Please don’t encourage Mathochist to dumb down his posts. (Well, you can if you want to, but I’m going to voice the opposite opinion.) First of all, your claim that his posts are indistinguishable from nonsense for anyone without doctoral training in mathematics is untrue. I haven’t had doctoral level math training (*), and I can usually understand a good bit of what he posts. Sometimes it goes over my head, but that’s OK – there are plenty of people here who will post things I’m sure to understand completely. It’s nice to have at least one or two people who will post the really technical stuff. That way, if I do completely understand what they’ve said, I’ll know my understanding of the topic is really sound.
Anyway, if you’re suggesting that Mathochist is deliberately trying to confuse people to make himself feel smart (“On the other hand, it’s harder to impress the impressionable when you do it my way”), I don’t agree. I think it’s understandable that when someone asks a question on a topic you’ve been studying for years, you’d want to share what you’ve learned in all its glory.
Anyway, saying “Don’t post anything you need a degree in math to understand” seems like it would limit the benefit mathematicians get from this board without really benefiting anyone else – after all, there are already people attempting to explain tensors in manners of widely varying technical precision. And anyway, it was a technical thread to begin with. If someone posted a C++ question, should I avoid responding with code because the majority of the board (I’m guessing) doesn’t know C++ syntax?
(*) – I do have an undergraduate degree in physics/math, and have had a year of graduate training in physics. I hesitate to mention that, because I like to exploit my anonimity to ask dumb questions without feeling embarassed – for instance, I know how tensors transform, and could have probably told you if I thought a bit more about it, I just totally spaced on it, and wasn’t getting how the transformation matrix was different. Anyway, I shouldn’t worry – I trust no one here would be so obnoxious as to say things like “You have a B.S. in physics and you don’t know the answer to that?”
It’s funny how “trivial” often seems to mean "Difficult enough to prove that I don’t think it’s worth it. It’s as if the contiuum of triviality/non-triviality forms a closed loop – anything complex enough becomes trivial again.
I wonder if the desire for more illustrative examples and less abstract rigor depends upon age, or more exactly on when the scientific and technical education occurred.
When I took undergraduate schooling from 1946 to 1950, engineering training was not nearly as rigorous and mathematical as it is now. I went for 1 year of advanced engineering at UCLA in 1960 and was surprised and interested at how much more rigorous was even the undergrad work that I had occasion to observe.
My background is no longer a problem because I’m long retired but I think that someone starting out now with the educational method that I was trained in would have a hard time keeping up.
You’re likely confusing difficult with tedious. There are a great many results which are obvious (in the sense that anyone with a certain amount of mathematical sophistication would be floored if they didn’t hold) that are proved by long standard diagram chases or a lot of tedious algebra. Actually giving the proof in class would simply eat up the microcentury without enlightening the students in any way.
Examples:
Proving that given an associative algebra A, setting [a,b] = ab - ba defines a Lie algebra. The Jacobi identity is basically a long expansion of polynomials that fills half a page, which one then checks to see that each term cancels against another. It’s rote enough that a high school algebra student could do it, and doesn’t lead to any deeper insight of the theorem. It’s left as an exercise because (as my professor said to us), “I had to write it out when I was a graduate student. Now it’s your turn.”
At various points when I’m teaching multivariate calculus, there come lists of properties that just aren’t worth the time to prove. All the derivative properties of vector-valued functions of a single variable are proved in pretty much exactly the way the students have already seen in earlier classes, proving all six or seven results would take a whole class period, and working out these formulas provides valuable practice in manipulating these sorts of functions. So, I write the list up on the board, wave my hands at one of the more difficult ones (say, the Leibnitz rule for the dot product), and let them work out the rest on their own.
If you really can’t see how to do a “trivial” proof on your own, go to office hours. I really can’t emphasize too much how many people are content to sit back and assign blame to the professor for being unclear rather than take charge of their own education. This isn’t high school anymore and we’re not expected to spoon-feed you.
Let me add to the crowd that appreciates technical discussion. I don’t have a math degree, and admittedly, much of what gets posted by Mathochist and the other mathematicians on the board goes over my head. But there is also a fair bit that I do understand, and even if there were not, so what? If I were to go into a thread in Cafe Society, say, on guitar-strumming technique, I probably wouldn’t understand most of that, either, but that doesn’t mean that the folks who do understand should be forbidden to discuss it.
