Mathematics and Reality
How is mathematics related to reality?
Why is the universe described with mathematical laws?
The unreasonable effectiveness of math in describing the universe
Can everything in the universe be explained by math?
Plato suggested that everything in our world is just an approximation of perfection. He also realized that we understand the concept of perfection even though we never encountered it. He came to conclusion that perfect mathematical forms must live in another world and that we somehow know about them by having a connection to that universe.
No one has ever seen a perfect circle, but we still understand what a perfect circle is and we can describe it with mathematical equations.
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? -- Albert Einstein
Mathematics and reality
Prof. Raymond Tallis 2014
https://philosophynow.org/issues/102/Mathematics_and_Reality
Pythagoras' discovery of a connection between music and mathematics, that, doubling the length of a string on a musical instrument produces a note an octave lower, has influenced his declaration that "all is number". This idea had further resonated with Aristotle: "the principles of mathematics are the principles of all things". It has also been the rationale behind Plato's insistence that no one should enter his Academy without knowledge of geometry. Many centuries later, Galileo's assertation that "the book of nature is written in the language of mathematics" has been a guiding principle of science since the scientific revolution. In modern times, the idea that the universe is a gigantic computer, as well as the belief that everything, consciousness included, is information that is either digital or can be digitized losslessly, is just a recent echo of Pythagoreanism.
The resolution of the puzzling relation between mathematics and physical reality has become even more pressing in the last century as physics made spectacularly precise predictions at the quantum level. The pursuit is on for an all-encompassing theory that will unite the two pillars of the contemporary physics - relativity and quantum mechanics - in a single Theory of Everything. With physics being underpinned by math, many have found the Pythagorean ideals very appealing even today.
The Unreasonable Effectiveness of Mathematics
In his 1960 seminal paper, "The Unreasonable Effectiveness of Mathematics in the Physical Sciences", Eugene Wigner has noted how the mathematical models of various physical phenomena lead to an amazingly accurate descriptions in an uncanny number of cases. In the 1980's paper, "The Unreasonable Effectiveness of Mathematics", R.W. Hamming has reintroduced Wigner's idea, writing how "constantly what we predict from symbolic manipulations 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.
Hamming writes that "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".
An even more spectacular example 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 XIX 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 negative one) are real entities?
Or, the 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 his book "Our Mathematical Universe: My Quest for the Ultimate Nature of Reality" from 2014. What he calls his "mathematical universe hypothesis" or "mathematical monism" denies that anything else 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.
Quality vs quantity
Consider the most famous of all mathematical equations, E = mc². As with any law, it describes a mathematical relationship between values of variables (E for energy, m for mass, c for the speed of light in vacuum) 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 - that enables making predictions - is a world of quantities and it lacks 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 a man with a 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 during a 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". Here, 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.
Actuality
There is something else missing as well - 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 the so-called 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 (or just qualities)
particulars i.e. singulars
stuff
Got the time
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 one 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 (e.g. "here", "now").
The challenge of metaphysics must be to see how these different kinds of truths relate.
This doesn't mean, either siding with the deliverances of immediate experience against those of mathematical physics, or, dismissing immediate experience as unreal.
It does mean, however, that we should reexamine the greatest mystery: the world makes 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.
No, The Universe Is Not Purely Mathematical In Nature
The idea that the forces, particles and interactions that we see today are all manifestations of a single, overarching theory is an attractive one, requiring extra dimensions and lots of new particles and interactions. Many such mathematical constructs exist to explore, but without a physical Universe to compare it to, we're unlikely to learn anything meaningful about our Universe.
At the frontiers of theoretical physics, many of the most popular ideas have one thing in common: they begin from a mathematical framework that seeks to explain more things than our currently prevailing theories do. Our current frameworks for General Relativity and Quantum Field Theory are great for what they do, but they don't do everything. They're fundamentally incompatible with one another, and cannot sufficiently explain dark matter, dark energy, or the reason why our Universe is filled with matter and not antimatter, among other puzzles.
It is true that mathematics enables us to quantitatively describe the Universe; it is an incredibly useful tool when applied properly. But the Universe is a physical, not mathematical entity, and there's a big difference between the two. Mathematics alone is doubtfully sufficient to support The Fundamental Theory Of Everything.
Kepler first came up with a very elegant mathematical model describing the orbits of planets that turned out incorrect. He then threw that model and started crunching the empirical data attempting to find what types of orbits would match the way planets actually move, and came away with a set of scientific (not mathematical) conclusions.
So, Kepler first tried to fit reality into his mathematical model (massaging data to fit the model), but when that showed futile, he instead searched for a mathematical model that reality fits (massaging model to fit the data).
Kepler's 3 laws: 1) that planets move in ellipses with the Sun at one focus, 2) that they sweep out equal areas in equal times, 3) and that the square of their periods is proportional to the cube of their semimajor axes, apply just as well to any gravitational system as they do to the Solar System.
This was a revolutionary moment in the history of science. Mathematics wasn't at the root of the physical laws governing nature; it was a tool that described how the physical laws of nature manifested themselves.
The key realization that ensured further progress was that science needed to be based on the observable and measurable data. Any experimental data is considered consistent only if repeated experiments produce exactly the same result, every time.
By the early 1900s, it was clear that Newtonian mechanics was in trouble. It could not explain how objects moved near the speed of light, leading to Einstein's special theory of relativity. Newton's theory of universal gravitation was in similarly hot water, as it could not explain the motion of Mercury around the Sun. Concepts like spacetime were just being formulated, but the idea of non-Euclidean geometry (where space itself could be curved, rather than flat like a 3D grid) had been floating around for decades among mathematicians.
Developing a mathematical model to describe spacetime (and gravitation) required tweaking the involved mathematics in a way that would agree with observations. It's the reason why Albert Einstein is so much more famous than David Hilbert. Both men had theories that linked spacetime curvature to gravity and the presence of matter and energy. Both of them had similar mathematical formalisms. Today an important equation in General Relativity is known as the Einstein-Hilbert action. However, Hilbert, who had come up with his own independent theory, pursued bigger ambitions than Einstein - Hilbert's theory targetted both matter, electromagnetism, and gravity as well. And that simply didn't agree with nature. Hilbert was constructing a mathematical theory he thought it ought to apply to nature, but he couldn't produce equations that correctly predict the quantitative effects of gravity. Einstein did, and that's why the field equations are known as the Einstein field equations.
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