Prompted by an X post by anna kw, I’ve been reading “Logos and Alogon” [Plotnitsky 2023], by Arkady Plotnitsky. I’ve also been reading the author’s previous work “Reality Without Realism” [Plotnitsky 2021].
The central thesis of Plotnitsky is that the unthinkable, the inconceivable, or to use his favorite term the alogonal, is indispensable in mathematics and science. The alogon, “that which is strictly inconceivable or incomprehensible, unthinkable,” is part of the logos. Mathematics and science would grind to a halt without unthinkable concepts and inconceivable thoughts.
The early mathematicians of the Greek school of Pythagoras could not conceive of irrational numbers as numbers, because their concept of number was a rational number that is the ratio of two integers. Yet the Pythagoreans discovered that the square root of two is not a rational number and made room for irrational numbers in their thinking. This is a “radically Pythagorean” stance.
Another simple example of alogonal, radically Pythagorean thinking in mathematics is the origin story of complex numbers. Renaissance mathematicians like Girolamo Cardano used the inconceivable square root of minus one in their algebraic calculations. First reluctantly - Cardano himself called it “sophistry” [Brooks 2017] - then more and more confidently.
The square roots of minus one and other negative numbers are inconceivable as numbers, because the square of any number, positive or negative, is always positive. Yet these numbers, which mathematicians came to call imaginary numbers, played a more and more important role in mathematics and physics. Today, imaginary numbers are not only used in all sorts of scientific calculations, but also play an absolutely vital and inescapable foundational role in quantum mechanics [Chapters 9-11 of Prisco 2024].
We now see imaginary numbers as complex numbers. Complex numbers (for example, two plus pi (3.14…) times i, where i is the imaginary square root of minus one) are ordered pairs of real numbers (in the example, two and pi) with a certain algebra. This is conceptually clear, but imaginary numbers are still unthinkable as numbers: an ordered pair of numbers is not a number.
Yet, the unthinkable complex numbers are often very useful and at times indispensable to work with thinkable numbers. Studying the complex zeta function of Bernhard Riemann has opened the door to important insights on the distribution of prime numbers among the integers (what could seem more unrelated to complex numbers than simple integer numbers?). The Riemann hypothesis that the interesting zeros of the zeta function are confined to a certain line in the complex plane is unproven (perhaps it is undecidable in the sense of Gödel?), but a proof could shed more light on the distribution of prime numbers.
Riemann is one of the heroes of Plotnitsky’s book. He was one of the mathematicians that jumpstarted the study of non-Euclidean geometries upon which Albert Einstein’s general relativity is based, as well as much of contemporary physics. Warped non-Euclidean geometries are not strictly inconceivable because there are analogies with surfaces in Euclidean three-dimensional space to help our intuition, but some higher-dimensional warped geometries are totally beyond our intuition.
To Plotnitsky, physics is the study of reality without realism, by which he means that ultimate reality is beyond our ability to conceive and that’s that. Quantum theory, he says, “made this reality alogonal.”
The origin story of quantum mechanics is similar to that of imaginary numbers. A new and unthinkable type of particle was introduced to make sense of what particles like electrons do in the real world. The new imaginary particles called quantum particles are strictly inconceivable because they must be allowed to behave in unthinkable ways, like being in two places at the same time, but they help get the job done with calculations that have always given the correct answer (that is, predicted actual experimental results) so far.
Plotnitsky dives into the early (but perhaps not yet fully understood) interpretations of quantum physics proposed by Werner Heisenberg and Niels Bohr. Physics must limit itself to talk about what can be actually observed (measured), and refrain from introducing concepts and pictures of what happens between observations.
Plotnitsky often cites Heisenberg’s book “Physics and Philosophy” [Heisenberg 1962]. A key quote is “There is no description of what happens to the system between the initial observation and the next measurement.…The demand to ‘describe what happens’ in the quantum-theoretical process between two successive observations is a contradiction in adjecto, since the word ‘describe’ refers to the use of classical concepts, while these concepts cannot be applied in the space between the observations; they can only be applied at the points of observation.”
So what is a particle, really? The question must remain unanswered.
The best we can do is to think of elementary particles as mathematical abstractions like “irreducible representations of the corresponding symmetry groups,” says Plotnitsky, “new mathematical ‘atoms,’ which I shall call ‘Galois atoms.’” Évariste Galois, a 19th century mathematician, was one of the originators of group theory, a branch of mathematics that helps deal with the weirdest aspects of physics.
