The Unreasonable Effectiveness of Mathematics

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    Colyvan, Mark

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    The Miracle of Applied Mathematics

    "Mathematics has a great variety of applications in the physical sciences. This simple, undeniable fact, however, gives rise to an interesting philosophical problem: why should physical scientists find that they are unable to even state their theories without the resources of abstract mathematical theories? Moreover, the formulation of physical theories in the language of mathematics often leads to new physical predictions which were quite unexpected on purely physical grounds. It is thought by some that the puzzles the applications of mathematics present are artefacts of out-dated philosophical theories about the nature of mathematics. In this paper I argue that this is not so. I outline two contemporary philosophical accounts of mathematics that pay a great deal of attention to the applicability of mathematics and show that even these leave a large part of the puzzles in question unexplained."
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    Dembski, WIlliam A.

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    The Last Magic: Review of The Applicability of Mathematics as a Philosophical Problem by Mark Steiner (Harvard University Press, 1999)

    "According to Steiner, the reigning 'ideology' or 'background belief' for the natural sciences is naturalism. Typically naturalism is identified with the view that nature constitutes a closed system of causes that is devoid of miracle, teleology, or any mindlike superintendence. An immediate consequence of naturalism is that it leaves humanity with no privileged place in the scheme of things. It's this aspect of naturalism that Steiner stresses. Naturalism gives us no reason to think that investigations into nature should be, as Steiner puts it, 'user-friendly' to human idiosyncrasies. And yet they are."
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    Howell, Russell W.

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    Does Mathematical Beauty Pose Problems for Naturalism?

    "Numerous events occurred in 1960 whose effects could hardly have been predicted at the time: several African Americans staged a "sit-in" at a Greensboro lunch counter, the Soviet Union shot down Gary Powers while he was ying a U2 spy plane,..and four Presidential debates between John Kennedy and Richard Nixon aired on national television. Less well known was the publication of a paper by the physicist Eugene Wigner. Appearing in Communications in Pure and Applied Mathematics, a journal certainly not widely read by the general public, it bore the mysterious title The Unreasonable Effectiveness of Mathematics in the Natural Sciences. (Wigner, 1960) Like our cultural examples of the 1960's, it has had effects beyond what most people would have imagined. Our purpose here is to tease out some strains of an important question that has emerged from Wigner's work."
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    Polkinghorne, John

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    The New Physics and Opportunities for Ontological Initiatives

    "The universe has proved to be astonishingly rationally transparent, and the human mind remarkably apt to the comprehension of its structure...It has also turned out that it is mathematics that is the key to unlocking these scientific secrets. In fundamental physics it is an actual technique of discovery to look for equations that have about them the unmistakable character of mathematical beauty...It seems to me that a purely naturalistic metaphysics is unable to cast light on this deep intelligibility, for ultimately it has to treat it as a fortunate but fortuitous fact."
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    Wigner, Eugene

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    The Unreasonable Effectiveness of Mathematics in the Natural Sciences

    "...[M]athematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors."