Online research in philosophy

We describe the Ax-Kochen definable subsets of the value group of a Hensel field and apply our results to a problem on identifying invariant factors in Hecke algebras.

_This is an account of his present thinking by an excellent philosopher who has been_ _among the two or three foremost defenders of the doctrine that determinism and_ _freedom are incompatible -- that logically we cannot have both. In his 1983 book,_ _An Essay on Free Will_ _, he laid out with unique clarity and force a fundamental_ _argument for this conclusion. What the argument comes to is that if determinism is_ _true, we are not free, since our actions are effects of causal circumstances in the_ _remote past, and those circumstances are certainly not up to us. To that line of_ _thought, in the article below, by way of the supposition of a world of angels, he adds_ _something new. This is a fundamental difficulty with the freedom that we cannot_ _have if determinism is true. The difficulty, indeed a mystery, is one having to do with_ _the opposite of determinism -- indeterminism._.

Incompatibilism, the view that free will and determinism are incompatible, subsists on two widely accepted, but deeply confused, theses concerning possibility and causation: (1) in a deterministic universe, one can never truthfully utter the sentence "I could have done otherwise," and (2) in such universes, one can never really take credit for having caused an event, since in fact all events have been predetermined by conditions during the universe's birth. Throughout the free will.

Kochen and Specker's theorem can be seen as a consequence of Gleason's theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason's theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated.

Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated. r 2003 Elsevier Ltd. All rights reserved.

Of liberty and necessity, by D. Hume.--The doctrine of necessity examined, by C. S. Peirce.--Determinism in history, by E. Nagel.--Some arguments for free will, by T. Reid.--Has the self free will? by C. A. Campbell.--Dialogue on free will, by L. de Valla.--Can the will be caused? by C. Ginet.--Free will, by G. E. Moore.--A modal muddle, by S. N. Thomas.--Determinism, indeterminism, and libertarianism, by C. D. Broad.--An empirical disproof of determinism? by K. Lehrer.--Free will, praise and blame, by J. J. C. Smart.--Bibliographical essay.

Determinism is a claim about the laws of nature: very roughly, it is the claim that everything that happens is determined by antecedent conditions together with the natural laws. Incompatibilism is a philosophical thesis about the relevance of determinism to free will: that the truth of determinism rules out the existence of free will. The incompatibilist believes that if determinism turned out to be true, it would also be true that we don't have, and have never had, free will. The compatibilist denies that determinism has the consequences the incompatibilist thinks it has. According to the compatibilist, the truth of determinism does not preclude the existence of free will. (Even if we learned tomorrow that determinism is true, it might still be true that we have free will.) The philosophical problem of free will and determinism is the problem of understanding, how, if at all, the truth of determinism might be compatible with the truth of our belief that we have free will. That is, it's the problem of deciding who is right: the compatibilist or the incompatibilist.

The interpretation of quantum mechanics has been a problem since its founding days. A large contribution to the discussion of possible interpretations of quantum mechanics is given by the so-called impossibility proofs for hidden variable models; models that allow a realist interpretation. In this thesis some of these proofs are discussed, like von Neumann’s Theorem, the Kochen-Specker Theorem and the Bell-inequalities. Some more recent developments are also investigated, like Meyer’s nullification of the Kochen-Specker Theorem, the MKC-models and Conway and Kochen’s Free Will Theorem. This last one is taken to suggest that the problems that arise for certain interpretations of quantum mechanics are not limited to realist interpretations only, but also affect certain instrumentalist interpretations. It is argued that one may arrive at a more satisfying interpretation of quantum mechanics if one adopts a logic that seems more compatible with the instrumentalist viewpoint namely, intuitionistic logic. The motivations for adopting this form of logic rather than classical logic or quantum logic are linked to some of the philosophical ideas of Bohr. In particular a new interpretation of Bohr’s notion of complementarity is proposed. Finally some possibilities are explored for linking the intuitionistic interpretation of quantum mechanics to the mathematical formalism of the theory.

The two theories that revolutionized physics in the twentieth century, relativity and quantum mechanics, are full of predictions that defy common sense. Recently, we used three such paradoxical ideas to prove “The Free Will Theorem” (strengthened here), which is the culmination of a series of theorems about quantum mechanics that began in the 1960s. It asserts, roughly, that if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity. More precisely, if the experimenter can freely choose the directions in which to orient his apparatus in a certain measurement, then the particle’s response (to be pedantic—the universe’s response near the particle) is not determined by the entire previous history of the universe. Our argument combines the well-known consequence of relativity theory, that the time order of space-like separated events is not absolute, with the EPR paradox discovered by Einstein, Podolsky, and Rosen in 1935, and the Kochen-Specker Paradox of 1967 (See [2].) We follow Bohm in using a spin version of EPR and Peres in using his set of 33 directions, rather than the original configuration used by Kochen and Specker. More contentiously, the argument also involves the notion of free will, but we postpone further discussion of this to the last section of the article. Note that our proof does not mention “probabilities” or the “states” that determine them, which is..

Conway and Kochen have presented a “free will theorem” [4, 6] which they claim shows that “if indeed we humans have free will, then [so do] elementary particles.” In a more precise fashion, they claim it shows that for certain quantum experiments in which the experimenters can choose between several options, no deterministic or stochastic model can account for the observed outcomes without violating a condition “MIN” motivated by relativistic symmetry. We point out that for stochastic models this conclusion is not correct, while for deterministic models it is not new. In the way the free will theorem is formulated and proved, it only concerns deterministic models. But Conway and Kochen have argued [4, 5, 6, 7] that “randomness can’t help,” meaning that stochastic models are excluded as well if we insist on the conditions “SPIN”, “TWIN”, and “MIN”. We point out a mistake in their argument. Namely, the theorem is of the form deterministic model with SPIN & TWIN & MIN ⇒ contradiction , (1) and in order to derive the further claim, which is of the form stochastic model with SPIN & TWIN & MIN ⇒ contradiction , (2) Conway and Kochen propose a method for converting any stochastic model into a deterministic one [4].