There is no infinity.

I was recently reading “Statistics for Dummies” by Deborah J. Rumsey (yeah, I know, but I'm on the road and grabbed something statistics related from the library I could read on my Kobo), and ran across this tidbit:

The probability that X is equal to any single value is 0 for any continuous random variable (like the normal). That’s because continuous random variables consider probability as being area under the curve, and there’s no area under a curve at one single point.

Her example of a continuous random variable is the length of fish caught in a fishing contest. Now I know I am just Joe Schmuck, with almost no knowledge of statistics, and even less credentials, but that statement, to be charitable, is just silly. I am sure if I were to challenge Dr Rumsey to a public debate on this, she would crucify me, as she would be well prepared with mathematical theorems backed by centuries of publications in peer reviewed journals, but that doesn't change the fact that she is now qualified to write speeches for George W.

Any half-wit, unencumbered by accepted truisms in statistics, can tell you that her statement is utter nonsense. This here fish I have on the end of my fishing line happens to be X cm long, but Dr Rumsey tells me there is a zero chance of that fish existing, so it can't exist, so that must be some very unusual crocodile wiggling on the end of my line. The other obviously troublesome problem with her statement is that it applies equally to any other length of fish you'd like to consider, so by definition there are no fish at all in that lake, or any other lake for that matter. And if there are no fish, then the population (or sample) size is zero, so there is no mean and no sigma, and Dr Rumsey doesn't have a probability distribution, and thus can't actually say anything about the probability of catching a fish of a certain length – her statement is internally inconsistent.

I've used the concept infinity in my previous life as a scientist. I always thought of an infinite value as something that was so big, that if it was any bigger it wouldn't make any difference. I get it – it can be useful. Integrating from -∞ to ∞ can make for some clean, pretty math. Modern physics relies heavily on mind-bending mathematics (the race is on to add more dimensions to M-theory), and so infinity can be a useful tool for modelling the real world.

Things start to go haywire when you begin to think of infinity, literally, as a real thing. Some physicists have suggested that the universe is infinite, and they're not invoking some trickery like warping space-time so that a traveller on a space ship would travel in a big circle around the universe and eventually arrive back where they started – they mean that the extent of the universe is truly without bound. I know that at very large and very small scales, human intuition GPF's, but “say what?” I have to ask. Infinite you say? I just can't buy that. A quantum fluctuation on steroids produced the Big Bang (which, interestingly, was silent) and somehow that produced an infinite amount of space and matter? No way.

Following this to its logical (illogical?) conclusion, leads to all sorts of problems. An infinite universe implies an infinite number of stars, an infinite number of planets earths, and an infinite number Barack Obamas. You see, if the astronomical bodies of the universe formed essentially at random (which all the evidence supports), and you have an infinite number of planets, that means an infinite number of them will be exactly like earth. No all of them mind you, but still an infinite number. Thinking about that idea alone is enough to make you bleed through your eyes – take the infinite number of planets and divide them by 26 wappagazillion, and you're left with ... an infinite number of planets. So even though planets exactly like earth are extremely rare, there are just as many earths as all the rest of the planets combined: infinitely many. I would have thought that simple observation was enough for some philosopher to disprove infinity once and for all [maybe one has – I didn't check – but this Schmuck's blog is about writing, not solving great mathematical problems]. But I'm getting side tracked yet again. So we've got an infinite number of planets exactly like earth, and on each one we've got life evolving randomly, and since we've got an infinite number of them ... well the result in a infinite number of earths with Barack Obama and Vlad Putin getting their knickers in a knot over a small, impoverished country in eastern Europe, and making life hell for millions (a small price to pay for their ego games, and darn if I can't stay on topic today). So, there are an infinite number of Joe Schmucks out there too. How can anyone really believe that?

The flaw in Dr Rumsey's statement is that she has fallen into the infinity trap. Her implicit assumption is that there are infinitely many possible fish lengths. If you accept this, then it follows that the width of the area under her probability curve for any given fish length is infinitely small (a pompous way of say zero), and so the “area under the curve” for a given fish length is zero, and so is the probability of me having a fish on the end of my line. It is all so sensible when you think about it like this, but it is also a pile of rubbish.

To have an infinitely large number of fish lengths, the smallest possible difference in fish size must be infinitely small, but in real life that just isn't so. Fish are made of molecules – proteins and fats and those sorts of things – and molecules are made of atoms. Atoms are only so small – free quarks don't exist in nature, and that includes fish bodies. I am guessing that the smallest possible difference in the size of two fish can be no smaller than the size of an atom, and probably larger than that. So the smallest difference in fish lengths may be really, really small, but it isn't zero.

But remember, we are talking about the lengths of all fish caught in a fishing contest. The lengths of the fish need to be measured, and in practice, it's not possible to measure the length of a real fish to anything like atomic scales. Think about it, we need some repeatable process so we can reliably say what the length of a given fish is, and attempting to measure the length down to scales where Brownian motion and Heisenberg uncertainty are important just isn't going to generate reliable, repeatable results – not to mention a slimy, floppy (or flopping if it is freshly caught) fish just doesn't lend itself to atomic precision length measurements. Her probability distribution is a distribution of actual measurements, not hypothetical measurements, and we can't measure the length of something with infinite precision – this is not dependent on our technology, it just isn't physically possible. In practice, I think you'd be doing pretty well to manage +/- 0.1 mm, in which case the width of the area under the curve becomes small, but in some ways a long ways from zero.

Her statement is based on the assumption that the smallest difference in fish lengths is infinitely small and we can measurement with infinite accuracy – both those assumption are wrong because they rely on applying infinity to real life physical things. Her continuous random variable isn't really continuous, and neither is any other you care to dream up.

There is no infinity, and maybe that is a fish on the end of my line after all.