Is truth objective, eternal, and the same for everyone?#
Johannes Siedersleben, Oxford, July 2024
A bedrock of objective, eternal truth – sounds too good to be true? It was supposed to be true for over 1000 years, throughout the Middle Ages. The Church held that all conceivable human knowledge was already recorded in the Bible, in the writings of the Fathers (Ambrose, Jerome, Augustine, Gregory) or in selected Greek works (Plato, Aristotle: yes, Epicurus: no). What was there was considered ultimate, irrefutable wisdom. What wasn’t was wrong, useless, or both. Around 1500, the intellectual monopoly of the Church came to an end with the likes of Copernicus, Kepler and, of course, Martin Luther. Now, 500 years later, the world has changed. We sit on a vast, unfathomable mountain of knowledge, we fly to the moon, we replace hips and hearts, and computers are smarter than we are. At the same time, everything can be questioned, no cherished conviction is safe, and every idea, belief or theory, however absurd, has its adherents, its bubble. This can be amusing, as in the case of the flat-earthers, or dangerous, as in the case of QAnon. We seem to have gone from one extreme to the other, from the fiction of a bedrock of complete and unassailable truth to a world of arbitrariness, where everyone is free to join their preferred bubble and cultivate their private variety of truth. Is there anything left of the bedrock of truth, is there anything binding, and to what extent are we entitled to create our private truth? There is no simple answer, if there is one. So I prefer a descriptive approach: rather than explaining what truth is, by whatever authority, I propose to explore some varieties of truth that we encounter in history, science and everyday life. I just mentioned two of them: the view of the Church in the Middle Ages and the individualistic view of some contemporaries. Now I’m going to look at four more: logic, chess, mathematics, and physics. Let’s see where that takes us.
Logic#
“Is in italics” is not in italics. “The least integer not nameable in fewer than nineteen syllables” is being named in eighteen (Berry’s Paradox). The statement “There is no absolute truth” gives a contradiction when applied to itself, and the same is true of “Everything has a cause”. “All Cretans lie. I am a Cretan.” gives us a lot to think about Cretans. Can an almighty God make a stone so heavy that he cannot lift it himself? The barber “who shaves all the men who don’t shave themselves” doesn’t tell us who shaves himself. He is the popular version of “the set that contains all the sets that do not contain themselves”. This innocent set, known as Russell’s Paradox (1903), is at the origin of a revolution that has shaken mathematics and led to a complete renewal of its foundations. Here is my message: there is more to logic than just true and false, it is a sea of troubles, and beware of self-reference! Aye, there’s the rub! What is the truth of this paragraph I am writing? What would be the truth of an answer? What would …? It’s hopeless, let’s go back to calm waters.
Chess#
The game of chess (or any other game with fixed rules) is a small realm of indisputable truth: no doubt about who wins, or whether a given move is legal. So the truth of chess is objective, eternal, and the same for everyone. But it’s relative to arbitrary, man-made rules, and therefore a bit of a cheat.
Mathematics#
Let’s start with Peano arithmetic: natural numbers populate the set N of natural numbers, and are subject to Peano’s axioms, which are arbitrary and man-made. What do numbers have in common with chess? They are the chess pieces, the set N is the chess board, the axioms are the chess rules, and the logical steps from one statement to the next are the moves. Peano arithmetic is just another game. Alternative games could be Robinson or Presburger arithmetic, where some of Peano’s axioms are dropped – a bit like chess without knights or bishops. Do they agree that 1 + 1 = 2? They do: according to an axiom shared by all three systems, every natural number has exactly one successor (that’s what “+ 1” means), and the successor of 1 is called 2. To say that 1 + 1 = 2 is like saying that a triangle has three angles, full stop. So, mathematical truth is basically as disappointing as chess truth, up to two points: (1) All chess games start from the same initial position, while mathematical theorems are stacked to form a giant building. (2) Chess has no counterpart in the real world, while mathematics models part of it. Numbers, for example, are a model of everything that can be counted: five apples, five fingers and five pounds share the property of fiveness, and 7 + 5 = 12, no matter what is added. Vector spaces, as another example, are a model of the space we live in and of much more. It should be noted that the rules of the mathematical game are not entirely arbitrary. Rather, they have been adjusted until mathematics matches the world. This is one of the reasons why it works so well [Yanofski, 2016].
The bedrock of mathematical truth is not as solid as you might think. Some theorems can be approximately verified by counting or measuring, such as the Pythagorean Theorem, but most cannot: how would you verify that the number of primes is infinite? We have to trust our own judgement, or that of other mathematicians, that the proof is valid. Consider this: Andrew Wiles’ proof of Fermat’s Theorem is over 100 pages long and has been fully understood by only a few experts. How many of them, perhaps weighted by reputation, are needed to confirm a proof? And it gets worse: Around 1900, the foundations of mathematics were completely overhauled by scientists such as Russell, Cantor, Hilbert, and many others. The existing theorems remained valid, but many proofs had to be revised. In 1931, Gödel shattered mathematics with his incompleteness theorems [Franzén, 2005, Hoffmann, 2013]: many propositions can neither be proved nor disproved. When will the next Russell come along to revise the foundations again, the next Gödel to reveal even more incompleteness?
Physics#
This is how physicists work: they start with a handful of non-provable principles, such as the constant speed of light or the principle of least action. They then apply appropriate mathematical theories and test the results in the laboratory. This process is repeated until mathematics matches the measurements. Historically, things have been less straightforward [Kuhn, 2012], but that is the essence. Physicists have been extremely successful: relativity, quantum mechanics, quantum field theory have changed our world. Without physics, there would be no computers, no aeroplanes, no spacecraft. Truth in physics depends on empirical evidence, which by definition is limited, and more so than you might think: we learn from quantum theory that very small particles have no definite properties, such as position or momentum. Electrons are very different from tiny specks of dust. So what are they? The debate goes on. Our perception is restricted to a tiny part of our four-dimensional universe. So how can we ever hope to explore other universes?
There is no bedrock of indisputable physical truth. Greek physics was interesting, but often wrong. Newtonian physics, long thought to be ultimate and unassailable, turned out to be only an approximation of more refined theories. Quantum mechanics and general relativity have both been amply confirmed by experiment, but have so far resisted all attempts to unify them. New theories have always been challenged, not embraced. Newton was quickly accepted by his fellow physicists, but it took until 1822 for the Church to admit that the Earth orbits the Sun. Einstein’s theory of relativity was met with much scepticism, and it took about 20 years for the world to really appreciate it. New theories enter a Darwinian battlefield, and even if they survive, they are forever subject to competition from new theories, as well as to ignorance and malice (Einstein was a Jew). Physics can never offer more than the best theories at hand. That’s not bad, but it’s not what we mean by truth.
Tentative Conclusion#
After this brief glimpse at just four of the many varieties of truth, my conclusion can only be tentative. Here it is: not in chess, but at least in logic, mathematics, and physics truth seems shaky. This, however, is a feature, not a bug! Science has been so successful because it has never rested on its laurels, because it has always questioned its results, and because even the big shots have always been subject to criticism. Science has given up the idea of absolute truth: yes, indeed, everything can be questioned, no cherished conviction is safe! Depending on the subject, the debate is restricted to experts or open to a wider audience. I’ll give four examples: (1) Russell and Whitehead’s Principia Mathematica is hard reading. It never made the headlines, however revolutionary it may have been. (2) The question of whether the earth orbits the sun was quickly settled among the experts, but it took the rest of the world 200 years to accept it. And you cannot blame them: imagine you are 400 years back in time and someone told you that as you sit here, the Earth is moving through space at 67,000 miles per hour. What would you have said? (3) After a rocky period of heated debate, Darwin’s theory of evolution is now widely uncontested among biologists, but there are still vocal groups who disagree. (4) Ten years ago, you could still find a few meteorologists who denied anthropogenic climate change. Today, there are virtually none. And yet there are influential organizations that call climate change a Chinese hoax and oppose all measures to reduce CO2 emissions. Can we blame the opponents in examples (3) and (4) for ignorance, narrow-mindedness, or stupidity? Should we require every debater to have a PhD or a minimum Hirsch index? No. You might just as well restrict the vote to an arbitrarily defined elite. Freedom of speech covers all nonsense, including denying accepted knowledge, which after all is not necessarily final. Flat-earthers, homeopaths and the like are friendly and do no harm. Freedom of speech does not cover the hatred, slander, and incitement to crime spread by less benign groups. What if they win the next election?