I agree with Plotnitsky that our science needs unthinkable concepts and inconceivable thoughts to advance. Our mind is the product of evolution, and we didn’t evolve to eat, or escape from, Galois atoms. Our mind can only think of certain things in certain ways, because evolution has selected for us those thinkable things and those ways to conceive them.
Reflecting on Plotnitsky’s ideas on mathematics and physics, I find parallels with what I call irrational mechanics [Prisco 2024]. Should I call my ideas alogonal mechanics?
But I’ll stick with my favorite label because there are differences as well.
I think what is beyond our mind today will not stay beyond our mind forever. In particular, we are developing artificial intelligence (AI) [Chapter 12 of Prisco 2024] that could soon vastly exceed our cognitive abilities and think the unthinkable (from our current perspective). Those AIs are likely to eventually replace us, but I include them and all sorts of conceivable (or inconceivable) human/AI hybrids in my concept of “we.”
Irrational and imaginary numbers were first introduced as inconceivable entities but then they have been gradually brought into the realm of the thinkable and today we introduce them to students in simple ways without conceptual problems. Today, we have adapted to and are prepared to think of numbers beyond integers, rational, and real numbers.
A similar adaptation hasn’t happened yet for quantum mechanics, and many people including experts still find quantum particles inconceivable and impossible to think of. Quantum mechanics seems hopelessly weird, and alogonal.
But is quantum mechanics so weird, really? We think of things called particles and conceive them as little marbles too small to be seen. We see big marbles and our experiments detect little particles that are in one place only. But the most accurate and useful mathematical description of reality that we’ve found so far is based on entities called wavefunctions. The wavefunction associated with one or more particles is a (complex) function that is best thought of as a probability amplitude, whose (real) absolute value is a probability. Quantum mechanics predicts the probability of finding the particles in certain places, and there’s nothing weird.
OK, but what about entanglement and spooky actions at a distance [Chapters 9-11 of Prisco 2024]? The clearest answer is that amplitudes and probabilities are not things that propagate in space. Like, if one ticket wins the lottery, the probability that another ticket wins the lottery becomes zero wherever the other ticket is, instantaneously, and there’s nothing weird in this.
According to Heisenberg and Bohr, this is all we can say and that’s that. Perhaps Heisenberg and Bohr are right? I’m a less-is-more thinker, and I find their minimalist positivism appealing.
Yet, I think quantum mechanics is not the final word and consider the possibility that it might be replaced by new physics of sub-quantum reality very likely. I also think intuitive concepts are essential to advance. So I like to think of the unconceivable, in quantum physics and elsewhere, with simple analogies that give a certain flavor of how it could be conceived.
Interpretation of quantum mechanics like those of Everett [Carroll 2019] and Bohm [Kay 2024, Bricmont 2016, 2017], assisted by the mathematics of decoherence, suggest thinkable pictures of what happens behind the scenes. So I keep an open mind, and time will tell.
References
[Bricmont 2016] Jean Bricmont. Making Sense of Quantum Mechanics. Springer, 2016.
[Bricmont 2017] Jean Bricmont. Quantum Sense and Nonsense. Springer, 2017.
[Brooks 2017] Michael Brooks. The Quantum Astrologer's Handbook: a history of the Renaissance mathematics that birthed imaginary numbers, probability, and the new physics of the universe. Scribe, 2017.
[Carroll 2019] Sean Carroll. Something Deeply Hidden: Quantum Worlds and the Emergence of Spacetime. Oneworld, 2019.
[Heisenberg 1962] Werner Heisenberg. Physics and philosophy: The revolution in modern science. Harper & Row, 1962.
[Kay 2024] Adam Forrest Kay. Escape from Shadow Physics: The Quest to End the Dark Ages of Quantum Theory. Basic Books, 2024.
[Plotnitsky 2021] Arkady Plotnitsky. Reality Without Realism: Matter, Thought, and Technology in Quantum Physics. Springer, 2021.
[Plotnitsky 2023] Arkady Plotnitsky. Logos and Alogon: Thinkable and the Unthinkable in Mathematics, from the Pythagoreans to the Moderns. Springer, 2023.
[Prisco 2024] Giulio Prisco. Irrational mechanics: Narrative sketch of a futurist science & a new religion. Giulio Prisco, 2024.