I wasn't sure where to post this. It originated from an ongoing discussion in the Q&A, but since it's not strictly an astronomy topic, and not a question, but more of a statement, I thought I'd include it here (besides, OTB has double the readership, he, he, he... )
I had a math professor (who'd initially studied physics, along with a masters in physics, but she (yes, she) got her PhD in math so she could teach). She once told the class something I'll never forget. well, perhaps not word for word. One day, after a student had made several errors trying to solve a partial differential, she sat on the edge of her desk (she was cute! Only about 28, but married...), and said, "Intuition is a wonderful thing. Yes, we all have to learn how to do the math properly. But if we understand what's behind the math, the physical properties of the world that the tools of math can describe, whether it's physics, engineering, or chemistry, then that intuitive understanding can help keep us on track, it can help us spot our mistakes. Without that intuition, that sense of "that's not right" or "that can't be right..." we will go blindly down the road, ignorant of our mistake.
I have a minor in math. However, since I haven't used anything beyond calculus and diffy q's (eigenvectors/values) since graduation, much of anything beyond that, including the partials and transforms are lost on me, today. I know what they are, what they're for (heat and mass transfer, fluid flows, etc.). I very strongly remember most of my physics and chem, including quantum mechanics, orbitals, excitation, etc. I have a strong understanding of SR and GR, and understand the conflict between them and QM, just as the old lady in the ancient commercial said, "where's the beef."
I do not believe for a second that there is anything in the entire universe that one can not explain/educate/inform using English. Math may take 100 characters, whereas English may take 1,000, or even 10,000. But it can be done.
I'll be I can type a 1,000-word explanation in English faster than most of you can insert 100 strange characters using XP's CharMap...
And far more neophytes would be able to understand what I'm trying to say than what you're trying to say, even if we're both trying to say the exact same thing (such as, "within the event horizon, the gravitation gradient of a black hole is so strong, warping space-time so severely, that even light cannot escape.")
Twelve seconds. I timed it. I'm sure an astrophysicist could replicate that in due time. It would take me at least half an hour to dig up the math to begin with, then probably another 45 minutes to post the annotation in a correct format.
Given my original explanation, the experts understand it in a heartbeat, and the neophytes in short order, whereas a math approach is totally lost on 95% of this board composed of neophytes.
I agree that if one can learn (and retain) the language of math, that it is, in many situations, far more elegant. But when one cannot, English can make do, provided it's properly stated. I will say this - English will not allow for advancement in the field, except to propose new ideas which may not have been thought of before. In that case, we must revert to rigorous study in math, physics, and the other scientific disciplines.
My first introduction to science occurred when my father sat me down at the kitchen table when I was just five and said, "If we took a string that we couldn't stretch, wrapped it around the Earth such that it was taught, then added 1 inch to that string, assuming a perfectly spherical Earth, how far up from the surface of the Earth would that string rise?"
I thought for several seconds, then said, "six inches."
My intuitive answer as a five year old was close to the truth and I'll personally send a PayPal buck ($1 USD) to anyone who correctly solves this problem. PM me.
Intuition. It's a wonderful thing. To know the math is one thing. To understand the physics behind the math is like having a tour guide, or a tracker, hunting for sign. They may not know the biochemistry of what keeps either themselves or the game they seek alive, but a good tracker will find the game and they'll be feasting before nightfall.
I believe the same is true with all scientific disciplines. I picture orbitals in my head. I picture pdfs (probability distribution functions) in my head. I picture what happens near Schwarzschild radii in my head. I picture what happens with accretion disks, the generated magnetic fields, etc., in my head.
Why? It's not because I cannot understand the math any more (although that's becomine more of a factor the less I use it). It's because I've been so long out of practice that I cannot actually do the math any more (well, on rare circumstances I've forced myself to, pulled out the texts, and have made a decent stab at it, with acceptable results - not like some of you, though).
What I can do is take a look at an equation and most of the time say, "that's a LaPlace transform," or other stuff. What I can no longer do is solve for higher math problems. Give me six months in a full-time refresher course, and I could be back up to speed, but I have to work for a living, and don't have a lot of time for it (I could sacrifice the time I spend on BAUT, yes. However, I was never more than a passable mathmetician, and to me, BAUT is both recreation and learning, but not in a mathmatical sense, but in an intuitive sense).
Back to Math vs Intuition vs English...
1. Math is the appropriate language used to describe the precise relationships between physical (including SR/GR, QM, and hopefully, one day, GUT).
2. English can be used to describe the various phenomena of the Universe. However, to do that effectively takes a level of skill that escapes most people, and requires approximately 10x as many characters.
3. Intuition. Ahhh... This is what ties the two arenas together, but not as a middleman. Rather, as an interpreter, one who understands both realms.
You see, the following axioms hold true:
1. One can learn the math (but not how to convey the meaning in English) and understand (intuitively) nothing.
2. One can learn English (but not how to convey the meaning in Math), yet still understand (intuitively) everything.
3. One can learn both English and Math, and while remaining proficient in the former, but lacking in the latter, continue to retain a complete understanding of what's really going on throughout the universe.
So which is the greater language?
I would say neither, although I've known some very non-math people to have a very tight grasp of what's really going on.
