Does ‘Almost Infinite’ Mean Anything, Really?

6Feb - by Alan - 0 - In Uncategorised

It’s a common enough phrase, and I think most of us have no trouble understanding that the speaker or writer is referring to a quantity of something that is mind-bogglingly huge. Infinity is a slippery concept; you may suppose that you understand its essential quality of being really big – like space, but then erroneous statements about it tend to be depressingly common. The commonest being that it is a kind of number so that something can be ‘almost infinite’. But the nerds and mathematicians among us may well pause and wonder ‘how does that make any sense?‘. If you are halfway, or 90%, or 99.99% of the way to Infinity – you still have an infinite way to go! There is never ever a Yes answer to the question ‘Are we almost there yet?’. We are never going to get there, so there’s no ‘almost’ about it. Yet ‘almost infinite’ is a common enough piece of nonsense, as you can see if you Google it (6,000,000 results, not quite infinite).

Charles Darwin was quite fond of the phrase, such as ‘almost infinite number of generations‘ in The Origin of Species, and even using it doubly as “beings almost infinite in number, during an almost infinite lapse of time” in The Variation of Animals and Plants under Domestication. And: “The mind cannot possibly grasp the full meaning of the term of a hundred million years; it cannot add up and perceive the full effects of many slight variations, accumulated during an almost infinite number of generations” (last chapter of The Origin of Species).

At the time of writing, Google lists almost half a billion results with “almost infinite”

Clearly, in most contexts of common usage, ‘almost infinite’ simply means really really big – like space. Or is space itself infinite? We often hear that the Universe is infinite. But is it really? Is it more a poetic notion of grandeur and incomprehensibility than a genuine scientific statement of fact? Could any physical quantity be infinite? For any quantity to be physically meaningful, there has to be a way to measure it, directly or indirectly. But how could any finite device record an infinite value? Well, we might try a mathematical transform that maps an infinite range to a finite range. Escher is well-known for his artistic accomplishments using hyperbolic geometry. The problem is, that any such measurement would need to be infinitely precise…

Some people believe that if space is Euclidean, then it must be infinite in extent. Here there are 2 problems. First, the measurement problem already mentioned. We may determine that space is flat very accurately, but any uncertainty in the measurement leaves room for space to curve oh-so-slightly. Secondly, a flat space can be achieved by other geometries, e.g. a torus.

Space is almost infinite. As a matter of fact, we think it is infinite. — Dan Quayle

Is the Universe infinite? People often say it is, but how could you ever measure or prove that? Even if you measure space to be Euclidean, your measurements have limited precision and accuracy (not to mention quantum indeterminacy), and so non-Euclidean at some very large distance can’t be precluded. Perhaps you could say that the Universe is almost infinite, within the limits of experimental accuracy?

Everybody’s deity is supposedly infinite. This is the end result of a pissing contest between religions. If you said your god is as big as N, then I’d say mine was as big as N+1, so you’d reply with N+2, …

Apparently, given an infinite number of monkeys with typewriters, at least one of them would perfectly reproduce the complete works of Shakespeare. Alternatively, one monkey with an infinite lifespan would eventually do it (before typing random nonsense).

Wikipedia: The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.

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