Thursday, October 16, 2014

Zeno's Paradoxes


My opinion on Zeno's paradoxes is that they are not real paradoxes, that they are not deep, and that they are in need of dissolution, not resolution. Usually math teachers in high school, and sometimes middle school, present a narrative to their students in which Zeno's paradoxes were long-standing mysteries that differential and integral calculus finally solved. This narrative is echoed in various recesses of pop-math culture. But it isn't true.

The paradox of the arrow, for instance, is claimed to be resolved by differential calculus and the concept of instantaneous velocity. However, it is actually dissolved if we carefully look at what it is that instantaneous velocity means—and what motion at all means for that matter. We say that an object is in motion if it has different locations at different times. It is at rest if it does not. Of course, in defining rest, we must be careful that the object in question may return to a location; it might be better to say that an object is at rest between times A and B if its location remains unchanged over the time interval (A,B). But the point is that saying an object is in motion, or that an object is at rest, requires a consideration of its location over a finite interval of time. It is simply incoherent to say that an object is motionless at a single instant because it's at the same location for that 'entire instant'—yet this is exactly what Zeno does.

Instantaneous velocity, to be sure, actually involves comparing the location of an object at different times, and considering its average velocity over the time interval delimited by these distinct moments. One of the moments is left fixed, and the other varied, so that it is taken arbitrarily close to the fixed time. If the average velocity for this arbitrarily small span of time has a limit, this is called the instantaneous velocity at the fixed moment. However, introducing the concept of instantaneous velocity does not resolve the arrow paradox, but rather carefully avoids making the mistake that Zeno made of comparing an object's location at a single instant of time to itself. But one does not need to correctly conceptualize instantaneous velocity to avoid making Zeno's mistake.

By the way, this dissolution of the paradox is quite ancient; Aristotle offered it in his Physics. We did not need to wait for Newton (as the narrative usually goes) to be confident that things can move.

The other popular paradox has two forms, though they're really variations on the same theme: Achilles and the tortoise, or the dichotomy paradox. Essentially, either can be put as: there is an infinite number of ordered tasks, A, B, C, ..., each (aside from A) a successor to another, and then there is a single event Ω which occurs after all the others. A, B, C, ... are Achilles catching up to where the tortoise previously was, or Achilles being a distance 1 m, 1/2 m, 1/4 m, ... from the finish line of a race; Ω is Achilles passing the tortoise or finishing the race. The claim is that Ω can never happen, since an infinite number of events must occur first.

This has always struck me as a plain non-sequitur. Why the conclusion should follow is never cogently argued, and it's difficult to offer a counter-argument when no real argument is presented. Perhaps it should suffice to point out that the scenario outlined above is mathematically consistent, in case there was any doubt. We can associate the events with numbers: 1, 2, 3, ... aside from Ω, which is just called Ω. The numbers themselves are ordered the usual way, and we have a rule that for any number n, n < Ω.

The ordering relation < (called "happens before") has to satisfy two axioms, which are already satisfied for the numbers. First, it is trichotomous: exactly one of A < B, B < A, or A=B is true. If we consider n and Ω, it is clear that only n < Ω is true, by definition; and if we consider Ω alone then only Ω=Ω is true. Second, the ordering relation is transitive: if A < B and B < C then A < C. This has no implications for a triple with two or more identical elements; and if we consider a triple of distinct quantities (m,n,Ω), with m < n, then the fact that n < Ω requires m < Ω, which is also true by definition. We have consistency.

In fact, the collection of all points on a line segment—such as the last meter of Achilles's race—obeys a strict ordering relation, with absolutely none of the points having an immediate successor or predecessor. Perhaps, for some reason, Zeno had a problem with this. Of course, I can only speculate what he was thinking, but he may well, like Aristotle, have over-anthropomorphized nature, and imagined that it must operate as we humans do in our daily lives: by executing a discrete collection of tasks, one-after-another, each punctuated by the next. But nature does not work like this.

