So Irritatingly Indifferent

It is difficult me to get through any paragraph of W. D. Hamilton’s introductions to his collected papers in Narrow Roads of Gene Land without wanting to quote something or other. To think that these are the only writings of his in print besides his scientific papers! As a writer, I believe he is comparable to Huxley and Dawkins, and needless to say he is a better scientist than either of them.

An excerpt from his preface to Vol. 1 of Narrow Roads of Gene Land:

Believing in the explanatory power of evolution by natural selection is like migraine, or perhaps still more like being, as it was in the old days, a ‘wise woman’. The majority of humanity seem to have difficulty in accepting that the ‘oddness’ of such a believer can be real — that is, simply an oddness and nothing else. As the migraine sufferer is suspected of malingering, and the woman who is merely literally wise, of witchcraft, so the evolutionist is always suspected of covert agendas unconnected with reality or the search for truth. In despair over the unending bemusement in friends and relatives and over the stream of articles and books that still pours forth stating Darwinism to be wrong, dead, right except for natural selection, superseded by this stale or ridiculous notion or that (all of which, evidently, the public eagerly buy and read, no matter what the competence of the writer or his knowledge of his evidence); puzzled, in short, by resistance to ideas that seem vastly more obvious and intuitive than, say, relativity or quantum mechanics, which every one accepts blithely or without understanding, the evolution sufferer sometimes comes to believe it must be he who is mistaken… At other times the evolutionist may feel like one of the stranger ‘genetic morphs’ of his own theories — mutant carrier, say, of a fourth intellectual pigment of the retina capable of raising into clear sight patterns of nature and of the human future that are denied to the majority of his fellows, or perhaps just a person bewitched in babyhood to have revealed to him through blind sight, through such X-ray eyes, all the ravishing and foreboding beauty of the world that he now endures.

Since this was written in 1996, clearly he cannot mean that there is still widespread resistance to the idea that evolution explains the diversity of life as we know it, and that man evolved from animals. He is most likely referring to his own travails with opponents of sociobiology.

I could go on discussing Hamilton’s introductions indefinitely, but a tedious problem set beckons, so I will end by quoting the final paragraph of the abovementioned preface:

Having listed friends — with all too many still unmentioned since the net eventually spreads too wide — I am tempted to begin listing my enemies, those who have spurred my work on by being so outstandingly wrong, so critical, and so irritatingly indifferent. But this multitude of patrons, beyond those broad classes already hinted at, had better be left to be implied in the papers themselves.

Wanted textbooks

These may not even exist. At least, I haven’t found them through the library or Google.

1. A text that treats thermodynamics on manifolds. This one by Agricola and Friedrich does briefly, and rather nicely too, but hopefully there is a more comprehensive treatment out there.

2. An introductory text to measure theory that doesn’t involve slogging through sigma algebra beforehand. Again, I have found an example of this, namely Kolmogorov and Fomin’s Measure, Lebesgue Integrals, and Hilbert Space, but they start out with generalised rectangles, which is rather specific to R^n and not as general as the approach we are taking in class. Not that I don’t want to learn sigma algebra eventually. It’s just not something I can fit in and learn everything else I’m supposed to learn as well in the next five weeks.

Leibniz’s Space/Time Relativism

From Leibniz’s Fourth Paper of the Leibniz-Clarke Correspondence:

6. To suppose two things indiscernible, is to suppose the same thing under two names. And therefore to suppose that the universe could have had at first another position of time and place, than that which it actually had; and yet that all parts of the universe should have had the same situation among themselves, as that which they actually had; such a supposition, I say, is an impossible fiction.

Clarke, who in this correspondence is defending Newton’s ideas of absolute space and time, replies:

Two things, by being exactly alike, do not cease to be two. The parts of time, are as exactly like to each other, as those of space: yet two points of time, are not the same point of time, nor are they two names of only the same point of time.

Until I read this, I had been gravitating towards Leibniz’s relativism, the idea of some sort of absolute grid of coordinates of space and time seeming contrived and unnecessary. Clarke’s reply, though, alerted me to some potential problems in Leibniz’s stance.