And Feynman once quipped that to a mathematician, any statement which has already been proven is trivial, which leads to Feynman’s Theorem: “Mathematicians can only prove trivial theorems”, since once a theorem is proven, it is then trivial :).
But on topic: I’m a physicist, specifically a relativist, which means that I use tensors a lot. But I don’t have a rigorous notion of what exactly they are. I do know a lot of things about how to use them, however, and I’m apt to define a tensor as any object which can be used in those ways. In physics, a tensor of rank higher than 1 is itself almost never of interest. What is of interest is particular combinations of such tensors which produce scalars. Even something as simple as a displacement vector is really only useful in terms of its components (the contractions of the vector with the dual basis vectors you’re using), or its magnitude (the square root of the contraction of the vector with its own dual) or the like. With a tensor, I can perform these contraction operations by means of operations on the components, generally expressed using the Einstein summation convention, and eventually get a scalar result. And despite the fact that I used the components to perform that calculation, I would get exactly the same result if I used any other coordinate system to do my calculation. So I would say that any object which displays that property is a tensor.
You don’t think a tensort of higher than rank 1 is interesting! I understabnd what you mean, i.e. that in our heads we like to think about what’s going on in terms of concepts such as speed, distance, time, etc, not stress-energy (or some other tensor), even though we can certainly give a direct physical interpretation to the (components of) the stress-energy tensor. Personally relativity is one of my favourite areas of physics, but the abstract mathemtical side appeals to me in patricualr (wich is why i suppose I like looking at alternative ways of tackling it such as rapidity and Bondi k-calculus) , so I find the metric tensor just as interesting as the proper time.
“Dumbing down” were his words, not mine. There are better ways of introducing the notion of a tensor than through category theory. Tensors predate category theory by at least fifty years.
Anyone who believes that studying math is masochistic must also believe that teaching math is sadistic.
Sure, but where’s the harm in seeing as many different ways as possible of defining a tensor. Especially since the original poster said: “I’m interested in how different people define a tensor” Not, “please give me an introduction to tensors.”
To devote oneself to pure mathematics is a harsh proposition. It can be a grueling process (physically as well as mentally) to push forward and find something that is true, that you can prove is true, that nobody else has proved true before, and that you can prove nobody else has proven true before*. The hours, while on paper are short, in reality are long since a problem has a tendancy to move into your mind and sit there until you finally see the solution (if ever). The pay, compared to what you could be making and what your friends do make, is awful. The job market is contracting. Nearly every single person in this country who hears what you do will say the exact same thing, “I was always horrible at math”, with the subtext of, “everyone else was horrible and you’re not, so you’re weird and possibly to be avoided”.
What would you call someone who willingly put themselves through this because they derived some inexplicable pleasure from it?
speaking of which, I’ve found that my generalized notion of a tensor product has been almost discovered before by Peter Freyd.
On second thought, I believe you are simply suffering from an acute case of being a late-stage doctoral student. Hang in there and good luck with your career; I’m sure everything will work out. Remember, if you don’t get your dream faculty job, you can always go into finance. Then you can get rich enough to retire at 40 and do math in your free time.
No, you miss my point entirely. The handle is a wry commentary on the amount I put myself through for the sake of my (a)vocation.
Further, even if I did regard mathematical research as masochism, it hardly follows at all that I regard mathematical education as sadism. If you can’t tell the difference between research and teaching…
Not exactly. I think that, for purposes of physics, a tensor of rank higher than 1 is uninteresting in and of itself. The higher-rank objects most assuredly do exist, and the interrelationships between them are fascinating. But when it comes time to actually do anything with one, you’ll almost always reduce it down to a scalar by some means or another first. Of course, it’s highly significant that these scalars are made out of higher-rank tensors, and the tensors, like all mathematical concepts, are an interesting object of study in their own right, but that study isn’t physics.
Isn’t this true of the application of all mathematics to physical problems? For example, the end result of the most complex integration is an algebraic formula that you can plug numbers into and get a number that tells you the stress, gallons/sec, amperes, or whatever.
You forget that the Masochist enjoys pain. Hell, when I was doing math/physics research at college it was a pain, but it hurt so gooooood. Then I discovered I liked money and job security so I became a database admonisher. In some ways I failed the chance of doing science no one had done before, but I don’t miss it too much.
P.S. I have never heard of a (serious) post grad doing short hours, when you are driven by your vocation 12 hour days are the norm.