As for me, the reason is that until I first learned the math, I didn't understand the physical universe. However, I internalized most of what I learned, as either English or mental/visual/pictorial/animated pictorial representations of what I was learning (I'm a very visual person, not a particularly math-oriented person like some of the others on this board). I've also worked as a word smith (technical writer); I find it easy to translate complex topics into paragraphs and sentences that most people understand (I'm not always right, but hey! That's one of the reasons I joined this board, both to learn something new and because I actually invite others who're more in the know to keep me straight!).
Back to the OT.
Math vs Intuition vs English
I stipulate that all three are equally important, if not vital (and feel free to substitute the word "Language" for English).
Without at least a basic understanding the math involved with any given topic, we have no guide. We're blind. My favorite example of this is the 3-4-5 triangle. During one test in Statics, the prof had inadvertantly set up problem as a 3-4-5 triangle. (3 is one side, 4 is the other, right triangle with the first, and 5 is the hypotenuse). My answer was, "3-4-5 triangle, but with initial values of 9 and 12, thus the answer is 15." He flunked me on that test, giving me a 66% (there were only 3 questions). I took it to the dean, and thankfully, the dean said something like (actually, remotely distant), "Golly, gee whiz, <prof>! He got the answer right, and showed his work both on the other questions. You put such a simple question/solution before him that the answer is obvious. How many other students recognized that it was a simple 3/4/5 solution?" The prof said, "None, but..." at which point the dean said, "he gets an A for recognizing a valid shortcut, especially since he wrote it as such on his response."
I wish I had shortcuts for some of the higher math I had to learn. I don't, as I didn't understand it as nearly as well as basic algebra and trig, which I can visualize.
Without intuition, we have no innate understanding of what we're either examining or doing. Hopefully, math leads to an intuitional understanding of the world around us, whether it's here on Earth (usually standard Newtonian mechanics, statics, and dynamics), fluid flows, etc. - everything that has nothing to do with either SR or GR. Everything is macro.
I believe everyone, by the time they graduate high school, should have a basic understanding of these macro, non-SR/GR issues, including basic chemistry.
Intuition is nothing more than being able to picture in one's mind's eye, what's actually going on. If you do that by visualizing math symbols while equating those symbols to reality, more power to you! I lost the math part of that years ago. If it's simply having a firm grasp on the concepts, and you can still picture it, hey, I'm with you there, as that's where I am these days.
Now we get down to English, which has absolutely nothing to do with knowing, either mathematically or otherwise, what's really going on. Rather, it has to do with communication.
I've seen a LOT of what I, as an editor (yes, I've worked as an editor) would call "stilted communication." I can read between the lines and see that there's a message that you want to put forth in English but in your attempts, you just... ...can't... ...quite... ...get... ...there... ...from... ...here.
People mock you. They misunderstand you. They may discount you because of the way you word your posts (hey, been there, done that, been there and have received that - it goes both ways).
Largely, this board is in English, not Mathematica or other notation. We discuss ideas, in English.
Thus, it is absolutely paramount that we do just two things:
1. Speak clearly and concisely when discussing the details of the topic at hand. Feel free to add anecdotes (as I have, above). Sometimes they help.
2. Rambling on ad nauseum does not constitute an "anecdote." It constitutes "rambling on."
As an editor for some major 'zines, many of you have read, I'll further subdivide this, translating what we do in the publishing department into board management (and I've been involved in board management for more than a decade before I was ever published as an author or became an editor):
2. a. Include content pertinant to the OP.
2. b. If it's not pertinant to the OP, separate into a new thread. Caution is advised, however, as if it has any substantial relation to the OP, either splitting off into an OP or creating a new OP is probably not in the best interests of the board.
2.c. If at all possible, attempt to find (through the search window in the upper right-hand corner) a thread which has already discussed what you wish to discuss. Take the time to read through the threads, and then post in reply to the previous post.
It's been my experience that 95% of the posts asking detailed questions (rough wag) here on BAUT are "first-timers." Someone, somewhere, began terming them "homework questions."
Sounds about right.
You folks are paid professionals, as am I.
Therefore, before you start a new post, please take the time to dig/research previous posts which have already covered the same topic.
Back to the OP....
1. One not need to understand the math behind a phenomena to understand the phenomena itself. Most of the greatest discoveries of the modern age from DaVincie to modern times originated from an intuitive understanding to experiment, and the math followed.
2. One MUST have at least an understanding of the phenomena itself (intuitively) to be able to do anything with the knowledge one has gained, regardless of how well one understands the math behind it.
Therefore, I propose the following guidelines:
1. Learn the math to the maximum extent possible.
2. Learn what's really going on behind the math to the maximum extent possible.
3. Learn English (or whatever language on whatever you wish to communicate) to the maximum extent possible, so that you can accurately and succinctly communicate your findings with others (rather than blabbering all over the pages, as I see here all too often on BAUT). I won't point fingers, except to say that if you have more than two lengthy posts back to back in the same thread... You're one of them! (yes, on one thread that I began, I'm guilty. In my defence, I thought I had a solution so something, but was incorrect in my assumptions).
To counter that, get off your high horse, through out your idea in a single post, and wait a while. You might actually learn something!
That's all I have for today. Perhaps more tomorrow.