And of course this is exactly what she did. She was inspired to solve this problem through collaboration.
I studied years of advanced university math and was often surprised how much of it was discovered before the 1870s. Those guys rarely had PhDs, and a few were amateur French aristocrats who seemingly had a lot of time in their hands.
…one of which, fittingly, appears to be a purple rose.
Damn right about that. When I started teaching, the department had full control over admission to grad school. Over the years, the effing administration gradually took more and more control.
I know a couple stories though that I’d like to tell. A woman I used to know solved a long open problem when she was in HS. She eventually got her degrees and she was hired by McGill (she has since gone on to NYU). She was a fine productive mathematician but not a transcendent genius and never did anything else as startling. The point is that one bright new idea can give you a wedge to solve an old problem.
I also wanted to say that it is extraordinarily unlikely that she missed anything mathematical while skipping an undergraduate degree. She did miss all the other aspects of a college degree.
I looked up the conjecture and I could not understand what it was about. Far from my field of abstract algebra.
You’re not giving the rest of us much hope, here.
“This guy” was Richard Feynman.
Well, I would hope her advisor (or mentor or whatever the title is) suggests she take some intro classes in other subjects like biology or history, just to give her a better rounded education. And give her interaction with other kids of similar age.
She will not be able to get a teaching credential in Colorado. No matter how many master’s degrees and doctorates you have, by law you need a bachelor’s degree to be a public school teacher here.
Somehow I don’t think that is currently in her career plans.
Sounds like my experience with taking Latin for two years as a high school language requirement. Prior to that, I was a marginal-to-poor English grammar student. After my second year of that dead language, I was an ‘A’ student in English from then on out.
Probably not, but I wonder how many bureaucracies will not accept a PhD in lieu of a bachelor’s degree requirement.
That situation probably rarely occurs, if ever. There must be damn few individuals with a PhD that don’t have a bachelor’s degree.
More power to her. I tried to review Algebra 1 (which I got a low C in when I was in 9th grade) and it started giving me a headache withing 10 minutes or so.
Fittingly, because Mia Farrow played the character “Hannah” in the movie Hannah and Her Sisters, and she also starred in The Purple Rose of Cairo?
Around here, the standard would be to require a “bridging” year, a pre-qualification of some sort. Admission into the post-grad stream would be easy, but it would carry some kind of rider on it that you have to complete the pre-qual first.
You can call that “outright rejection” or you can call it “overturned”, but it amounts to the same thing: supervisors and admission committees don’t have unqualified power.
You probably didn’t recognize that as a quotation from American theoretical physicist (and Nobel Prize winner) Richard Feinman, who did continue teaching, and doing other stuff. And he was talking about himself: every man is the hero of his own story.
Believe it or not, in the inane questions asked repeatedly by undergraduates are pearls of inspiration in questioning the established paradigm. Having to iteratively refine your own explanation of basic principles until they are capable of being interpreted through the intuition of a novice also means having to express them in the clearest possible terms, first in your own head and then to an audience in some kind of foundational framework. Now, undergraduates aren’t generally going to ask questions about or be familiar with cutting edge research in a field (at least, beyond the pop-sci level) but they are going to keep asking ‘dumb’ questions like “Why do things have inertia?” or “Why does the Second Law of Thermodynamics prevent reversible systems?”, and even why there are answers (of a sort) to these questions, having to express them to neophytes means having to think about why these are accepted principles and the assumptions behind them. The ‘peer community’, on the other hand, doesn’t generally ask these questions because they’ve been indoctrinated in the precepts of the field sufficient that these principles are internalized and you just don’t ask ‘silly questions’. Thinking about more advanced problems also means having a very firm understanding and application of fundamentals, and that comes from having to think about them frequently, which you aren’t challenged to do so much when you are in an idealized environment with other people who also don’t spend much time thinking about basics.
I worked as a teaching assistant (TA) for engineering statics and mechanics of materials courses for a summer and semester, and that (plus writing tutorial software for the same for three years) really entrenched basic principles and let me anticipate what people do and do not understand about fundamental rigid and flexible body mechanics. It’s been over a quarter of a century and I can explain to anyone with basic algebra skills how to solve a combined loads on rigid body or generate a shear/moment/deflection diagram, and why the procedure works that way from basic principles. (Technically you do need some basic calculus for more complex deflection calculations but that can generally be handwaved away for textbook-type problems.) This has proved to be a very beneficial experience because I always start looking at an analysis from the fundamentals rather than just looking at the results, and have not just occasionally poked holes in an ‘advanced’ simulation by observing basic errors in boundary conditions or constraints that were not physically realizable and impacted the analysis.
Stranger
Feynman (linked above) put the same idea like this:
The questions of the students are often the source of new research. They often ask profound questions that I’ve thought about at times and then given up on, so to speak, for a while. It wouldn’t do me any harm to think about them again and see if I can go any further now. The students may not be able to see the thing I want to answer, or the subtleties I want to think about, but they remind me of a problem by asking questions in the neighborhood of that problem. It’s not so easy to remind yourself of these things.
No, I didn’t recognize the quote as being Feynman, but I absolutely stand by what I said in post #18, and particularly reject Feynman’s implication that it was problematic that the august faculty at Princeton’s Institute for Advanced Study had “no classes to teach, with no obligations whatsoever”. Feynman must surely have been aware of the spectacular record of accomplishments of the IAS, perhaps one of the greatest research institutes in the world.
Feynman was really talking about himself. Unlike most distinguished researchers, Feynman was a devoted popularizer of physics in the tradition of Carl Sagan popularizing cosmology. He loved to teach, and loved talking to the layman in layman’s terms. His devotion to teaching was commendable, but most accomplished researchers aren’t like that, and literally don’t have time for it.
This is an old trope in academia and Feyman clearly believed it, but I question how often it actually happens. In reality, who is more likely to ask the most probing questions, your doctoral candidate or post-doc, or a first-year undergraduate? Feyman’s field of quantum physics may have been more amenable to this claim because, as Feynman himself famously said, nobody really understands it, so in a sense there are no “dumb” questions.
More pragmatically, where is a talented researcher’s time better spent, working with his team doing research, or preparing lecture slides for undergraduate classes?
As I said, Feynman’s dedication to teaching and popularizing physics was wonderful and noble, but those who dedicate their lives exclusively to research are no less noble.
My own graduate department hired professors who did both - the belief was that, on average, a person who was only good at one and not the other probably wasn’t actually all that good at the one. Clearly, there are exceptions, but, on average, I’ve found it to be true.
And not just in academia. Here in the real world in applied science, I find the folks who are actually the best at what they do are also the best at iteratively refining their own understanding and tend to be the best at mentoring the new staff, many straight out of their own graduate programs.