As I mentioned Friday, I’m going to begin a short discussion about this paper on some consequences of special relativity and quantum mechanics on our view of determinism and the Universe. The authors, John H. Conway and Simon Kochen, establish three “axioms” (and a “paradox”) and from these statements establish consequences which have wide ranging implications. All of these measurements and the following discussion regard the behavior of a spin 1 massive particle. Spin 1 massive particles can have three possible measured values of quantum mechanical spin, namely -1, 0, or 1. Part 1 in which the axioms (and the Kochen-Specker paradox) are discussed can be found here.

In this installment of my discussions of this (which will have at *least* one and perhaps two more parts) I will examine the theorem which is at the heart of this paper. Blog neighbor Jim Anderson, noting my “homework assignment” finds the third paragraph daunting. The statement of the (strong) Free Will theorem is:

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.

Conway and Kochen prove this theorem by contradiction, that is they assume the theorem is not true and show that leads to a problem, in this case the the contradiction comes in the form of the Kochen-Specker paradox.

The basic form of the proof is to take two TWIN particles subjected to the SPIN measurement and begins to follow the consequences that these particles are “not free”. What is meant by free? This takes a particular meaning. If this measurement is free it means that the result of this measurement is not the consequence (a function of) of anything which has occurred earlier in any reference frame.

So, the authors express this measurement in terms of a collection of parameters denoted as alpha. In brief, the method employed in the proof is to pare down that unconstrained parameters sets (axis or other prior settings) via group arithmetic and MIN (one of the axioms from yesterday) to be able to finally express the measurement as a function which is recognizable as the same function which by the Kocken-Specker paradox cannot exist. Then, since the function cannot exist then the prior constraints on the particles measurement cannot exist either.

The paragraph quoted by Mr Anderson as less than transparent to the worlds most competent reader are placed there largely, I think, are included to these results to bear on a more recent proposal (called GWR and rGWR in the paper) which attempt remove by stochastic arguments the “measurement/collapse” of quantum wave functions which is philosophically speaking, uhm, difficult. I have not read any of the rGWR papers or any discussions of them so I will leave that for another time.

Mr Anderson (and his commenter) remark that this paper perhaps goes too far, offering

From what I can tell, it’s an attempt to demonstrate free will by noting that at least one property of elementary particles is nondeterministic. This still doesn’t prove the philosophical idea of free will, however. It appears only to impute it to an object, with a lot of anthropomorphizing to make it all work.

I don’t think that’s the case at all, however. The notions of free will which they think this offering lacks “intentionality, “responibility” and so on are not being discussed here. In any discussions of free will and compatabilism see for example wiki or the Stanford Encyclopedia, there is indeed a lot of discussions over whether *determinism* and free will are can co-exist. Yet, the universe *in which we live* is not deterministic. So the compatibility problem shifts. It is not a question of whether free will and determinism can exist but how free will arises in a fundamentally non-deterministic universe. The usage of the term “free will” for the theorem is to point that the freedom of the elementary particle to choose it’s “101″ (squared) spin statistics result is equivalent and indistinguishable from the *experimenters* free will to determine the axis by which the measurement will be taken. No the axis of measurement (and the particles choice of 101,011, or 110) is not a moral choice obviously. But glancing through the compatiblism articles cited above, little space is seemingly granted to the considering consequences of a non-determinstic universe … or if incompatiblisim may be possible, i.e., “or that free will is true, therefore determinism is not” … and since determinism is not might free will be a possiblity?

The point is much discussion within the philosophical community grounds itself on the notions of whether or not determinism is true, i.e., whether the universe is really or is really not deterministic. Physics insists that there is an answer to *that* part of the question. The universe is *not* deterministic. So however you argue about free will that part of the argument should be settled.

Both people (A and B) have a large set of experimental data and theories which support continuityI think person B is just being a bit more precise with language. There’s a large set of experimental data that supports continuity on the macro scale. (Macro here meaning the smallest resolution we are able to theoretically examine).

Relate that to the hypothetical cartoon.

A: “Of course it is continuous, look how the continuous equation y = -x^2 -1 describes the cartoon”

B” “The cartoon is continuous on the scale we are able to measure it. we should make no assumptions about the cartoon below that scale either way”

Think of linear regression. If some relationship appears very linear over a particular range, standard practice is to use linear regression to estimate the function. Even if this function proves to do a very good job of producing good results over the range, though, it would be a heroic assumption to assert that the relationship remains linear even when you are far removed from the observed range.

Boonton,

Did you even

lookat the Wigner paper? “It would be heroic” … and yet it has proven so far to be true (heroically in fact).And this isn’t just one equation and one measurement. It’s large libraries and centuries of data, theory and experiment.

Boonton,

I don’t think “B” is being more precise. “A” is correct as well, note he is willing to abandon the continuity if evidence to contradiction arises. I think the “A” position is the default taken by most people and scientists.

I don’t know, B is being more precise IMO. B is correctly pointing out that the evidence so far collected applies to the area it was collected from and it’s only a hypothesis that areas where evidence hasn’t (or cannot) be collected from would follow the same pattern. If you had marketing data for France that a new soft drink was very popular would you trust, absent any marketing data from the US, that it would be safe to just assume the US would follow the same pattern?

Boonton,

No. The

evidencepoints to continuity as all the mathematical machinery which requires continuity and which does not hold true in the absence of continuity correctly describes what we see. Theevidencewe have supports continuity. If you want to be “careful” and follow “B” then you should not use mathematical methods that require continuity … which is problematic for performing calculations.