Can science tell us everything?#
Johannes Siedersleben, Oxford, July 2024
Calculemus! said Leibniz (1666). A salient feeling of the Enlightenment was that mankind had finally succeeded in deciphering nature and that the deep questions of whether there is a God, what is good and what is evil, and so on, were just waiting to be solved by calculation. This is what Spinoza tried to do in his Ethics (1677), and, much later, Einstein in his paper on The Laws of Science and the Laws of Ethics (1950). The mathematical approach failed in philosophy, which could never be reduced to mere calculations, and the deep questions are still waiting to be answered. But the new, bold way of thinking succeeded in science. Starting with pioneers like Descartes and Galileo, science has evolved like a series of rockets, each building on the previous one and gaining speed: Newton and Leibniz, Darwin and Wallace, Bohr and Einstein, Dirac and Schrödinger are just a few prominent examples. Will there be a day X in 100 or 1000 years when they say, OK, mission accomplished, all questions ticked? The answer is no. Gödel showed that many propositions can neither be proved nor disproved (1931), Church and Turing showed that many problems cannot be programmed (1936). The vision of the Enlightenment was illusory. With this in mind, and using physics as an example, I will describe in the first section what science tells us, and in the second what it doesn’t.
What science tells us#
Imagine the anonymous scientist who, long before CE, studied round objects like bowls, wheels, and the sun. He abstracted away properties such as weight, colour, and the third dimension, eventually arriving at the idea of roundness shared by these objects: the circle was invented. Much later, in 250 BCE, Archimedes calculated its surface, circumference, and the number pi. The circle is an abstraction or a model; bowls, wheels, and the sun are tangible instances, but they are never perfectly round. Archimedes’ formula is exact for an ideal circle (which isn’t real), but approximate otherwise. As Einstein sadly commented [Einstein, 1953]: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Imagine Newton (1687) studying falling apples or, more generally, how heavy objects move when subjected to gravity. He abstracted away properties such as volume, feel, and colour, eventually arriving at the idea of volume-less particles with positive mass, and calculated how they move [Susskind and Hrabovsky, 2014]. Raindrops, apples, and planets are tangible instances, but of course they do have shape, volume and many other properties. Newton’s equations of motion are exact for particles (which aren’t real), but approximate otherwise. Newtonian mechanics takes place in space-time, with four dimensions extending in both directions straight to infinity. This is the stage, and the actors are volume-less particles. You need at least one of them; if there are many, it’s called statistical mechanics. Newtonian mechanics is a model (in the mind, on paper, or on a computer) that describes reality and allows us to make predictions, but it is separate from nature. A falling raindrop is completely unaware of Newton’s laws, raindrops have always fallen in the same way. Newton’s laws have less effect on objects in motion than a thermometer has on temperature.
Imagine Einstein (1905) studying two or more particles moving at high relative speed, a case where classical mechanics fails [Susskind and Friedman, 2017]. He kept the Newtonian theatre (four-dimensional space-time and particles), and included the speed of light, which is constant and doesn’t depend on the observer. From his famous thought experiments, he concluded that actors travelling at high relative speed would perceive each other in a distorted way: to an observer, the distances in a passing spaceship would seem shorter, and a clock over there would seem to tick more slowly. But the traveller doesn’t shrink and ages at the normal rate. Relativity is symmetrical: the traveller sees himself as an observer, and for him the observers are the travellers. Observer and traveller are interchangeable roles, speed is always relative, and the notion of rest is meaningless. From the point of view of an electron, the observing physicist passes by at the speed of light. The equations of motion had to be adjusted so that all the actors agree, no matter how fast they are moving. This led, among other things, to the famous equation E = mc2 and to a new quantity called proper time, which is shared by all the actors. The theory of relativity is, in a sense, Newtonian mechanics with guardrails to keep events on the track of invariance of proper time, called Lorentz invariance. Relativity theory has consequences for everything that can be said about time and causality: the observer and the traveller each have their own time, and the temporal order of events depends on who is observing.
Imagine Schrödinger (1920) studying the hitherto unexplained double-slit experiment [Susskind and Friedman, 2014]. He didn’t care about relativity, and kept the Newtonian theatre (four-dimensional space-time and particles) up to a point: he replaced particles moving at known speeds along known paths with evolving probabilities called wave functions. In classical mechanics you knew exactly where a particle was going to be, in quantum mechanics you just get the probabilities. The equations of motion became Schrödinger’s equations, describing the evolution of wave functions rather than the motion of particles. So determinism was gone. Also gone was the idea of a continuum: on a small scale, what looks continuous (e.g. energy) breaks up into tiny pieces called quanta. What is the reality under the quantum mechanical bonnet, if there is one? The Copenhagen debate on this question lasted over 100 years and continues to this day. Is it conceited to assume that everything must be explainable by our limited experience of what we call reality?
Imagine Dirac (1928) sitting by the fire in his Cambridge study, and trying to bring relativity and quantum mechanics together. He kept the quantum theatre (four-dimensional space-time and wave functions), and did care about relativity theory. The formula he found on a memorable evening in 1928, and which bears his name, is the equation of motion of quantum field theory (QFT). QFT features a set with the frightening name of the Lie algebra of Lorentz invariants, which consists of four disjoint components. We live in number one, with time and space as we know them. In number two, time runs backwards and space is like ours. In number three, time runs forward and space is inverted (whatever that means), and in number four, time runs backward and space is inverted. So we end up with four varieties of physics: one is ours, and nobody knows if there are universes out there that are described by the other three.
OK, let’s call it a day. What have we learnt? Science has produced powerful models that describe the observable world and make good, but imperfect, predictions. None is definitive; each has refined the previous one, been more precise and often more complicated. Other models are around (general relativity theory, string theory, loop theory), playing on different stages (curved space, more than four dimensions), and the next Einstein may already be born. The relationship to reality can be obvious (circle), plausible (Newtonian mechanics), puzzling (relativity), unfathomable (quantum mechanics) or mind-boggling (QFT). While Archimedes and Newton drew their ideas from everyday experience, their successors were motivated by experiments that few of us have seen, let alone verified. Their results defy common sense and challenge our naive understanding of reality. We have learned more than we could have hoped for and are left with deep questions that may never be answered.
What science doesn’t tell us#
Mankind is an insignificant species. We are minuscule creatures, living for a few world seconds on a planet that is barely a grain of dust in the universe, equipped with the senses and the brain that evolution has happened to give us. Scientists, for all their cleverness, are like beetles exploring a few metres of motorway: this motorway is straight, they would say, and it goes on to infinity. Every discovery raises new questions and no matter how far we go, the list of unsolved problems will always grow faster than the list of solved ones. Mysterious forces could be out there, influencing our lives or not, that you can call God, gods, ghosts, or whatever. Science does not even claim to be useful beyond the perceptible world, and it is far from exhaustive within it: Our minds are confined to the limits discovered by Gödel [Franzén, 2005], our computers to those discovered by Turing and Church. The age of the universe is estimated to be about 13.8 billion years and its diameter to be about 93 billion light years. Because of the finite speed of light, we will never see more than a tiny fraction of this vast four-dimensional space. No one knows how many dimensions our universe really has: just four, a million, or infinitely many. No one knows how many universes there really are: just one, a million, or infinitely many. And what does real mean? Perceptible? Perceptible by whom? Aliens may perceive a reality very different from ours, and large parts of the universe have probably never been perceived by any being. There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.