Caution - deduction from infinity can lead to madness

I'm currently reading for review Max Tegmark's intriguing newish (well, new in paperback) book, Our Mathematical Universe. It's generally rather good, though it's a bit infuriating that they clearly haven't updated the text to reflect this edition, as Tegmark keeps referring to the image on the front of the book as showing the Cosmic Microwave Background - if the CMB really looks like that, cosmology truly has got exciting again.

However, that wasn't my point. Having set the stage with an explanation of the hot big bang with inflation theory, Tegmark begins launching off into the possibilities for multiverses, and there's a lot of deduction from infinity. (If this doesn't mean anything to you, I'll get there in a moment.) Georg Cantor, the great mathematician of infinity, ended up in a mental hospital - you play with this stuff at your peril.

What I mean by deduction from infinity is arguing along these lines. If eternal inflation holds, there are an infinite set of big bangs producing universes (of which ours is one). Each will be subtly different due to quantum fluctuations. So as they are infinite, every possible outcome will happen in one of these universes - for example, one where you read this blog and sneer, rather smile at its cleverness as you currently are doing. (Hopefully.)

The problem is that infinity can't be used like this to deduce things. Let's look at some simpler infinite sets to see why. First, bear in mind that every member of an infinite set does not have to be different. So, for instance, you can have the set 1, 0, 0, 0... where all the members are zero except the first. If 1 represents our universe, all the others could be devoid of life. (I do remember those quantum fluctuations, but that doesn't mean that you couldn't end up with all but one devoid of life.)

Here's another one that's a little more interesting: 1, 4, 9, 16... - the infinite set of the squares. One for every universe that has life in it. But what if the actual infinite set of universes only corresponded to the numbers that aren't squares with a different value to its square root: 1, 2, 3, 5, 6, 7, 8... - there's an infinite set of those, none of which has life, apart from no 1 - us.

Or again, think of the infinite set of positive integers: 1, 2, 3, 4, 5, 6... If one of these could be part of an infinite set but also demonstrably unique, that could like the condition for life. And guess what - 1 is unique. It's the only positive integer that is its own square and that when something else is multiplied by it, that something else doesn't change. So despite there being an infinite set of subtly varying possibilities only one is in the 'life' state.

My examples here don't prove that there aren't all those different variations of you in parallel universes, but rather they demonstrate that you can establish pretty well anything you like if you try to deduce things from an infinite set - and my suspicion is that the deductions made by cosmologists are equally suspect.

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

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

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

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