As it happens, one cannot become a mathematician without learning from other mathematicians.
Well, I dare say that there are occasional weird geniuses who could manage it simply by reading, but these people are extraordinarily rare. Math departments often have one or two unaffiliated but tolerated kooks hanging around the place who fancy themselves members of this rare breed, peddling their theories of quantum marshmallow gravity or whatever. They are tolerated, I think, because the main difference between these kooks and the Real Mathematicians is that the latter agree on their underlying system of abstract craziness, and can generally reproduce one another's fancies. Occasionally, some fancy or another manages to match up with the real world and someone can build the Hydrogen Bomb which always exciting.
In any case, mathematicians are aware, I think, that the veil between kookery and mathematics is fairly thin, so the quantum relativity guys are not dragged off by security as long as they don't pee in the staff lounge, or drink too much of the coffee.
Mathematicians are mildly obsessed with the family tree of master / apprentice relationships, who advised whose thesis, and so on. Also, with who has published papers with whom. Essentially all mathematicians can trace themselves back to Gauss, by one path or another. There is little in the way of "islands" in mathematics.
Hanging around with other mathematicians for many years is necessary to master the processes and techniques of mathematics. The kooks in general have no such mastery, but simply ape the general shapes of things, poorly.
A side-effect of hanging around and learning the methods, though, is that one also learns a surprising amount of aesthetics. The bearded fellow at the front of the lecture hall intones "A beautiful theorem of Banaschewski" and later your advisor, staring at your scribbled idea, mumbles "well, it seems right, but it's very ugly" and so on. A few years of this and you've got a pretty good notion of what pretty mathematics looks like, as opposed to the other sort.
The kooks may, interestingly, have some grasp on the aesthetics, but lacking the mechanics, being as it were an island of one worker with no connection to anyone else, they don't actually have anything interesting to say.
Painting is, I believe, similar. The methods and processes of painting are complex enough that if you want to be any good at it, you've got to hang around with someone who knows this stuff. You can struggle along with some books, but if your experience is anything like mine, you cannot really even learn to draw from a book, let alone paint.
In the process of learning how to get paint to stick to canvas, and how to mix green, and where to hold the brush, you're likely to learn something of aesthetics. Indeed, in the fine arts, it's likely that your teachers will be at some pains to pound some such notions into your head.
While Painting with a capital P may not be quite as thoroughly connected a family tree as mathematics, I do think that there are not too many small islands of thought, technique, and method. Mostly, everyone learned from someone else, and they learned from someone else, who collaborated with someone, and so forth. Perhaps not back to Leonardo, but many steps.
Consider now photography.
Photography is at its heart, as I have argued elsewhere, entirely a process of selection rather than creation. The mechanics of the process are so thin as to be negligible. The essential thing in photography consists of nothing but the aesthetic considerations, all those pesky issues of what it looks like, what it means, why is it good, or bad, or stupid, or monumentally important.
Photography distills fine art to exactly those details which, in other disciplines, are passed on more or less osmotically through a long apprenticeship, through long collaborations, through long associations with other workers in the field so necessary to master the techniques and methods.
Tragically, photography simultaneously removes all immediate need for any such associations. You can in fact master the processes and methods of photography simply by reading the manual. This is indeed the entire point of all commerce in the equipment and materials of photography -- to enable the amateur to more and more easily master the methods and techniques of photography without any kind of apprenticeship.
This has led to endless little islands of thought and method. I was taught, to the extent that I was taught by anyone which is not much, by my father who learned it as far as I know from the manual. I am one of an island of two. There are endless islands of one out there, the 'self-taught' photographer is practically a cliche.
This is not to say that apprenticeships are not available; but there are, and can be, no "schools" of any meaningful sort. God knows I can take a workshop from any number of assholes, none of who learned anything from anyone. Ming Thein sits at the top of a family tree, an island of photographers with, I dare say, hundreds of members. None of these photographers have much of a connection to any larger tradition, it's simply Ming's ideas which he got mainly from the manual, and perhaps a few videos on technique. Virtually none of his students will successfully have many students of their own. This island of workers might peak at 1000 photographers before they start dying off, and in 100 years there will be nobody alive who learned from anyone who learned from Ming Thein.
There are a few programs that produce little collections of bores (I'm looking at you, Düsseldorf) and the result is a few dozen or a few hundred students all doing more or less related work, all working from as well as building a Tradition of some sort. These schools are inevitably going to be drowned out. You don't have to go live with Thomas Ruff for five years if you want to take pictures, so mostly people won't. Whatever sense of aesthetics Ruff has learned, rebelled against, and expanded into his own will not last much past him. The Düsseldorf school might last 3 or 4 generations, but I predict that in due course there will be nobody alive in the chain of teacher / student with a connection to the Bechers.
In short there is no Tradition in photography, and this appears to be irrevocably the case.
I don't suggest that Tradition is the be all and end all, but it is the path through which more than technical knowledge passes. Ideas about what is beautiful, what is good, what is meaningful, and what matters are also passed down through Traditions.
Photography, unfortunately, lacks these things. For better or for worse.
Sometimes, to be sure, for better.