Leibniz maintains that the only thing determining spatial relations is the configurations of objects relative to one another. Thus, for example, a universe that in Newton’s absolute coordinates is a mirror image of ours is, for Leibniz, really the same universe. This does not seem particularly revolutionary when applied to space (I shall postpone all talk about the bucket experiment until I’ve properly understood Mach’s response to it). However, there does seem to be something strange about Leibniz’s relativism when applied to time. By his definition, a different moment of time corresponds to a different configuration of bodies in the universe. Leibniz claims that while two abstract units of time may be identical to each other, two real units of time will alway be different (5th paper, Section 27). Presumably, then, a cyclic universe with a closed spacetime curve will, for Leibniz, simply be repeatedly going through the same moments of time, whatever that means. This is not particularly problematic until we consider a non-deterministic universe. Conceivably, such a universe could undergo a dynamical trajectory from A to B (taking A and B to be points on its manifold of states), then at some later time return to point A, only to proceed this time to B’ instead of B. To Leibniz, both the moments of time in which the universe is in state A are really the same moment of time. Yet, intuitively, if the universe subsequently moves to B’ instead of B, we would be tempted to call them different moments of time. The crux is whether one should consider moments of time preceding and succeeding the moment in question, in addition to the spatial configuration of the universe at that moment, when distinguishing moments of time from one another. If one does not (and Leibniz doesn’t seem to, judging by his complete lack of references to neighbouring moments of time when talking about how particular “points” of space or time are defined), then Leibniz’s conception of time would be very strange indeed in a non-deterministic universe.

It is also not clear if Leibniz considers the rates of change of spatial relations (i.e. velocities) to be part of the configuration of a universe. He would have to, it seems, in order to avoid absurdities. If he considered only spatial configuration, then, using arrows to represent velocity and A and B to represent different objects, one could have
→ A
← B
be the “same thing” as
← A
← B
although the two are clearly different in physically crucial aspects.

The Two Cultures

Today’s physics colloquium was by Charles Falco, of the Hockney-Falco thesis, a rather notorious hypothesis in the art history community. The thesis, in a nutshell, is that some 15th century Flemish painters (Campin, Jan van Eyck, de Flandes amongst others) and their artistic descendants sometimes used concave mirrors to reflect parts of the subjects they painted onto their canvasses in order to help them paint them more accurately. I emphasise “parts of” because ignoring that part of their hypothesis has led to many wrong-headed arguments against their thesis, such as those here which discuss the difficulties artists would have had trying to draw self-portraits, imaginary objects, etc. with concave mirrors. Falco made it clear that he is not saying that all drawings were done with mirrors, and in fact most of the examples he showed were of drawings where only one small intricate part was done with mirrors, the rest displaying no tell-tale signs of inconsistent magnification. Therefore the problems of painting “inverted” frescoes if one uses mirrors and so on are quite irrelevant — if the artist had to paint a fresco or a self-portrait he just wouldn’t use a mirror.

Most of the audience, being physicists, found the evidence rather convincing, although everyone was highly amused when an experimentalist launched a barrage of questions at him about whether the artists used the technique consistently in all their works. Nonetheless, consistent with Falco’s accounts of his talks to different audiences (artists, scientists or art historians), most of the questions asked concerned historical evidence. In contrast, the questions asked by an art historian in the audience were, well, not really questions, but more of a lecture to the audience about how the phenomenon of art could not be “reduced” to optics. Falco, of course, agreed, and so would any sane person, but she went on and on about how Falco seemed to be presenting his research as though it was a reduction of art, and she was concerned that “especially for an audience like this” art should not be presented that way. People had already been sneaking dubious looks at one another as she went on at length about the ineffable, non-reducible aspect of the artistic experience. By the time she implicitly accused the audience of being crude uncultured reductionists for the fifth time, audible sniggers were bubbling up throughout the lecture hall. Falco somehow managed to pacify her and the session moved on to more interesting questions concerning historical evidence, but she wasn’t finished yet and had to get in another dig at the audience as the very last “question”.