This second paradox is usually claimed to be resolved by integral calculus, but it really has nothing to do with time elapsed or distance traversed. What Zeno had a problem with was one event occurring after infinitely many others, so the relevant mathematics is ordering relations, not integrals. And the 'paradox' turns out to be a transparently bad case of sloppy thinking.

There is another paradox that involves rows of bodies passing each other, but I can't make out what it's trying to say. Nobody really talks about it though, so it probably isn't worth addressing.

Wednesday, October 15, 2014

Quantum physics and realism

I'm going to get things started here by sharing a note I wrote in the past, on quantum mechanics and realism about the physical world. This was motivated at the time by several arguments I was having with folks who rejected the Copenhagen interpretation of quantum mechanics without really understanding it. To be sure, it might be that the "Copenhagen interpretation" is not a single well-defined set of beliefs, since several self-proclaimed adherents of it do differ in a few beliefs. However, I do think there are several principles that its adherents hold in common, and that attacks on the Copenhagen interpretation tend to be attacks on a caricature which is not believed by anyone.

The approach I've taken is to take physical theories as seriously as possible. There are two theories in particular that concern me. The first is quantum mechanics itself, which I take to be a complete theory, needing neither to be supplemented (pace Bell), nor to have any piece of it, such as state reduction and the Born rule, removed (pace Everett). The other is special relativity, which succinctly says that a physical theory must be covariant under Lorentz transformations, and which I take to mean that the objective aspects of a physical theory's ontology must be Lorentz invariant.

For the interpretation of quantum mechanics to be Lorentz invariant, the stateーa ray in Hilbert space if you've got a pure state, or a density matrix in generalーcannot be a physically real thing, inhering in the system; nor can reduction of the state (also called collapse of the wave function) be a physically real thing. The reason for this is that the state reduction of an entangled, spacelike separated pair of objects does not admit a Lorentz invariant description. In some reference frame, state reduction of one object would cause the state reduction of the other; in some other reference frame, the second would cause the state reduction of the first. This is a non-invariant, and thus unacceptable description of causation. States and state reduction are part of the theory, to be sure, but what this means is that the state must be subjective.

So this makes the first point I want to make:

1. Physical states are subjective

Since states are often associated with values of some measurable quantity, such as spin or polarization, this also means that the values of these quantities are subjective. In a Bell/EPR-type experiment with entangled photons, prepared in a state like |HH>+|VV> (both either horizontally or vertically polarized), this means that Alice can find her photon to be horizontally polarized while, for Bob, there is no matter of fact about how his photon is polarized. To be sure, Alice knows Bob will find his photon to be horizontally polarized, and cannot find his photon to be vertically polarized; and if he measures his photon polarization in the {H,V} basis, his determination will be consistent with Alice's expectation.

It's important to notice that subjectivity does not entail that contradictions will occur. Of course, if physical states were objective and observer-independent, then this would entail that contradictions do not occur. But it's a logical fallacy (denying the antecedent) to then say that subjectivity will allow contradictions. In fact, the formalism of quantum mechanics prevents contradictions from occurring, e.g. between Alice's expectation and Bob's finding. From Alice's point of view, Bob's photon is horizontally polarized, so he finds it to be horizontally polarized. From Bob's point of view, Alice's measuring device, and even Alice herself can be considered as part of a large quantum mechanical system that becomes entangled with the photon being measured. So after Alice performs her measurement, the state of the total system (as ascribed by Bob) is |HH;Alice finds H>+|VV;Alice finds V>. He either measures his photon to find H, and then speaks to Alice to find that she too measured H; or conversely, he measures V, and then speaks to Alice to find that she too measured V. There is no possibility, in this scenario, that Bob would measure H and find Alice to have measured V.

Now, I should say something about time ordering and Lorentz invariance of the description, in anticipation of a possible objection. If Bob and Alice are spacelike separated when they make their polarization measurements, then the order in which the measurements occur is frame-dependent. So rather than the steps:

|HH>+|VV> --> |HH;Alice finds H>+|VV;Alice finds V> --> |HH;Alice finds H>,

corresponding to Alice making her measurement first, it may happen in another frame of reference that

|HH>+|VV> --> |HH> --> |HH;Alice finds H>.

