Keith’s Conundrums: The Large and the Small

Recap

Sorry for missing November – hope you all had fun on my birthday!

This time…

The last book in Roger Penrose’s series about artificial intelligence is The Large, The Small and the Human Mind. It is by far the weakest of the three; one might even say it is just a bunch of favourable folks echoing back his controversial (or wrong – see the reviews of Shadows of the Mind by AI researchers and crucially, logicians) ideas.

Rather than do yet another bit about the last clause, as I have done many times in the AI context these last few years, I’d like to instead deal with “the small”. This also goes well with mole day, which some of us may have just celebrated in late October. Rather than give one concerned effort to one aspect of “small”, I’d like to instead raise (in a small way!) […] little puzzles that came to mind on this subject. But first, a note about the history. It is inevitable, these days, to encounter discussions of atomism when the subject of the small is broached. This is correct in my view, and as an author of a brief contribution to this subject (my BA Honours thesis was on this), I’m still more than up for discussions about that history as well. However, let’s get some Greek straight – a lot of little boxes of history in chemistry books, for example, get this wrong (and even noted philosopher of chemistry, Eric Scerri, does as well at one point). “ἄτομος” (atomos) does not mean “indivisible”. It means “not to be cut”, and first appears in the Greek corpus in the context of a sacred meadow, which assuredly is cuttable, it is just forbidden to do so. Only later does it seem to (perhaps) mean “uncuttable”. Similarly, “αδιαίρετος” (adiairetos) is unclear: all Greek adjectives ending in that way are modally ambiguous. Tradition has it that the correct translation is “invisible”, but there are just enough hints that “undivided” is at least not totally implausible, for the original Democritean/Leucippean understanding, that is. (There’s a very late testimonium, for example, that reports that “Democritus thought that an atom could be as big as the cosmos.” We’ll return to “cosmos” when we do “the large” sometime.)

That out of the way, let’s turn at last to some puzzles.

In classical field theory (a branch of physics), charged particles can be zero distance apart. Yet by (say) the Coulomb law, this would entail that the electric force between them is infinite, and hence the mutual acceleration is (F=ma) also infinite. This example is done with electromagnetic forces since they can be repulsive rather than merely attractive (as is the case in Newtonian gravitation). Does this point to a problem?

In mathematics (WARNING: math vs concrete world concerns!), the rational numbers are said to be dense: this means that for any two, there is one in between.  So there is therefore no rational number nearest zero: for any proposed one Q, take the number Q/2. This also suggests that there is no such smallest, either, in absolute magnitude. Yet in conventional mathematics, there are also real numbers, but so many of them that there is held to be a sense in which there are more of them than of rational numbers, and, moreover, there are non-rational real numbers between each pair of rational numbers. Does this suggest that some real numbers are smaller (“between” after all) than something that is said to have no smallest? 

In chemistry, the hydrogen molecule is one of the smallest molecules there are. However, chemists also point out that molecules have parts, so in particular, the parts can be at different relative distances at different times. We also know from quantum mechanics that the position of (say) one of the electrons that makes up part of the hydrogen molecule (there are 2) may be, from time to time very far from the nucleus of either atom. So how is it that we can (a) speak of a molecule’s shape and (b) (more importantly for the “small” theme) if there is a non-zero probability of the electron being arbitrarily far away, isn’t it correct to say that hydrogen molecules are sometimes arbitrarily large, just not most of the time larger than some size? So isn’t everything then small and large? (Or if you’re a particle physicist, I suspect that you’d claim that hydrogen molecules are always large, but …)

In biology, a cell is held to be the smallest living thing. Why is that? Is there a purely biological reason?

In “popular physical geography” (as opposed to, perhaps, the physical geography academic discipline), there’s a difference between a mountain and a hill. What is it? People from BC or Alberta who see Montreal’s Mount Royal often say: “Ok, hardly a mountain” and words to that effect. This does appear to be purely verbal, and should not be confused with the French distinction (often misunderstood) between “fleuve” and “riviere” (which appears to have nothing to do with size) and in fact encodes a piece of knowledge (if used correctly). On the other hand, the difference in French between “riviere” and “ruisseau” seems to be similar to the “mountain/hill” one.

Moving to languages: I’ve deliberately varied the discussion here across the languages in the previous one. Does the “small/large” distinction get consistently reflected across languages? There’s a Sesame Street segment/song “Big birds don’t fly”. Is this a definition or a law-like (?) generalization? (I realize I could have saved this one for the “large” column!)

People in my (working) profession are often told “that doesn’t need any security, it is so small” – referring to a computer program or application. What’s the relationship between size and danger? In chemistry, are larger molecules more likely to be explosive, flammable, or poisonous (etc.)?

And that’s it – this is supposed to be a small column! (Is it?)