I was just talking to Michael Weisberg, who is also visiting ANU currently, and he pointed me to this video showing a counterintuitive physics demonstration. I had seen the video before, so we started discussing how it might work. He pointed me to the explanation videos that the author of that video made, but they don’t really clarify things very much. When we tried to work it out ourselves, we came to the conclusion that it had to be impossible (unless there was slippage between the wheels or one of the surfaces) – until I realized one feature of the cart that I hadn’t noticed before.
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Wow, it’s been about four months since I’ve posted here! Anyway, I’ll try not to continue that pattern in the future.
In the February issue of the Notices of the American Mathematical Society, John Conway and Simon Kochen have a paper explaining their “free will theorem”, which I believe strengthens it slightly from earlier versions. I had heard rumors of this theorem for a year or two, but had never seen more than an abstract, or a discussion in the popular media, so I couldn’t see the actual content of the theorem in order to see what it really says. So this paper was nice to see.
It’s important to see the actual statement, because the way it’s been summarized is basically as they put it, “It asserts, roughly, that if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity.” Which is a tendentious statement, to say the least, given that it uses a term like “free will”.
Here is the full statement of the theorem, from their paper:
SPIN Axiom: Measurements of the squared (components of) spin of a spin 1 particle in three orthogonal directions always give the answers 1, 0, 1 in some order.
The TWIN Axiom: For twinned spin 1 particles, suppose experimenter A performs a triple experiment of measuring the squared spin component of particle a in three orthogonal directions x, y, z, while experimenter B measures the twinned particle b in one direction, w . Then if w happens to be in the same direction as one of x, y, z, experimenter B’s measurement will necessarily yield the same answer as the corresponding measurement by A.
The MIN Axiom: Assume that the experiments performed by A and B are space-like separated. Then experimenter B can freely choose any one of the 33 particular directions w , and a’s response is independent of this choice. Similarly and independently, A can freely choose any one of the 40 triples x, y, z, and b’s response is independent of that choice.
The Free Will Theorem. The axioms SPIN, TWIN and MIN imply that the response of a spin 1 particle to a triple experiment is free—that is to say, is not a function of properties of that part of the universe that is earlier than this response with respect to any given inertial frame.
The definition of “free” used in the MIN axiom is the same as that used in the Free Will Theorem – some event is “free” in this sense just in case multiple versions of it are all compatible with everything before that event in any reference frame. Mathematicians express this notion in terms of functions, and philosophers would say that the event doesn’t supervene on anything outside the future light cone.
When we note this definition of “free”, it seems that the initial summary of the theorem is trivial – if some human action doesn’t supervene on the past in any way, then of course this is also true for some subatomic particle, namely, the first one whose movement would be different under the different choices of action by the human.
However, the theorem points out something stronger than this – nothing in the axioms involved assumes that the experimenter is a physical being made up of subatomic particles. Even if you think it’s a conceptual necessity that the experimenter (or at least, the experimenting apparatus) is made up of subatomic particles, nothing requires that there be a first such particle whose motion is different in the choices of how to set up the experiment. So without the theorem, it’s conceptually possible that human movements are free in the sense described, even though the motions of any specific particle are determined by the motions at earlier times, because human actions are at least in part composed of chains of motions of particles with no earliest member. So the theorem really does prove that indeterminacy at the human level requires indeterminacy at the particle level.
However, it seems to me that Conway and Kochen go on to make some bad interpretations of what this theorem says about freedom, determinism, and interpretations of quantum mechanics. They say, “our theorem asserts that if experimenters have a certain freedom, then particles have exactly the same kind of freedom.” This is true for a very specific type of freedom (namely, non-supervenience on the past) but their theorem says nothing else about any other kind of freedom, or whether their freedom has anything to do with the kind of freedom that matters. It may be that this kind of freedom is an important component of free will in the ordinary sense, but it may be that free will essentially requires not just non-supervenience, but also some sort of complex structure that just isn’t possible for the motions of individual particles.
