There are people who argue strongly (and have done on this blog, among others) that the problem with fundamental physics is mathematicism – that is the philosophical standpoint that mathematics rules the universe. I have some sympathy with this point of view, but I see the problem slightly differently – it is not necessarily mathematicism per se that is the issue, it is the mathemysticism that goes with it. Physicists of a certain type tend to have mystical experiences of mathematics. When they encounter an exceptionally beautiful piece of mathematics, such as octonions, or E8, they find it hard not to be enraptured by the mystical experience it offers.
Mathematicians have these mystical experiences too, but there it does less damage, because (pure) mathematicians (according to G. H. Hardy) are completely irrelevant to the real world. A physicist who glimpses the extraordinary beauty of the octonions at a young enough age is liable to be afflicted for the rest of their life with a mystical feeling that the octonions are the key to all the mysteries of physics. Much the same applies to E8. Or string theory, I suppose, although to me string theory has no beauty and no attraction whatsoever.
Such mystical experiences, unfortunately, are not really a good guide to what mathematics is actually useful for physics. This is where people like Sabine Hossenfelder start ranting about physics being “Lost in Math” – with good reason. But “Math” doesn’t necessarily stand for “Mathematics”, or even for “Mathematicism”, what Sabine is really ranting about is the “Mathemysticism” that leads off in the wrong direction. Following dreams and visions instead of cold hard reality.
When I started to get interested in the idea of E8 in fundamental physics, around 15 years ago or so, I came from a mathematical point of view that “that would be interesting”, but quickly became troubled that “interesting” had become more important than “true”. I was disturbed that there was no appeal to experiment, but only an appeal to the “Standard Model”, which apparently “explained everything” and “is correct”. A new model could only be judged on whether it reproduced the Standard Model, not on whether it agreed with experiment. These two criteria were assumed to be equivalent, but of course they are not.
I lost interest in the “interesting” models, and became obsessed with experimental truth. From that point onward, I pursued “anomalies” and other “unexplained” features of reality. I lost my faith in E8, and all other graven images, and pursued “truth” on my own. I abandoned the (Standard Model) Church of England, and took up meditation on the Buddhist model. I gradually lost myself, and became one with the cosmos. (Interesting that people often say they take up religion to find themselves, but to me the purpose of religion is to lose oneself.)
I lost myself in many ways and in many places, and meditated on the essential unity of the cosmos. Not the standard physicists’ duality between gravity and particle physics, and the contradiction between the Yin and the Yang. But the deep unity that underlies the two points of view. How to reconcile the Yin and the Yang. The Yin-stone (Einstein) and the Yang-Mills. And I sought the answer within. Within the mathematics inside myself. I wandered for years through all the mathematics I knew. I searched for clues everywhere. Sometimes manna came dropping from heaven, sometimes I was afflicted by a dreadful thirst. Sometimes I gave up hope, other times I glimpsed the promised land.
And only then, after ten years wandering in the wilderness, did I find my way back to E8. Not as a mystical vision. Not as wishful thinking about the beauty of the universe. But as inner truth. With E8, I found a way to explain all nine of the mixing angles of the Standard Model. All in the same way. That isn’t mysticism, that is cold hard reality.