Keith’s Conundrums: The Larger and the Smaller

The Larger and the Smaller

Keith Douglas

Recap

I belatedly received a comment on “The Large and the Small” from Arif Uddin. Thanks, Arif, for writing. I agree that sometimes one should take action about injustice, etc., and your cases do present interesting challenges in that way. I’d like to think very sympathetic to the plight of those who suffer. However, although I have posed ethical questions in this column in the past, it is meant as a “head scratcher” – this does not dismiss, I think, the importance of dealing with the matters Arif raises, but simply contextualizes them. I’d like to think we (as a species) can do a bit of both. I also think epistemology and metaphysics are important for ethics, so I write about the first two for that reason as well. I hope, Arif, you can get the treatment for your diabetes you need.

Alex B. asks us to consider the disciplines involved in the problems of collective opinion I raised last month. Those of you who have been reading this column for a while may recall that I used to give hints about how to think through the puzzle by giving disciplines, which I think might have material to help (or further provoke perplexity). I have resumed that by answering him on the post itself; check it out.

The Small and The Large

We now look at “the large”, having dealt primarily with the small a while ago. I preface each subproblem with the disciplines that are used in stating the problem as a way of reference (as I mentioned, I once did this).

Set theory, number theory 

Which is larger, the first infinite cardinal, or the value of the Ackermann function evaluated at (5, 5)? Clearly the former. Yet, I for one (and I’m no set theorist) find it a lot easier to wrap my head around the first: I can calculate with it, whereas I have no idea what to do with the A(5,5). People have suggested that this is because there is another sense of large at play with this value of the function – namely it has more structure. “Large” therefore does not only apply spatially – which we should have known anyway, given that this is a math example. However, is this a metaphor?

Cosmology and mereology 

What’s the largest thing that exists? This used to be the use of the term “universe”. These days, people talk about the multiverse, or perhaps “the cosmos”. Regardless of what you call it, this is literally everything. But is literally everything a thing? Some mereologists say that literally pairs of “anything” have a mereological sum; others assert some sort of unity is required. I am unsure, though I am pretty sure that only things (with their properties) and events (separately!) engage in mereological relations. However, returning to physics (if that makes sense – I regard it as a matter of emphasis), this raises questions of temporal parts that are hard to grasp in light of relativity.

Chemistry

My father used to say that from his perspective DNA (and other naturally occurring nucleic acids) and many proteins were “large molecules”. But it occurred to me a while ago that I had been taught that table salt also (to the extent it was pure) consisted of many molecules of sodium chloride. But this is a very different sort of thing (or is it?) because it contains ionic bonds (no electron sharing) rather than covalent bonds of the biomolecules just mentioned. Yet ionic vs. covalent is a matter of degree, maybe? So is it possible that a macroscopic crystal of table salt is in fact a macroscopic molecule, and hence far larger than even the largest of proteins? And then I got even more confused by considering polymers, which are now not even ionic in the relevant way in general. The PVC of a ZipLoc (or whatever), the nylon of a stocking, Plexiglas … could these be …? If so, Democritus’ supposed saying (that an atom could be as big as the cosmos) may yet hold true in the area of combinations and systems of atoms. Yet …

Stellar particle physics

Is a neutron star a large atom? I’ll leave that one by itself.

Biology

My grade 11 biology teacher told us to not worry too much about those monster movies where they show an ant capable of eating a cow whole, as such an insect would never survive. This was in the context of teaching about open and closed circulatory systems. Similarly, Godzilla would be known to be impossible even to Galileo, who arguably invented (or at least popularized) the science of biomechanics, exploring the strength of materials in not only building materials and (siege and civil) engines but also in living things. Godzilla, however, could exist if it had the right limb proportions. So I ask, what’s the largest possible living thing? I’ve heard it claimed that both the redwoods in California (or Endor, to Star Wars fans) and some fungi networks have the title. Which is it? Could there be something still larger? Are the Gaia people right? What then about something still larger than a planet?

Demography and sociology

What’s the largest (by population) city in the world? I learned within a few years of my work at Statistics Canada that different producers of official statistics disagree on what counts as a city boundary, so comparing is hard, when matters like “urban area” rather than the incorporated city (e.g., Pittsburgh) are at stake. The question needs a stipulative definition of “urban area” to make sense and that usually uses a population density cut off. However, this creates other puzzles. For example, the population density obviously falls to zero if the area is small enough, but even with that aside – some cities appear to become very sparse indeed yet resume … so can cities be in pieces? Arguably yes, since (for example) Istanbul or New York are divided by water. So, can this happen by land, like here in Ottawa, where Barrhaven and some of the rest of the city involve a massive drop in population density? Yet there is of course, the political matter – Ottawa-Gatineau is continuous at least transitwise, yet is not one city (similarly, College Park and Washington, for example). Why?

Social cognitive psychology

How do we learn what “large” in “large coffee” amounts to? Of course from examples, but why do we come to expect that they are always more or less (within an order of magnitude) the same size? Has anyone been to a place where the “large” food item was (say) 100x times larger than the one they were used to? I used to say as a joke that the “very large” soups in the typical Vietnamese restaurant here must be the size of a bathtub, since the “mediums” are larger than usual. But even they are really only twice or so the size expected elsewhere. Are we ever corrected when we say something “ordinary” is large or small? I don’t remember anything much like that until my teenage years, when we discussed what a large ski hill was – many of my friends had only been to (at most) Mont St-Anne, which is small relative to Whistler (say) but large for Quebec. There’s no “absolute” large in general, of course.

Cosmology (redux/history of)

In antiquity, it was often thought (by some) that the cosmos must be infinite because one would be confronted by the following dilemma. Suppose one goes to the edge of the finite cosmos and throws a spear. Either the spear continues, and therefore the supposed barrier isn’t, or it stops, but that would require something outside to stop it, so there is therefore something beyond. Most contemporary physicists just dismiss this thought experiment by saying the size of the cosmos cannot be deduced by thinking about it. I’d like to offer the somewhat whimsical idea that there’s a far more interesting task here: what are all the “loopholes” in the thought experiment? Can you think of ways in which something could be very small and yet have “no outside”? (Those who have read Hawking’s A Brief History of Time may remember one of them. There are more.)

Science fiction

In the Star Trek: The Next Generation episode “Remember Me”, Dr. Crusher is forced to ask the ship’s computer, “What is the nature of the universe?” The computer tells her that the universe is a spheroid region some 705 meters in diameter. This seems small, but it is also incorrect if “universe” is taken to mean “everything”. In the context of the show, however, “universe” is taken in the “we expect it to be isolated” (like a hubble volume) sense, but of course for the sake of the plot, these causally isolated regions are also not what they are talking about. All this is by the by, only to ask: If Dr. Crusher were to have asked the computer whether this seemed “large”, what do you think would have happened?