The unreasonable monopoly of Riemann integrals

The tradition of defining integration in all introductory calculus and analysis classes as a limit of Riemann sums is, in Bob Geroch’s words, a “horrible historical accident”. This was by way of an introduction to Lebesgue integration, which is a more reasonable way to define integration because it does not depend on the ordering of the real numbers. Instead of choosing arbitrary rectangles or intervals in the domain to define Riemann sums, one instead chooses the appropriate sets of intervals (or rectangles) based on the range of the function. This way the sets over which we calculate the volumes that we take a limit of are defined by the characteristics of the function and not by the quirks of the real number line, which are irrelevant to the integral (since one can translate the function any distance along the real line with no effect on the value of the integral). All we need is a suitable notion of the size of the sets in the domain delimited by the chosen ranges of the function, and we can happily evaluate the integrals of strange cats like the Dirichlet function, which the Riemann integral, with its nitpicky concern over where on the real number line its domain is, dismisses as undefined.

In Geroch’s typically colourful language:

I have no idea why they teach people integration using Riemann integrals. Riemann must have had a very good press secretary.

The parallel between using manifolds instead of R^n, and using Lebesgue integrals instead of Riemann integrals, is obvious. In both cases one is abandoning meaningless coordinate systems for something more general and essential.

Darwin and His Bulldog

Thank god Darwin was forced, by Wallace’s letter from Malaya, to publish The Origin of Species in a rush. If he had more time it might have been written in the style of The Descent of Man, which I find an absolute pain to read. Every point he makes is embellished with a forest of tangential remarks, which would not be a problem if not for the fact that he did not organise them clearly enough to allow readers to see the main point without having to rummage through his flowery discursive language. The Origin makes its points far more efficiently, without losing anything in terms of beauty or persuasiveness. It is this very efficiency which makes it so much more elegant than The Descent.

Moving on to Huxley after The Descent is positively thrilling. In the essay Evolution and Ethics, he spends the first ten paragraphs or so lyrically expounding on how the beanstalk’s evolution in Jack and the Beanstalk is representative of evolutionary processes in the cosmos. Complete fluff in terms of making an intellectual point, but his prose flows so smoothly that one does not curse him for wasting one’s time. Darwin, by contrast, will attempt to use flowery descriptive language, but in comparison with Huxley, he sounds like he’s trying to talk with his mouth full and having to take a break every now and then to breathe or to chew some particularly difficult bit. One of my favourite bits in Huxley’s essay:

The one supreme, hegemonic, faculty, which constitutes the essential “nature” of man, is most nearly represented by that which, in the language of a later philosophy, has been called the pure reason. It is this “nature” which holds up the ideal of the supreme good and demands absolute submission of the will to its behests. It is this which commands all men to love one another, to return good for evil, to regard one another as fellow-citizens of one great state. Indeed, seeing that the progress towards perfection of a civilized state, or polity, depends on the obedience of its members to these commands, the Stoics sometimes termed the pure reason the “political” nature. Unfortunately, the sense of the adjective has undergone so much modification, that the application of it to that which commands the sacrifice of self to the common good would now sound almost grotesque.

One of the starting points of last week’s class discussion was whether we thought Darwin or Huxley was more intelligent. The split went about 2/3s Darwin and 1/3 Huxley. I must admit that I voted for Darwin only because I had read The Origin; I had not been particularly impressed by what I’d read in The Descent. If Huxley had indeed come up with the idea of epiphenomenalism on his own, I would be impressed, but I have not so far found any evidence that he was not just describing a theory that was already in circulation amongst his fellow biologists and philosophers. His famous essay, in which he likens consciousness to the whistle on the train that does nothing to help the train move, is well written as always, but relies on dubious points about the behaviour of lobotomized frogs and possibly deliberate misinterpretations of what Descartes said about the role of the nervous system. Darwin, on the other hand, builds up his points in a more measured, ponderous, but ultimately rigorous manner. Huxley gives us flashes of unsupported brilliance; Darwin reveals the unity of his theory step by step, so that the brilliance of his proposal is only fully revealed after the whole structure has been meticulously erected. That, I suppose, is why Huxley was ultimately only a dog to his master.

