Category Archives: mathematics

26 Apr

The things clever people say!

It’s surprising how often those with the greatest reputations for intelligence say the stupidest things.

Bertrand Russell (1872-1970) for instance, following his 18th century predecessor David Hume, said that the fact the sun has risen every morning in the past gives us “no reason” to believe that it will rise again tomorrow.

But surely if an event has always happened before, we might think we have every reason to expect it to happen again?

Let us find another example from our ordinary experience, not so far-fetched, and see if we can use it to alleviate some of Bertie’s bizarre scepticism. Suppose every Thursday evening for the last ten years just after seven o’clock, I have seen Cedric Buggins leave his house and go into The Rose & Crown on the corner. Further suppose that today is Thursday and I have just seen Mr Buggins leave his house, dressed in his usual jacket and slacks, and head off in the direction of the pub.

According to Bertie’s way of thinking, I would have to say that I have no reason to suppose Mr Buggins is going to the pub. And that, I would say, is downright perverse!

What else do I think Mr Buggins is going to do if not go to the pub (as we say) “as usual”? He must be going somewhere. Am I suddenly, on this Thursday of all Thursdays, to think, “Ah I see Buggins is on his way to the Methodist chapel”?

I think that Russell (and Hume) is here using the word reason inappropriately: he is using it rather as we would use the word proof in logic or mathematics. But conjecture about what’s going on in the empirical world – the sunrise or Buggins’ going to the pub – is not the same sort of thing as reasoning in mathematics and logic.

We might say that Russell is guilty of making a category mistake in applying the mode of reasoning which appertains to the a priori  realms of maths and logic to the empirical world of our daily experience.

Formally, we might wish to add that here Russell is committing the fallacy of ignoratio elenchi by high redefinition of the words no reason. He is applying the sort of reasoning we use in deduction to the happenstance world in which the appropriate mode of reasoning is induction. Deductive logic deals in certainties. Induction is to do with probabilities. Of course I can’t prove that Buggins is going to the pub again tonight or that the sun will rise tomorrow in the same way that I go about proving 7 + 5 = 12. But that doesn’t mean in these cases I have no reason to believe as I do

What else should I believe in this case? There is a plainer way of stating this: it means Russell is missing the bloody point!

While I’m on about Bertie, there’s something else he said that worries me. He complained, “All my life in my study of mathematics, I have been disappointed that I cannot prove its axioms.”

Examples of axioms are, for instance, that the internal angles of any triangle add up to 180 degrees and that parallel lines never meet.

Of course neither I nor Russell can prove the axioms, because proof is something which is applied to propositions. And the axioms are not propositions but definitions. When I say that the internal angles of a triangle add up to 180 degrees, I offering a definition of what I mean by the word triangle. If any plain figure has angles which do not add up to 180 degrees, then it is not a triangle. And if ever we notice two lines converge, we must say these are not parallel lines – because parallel lines are defined as lines which never meet

Bertie, thou art the cleverest man in Cambridge. Knowest thou not these things?