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Tallis in Wonderland
Mathematics & Reality
Raymond Tallis on maths’ unreasonable effectiveness.
The belief that mathematics is the surest path to the truth about the universe because the latter is at bottom mathematical has been very influential in Western thought. It goes back to Pythagoras’s assertion that “All is number,” or, as Aristotle paraphrased it, “The principles of mathematics are the principles of all things.” It is the rationale behind Plato’s insistence that no one should enter his Academy without knowledge of geometry. Pythagoras’s discovery of the mathematical ratios underlying pitch, such that doubling the length of a string on a musical instrument produces a note an octave lower, has resonated long and loud through human consciousness. Galileo’s assertion that “The book of nature is written in the language of mathematics” has been a guiding principle of science since the scientific revolution to which he contributed so much. The idea of the universe as a gigantic computer, and the belief that everything (including conscious experience) is information that is either itself digital or can be digitised without loss, is but a recent manifestation of Pythagoreanism.
The question of the relationship between mathematics and reality has become increasingly urgent over the last century as physics, led by ever more abstruse calculations, makes spectacular advances. This has enabled extraordinarily precise predictions about every aspect of the material world from the very small (Higgs bosons, etc) to the very large (gravitational waves, etc), and, through the technology underpinned by it, a massive amplification of our agency. This has persuaded some thinkers that mathematical physics has, or will have, the last word on many, perhaps most, aspects of the world in which we live. The pursuit of a theory which combines the two pillars of contemporary physics – relativity and quantum mechanics – is advertised as the search for ‘a Theory of Everything’. If you want to know the time, you ask a policeman: if you want to know what time (or space, or the basic stuff of the universe, or even consciousness) is, you ask a physicist.
This is disconcerting because of the increasing divergence between the world as physicists describe it and the world as we experience it. The quantum mechanics underpinning electronic communication systems – including the laptop on which I am writing this, and the email by which I will send it to the Editor of Philosophy Now – is entirely at odds with anything that could be remotely described as common sense. Yet the ideas that the universe consists not of discrete objects but of a field corresponding to a wavefunction, or that it has countless parallel siblings (‘many worlds’), are close to being mainstream.
For many, the incomparable predictive and technological powers of physics have furnished incontrovertible evidence in support of its claim to grasp some, perhaps the, fundamental truths about the world; that it produces not merely models of reality but reveals reality itself. As Hilary Putnam expressed it, “scientific realism is the only philosophy of science that does not make the success of science a miracle” (Philosophy in an Age of Science, 2012). And since the power of physics lies in the employment of mathematics, some have found it difficult to resist the Pythagorean conclusion that reality is mathematical.
In a famous paper, ‘The Unreasonable Effectiveness of Mathematics in the Physical Sciences’ (in Communications in Pure and Applied Mathematics, Vol. 13, #1, 1960), Eugene Wigner noted how “The mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.” And R.W. Hamming, in a paper that consciously echoed Wigner, ‘The Unreasonable Effectiveness of Mathematics’ (in The American Mathematical Monthly, Vol 87, #2, 1980), noted the strangeness of how “constantly what we predict from the manipulation of symbols is realised in the real world.”
One of the most familiar examples of this unreasonable effectiveness of mathematics is also one of the most striking: Newton’s law of gravitation. The numerical coincidence Isaac Newton noted between the speeds of falling bodies on Earth, and between the parabolic pathways taken by thrown rocks and the elliptical orbit of planets, led to a mathematical law with a universal application. As Hamming puts it, “The law of gravity which Newton reluctantly established and which he could verify to an accuracy of about 4% has proved to be accurate to less than a ten thousandth of a per cent.” Wigner makes the general point: “while science is composed of laws which were originally based on a small, carefully selected set of observations, often not measured very accurately, [these] laws have later been found to apply over much wider ranges of observation and much more accurately than the original data justified. Not always… but often enough to require explanation” (italics mine). An even more spectacular example, cited by Wigner, was the importation of matrix algebra into quantum mechanics. This has proved extraordinarily powerful in predicting what is going on at the sub-atomic level. Matrix algebra was originally invoked in response to the observation that some rules of computation Werner Heisenberg was using to understand quantum results were formally identical with the rules of computation using matrices that had been established in the nineteenth century. Applying the rules of matrix mechanics to situations beyond those in which Heisenberg’s rules applied – or were meaningful – allowed predictions to be made that agree with experimental data to within one part in ten million.
There is clearly more to mathematics in physics than a convenient notational system. Should we therefore conclude, along with some physicists, metaphysicians, and philosophers of science, that mathematics does not merely offer the most effective ways of modelling the universe, but that it is the most faithful portrait of what the universe really is like? Or, even more radically, accept the Platonic claim that mathematical objects (even non-real items like the square root of -1) are real entities? Or, most radically of all, embrace an industrial-strength Pythagoreanism and conclude that “All is number”? This last is a view advanced by Max Tegmark in a recent book, Our Mathematical Universe: My Quest for the Ultimate Nature of Reality (2014). What he calls his ‘mathematical universe hypothesis’ or ‘mathematical monism’ denies that anything exists other than mathematical objects: even conscious experience is composed of ‘self-aware’ mathematical substructures. According to this view, mathematics is not merely the best guide to reality, it is reality.