This appears at first to violate Lorentz invariance, since the description of how the state changes with time is frame-dependent. But this comes from the mistake of reifying the state. Lorentz invariance is a symmetry that observer-independent descriptions should satisfy. To be sure, even in a Lorentz invariant theory, since the temporal order of spacelike separated events is frame-dependent, "A occurred, and then B" is not a Lorentz invariant statement. In a sense, it's too a subjective rather than an objective statement since it is attached to a particular frame of reference. And this is really the point: descriptions that are not Lorentz invariant are subjective. Spacelike separation of events is an objective relationship between them, but their temporal ordering is not.

Likewise, what exactly the state is of a physical system is also subjective. However, just as special relativity ensures consistency between frame-dependent measurements (such as of energy or time elapsed) through Lorentz transformations, quantum mechanics ensures consistency between observer-dependent assignment of states. The consistency partly owes to entanglement, as I described above. But there's also another important rule, which is that the temporal order in which independent (especially spacelike separated) measurements occur should have no bearing on the outcomes. This has been translated into a formal rule. State reduction is effected through a projection: there is a matrix E which sends any photon state to |H>, for instance, or at least |H> times the length of the state vector in the direction of |H>. (It's possible for this to be zero, but this means that the probability of finding the photon to be horizontally polarized is zero. In particular,, E|V>=0.) Bob and Alice have their own set of projection matrices corresponding to possible measurement outcomes. The independence from measurement order means that any of Alice's matrices must commute with all of Bob's; the order in which they're applied produces the same result.

The Lorentz-invariant, observer-independent description would be along the lines: Alice measures her photon polarization in spacetime region A, and Bob measures his photon polarization in spacetime region B. You can then use Alice's and Bob's projection operators together to determine joint probabilities for their measurements. You'll find that the probability that Alice and Bob find H and V, or V and H, is zero; the probability both find H or both find V is 50%. The state reduction occurs for a particular observer, in a particular frame of reference; and since the state is subjective, the way that it changes doesn't have to admit a Lorentz invariant description.

However, despite the ultimate subjectiveness of physical states...

2. The world is pragmatically real

One of the major objections to the Copenhagen interpretation of quantum mechanics was, I believe, posed by Albert Einstein: that he would like to think that the moon is there when he is not looking. To be sure, I agree with this, and would say that the moon is indeed there when we are not looking. However, I would call this a subjective statement, in the sense that I would call statements about the states of microscopic physical objects subjective. That is to say, where the moon is (and that the moon exists at all) is true for us, rather than true in some absolute, observer-independent way. However, because quantum mechanics admits no contradiction between observers' experiences, we will all agree on where the moon is when we do look.

Moreover, for an object as large as the moon, subject to conditions that do not bring about chaos (n.b. there's a controversy about the classical/quantum transition when chaos is involved, but that's far too technical for me to deal with here), the trajectory is almost exactly as classical physics would predict. This means that we can say, to quite reasonable certainty, where the moon will be when we decide to look. For all practical purposes, this means we can say the moon is there when we're not looking. When we do look, we'll only affirm this claim. This is in stark contrast to microscopic objects such as electrons, where the indeterminacy in its physical properties is too great to allow such a description. There is no particular place around the nucleus of a hydrogen atom that we can expect to find its electron; we can at least say that its electron is somewhere, and even give a probability density, but we can say no more. For the moon, howeverーit will be exactly in the sky that any astronomy-savvy person (or an online moon calendar or sky map) will tell you.

While formally subjective, the moon's position is effectivelyーfor all practical purposes, anywayーobjective. The same can be said of anything else that falls within the effective domain of applicability of classical physics. Most importantly, I think this allows pragmatic realism about the objects of most of the natural sciences. We can say that the common cold is really caused by a virus, for instance, rather than having to consider the virus as an instrumental, unreal tool that summarizes the observable symptoms of the common cold. We can say that Earth really has magma and an iron core under its crust, rather than these being purely theoretical fantasies. And so on.

