Thanks to everyone for supporting our survey! Within a day of my asking for my subscribers’ help, the number of respondents doubled; it’s now shot up past 250—enough to draw conclusions even about weak effects with some confidence. At the end of the month, I’ll conclude data collection, and begin analyzing the responses to report the findings for you here.
Mathematics, it has been said, is the language of the universe.
Some have taken this claim in a rather geometric sense, as Galileo Galilei did in his Assayer:
Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.1
Others have taken the critical importance of mathematics in a more literal sense, claiming that mathematics is the very essence of the universe itself. Such is the view of Max Tegmark, a cosmologist:
I argue that with a sufficiently broad definition of mathematics… our physical world is an abstract mathematical structure… that our universe is mathematics in a well-defined sense.2
Others, going back to the Pythagoreans, took the idea of a mathematical universe in a wide-eyed, idealistic sense. The Pythagoreans scarcely understood mathematics as we do today, and seem to have been profoundly disappointed to discover the existence of irrational numbers—numbers which, like √2 or π, cannot be expressed as one integer divided by another:
The discovery of such irrationality was disquieting because it had fatal consequences for the naive view that the universe is expressible in whole numbers; the Pythagorean Hippasus is said to have been expelled from the brotherhood, according to some sources even drowned, because he made a point of the irrationality.3
Yet there are stranger numbers even than those which are merely irrational.
I’ve mentioned before that there is a number known as i. This number, i, is defined as the square root of -1. Clearly (so they say) no such number could possibly exist, so we call i an imaginary number. Yet, however paradoxically it may seem, i has straightforward applications in quantum mechanics—our universe runs on imaginary numbers.
Most of my readers are probably sufficiently urbane that they can smile at the naïveté of those who have never heard of imaginary numbers. But I don’t want to merely amuse my readers at this point. I would prefer that you have some emotional clarity of how strange the world of mathematics can be. And so as long as we are on the subject of quantum mechanics:
Know now, that there is a distribution, commonly useful in physics, invoked as δ(x)—the Dirac delta function.
The Dirac delta function has the property that it is zero everywhere, except infinitely close to the origin, where it smoothly yet rapidly rises up toward infinity.
Despite this, know that the Dirac delta function encompasses a finite area between itself and the x-axis, and that this area is precisely one:
(A more conventional way of understanding δ(x) is to view it as the extension of a family of Gaussian curves which are stretched vertically and compressed horizontally so that their area never changes from 1, even as they become taller and thinner, and in the limit, approach infinite height and zero width. The usefulness of this distribution is that it picks out the value of a function f(x) at any point x = a when you slip δ(x - a) into integrals containing that f(x).4)
Does This Really Matter?
Maybe it doesn’t! After all, perhaps such mathematical curiosities merely have relevance to physics classrooms, somewhere far away. Or perhaps they only have relevance to things at a minuscule level no one cares about.
Or perhaps, these concepts are absolutely critical for understanding numerous weird phenomena, like…
The Casimir Effect, where a detectable attractive force exists between two uncharged plates, many orders of magnitude stronger than gravity could act,
Superfluid Helium, where an extremely cold (near 0K) liquid climbs the walls of its container or simply slips through it due to quantum effects,
The End of Moore’s Law, which is occurring in part due to electricity basically teleporting (i.e. tunneling) through the diodes as transistors get smaller and smaller.
And it turns out that even chemical reactions are facilitated by electrons tunneling past barriers they don’t actually have the energy necessary to surmount.5
You don’t have to understand this to appreciate what mathematicians and physicists accomplished in the previous century, to see that we live on a tiny island of reality where temperatures and pressures are such that things don’t flow through each other like supercritical fluids, or communicate with one another without any obvious force being at play. But in case its hard to appreciate, well:
We live on a tiny island of reality where temperatures and pressures are such that things don’t flow through each other like supercritical fluids, or communicate with one another without any obvious force being at play.
All the intuitions we form as human beings arise from experience with physics on the solid surface of a rocky planet, protected by an oxygen-rich atmosphere, with temperatures between the freezing and boiling point of water, and pressures in the neighborhood of 100 kPa. Much hotter and we would burn; much colder and we would freeze; much more pressure and we would be crushed; much less, and our saliva would boil.
But that doesn’t mean that the physics of our cradle, formed of friendly solids, liquids, and gasses is the physics of the cosmos. The majority of the matter in the universe exists at such high temperature that it isn’t even solid, liquid, or gas, but plasma, the motion of which can influence magnetic fields, and thus other plasma, at a distance much father than ordinary matter touches. If we were made of plasma, we wouldn’t assume that touch happens over short scales. We’d take for granted that if you were to dance in a circle, you would create a magnetic field that we could feel from across the room—not just an emotional impulse to join in, but literally a magnetic push or pull.
And then what happens when the largest stars expire? What’s it like after that? Somewhere near around 1/1000th of the matter in the universe is locked up in the centers of black holes, where density is virtually infinite, and the idea of touch loses meaning, since (so far as anyone knows, as there’s no experimental evidence to rely on) everything is touching everything else simultaneously at the heart of a black hole.
The End, or the Beginning?
All of this was viewed as totally impossible before modern physics. But whatever seemed reasonable at the time, the experimental evidence was overwhelming—reality is quantum reality.
I wouldn’t bother writing about this under normal circumstances; it’s all settled science, and you can learn about it on YouTube at any university.
My concern is that most of us still don’t appreciate what happened.
We don’t appreciate that the intuitions we’ve built working with convenient numbers that we add or multiply don’t hold in the imaginary realm, or the world of infinitesimals. We don’t appreciate that whatever makes sense is not the same as what the experiments show.
And it is with this is mind that I would like to introduce you to a large body of research that carries on in the same vein as quantum mechanics did, but has, thus far, been woefully neglected. I will cover this topic in detail in my next post.
Galilei, G. (1957) [1623]. "The Assayer". In Drake, S. (ed.). Discoveries and Opinions of Galileo (PDF). Doubleday.
Tegmark, M. (2008). The mathematical universe. Foundations of physics, 38, 101-150. Available online.
Thesleff, H. (2023, September 25). Pythagoreanism. Encyclopedia Britannica. https://www.britannica.com/science/Pythagoreanism
Weisstein, Eric W. (n.d.) "Delta Function." From MathWorld--A Wolfram Web Resource. Retrieved Oct 26, 2023 from https://mathworld.wolfram.com/DeltaFunction.html
Schreiner, P. R. (2020). Quantum mechanical tunneling is essential to understanding chemical reactivity. Trends in Chemistry, 2(11), 980-989.