Russell on Zeno

There is something about Bertrand Russell’s style of philosophising that grates on me. His proposed solutions to Zeno’s paradoxes, in “The Problem of Infinity Considered Historically“, can be summed up by “modern mathematics has solved everything”. However, he commits such a logical blooper in the process of this smug dismissal of Zeno’s reliance on “prejudices instilled by the arithmetic which is learnt in childhood” that I wonder if I am misinterpreting the blooper and hence being overly smug about it.

In his interpretation of the paradoxes, he claims that the paradoxes may be escaped by, amongst other options, maintaining that any finite interval of space or time consists of an infinite number of points and instants. In the same paragraph, he then writes “as we saw, the difficulties can also be met if infinite numbers are admissible.” He then goes into a long spiel on how there is no paradox if we allow for an infinite number of numbers in each finite interval of numbers.

The obvious blooper is that the admissibility of an infinity of numbers in a finite mathematical interval does not imply anything about the physical question of whether a finite interval of space or time can consist of an infinite number of points. Physical reality need not reflect all the abstract entities that mathematics invents. In any case, Zeno himself must have been perfectly aware that a finite interval of substance can contain an infinite number of points, for otherwise he would not have been able to formulate his paradox of plurality. And, on closer thought, Zeno must have needed this awareness in any case to even think that the Dichotomy was paradoxical, for the paradox lies in Achilles having to traverse an infinite number of points between two points enclosing a finite interval. If Zeno had, as Russell claims, only a schoolboy’s notion of arithmetic, and did not think there could be an infinite number of points between 0 and 1, then he could not think that there is anything paradoxical in the Dichotomy, because obviously a finite number of points can be traversed in a finite amount of time.

After this blooper, Russell digs himself into a deeper hole when he classifies some difficulties of infinity as “sham”. He speaks of the idea that the infinite has no end as a confusion that philosophers who take the sham difficulties of infinity seriously have. He never explicitly says what he means by “end”, but since he does say that

The series of instants from any earlier one to any later one (both included) is infinite, but has two ends; the series of instants from the beginning of time to the present moment has one end, but is infinite.

In this quote, he clearly means “end” to be a boundary point delimiting the set of instants. The existence of such “ends” of infinite series shows, he thinks, that it is not the case that “there cannot be anything beyond the whole of an unending series”. He then accuses Zeno of relying on this principle in his paradoxes.

Once again, I think it is clear that Zeno does understand that there can be a thing beyond the whole of an unending series. The crux of the paradox is the clash between there being a thing beyond the whole of an unending series (namely the end of the finite interval that Achilles is supposed to traverse) and Achilles having to do “infinitely many things” to reach that thing. Thus, in order to even think that it is a paradox, Zeno has to accept that there can be a thing beyond the whole of an unending series.

After this poorly aimed attack, Russell finally gets to a point that might be a more substantive “solution” to the paradoxes. He blames the paradoxes on the notion of counting, and counters this by arguing that one need not be able to pass the components of an infinite collection one by one for the collection to exist. This, I think, lies closer to the crux of the paradoxes. Max Black, too, makes use of the notion of counting to argue that the impossibility of the supertask he describes implies the impossibility of all supertasks, because they are all “like counting”. More on this later.

From the notebook: Die Fledermaus at the Lyric Opera

Jan 19 2007, Lyric Opera of Chicago, conductor Asher Fisch
Reference recordings:
Bayerische Staatsoper/Carlos Kleiber
Wiener Staatsoper/Andre Previn

Eisenstein (Bo Skovhus): Started out weak and unconvincing, got better towards the end. Tends to be drowned out in duets. A bit disappointed in the much-hyped Skovhus really. On the other hand, this is, I think, a difficult role to play convincingly. I’m not entirely convinced by the Eisensteins in the Previn or Kleiber recordings either. Skovhus was probably about the same level as Wolfgang Brendel in Previn’s, Eberhard Waechter in Kleiber’s is somewhat better (good acting).

Alfred (Bonaventura Bottone): I disliked the “Italian” feel he imbued his character with (from the flamboyant gestures to the hairstyle). Yes, Alfred should be flamboyant, but the Alfreds I’ve heard/seen elsewhere sound more sincere than Bottone did. One was never entirely convinced by his love for Rosalinde, and I probably had more sympathy for Eisenstein in this production. Decent singing.

