The Reluctant Platonist: On Math and Beauty

“I confess that I find Aristotle frequently tedious, in a way that Plato is not,” writes Steven Weinberg, “but although often wrong Aristotle is not silly, in the way that Plato sometimes is.”

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The first couple of chapters of Steven Weinberg’s To Explain the World— I’m only a couple of chapters in– are extraordinarily helpful when you spend a lot of time reading about how modern scientism assumes that only the kinds of questions that modern science asks are legitimate questions.  It’s breathtaking how true this is: Weinberg genuinely sees Aristotle as a kind of failed Newton, and Plato as sinister and as having NOTHING to do with Aristotle.

Aristotle in this narrative ditched the forms entirely, and his commitment to entelechy was to do with his natural prejudices because he was a biologist before Darwin; if it hadn’t been for this it would not have occurred to him to ask what things are “for.”

It just bothers Weinberg so much, the undisciplined asking what things are “for,” — asking “what is best” is equally bad but somehow not quite as embarrassing. I picture him at a party trying to introduce Aristotle to Stephen Hawking.  He would have the quote he cites from RJ Hankinson in his head: “We must not lose sight of the fact that Aristotle was a man of his time– and for that time he was extraordinarily perspicacious, acute, and advanced.”

Weinberg has misunderstood the philosophers who have tried to explain his mistake to him; he reports on these conversations without understanding what he is describing, I think.  He complains that someone had told him that he should not expect Aristotle to be answering the kinds of questions that he is asking, but that he should allow Aristotle’s questions to stand on their own terms–  but he sees this as simply an appeal to relativism; “Aristotle’s questions and answers were right for Aristotle; don’t judge him by your standards, Weinberg”– and that, he quite rightly rejects.

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But that’s not what his philosopher friend was saying at all.  His friend was gently alerting Weinberg to the existence of a universe of legitimate questions, and thus a much bigger universe in general, than those that physics permits.

There’s one place in particular that Weinberg seems almost to consider the bigger universe.  He writes,


“In commenting on the prejudices of the pre-Socratics, I don’t mean to say that a priori reasoning has no place in science.  Today, for instance, we expect to find that our deepest physical laws satisfy principles of symmetry, which state that physical laws do not change when we change our point of view in certain definite ways. Just like Parmenides’ principle of changelessness, some of these symmetry principles are not immediately apparent in physical phenomena.  They are said to be spontaneously broken.  That is, the equations of our theories have certain simplicities, for instance treating certain species of particles in the same way, but these simplicities are not shared by the solutions of the equations, which govern the actual phenomena.  Nevertheless, unlike the commitment of Parmenides to changelessness, the a priori presumption in favor of principles of symmetry arose from many years of experience in searching for physical principles that describe the real world, and broken as well as unbroken symmetries are validated by experiments that confirm their consequences.  They do not involve value judgments of the sort we apply to human affairs. 

“…There remains a poetic element in modern physics.  We do not write in poetry; much of the writing of physicists barely reaches the level of prose.  But we seek beauty in out theories, and use aesthetic judgments as a guide to our research.. Some of us think that this works because we have been trained by ventures of success and failure in physics research to anticipate certain aspects of the laws of nature, and through this experience, we have come to feel these features of natures laws are beautiful.  But we do not take the beauty of a theory as convincing evidence of its truth.

These are amazing passages.  A good empiricist, Weinberg is observing a strange behavioral and psychological and aesthetic phenomenon in physicists: that physicists look for beauty and use it as a guide for which theories to pursue.  He’s clearly uneasy about this, puzzled.  It’s not the unease and puzzlement he might feel in thinking about the universe itself as beautiful– the objects under examination are physicists, not physical objects or even theories in physics.  He’s puzzled by this unaccountable behavior, which he observes in himself and others, this use of beauty.  And it is a use of beauty that shapes his science.  From an interview on NOVA: 

NOVA: Do you think that string theory could turn out to be just plain wrong?

Weinberg: I don’t think it’s ever happened that a theory that has the kind of mathematical appeal that string theory has has turned out to be entirely wrong. There have been theories that turned out to be right in a different context than the context for which they were invented. But I would find it hard to believe that much elegance and mathematical beauty would simply be wasted.

It’s data, this behavior that he sees in himself and in other physicists.  And Weinberg offers a hypothesis: maybe physicists have been “trained” in a sort of Pavolvian way to respond to successful theories as though they were beautiful.  This gets him out of the trap of thinking that there is something real called beauty in the world; there’s just what’s successful, which physicists learn to “taste” as beautiful.  In that NOVA interview, he compares this sense of beauty to what a horse-breeder means when they say that a horse is beautiful: this is the kind of horse that, they have learned, tends to win races; it’s an “aesthetic sense that’s been beaten into us [the community of mathematical physicists] by centuries of interacting with nature.”  

I will only ask this, if there are any physicists reading: do you remember the first time you perceived a physical or mathematical theorem as beautiful?   Was it after any kind of training, conscious or subconscious, in physics-success perception as beauty could have taken place? Or were you younger than that?  Did you only start perceiving this beauty after you had studied the history of failed predictions in physics, or made a lot of failed predictions yourself, such that this training could have taken hold?  Or is this just not the way it works?

Of course, there’s always the possibility that there is a sort of physicist-mind existing outside all physicists, actually; that is what must be trained over centuries of success and failure; perhaps individual physicists participate in it in a Jungian sort of way. That ability for a discrete community’s intellectual traditions– quite unnatural ones; the quantification of nature; the development of general theories– to be shaped in a certain way so that the community perceives successful theories as beautiful– and then for new members of that community to sort of tap in to that perception, on a direct aesthetic level, would require something like this Physicist Meta-Mind, the Archetypal Physicist in whom all physicists participate. But I rather doubt that Weinberg would be any happier about this prospect than I am.  

I’m also amused that he– who is so severe against taking the methods of one science and applying them to another– has taken the most primitive kind of memetic/evolutionary approach to explaining this beauty perception– he’s using the methods of biology to describe an aesthetic perception in physics that doesn’t have to do with the physicists’ bodies or brain structures at all.

What if there’s a world of forms, Weinberg, in one version or another; and what if transcendentals are unified?  You can see what’s in front of your face.  Can’t you see what’s in front of your mind’s eye?

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