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Unformatted text preview: arXiv:math.HO/9911150 v1 19 Nov 1999 Machines, Logic and Quantum Physics David Deutsch and Artur Ekert Centre for Quantum Computation, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, U.K. Rossella Lupacchini Dipartimento di Filosofia, Universita di Bologna, Via Zamboni 38, 40126 Bologna, Italy. 19 November 1999 Abstract Though the truths of logic and pure mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics. Recent progress in the quantum theory of computation has provided practical instances of this, and forces us to abandon the classical view that computation, and hence mathematical proof, are purely logical notions independent of that of computation as a physical process. Henceforward, a proof must be regarded not as an abstract object or process but as a physical process, a species of computation, whose scope and reliability depend on our knowledge of the physics of the computer concerned. 1 Mathematics and the physical world Genuine scientific knowledge cannot be certain, nor can it be justified a priori. Instead, it must be conjectured, and then tested by experiment, and this requires it to be expressed in a language appropriate for making precise, empirically testable predictions. That language is mathematics. 1 This in turn constitutes a statement about what the physical world must be like if science, thus conceived, is to be possible. As Galileo put it, “the universe is written in the language of mathematics”[5]. Galileo’s introduc tion of mathematically formulated, testable theories into physics marked the transition from the Aristotelian conception of physics, resting on supposedly necessary a priori principles, to its modern status as a theoretical, conjec tural and empirical science. Instead of seeking an infallible universal math ematical design, Galilean science uses mathematics to express quantitative descriptions of an objective physical reality. Thus mathematics has become the language in which we express our knowledge of the physical world. This language is not only extraordinarily powerful and precise, but also effective in practice. Eugene Wigner referred to “the unreasonable effectiveness of mathematics in the physical sciences”[12]. But is this effectiveness really unreasonable or miraculous? Consider how we learn about mathematics. Do we – that is, do our brains – have direct access to the world of abstract concepts and the relations between them (as Plato believed, and as Roger Penrose now advocates[8]), or do we learn mathematics by experience, that is by interacting with physical objects? We believe the latter. This is not to say that the subjectmatter of mathematical theories is in any sense part of, or emerges out of, the physical world. We do not deny that numbers, sets, groups and algebras have an autonomous reality quite independent of what the laws of physics decree,...
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