Alogonal mechanics

Unthinkable concepts and inconceivable thoughts in mathematics and quantum physics.

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].

Niels Bohr, standing at blackboard. Picture from Picryl.