Consider Berkley's tree that falls in a forest without anyone to hear the sound of it crashing into the ground. One may say that it does not make a sound because sound is a sensation, and no-one is present to experience sound. However, if someone were there, and this person's ability to perceive sound were not impaired in any way, they certain would have heard the tree fall. We may well say, by fiat, that a sound occurs whenever, if someone who could experience sound were present, they would hear a sound. We're defining the presence of sound, in this case, by a counterfactual. Likewise, while position in quantum mechanics is the result of a measurement rather than something that inheres in a physical system, we can say that if a measurement of an object's location will definitely result in a position q, then we may as well say that the object is located at q, even though this is fiat. It's a pragmatic definition, and unlike the case of sound or other sensations, the means of measuring position is standardized enough that there will be no arguments analogous to whether a turquoise stone is blue or green.

Another related objection is that if our description of the world is observer-dependent, then there can be no description of what the world was like before observers. This is wrong. There can be no description of the world without observers to describe it, but anyone's description can include details of what went on before the one's own life, and indeed before any observers existed. The world at large is pragmatically real, and a realist description of the macroscopic world does allow a historical narrative that includes a Hadean eon, a giant impact, a Cambrian explosion, etc. It's our history in that it's true for us, as observers, in a manner that's ultimately subjective but pragmatically objective.

It's not right to say that the world existed in some weird superposition or statistical mixture until an observer appeared somewhere inside of it, suddenly collapsing the whole thing. To say as much commits the error of reifying the state, since a state can only be ascribed to a system by an observer, based on the information available to them. It's also not right to say that the world did not exist before any observers did, and suddenly came into existence with the first observer. To say as much contradicts the pragmatically realist historical narrative of the world (as explicated by our best geological and cosmological knowledge) that is consistent with an observer-dependent description of the world (since no principle forbids a subject from describing the past).

This leads me to my last point:

3. Consciousness is irrelevant

At the very least, there's nothing in the formalism of quantum mechanics that tells us who or what qualifies as an observer. The theory allows anything that is capable of extracting information from physical systems and constructing a classical description from them to act as an observer, however. Anything that can tell where the moon is qualifies as an observer as far as the moon's location is concerned. Machines and other unconscious objects may qualify as observers, too.

Consciousness was speculated to effect state reduction by some of QM's founders, most notably John von Neumann and Eugene Wigner. They believed that the physical state is a physically real thing, and that state reduction is a physically real process, so they needed to pick out special circumstances when state reduction actually happened. They also suspected that consciousness did not admit a physical description, so it wouldn't make sense to describe a conscious being as existing in a physically real superposition (or statistical mixture) of distinct conscious states. So they proposed that state reduction occurs when at last a physical system interacts with a conscious person. To be sure, it was on the face of it a reasonable conjecture, but probably wrong. I want to be sure I say, however, that the association of quantum mechanics with consciousness was not so asinine as "quantum mechanics and consciousness are both mysterious, so maybe they're related," and that it's both wrong and uncharitable to suggest that anyone proposing a relationship between the two has such shallow motivation. (Unless they're Eben Alexander or Deepak Chopra.)

Anyway, there are two reasons I'd give for rejecting this approach. The first is that, as I described above, state reduction does not admit a Lorentz invariant description. If descriptions that are not Lorentz invariant are subjective, then the state (and its reduction) are subjective anyway, so there is no need to find a special point at which state reduction really happens. The second is that there really is no reason to believe that conscious beings don't admit physical description. To be sure, conscious experience is described in an entirely different language than physical processes, but that's not sufficient reason to believe that consciousness is made of unphysical soul stuff. I would point to Daniel Dennett's writings on consciousness (especially Consciousness Explained) if you'd like to read something on this topic, but for me it will suffice to say that I don't think consciousness has any place in physics, and that it has nothing to do with quantum mechanics.

And that is how the cookie crumbles.

Hello, cruel world!

Behold, I have made a blog! This was done at the suggestion of one Kaveh Mousavi, whose own blog is On the Margin of Error. I plan to post things here occasionally, but will not take blogging too seriously. I've attempted a blog before with too serious and overwhelming a plan, but perhaps a few simple, standalone posts just when I feel like it will fare better.