They do make some good points about how the sort of freedom allowed for the particles is merely “semi-freedom” – that is, it is really spacelike separated pairs of particles whose motions are free, because the TWIN axiom says that the motions are in fact correlated in certain ways. They are right to point out that this means the freedom is different from “classically stochastic processes”, which clearly don’t provide any help in explaining free will. However, it really isn’t clear to me that this semi-freedom is any more help – correlations between twinned particles seem exceedingly unlikely to be relevant to the notion of free will.
“Granted our three axioms, the FWT shows that nature itself is non-deterministic. It follows that there can be no correct relativistic deterministic theory of nature. In particular, no relativistic version of a hidden variable theory such as Bohm’s well-known theory can exist.”
I agree that their axioms entail non-determinism. However, I don’t see why this should cause any trouble for the proponent of Bohm’s theory. It seems to me that a proponent of Bohm’s theory would just never grant the MIN axiom. Since the theory is deterministic, it entails that the choices of experimenters (assuming they are part of the physical world) aren’t free in the sense required by the axiom. Presumably, Bohmians are either compatibilists about free will (so that it doesn’t require freedom in the sense of the theorem) or insist that apparent free will is just an illusion. In either case, the seeming freedom of experimenters to set up their apparatus how they like gives us no evidence that this process is non-deterministic.
I suspect that a similar move can be made by the proponent of GRW theory, but I am unfamiliar with the details. They spend the last page or so of this paper engaged in a dialectic with a proponent of GRW theory who responded to some earlier papers of theirs, and give a modified version of the MIN axiom that they claim should be acceptable to the defender of GRW, but I suspect that a lot will depend on the interpretations of the words “independent”, “free”, and “choice” that they use.
In summary, I think the Free Will Theorem does a nice job of showing that a few facts about quantum mechanics (SPIN and TWIN) show that a certain type of macro-scale indeterminacy (MIN) entails a certain type of micro-scale indeterminacy. Additionally, the micro-scale indeterminacy is required not to be like most standard stochastic processes (because of the correlations over distances), so it may well be a place to look for interesting explanations of incompatibilist free will.
However, the theorem tells us nothing about compatibilism itself (which, contra Conway and Kochen, is not “a now unnecessary attempt to allow for human free will in a deterministic world”), because the theorem does nothing to prevent deterministic interpretations of quantum mechanics, whether Bohmian or otherwise. It may do something to constrain the shape that GRW-style theories can take, but this is less clear to me.
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When I was an undergraduate, I remember being very struck by some of the early results in the class I was taking on abstract algebra. Of course, I was eventually very struck by the results from Galois theory when we got there, but in the early parts of the class I was struck by the results proving the existence of the algebraic closure of a field, and proving the existence of a field of fractions for every integral domain. In particular, these seemed to me to validate the use of the complex numbers (once the reals were given) and the rational numbers (once the integers were given). I was still vaguely dissatisfied that we hadn’t yet had a proof of the existence of the integers, but I became happier when I saw the definition of the natural numbers as the smallest set containing 0 and closed under the successor operation, especially because this let proof by induction be a theorem rather than an axiom.
However, earlier this week (in conversation with Zach Weber, while visiting Sydney), I started realizing what I should have realized long ago, which is that these theorems really can’t be doing as much work in justifying our use of the various number concepts as I had thought when I was younger. Of course, these theorems are quite useful when talking about abstract fields or rings, but when we’re talking about the familiar complex, real, rational, and integer numbers, it’s no longer clear to me that these theorems add anything whatsoever. After all, what these theorems show is just that, by using some fancy set-theoretic machinery of ordered pairs and equivalence classes, we can create a structure that has all the properties of a structure that we already basically understood. Perhaps in the case of the complex numbers this mathematical assurance is useful (though even there we already had the simple assurance in the form of thinking of complex numbers as ordered pairs of reals, rather than as polynomials over R modulo the ideal [x2+1]), but for the rationals and negative numbers, our understanding of them as integers with fractional remainder, or as formal inverses of positive numbers, is already sophisticated enough to see that they’re perfectly well-defined structures, even before we get the construction as equivalence classes of ordered pairs of integers or naturals.
