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In classical physics this would seem rather evident. In quantum mechanics however the situation differs, and it is accepted that it is impossible to tell anything about a number of properties in between two measurements, because their value is at least partially determined through the measurement itself. In the case of the electron spin, e.g., a measurement does not only give one of two values, +1/2 and -1/2, but has a 50% possibility to reverse the former situation of the electron, something which can only be tested by a new measurement. Schröndinger firstly formulated the cat paradox in a 1935 article. It is a hypothetical experiment, where another quality is taken in account, namely radioactive decay. We imagine a vessel containing a radioactive material, a detector that records the emission of particles, a bottle with a deadly poison and a living cat inside the vessel. The detector's operation is so arranged that there is a 50% possibility for an atom to decompose while it is functioning and that a particle is consequently emitted and 50% it will not. If the detector records an event, the bottle is crashed and the cat dies, as the poison is liberated, otherwise it lives. We cannot know what happens until we open the vessel and take a look inside it. According to the classical view, the cat inside the vessel can be in one of the two states, dead or alive, the possibility being 50% each. Its actual state is ascertained when we open the box, but determined at the moment the detector is activated (or not activated). In the interval between that moment and the moment we open the box, the cat is in the finally observed state. Quantum mechanics also admits an equal possibility of the emission or not emission of a particle. Here, however, the two possibilities overlap, producing a superposition of states, a kind of limbo where the cat is neither dead, nor alive but "dead and alive" at the same time. This is expressed mathematically in the proposition that the state of the cat is a linear combination of the two states, "dead" and "alive". The experiment is governed by the rule that the superposition is preserved until the moment of observation, i.e., until we look somehow inside the vessel. With the act of observation the wave function of the cat collapses into one of the two states: we finally observe that the cat is either dead, or alive. Yet, until that moment, as Gribbin puts it, "there is a radioactive sample that has both decayed and not decayed, a glass vessel of poison that is neither broken nor unbroken and a cat that is both dead and alive, neither alive nor dead" (In Search of Schrödinger's Cat, London 1990, p.205) Given the centrality of the element of history and change in chess, one should not be surprised that certain ideas of quantum mechanics have found concrete expression in it. This applies specifically for Heisenberg's uncertainty principle, the philosophical implications of which Schrödinger's paradox sought to check. In fact, this limbo type of situation appears quite concretely in a number of chess positions. A comparison will help us clarify the matter, first pointed out and elaborated by the Greek IM I. Kourkounakis, in his article The Art of Logic (Athens 2002). B. Langstaff, Chess Amateur, 1922 Mate in two White: Kf5, Rd5, Bf6, pawns h5, h6. Black: Ke8, Rh8, pawn g5
The paradox of the specific problem lies in the fact that it has two solutions only one of which can be correct, but there is not way to decide which one. If white tries 1.Ke6 with the threat 2.Rd8#, black defends with 1...0-0. This proves that black's last move was 1…g7-g5 and consequently white has the right of taking the pawn en passant. If however white differs his choice, starting with 1.hxg6 (so that he answers 1…0-0 with 2.h7# ), then black protests that his last move was 1…Rh7-h8 (or perhaps 1…Rf [or g ] 8-h8 or even Kf7-e8. But this means in its turn that white can from the initial position mate in two starting with 1.Ke6, since black has lost the right to castle by moving either the rook or the king. But if white attempts this, then black answers again 1…0-0, claiming his last move was …g7-g5 etc. In this way a vicious circle is created with no escape in sight. We reach, as I. Kourkounakis observes, "the curious conclusion that the problem has a solution of mate in two, which however cannot be precisely determined. If the key is 1.Ke6, then it won't be 1.hxg6 and if it is 1.hxg6, then it won't be 1.Ke6. But when white tries to select one of the two, black is able to present a defense, which, if true, proves the other choice as the correct one" (The Art of Logic). Since then, a number of variations of the same theme were created, including a similar "mate in two" problem by Moravec. It is not difficult to discern the deep analogy of the situation in Langstaff's problem with the cat paradox. As in it, we can also choose here between two possibilities, but logic says they cannot both simultaneously be true. One solution corresponds to the dead cat and the other to the living one. However, until the moment of observation, i.e., until we see the solution evolving in an actual game, we are in an intermediate situation of uncertainty similar with the "dead-and-alive" cat: both solutions prove successively correct and at the same time none of them is correct. The analogy is indeed very impressive to be coincidental and a number of remarks can be made, useful for the understanding of the scientific problem as well. Firstly, an element of uncertainty seems to exist not only in our knowledge, but also inherently in things. And secondly, this uncertainty is dissolved in a historical manner, through participating in the evolution of the system. In both cases an abstract approach cannot reveal the “true situation”: being “outside the box” or looking just to the position without knowing the developments you cannot verify anything, but someone belonging to the system will be able to do so. There are of course deep differences, which cannot be dealt here. Suffice to mention the wave-particle dualism in the microworld, which makes uncertainty an organic element of its processes. Yet the very existence of the analogy emphasizes the wealth and many-sided use of chess. Christos Kefalis ************ If you like music, you may choose now a fine background Music:
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