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This chapter examines in some detail different characterisations of the problem of mathematics and world, situating it historically with examples treated at varying levels of detail. The first section begins to make the point that there is, indeed, an acute philosophical problem of mathematical applicability—of mathematics and world—and that this problem has exercised scientists and thinkers through the ages. I quote Einstein, Kepler, Galileo, and several more recent distinguished scientists such as Herz, Feynman, Wigner, Weinberg, all expressing their own mystification at how mathematics is so useful, particularly for physics. The section ends with some details of Mark Steiner’s characterisation of the problem of relating mathematics to the physical world, referencing more directly philosophical thought such as that of Plato, Descartes, Spinoza, Berkeley, Kant and Mill. Following this, some examples are treated at rather more length. Ancient history Pythagoras and the Pythagoreans: the beginning of the idea that the world (or cosmos as Pythagoras first taught us to say) is ordered by mathematics, the order notably exemplified in the relationship between musical harmony and arithmetic. Ancient to modern Plato to Descartes: comparing and contrasting Plato’s description (in the Parmenides, particularly, but with echoes elsewhere) of what he himself describes as the ‘greatest difficulty’ for his theory of Forms, with what Descartes calls the ‘objection of objections’ to his metaphysics (in Replies to Objections to his Meditations). Each of these philosophers’ concerns investigated here gives us a striking example of the ubiquity of our problem and its philosophical ramifications. Renaissance How did complex numbers and the square root of minus one enter into the city states of Renaissance Italy? And how did complex numbers turn out to be so useful centuries later? The story is told with details of the algebra involved, in both sixteenth century and contemporary terms. A contemporary example The physicist Eugene Wigner’s famous paper on ‘The Unreasonable Effectiveness of Mathematics in the Physical Sciences’ expresses its point of view on its face. Wigner’s ideas are positioned in relation to Pythagoras’ and others’. Dirac’s equation and the positron Paul Dirac predicted the existence of the positron mathematically. How? Some of the details of Dirac’s derivation of his eponymous equation are explained so we can get more of a grip on this intellectual feat. This is more detailed than much of what has gone before; it is also an excellent example of what many have found so mysterious in the power of mathematics to inform us about the physical world. Steiner and homomorphism Mark Steiner has written extensively about mathematical applicability. In conclusion to Chap. 1, Steiner’s characterisation of the problem of mathematics and world is delineated, together with his proposed solution. In broad terms, I adopt important aspects of his characterisation, relating it to the examples I have treated; I explain why his solution is unsatisfactory, and offer some general outlines of what is required for an acceptable solution. |