Adele (Marlis Petersen): Outstanding. Best Adele I’ve seen or heard. Improvisations (in song to Frank, and in conversational bits) worked surprisingly well.

Rosalinde (Andrea Rost): So-so. more straight-laced, less playful, less intelligent, less devious than Coburn in Kleiber’s production. Not alpha woman — unlike Coburn.

Frank (Andrew Shore): pretty convincing and in-character.

Falke (Martin Gantner): I think I prefer Wolfgang Brendel in Kleiber’s. This one seemed a bit dull; not flamboyant enough

Orlofsky (Alice Coote): Fassbaender ueber alles! To be fair it was Alice Coote’s first time in this role, but she was rather dull and did not exude smug arrogance as well as Fassbaender does.

Frosch (Fred Wellisch): As usual. I’m not a fan of the Frosch role.

Orchestra: Sound was rather muffled from my position in the upper balcony, but not their fault. Also a bit too straight-laced for me (as expected, did not master Viennese rubato).

Dancers: Excellent. I usually lose my concentration at this point in the opera, but this was different. Not thunder-and-lightning polka, which is nice. Series of other polkas instead, including the “Hungarian” one (appropriately for the mystery Hungarian countess!) Dancers dressed as Harlequins, with comic yet graceful movements and crowd-pleasing tricks in abundance (tossing and catching one another, etc.)

Other comments:

I still don’t understand why Die Zauberfloete is considered more “serious” art than Die Fledermaus. Perhaps I’m blinded by my distaste for cute Mozartean tunes and my embarrassing susceptibility to Viennese waltzes and polkas.

Overall, amazed at how I enjoyed it, given how near perfection the Kleiber production is.

For the first time, realised that the triplet singing about a reunion just before Eisenstein leaves for the party is an ironic reference to their meeting at the party!

Translations sacrificed fidelity to the meaning for humour and English rhyming. In particular, I thought the translation of “Glueckich ist, wer vergisst/Was doch nicht zu aendern ist” was mangled to no positive effect. It was translated as something like “he who is wise accepts things as they are”, which has a much flatter and duller feel than “happy is he who does not worry about what he cannot change”. Since when did “glueckich” have an implication of wisdom, anyway?

Adele stole the show (by far the loudest applause for her), and this I find rather unhealthy. I think the story is much more balanced with Rosalinde is the central strong character. Adele being the star of the show imbues it with a rather lopsided feel. Eisenstein as the star would be fine too. Just not anyone else besides Eisenstein and Rosalinde.

Interesting that DF was first performed at the Lyric only in 1982. Rather late, no?

Listening to the overture after coming home from the performance, I realised for the first time that the opening notes are from the “revelation” scene in the jail.

Some jokes that appear in both the Kleiber production and this production:

1. Frosch attempting to stick Frank’s hat on the wall, several times unsuccessfully, and goggling in astonishment when he finally manages to do it on his umpteenth drunken stumble out of the jail.

2. Adele, upon hearing Eisenstein tell the assembled company that she looks like his parlourmaid, exclaiming “I’m going to have my margarine“. I suppose this is a pun, but on what, I have no idea.

3. Are there stage directions in the libretto that Eisenstein must engage in some vaguely homosexual bonding behaviour with Falke after he agrees to go to the supper? That part has never seemed particularly funny to me.

Tactful Darwin

From The Descent of Man, Chapter 2:

The question [of whether man is innately endowed with a belief in an ominpotent God] is of course wholly distinct from that higher one, whether there exists a Creator and Ruler of the universe; and this has been answered in the affirmative by the highest intellects that have ever lived.

The careful phrasing, in which Darwin avoids saying that he believes that is the correct answer, allows him to avoid either revealing or contradicting his own agnosticism.

Zeno’s Arrow Paradox

This sounds straightforward on a casual hearing or reading, but once I sat down to parse its logical structure, I just couldn’t get it. The clearest description of it I’ve found so far is the one in Wesley Salmon’s introduction to this book:

In this paradox, Zeno argues that an arrow in flight is always at rest. At any given instant, he claims, the arrow is where it is, occupying a portion of space equal to itself. During the instant it cannot move, for that would require the instant to have parts, and an instant is by definition a minimal and indivisible element of time. If the arrow did move during the instant it would have to be in one place at one part of the instant, and in a different place at another part of the instant. Moreover, for the arrow to move during the instant would require that during the instant it must occupy a space larger than itself, for otherwise it has no room to move. As Russell says, “It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever”.

It seems that the following is sufficient for the paradox:
P1. An instant is a minimal and indivisible unit of time, i.e. it has no parts.
P2. If an arrow moves in an instant of time, it would have to be at different places at different parts of the instance.
C1. Therefore the arrow cannot move in an instant of time.

Everything after “Moreover” seems superfluous. Moreover, I have no idea how to make sense of that part. What could Zeno mean by an arrow occupying, or not occupying, a space equal to itself? It seems to make a mockery of the concept of occupancy to speak of something not occupying a space equal to itself, as though there is some sort of bubble around its physical shape that it occupies even though its physical shape does not completely fill that bubble. It seems to me that by the definition of occupancy, a body must occupy a space equal to itself, whether it is in motion or at rest. At any rate, how would occupying a space “larger than itself” allow a body room to move? It can hardly move into the space that it occupies, so having it occupy a space larger than itself doesn’t seem to give it any extra room. In any case, does Zeno expect that objects expand when they are in motion so as to give themselves room to move? I suspect this concept of occupancy, as it was originally intended, probably involves some inconceivably strange physics.

Did Zeno really mean that a body must occupy a space equal to itself? Aristotle’s account of the paradox in the Physics is:

Zeno’s reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.

Since I don’t have a copy of the Physics at hand, this was from a free online edition. I wonder if other editions have a clearer phrase than “an equal space”. It’s not clear to me that “an equal space” necessarily means “a space equal to the object itself”. Not that I can think of something else that it could plausibly mean, but as I explained, “a space equal to the object itself” is not plausible either.

Back, then, to what I consider the “main” part of the paradox. My first reaction, or at least my first reaction after reading the paradox closely, is that there is nothing paradoxical about C1. An instant of time has zero length, so of course an arrow does not move in an instant of time. Motion only applies to periods of time, not instants. It would not, for example, make sense to speak of an object that is at rest at all times except t=1, because that physical situation would be indistinguishable from the case when it is at rest at all times including t=1.

I’m cheating. There’s a further component to the argument. Namely:

P3: If the arrow cannot move in any instant of time, then the arrow is at rest at all times.
C2: Therefore the arrow is at rest at all times.

The obvious strategy is to attack P3. One way is to deny that there are such things as instants of time — in short, to deny an atomistic view of time. I can’t imagine how Zeno could salvage his paradox once it is allowed that time is infinitely divisible (of course, this itself he thinks is paradoxical, as he tries to show in his other paradoxes). If instants are denied, then nothing we say about these hypothetical instants can lead us to conclude anything about time in general.

P3 would also be undermined if it were true that motion is a concept that can only be applied to periods of time, per what I argued above. While it is true that we speak of motion at a certain moment, that is arguably a quirk of our language, and need not imply that we mean that an the object moves within the moment in question. What we mean when we say that is that at an arbitrarily small time interval after the moment in question, if we observe the object again, we will find it to be in a different position from where it was at that moment in question. The only way in which we detect bodies to be in motion is by observing the change in their positions at different moments of time, and hence over a non-zero period of time.

It is notable that even in modern calculus, all talk of rates of change is carefully formulated so as not to deal directly with change confined to one point on a function. The standard epsilon proof of a limit (and a derivative at a point is simply a special kind of limit) carefully avoids saying anything about what actually happens at the limit point, instead only referring to what happens as one gets arbitrarily close to the limit. This is why I don’t think calculus offers a solution to Zeno’s other paradoxes concerning the summation of an infinite series — it is just as clueless about what happens at the limit point as Zeno was. But that’s a topic for another post.