Skip to main content

Is there a law of the excluded middle?

I have just finished reading the excellent Inventing Reality for review, a great book that both outlines the essentials of physics and looks into what physics is really doing. It has a fascinating argument about the law of the excluded middle, which I just have to pass on.

The law of the excluded middle essentially says that a statement must be either true or false. Apart from tricksy statements like 'This statement is false', most mathematicians and all physicists seem to assume that this is true, that a statement has to be either true or false. But is really the case?

Bruce Gregory uses a simple, and apparently non-tricksy statement as an example. Here's a conjecture: 'A woman will never be elected president of the United States of America.' Is that true or false? As Gregory says, 'if we insist it must be one or the other, we seem to be committing ourselves to a future that somehow already exists, for the truth or falsity of the statement depends on events that have not yet occurred.'

Just think about that for a moment. We can certainly say it is very likely that there will be a woman president elected at some point in the future, but I can also posit a range of scenarios where it never comes to pass. We can't say that this statement is definitely true or false. And if that's the case, argues Gregory, might it not also be true of a mathematical conjecture, like the Goldbach conjecture (saying that every even number larger than 2 can be written as the sum of two primes)? Obviously this could be proved false, all we need to do is come up with a single even number that isn't the sum of two primes, but it is entirely possible that the conjecture will never have a definitive outcome.

The same, Gregory suggests, could be true of some aspects of physics. We can't assume that, say, String theory will ever be proved true or false. (Where 'true' or 'false' does not mean matches reality, as we have no idea what reality truly is. Rather it means 'successfully predicts measurable outcomes'.) Which makes for some interesting thinking about physics, the universe and everything.

Comments

Popular posts from this blog

Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5.  However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor

Why I hate opera

If I'm honest, the title of this post is an exaggeration to make a point. I don't really hate opera. There are a couple of operas - notably Monteverdi's Incoranazione di Poppea and Purcell's Dido & Aeneas - that I quite like. But what I do find truly sickening is the reverence with which opera is treated, as if it were some particularly great art form. Nowhere was this more obvious than in ITV's recent gut-wrenchingly awful series Pop Star to Opera Star , where the likes of Alan Tichmarsh treated the real opera singers as if they were fragile pieces on Antiques Roadshow, and the music as if it were a gift of the gods. In my opinion - and I know not everyone agrees - opera is: Mediocre music Melodramatic plots Amateurishly hammy acting A forced and unpleasant singing style Ridiculously over-supported by public funds I won't even bother to go into any detail on the plots and the acting - this is just self-evident. But the other aspects need some ex

Which idiot came up with percentage-based gradient signs

Rant warning: the contents of this post could sound like something produced by UKIP. I wish to make it clear that I do not in any way support or endorse that political party. In fact it gives me the creeps. Once upon a time, the signs for a steep hill on British roads displayed the gradient in a simple, easy-to-understand form. If the hill went up, say, one yard for every three yards forward it said '1 in 3'. Then some bureaucrat came along and decided that it would be a good idea to state the slope as a percentage. So now the sign for (say) a 1 in 10 slope says 10% (I think). That 'I think' is because the percentage-based slope is so unnatural. There are two ways we conventionally measure slopes. Either on X/Y coordiates (as in 1 in 4) or using degrees - say at a 15° angle. We don't measure them in percentages. It's easy to visualize a 1 in 3 slope, or a 30 degree angle. Much less obvious what a 33.333 recurring percent slope is. And what's a 100% slope