Suppose you had a rubbish that calculator that could only deal with numbers below 100. Assuming that, apart from that limitation, it behaved as calculators normally behave, you’d naturally interpret it as performing the addition function when you type in ‘1+1’ and get the answer ‘2’. But is it? Consider the following mathematical function, which, following the philosopher Saul Kripke, I will call ‘quus’:
QUUS: The quus function is just like the plus function when the numbers inputted are below 100, but when any of the numbers inputted are above 100, the output of the function is 5.
Given that the rubbish calculator can’t deal with numbers above 100, there’s no fact of the matter as to whether it’s performing the plus function or the quus function. So we can’t really say that the calculator is adding. Rather, it’s indeterminate whether it’s adding or quadding.
Okay, but we can still say that normal calculators add, right? I’m afraid not. Once we’ve conceded the above, essentially the same point is going to apply to all calculators. That’s because, for any calculator, there’s going to be some number N that is too huge for that calculator to deal with. And so we can just define the quus function in terms of N, yielding essentially the same problem:
QUUS*: The quus* function is just like the plus function when the numbers inputted are below N, but when any of the numbers inputted are above N, the output of the function is 5.
For the calculator in question, given that it can’t deal with numbers bigger than N, there’s no fact of the matter as to whether it’s performing the plus function or the quus* function. We cannot definitely say that the calculator is adding rather than quadd*ing!
Does any of this matter? As long as we get the right answer when we’re doing our accounts, who cares about the deep metaphysics? I don’t think the above would be of deep interest if it was just a problem for calculators. Things start hotting up when we ask whether the same problem applies to us. Can our brains add?
You might see where this is going. There is going to be some number so huge that my brain can’t deal with, and if we define the quus function in terms of that number, we’ll reach the conclusion that there’s no fact of the matter as to whether my brain is performing the plus function or the quus function.
The trouble is there certainly is a fact of the matter as to whether I perform the plus function or the quus function. When I do mathematics, it is determinately the case that I’m adding rather than quadding. It follows that my mathematical thought cannot be reducing to the physical functioning of my brain. We could put the argument as follows:
- If my mathematical thought was reducible to the physical functioning of my brain, then there would be no fact of the matter as to whether or not I performed the plus function or the quus function when I do maths.
- There is a fact of the matter as to whether or not I perform the plus function or the quus function when I do maths.
- Therefore, my mathematical functioning is not reducible to the physical functioning of my brain.
The problem is that conscious thought, e.g. about mathematics, has a specificity that finite physical mechanisms cannot deliver.
This is just one of the deep problems raised by what philosophers call cognitive phenomenology: the distinctive kinds of experience involved in thought. It’s broadly agreed that there is a ‘hard problem’ of consciousness. But when people think about consciousness, they tend to think about sensory consciousness, things like colours, sounds, smells and tastes. But consciousness also incorporates conceptual thought and understanding, and these forms of experiences raise distinctive philosophical challenges of their own. I believe that we’re not even at first base in appreciating, never mind addressing, the challenges raised by cognitive consciousness. If you thought it was hard to explain the feeling of pain, you ain’t see nothing yet!