But this is all a sort of prelude to thinking about the two more famous set-theoretic reductions, that of the reals to Dedekind cuts (or Cauchy sequences) of rationals, and that of the naturals to the finite von Neumann ordinals. Unlike the others, I think the Cauchy and Dedekind constructions of the reals are quite useful – before their work, I think the notion of real number was quite vague. We knew that every continuous function that achieves positive and negative values should have a zero, but it wasn’t quite clear why this should be so. Also, I think intuitively there remained worries about whether there could be a distinct real number named by “.99999…” as opposed to the one named by “1”, not to mention the worries about whether certain non-convergent series could be summed, like 1-1+1-1+1….
But for the reduction of the naturals to the von Neumann ordinals, I think it’s clear that this should do no work in explicating the notion at hand. To prove that enough von Neumann ordinals exist to do this work, you already need a decent amount of set theory. (John Burgess’ excellent book Fixing Frege does a good job investigating just how much set theory is needed for this and various other reductions.) And while some of the notions involved are basic, like membership and union, I think the concept of sets of mixed rank (for instance, sets that have as members both sets, and sets of sets) already strains our concept of set much more than any of this can help clarify basic notions like successor, addition, and multiplication. One might even be able to make a case that to understand the relevant formal set theory one must already have the concept of an ordered string of symbols, which requires the concept of finite ordering, which is basically already the concept of natural numbers!
In some sense, this was one project that Frege was engaged in, and his greatest failure (the fact that his set theory was inconsistent) shows in a sense just how unnecessary this project was. At least to some extent, Frege’s set theory was motivated by an extent to show the consistency of Peano arithmetic, and clarify the concept of natural number. However, when his explanation failed, this didn’t undermine our confidence in the correctness of Peano arithmetic. The same thing would be the case if someone today were to discover that ZFC was inconsistent – most of the mathematics that we today justify by appeal to ZFC would still stand. We wouldn’t abandon Peano arithmetic, and I think we wouldn’t even abandon most abstract algebra, geometry, analysis, and the like, except perhaps in some cases where we make strong appeals to the Axiom of Choice and strange set-theoretic constructions.
Of course, Frege’s actual attempted reduction of the number concepts to set theory would have been a very nice one, and could help explain what we mean by number, because he reduced each number to the collection of all sets with that many elements. However, modern set theory suggests that no such collections exist (except in the case of the number 0), and the proposed reductions are much less illuminating.
So I wonder, what role do these proofs play, that demonstrate the existence of structures that behave like the familiar natural numbers, integers, rationals, reals and complex numbers? I’ve suggested that in the case of the reals it may actually do important work, but I’m starting to be skeptical of most of the other cases.
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Joe Shipman recently posted an interesting e-mail on the Foundations of Math e-mail list:
I propose the thesis “any mathematics result more than a century old is suitable for undergraduate math majors”.
Note that the original proofs may be too difficult for undergraduates, I am only requiring that today a “boiled-down” proof (which may be embedded in a much larger theory than existed at the time of the original proof) could be taught.
So far I have only found one significant counterexample, Dirichlet’s theorem (which, in its logically simplest form, states that if a is prime to b, there exists a prime congruent to a mod b).
Can anyone think of better counterexamples? Does anyone know of a proof of Dirichlet’s theorem that does not require prerequisites beyond the standard undergraduate curriculum?
(Two other possible counterexamples, the Prime Number Theorem and the Transcendence of Pi, are proven sufficiently easily at the following links that they would, in my opinion, be appropriate for a senior seminar:
Another version of the thesis is “any mathematics result more than 200 years old is suitable for freshmen” (note that most high schools offer a full year of Calculus). Results that were merely conjectured more than 200 years ago but not really proved until later don’t count.
