century (in fact he’s still alive, so possibly the 21st century too). He presents the argument as an interpretation of Ludwig Wittenstein’s work, but most philosohers don’t accept the attribution.