The easiest way to see what is wrong with this extreme mathematical realism is to examine actual examples of mathematical physics. Consider the most famous of all mathematical accounts of the world: E=mc2. As with any law, it describes a mathematical relationship between values of variables (energy, mass) that in the context of the equation have no other properties than quantity. (More generally, and technically, physical laws are about the co-variance of quantitative parameters.) The energy in Einstein’s equation is not warm or bright or noisy, and the matter is not heavy or sticky or obstructive. The world of physical laws – which enables predictions of quantities – is a world of quantities without qualities.
This is not an accidental oversight. Galileo, who kick-started the scientific revolution, argued that colours, tastes, sounds, odours, had no place in the material world, whose book was written in mathematics. The qualities we experience were introduced by sentient beings. By contrast, physical reality itself was comprised of ‘primary qualities’, such as size, shape, location and motion, which can be expressed in mathematical terms without remainder. So the mathematized universe of physics lacks what are now (after John Locke) usually called by the slightly derogatory term ‘secondary qualities’. These resist being mathematised. The mathematics of light does not get anywhere near the experience of yellow, nor does the mathematical description of patterns of nerve impulses reach pain itself. This is sometimes seen as evidence that neither the colour nor the pain are really real – although it might be difficult to sell this claim to the man looking at a daffodil or a woman with toothache.
Not all physicists are entirely comfortable with the exclusion of (‘secondary’) qualities. A surprising example is Richard Feynman. “The next great awakening of human intellect” he said in a late lecture “may well produce a method of understanding the qualitative content of equations… Today we cannot see whether Schrödinger’s equation contains frogs, musical composers, or morality – or whether it does not.” Feynman is acknowledging that any world picture that claims to be comprehensive must incorporate qualities. Unfortunately, he does not appreciate how a quantitative account of the world must inevitably lack qualities. Better equations won’t restore qualities to the scientific accounts of the natural world.
And there is something else missing: actuality. Physical laws describe the most general relations between patterns of change. But a pattern of relations between events or objects is not an event or an object, which are singulars, realised through the qualities that characterise them. Wigner pointed out something along these lines when he remarked that “the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world… [they give] no information on the existence, the present positions, or velocities of these bodies” – in short, on actualities, which are all outside of the laws themselves, being the conditions on which the laws operate.
This lack of stuff has been celebrated by writers, such as the philosopher James Ladyman, who defend what they call ‘ontic structural realism’ (OSR). According to OSR, reality is an abstract structure, and, indeed, the mathematical structure of the world is that which is most truly real. For the rest of us, abstract structure without particular content is not merely impoverished; it is an impossibility. Mathematics, which in physics describes the general relations between changing variables, ultimately describes relations without relata.
So the mathematical world picture is one which lacks: secondary qualities (that is, qualities); particulars, or singulars; and stuff. There are other deficiencies. Its account of time (another purely quantitative parameter), remote from lived time, is one in which time can be squared, or placed under ‘distance’ as a denominator, or multiplied by the square root of minus 1 in order to be attached to space. Very unlike an hour in the garden. Mathematics lacks tenses, and all of those other things that make time important in our lives. It is a model of the world in which viewpoint – the necessary condition of observations, and hence of physics itself – and all the things that follow from viewpoint (a sense of now and here, and of the privileged reality of ‘present’ items), and even experience itself, are excluded. It was this that prompted Bertrand Russell’s observation that “Physics is mathematical not because we know so much about the world but because we know so little; it is only its mathematical properties that we can discover” (An Outline of Philosophy, 1927).
Even so, we still need to explain the ‘unreasonable’ effectiveness of physics and of the mathematics that lies at its heart. Clearly physics must be getting something very fundamental very right. Dismissing the contemporary Pythagoreans by arguing that mathematical physics grasps merely the quantitative aspects of reality – hence its extraordinarily precise quantitative predictions – leaves unexplained the fact that those predictions also enable technology that can shape the world – real experiences, events, objects, and stuff – in accordance with our wishes. Maths makes more than a passing contact with our lives. On the other hand, we cannot ignore the other kinds of truths, rooted in the actual experience of human beings that lie beyond mathematics: situational truths saturated with qualities and feelings and concerns, and differentiations of space and time (‘here’, ‘now’).
The challenge of metaphysics must be to see how these different kinds of truths relate. This does not mean either on the one hand siding with the deliverances of immediate experience against those of mathematical physics, or on the other hand dismissing immediate experience as unreal. It does mean, however, that we should re-examine the greatest mystery: that the world makes (growing) sense to us. Meanwhile, we reserve our judgement as to the relationship between mathematics and reality. Both the Pythagoreans and the anti-Pythagoreans have a lot of explaining to do.
© Prof. Raymond Tallis 2014
Raymond Tallis’s new book Epimethean Imaginings, is out now from Acumen.