Tag: Hopf fibration

Wheels Within Wheels: The Hopf Fibration And Physics II

Posted on by Jochen Szangolies

by Jochen Szangolies

Crucial step in the proof that 1 + 1 = 2, coming on page 379 of Russell and Whitehead’s formalist tour-de-force Principia Mathematica.

In the last column, I have argued against the idea that understanding in mathematics and physics is transmitted via genius leaps of insight into obscure texts rife with definitions and abstract symbols. Rather, it is more like learning to cook: even if you have memorized the cookbook, your first soufflé might well fall in on itself. You need to experiment a little, get a feel for ingredients, temperatures, resting times and the way they interact before you get things right. Or take learning to ride a bike: the best textbook instructions won’t keep you from skinning your knees on your first try.

Practical skills are acquired through practice, and doing maths is just such a skill. However, mathematics (and physics by extension) may be unique in that it likes to pretend otherwise: that understanding is gleaned from definitions; that the manipulation of symbols on a page according to fixed rules is all there is to it. But no: just as you need an internal, intuitive model of yourself on a bike, its reactions to shifts in weight and ways to counteract developing instabilities, the skilled mathematician has an intuition of the mathematical objects under their study, and only later is that intuition cast into definitions and theorems.

At the risk of digressing too far, this is a general feature of human thought: we always start with an intuitive conception, only to later dress it up in formal garb to parade it before the judgment of others. We are not logical, but ‘analogical’ beings, our thoughts progressing as a series of dimly-grasped associations rather than crisp step-by-step derivations. If we do find ourselves engaged in the latter, then as a laborious, explicit, and slow ‘System 2’-exercise, rather than the intuitive leaps of ‘System 1’.

Indeed, it couldn’t be otherwise: how should we know whether a definition is accurate, if we didn’t have a grasp on the concept beyond that definition? Read more »



Tuesday, August 20, 2024

Music Of The Spheres: The Hopf Fibration And Physics

Posted on by Jochen Szangolies

by Jochen Szangolies

The particles of the Standard Model (and gravity). Image credit: Cush, CC0, via Wikimedia Commons

Modern physics in its full mathematical splendor introduces an array of unfamiliar concepts that daunt the initiate, and often even bewilder the pro (or is that just me?). A part of it is just that it’s a complex topic, and its objects of study are far removed from everyday experience: a quark or a black hole or a glueball is not something you’re likely to find on your desk. Well, maybe the latter, if you’ve been sloppy while crafting recently, but as so very often, physicists further confuse things by giving familiar names to unfamiliar concepts (spin, I’m looking your way).

But saying ‘it’s complicated’ is merely a fig leaf. Lots of things are complicated, and we manage to navigate them with ease. Many jobs involve reams of specialist knowledge, from plumbing to hedge-fond management, and even just navigating our webs of social relationships comes with considerable overhead. So what is it that makes physics special?

There is, of course, the already mentioned issue of the remoteness of its central concepts. Many of the complicated tasks we solve are so ingrained to us that we scarcely notice their complexity—the act of throwing a ball, or catching it out of thin air in flight, involves calculations that, in a realistic setting, stymied the efforts of robotics engineers for a long time. Likewise, the acquisition of language—even present-day Large Language Models (LLMs) still need to ‘read’ tens of trillions of words to acquire a degree of language fluency a human child can pick up just from what is spoken around them in their first couple of years. By comparison, an average reader would take something like 80.000 years of continuous reading time to ingest the text on which an LLM is trained!

These are tasks that, in some manner, are performed ‘natively’ by the human brain, without us noticing their complexity. Such tasks are sometimes classed as ‘System 1’-tasks in the dual-system psychology popularized by Daniel Kahnemann in his bestselling popular science book Thinking, Fast and Slow. In contrast, solving a mathematical equation or reasoning through a logic puzzle are step-by-step, explicit ‘System 2’-tasks you have to concentrate on—they’re not performed ‘by themselves’ the way catching a ball is. Read more »