I’ve sometimes considered something like this. Can anyone else think of potential counterexamples? I wonder if there were some results known on solutions of differential equations in the 18th century that would be too advanced for first-years. And probably some particular calculations done in the 19th century that are just too large for an undergraduate to properly manage. I think it’s also possible that some of Cantor’s results on the possible Borel structures of the sets of discontinuities of real-valued functions might be too advanced, but it’s also possible that advanced seniors can manage them. Or perhaps the Riemann-Roch theorem? (I don’t actually even know enough to state that theorem myself.)
Another interesting corollary to this discussion – what’s the earliest result of the 20th century that is beyond the reach of an advanced undergraduate?
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How much philosophical sophistication does someone need to count as being “mathematically sophisticated” enough to follow a graduate course in algebra?
When reading through a draft paper by Colin McLarty (addressing different issues entirely), I came upon the following passage from Serge Lang’s canonical text:
I think it illustrates a lot of issues that often arise in understanding mathematical writing.
In actuality, mathematicians almost never write the statement that Lang wrote, except in the same sort of definitional statement. In particular, in place of “f(x)” they would write some expression in terms of “x” that one might use, such as “x3+2x-1″ or the like. Because this expression is just a placeholder, we might expect some neutral term, like “t“. But instead he uses a term that gives the reader the idea of what the overall expression is supposed to mean, at the expense of some abuse of notation.
Another issue of use and mention at work here is what the term “x” to the left of the arrow is doing. He doesn’t say whether “x” is a placeholder for a term denoting a specific element of A, or whether it is a sort of meta-placeholder, representing a variable that itself takes values in A.
In practice, I believe that both options are allowed. By a minor abuse of notation, one can write either “Under function f, 3 \mapsto 9″, or “Under function f, x \mapsto x2“. (I’m using “\mapsto” to stand for the arrow used in Lang’s statement.) In particular, the latter type of statement derives from the former by the standard practice of ignoring certain types of use-mention distinction, and allowing variables to stand either for elements of A or the names of elements of A. This abuse is allowed just about everywhere except in some parts of model theory, where it’s important to distinguish objects and their names.
So getting back to my original point, I think that an ability to know when a term is being used or mentioned, and whether it’s standing for itself, an expression that is partly composed in the way that it’s composed (this might relate to Lang’s famous statement that “notation should be functorial over the language”), or something totally different is important. I suspect that a non-sophisticated math student (or a sophisticated philosopher) would read the statement Lang wrote and suspect that the arrow would never be useful, because we’d always have to specify in some other place what f(x) was (that is, what expression “f(x)” refers to).
One aspect of mathematical sophistication seems to rely being aware of these different levels on some subconscious level, so that you can always jump to the right one, even through multiple abuses of notation.
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I recently stumbled upon the FPRAS webpage through a fortuitously placed ad in Gmail. It’s good to know that there’s a society devoted to this important intellectual figure, but it’s a bit distressing to know that they have such poor web design sensibilities. Also, the only description it has of the society suggests that it’s all about Ramsey Theory, ignoring his philosophical and economic contributions. Ramsey Theory is definitely very interesting stuff – on one level it basically says that if you’re looking at a big enough collection, then there’s bound to be some ordered substructure. (More precisely, for any positive integers n and k, there is an N such that any coloring of the edges of a complete graph on N vertices with at most k colors has some set of n vertices where all edges between them are the same color. For 3 and 2, the value is 6, so that if you have 6 people at a party, there are bound to be either 3 mutual acquaintances, or 3 mutual strangers.)
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Metaphysicians and ontologists may want to consult this list in doing their work. Or perhaps make contributions. I added “numbers”.
Math And Science Song Information, Viewable Everywhere – unfortunately, there are no songs about set theory it seems.
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