Keith’s Conundrums: Idealizations

Recap

Alex responded with a few answers to my probability related puzzles. I will make comments on a few of them myself before moving on to a topic very dear to me, though this will likely only come out as I put these last two topics together in a future (possibly the next) column.

#3 had a trick: us Canadians talk about “quarters” all the time. However, according to the Mint this is an Americanism. So some would say zero. But should be it rather be undefined, since the statement of the problem includes a failure of reference: think of “What’s the probability of throwing ‘mud’ on the toss of an Elbonian coin?”

#4 also has a trick, one I repeat but point out here – the usual model (that Alex alludes to) is that a coin has probability 0.5 of coming up heads, 0.5 tails. But is that so? Some stage magicians can make a coin land how they want! So what’s being assumed in that distribution? Ah, but “fair” coin is the phrase, right? Well, ok, but what’s that? Also, a coin has an edge. As we may notice, we can stand a coin on the edge – sometimes and for a while anyway. So can it land on the edge? Uh oh.

#21 I regard this question as in a way the most important, though aspects of it exist in 25 and 26. More in a future column. (Meanwhile, try to figure out why I say this.)

#28 illustrates the importance of domain knowledge – of biogeography. The only way the trip works is if somehow the hunter starts and ends at the north pole. There the only bears (?) are white. So 0? But yet, the north pole is in the ocean, far from land – (Russian and Canadian squabbles notwithstanding) so there are no bears, either. So…

I’d like to return to some of the questions of #4 as our subject for this month, addressed in a different way.

Idealizations

This time I would like to discuss at length the question of idealizations in our understanding of the world. It has been claimed that Aristotle and other ancient natural philosophers (and particularly, though I don’t know this area as well, the Daoists) tried to capture “everything” and hence failed because there was too much to do at once. Galileo’s innovation therefore was to select a tiny aspect of a “toy” system to understand simply and then complexify it with increasing “realism”. In this column, rather than one of Galileo’s cases, we’ll take up one of my perennial favourite, the gas laws and their seeming simplicity. Our discussion will be somewhat ahistorical and I do not see the need to mention names all the time, nor stick to a strictly chronological sequence. Your first question, therefore: Are these appropriate idealizations? Why? (Or why not.)

Amount of gas (n) is proportional to volume (V). This itself is a great innovation on several fronts. First: it might appear that volume and amount are really the same: if I have some pebbles it doesn’t matter how I arrange them: they still fill a given container if I shake them around. But then we notice that, with some substances, we can fill a leather bag and squeeze it smaller without it leaking. Or so it seems.

We’ve discovered gas, and its characteristic: compressibility. But how do we know nothing leaked? We can carefully measure (perhaps borrowing from Archimedes) the volume and it seems about right … Yet it appears we can’t be sure whether or not we’re dealing with a slight variation from cV=n or not. Also, amount of what? Are we dealing with something perhaps continuous, like a pebble seems to be, or like the heap of pebbles which has gaps? But the heap has gaps of something else – what would be between the bits of air? This raises questions of the vacuum, which was considered by ancient natural philosophers (void, empty, kenon in Greek) and there were debates over it again in the 17th century. It is interesting to reflect on why there is more work done on the history of atomism (the history of how we thought of stuff as being made of little bits of other(?) stuff) than on the history of the “empty” – which has always struck some – Descartes, for example – as being self-contradictory, even. A final complication: If we fill the bag, sometimes the bag seems to get no holes yet shrinks. What’s that about? And many noticed that the bag is more inclined to finally burst on hot days. Hot as in we feel something uncomfortable, for example, seemingly from the sun, not anything else at first…

Pressure (P) is inversely proportional to volume (V). This one is attributed to Boyle, and there’s at least two ways in which this textbook presentation is wrong. One is merely(?) the attribution: Boyle himself seems to attribute the discovery to Robert Hooke. Another is more interesting from the perspective of content, which is that the notion of pressure itself is complicated. We sort of have an intuitive understanding of a push squeezing something, but turns out that in order to “unify” the gas laws one actually needs a notion of force as well, given that pressure turns out to be a force per unit area. But this notion of force is due to Newton, more or less – though ideas about forces were discussed before and after and it took until the 19th century before we finally worked out the connections and differences between force, energy and related notions. Complexity: compressing the bag of gas (or the air pump Boyle and Hooke used) eventually seems to get harder and harder. Similarly at the other end: what happens when pressure goes to zero?

Sometime later than Boyle, people returned to the question of the bag bursting on a hot day, seemingly on its own. In modern terms, we have learned that volume is proportional to temperature. However, that takes a lot of work – much more than either previous case. For one, it takes inventing the concept of temperature – noticing the real property distinct from the sensation of heat and its objective counterpart. All three or four distinctions there are hard to make. Yet they were even done without even realizing that heat (not the sensation) was not a stuff! (This is where the famous caloric played a role.) But what is this temperature? Can we understand it in any other way? Is it “reducible” (the way pressure was) Why think that it is again just a linear relationship? Ultimately the latter comes from putting other things together, not related directly to what we can now compose out of what we have seen so far.

This combined expression, written these days as PV=nRT, has one more ingredient. What’s R? A constant to make the units work out nicely. Chemists like using “mole” as the unit of amount of substance, but one could use the dozen, the score or even just raw counts, at least in principle. The mole is used because of the scale of the amounts of stuff chemists usually deal with. So this gives a constant suitable to connect these properties on that scale. But one could work with units, as physicists do sometimes, where the proportionality constant is 1. Units (not dimensions – what’s the difference?) are “justified conventions” in Bunge’s phrase. Question: in what way justified?

It is often suggested that the “Galilean idealizations” (really: Galilean-inspired) used here are suitable because they don’t involve much error, and are therefore good enough. Why? What provoked scientists to complexify that neatly beautiful equation? Here to some extent is where realism may loom: I will claim that in order to understand why van der Waals and others studied further here is because they had a definite view: that the world is a given way (but conceptualizable in many different ways – as we have seen) and so we should try to find out what that is. We thus call the equation that is to be corrected under some circumstances the ideal gas law not because it is an ideal law but because it is a law (pattern) which describes an idealized gas. This requires having some understanding, a model of what that would be like.

Paradoxically(?), then, in order to be more realistic, one has to start with fiction(?). Each of the aspects we have discussed can be then taken up carefully and analyzed as to what it claims and what could be not quite right about it. I mentioned van der Waals – he’s given his name to the resulting next model’s equation. In this model, the gas now has internal structure: it is held to be composed of small little bits of finite (rather than infinitesimal?) size. There are other aspects, and work beyond even the van der Waals equation exists. What tells us what to use? How do we know we have the model right that we are complexifying? What happened when a model was completely rejected (e.g., that of Aristotle and Ptolemy by Galileo, Kepler, et al)?

Next time, I will attempt to discuss those last questions and relate them to some current AI (groan!) hype topics: are these systems capable of scientific discovery? Are they using the right model of cognition (or whatever else) themselves? Does my model (!) of starting simple, putting together complexity over time, etc. apply in the domain of psychology (?) where arguably the AI understandings should come from?