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.

Whoopee.

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...

Three axioms:

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.

__________________
I am Mugs, of the Alien clan of Usa, Nordamerica, a Terran, of Sol. Human.

Whoever says "perception is reality" is daft. It's merely an abstraction, and often not a very good one.

Gee Mugs... How long did this take to type?

Can someone please provide me with a "cliff notes" version of what mugaliens is trying to say?

__________________
"A long habit of not thinking a thing wrong gives it a superficial appearance of being right." -- Thomas Paine

Being intelligent is not a felony. But most societies evaluate it as at least a misdemeanor. -- Heinlein

Creationists make it sound as though a "theory" is something you dreamt up after being drunk all night. -- Isaac Asimov

DatGUm it Mugaliens, just give us a Link to Wikipedia!!!



Eta: Ok, I agree with the OP on this one.

The Cliff Notes version, I think.

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- Moose's one-line review.

"your going to regret that one. You are now a colonoscope...
- Chrissy, corrupting PraedSt's wish.

Cliff notes version:
To effectively communicate your idea you need good English.
To effectively get a good idea you need strong intuition.
To effectively develop that idea you need good maths.

So to successfully get an new idea accepted, use all three.

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And the "driving on the freeway on a scooter" analogy still holds true because the pilots are sitting in 7 to 30 ton aircraft o' doom and you are running around them in your very own Meatbody, Mark I. Beep, beep.
Trying to make sense of computers, The Error Log.

Well, I don't know* if Muggsie is right or not, but I have to say: Wow, nice work. You explained yourself quite well.

* OK, I do know, or at least I have an opinion.

Mugs, I think you've made your point. I don't think it's possible to convey what you're saying in this quote with math. But now I have one less thing to take you seriously about.

When you say 100 characters of math versus 1000 words of English, I'm thinking, for many things the ratio might be closer to 100 characters of math versus 100,000 words of English. I assume, for example if you have a commutative diagram of a categorical phenomenon, that each arrow is considered a character, for example--rather than the number of characters needed to specify the arrow's length, direction, etc. if you were to write it out in TeX or something. I.e, "characters" measures time to just draw it (where the relationships between the arrows is more important than the exact coordinates anyway, so you can draw the diagram pretty quickly) rather than typeset it.

__________________
-----
Todd (Bowie, MD, US, North America, Earth, Sol System, Vega region, Local Bubble, Orion arm, Milky Way Galaxy, Local Group, Virgo A Cluster, Virgo supercluster, the universe in which spock is clean shaven)

Quidquid latine dictum sit, altum sonatur.

personal page: http://blog.astrosketches.info

l=2*pi*r

l+dl=2*pi*r2

dr=r2-r1

solve the first two equations for r,r2

r=l/2pi
r2=(l+dl)/2pi

place in eqn 3

dr=(l+dl)/2pi-l/2pi

do algebra

dr= dl/2pi

dr=~1/6 in

Mug, you seem to have the wrong answer.

Or the intuitive/wordy one: Radius to circumference is a linear relationship where radius is circumference/(2Pi), so adding something to circumference adds that something/(2Pi) to radius.

I actually think mine's shorter

__________________
And the "driving on the freeway on a scooter" analogy still holds true because the pilots are sitting in 7 to 30 ton aircraft o' doom and you are running around them in your very own Meatbody, Mark I. Beep, beep.
Trying to make sense of computers, The Error Log.

I feel that some of my recent posts might have inspired mugaliens to post this, and I also feel that he is correct. I am one of those people who has little grounding in mathematics, but I am (relatively) successful at describing cosmological concepts in English. I have good days and bad days and recently I was a bit vague in some of my posts which led to my being misunderstood - this just wastes everyones time.

How would one represent the following sentence with mathematics?

The expansion of the universe has a cumulative effect where, at distances outside of our Hubble Sphere, the light from distant objects is actually receding from us from our point of view, as it makes its way towards us from the point of view of its source.

Now, is there enough information in that sentence to do the concept justice or does it leave the possibility of introducing misconceptions to the reader? This is where I sometimes have a problem, as I know what I mean and it is difficult to put myself in anothers shoes. I would appreciate any guidance here as I want to be a useful member of BAUT and not a pain in the backside!

I'm sure there is a way to express it mathematically, but I agree with you that sometimes there's no point in doing that. Conversely, sometimes it's difficult not to lose something important in the translation from math to plain language; for example in the most esoteric aspects of QM and relativity.

Still, when you do get lost for oversimplifying, be ready that there will be critics who will ask you for the math before they keep listening to what you're saying.

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Grant Hutchison

Depends upon what you mean by "rise". If you mean, the entire circumference would expand, and the distance from the string to the earth would be the same all the way around the (perfectly spherical) earth, it would rise up about a sixth (1/2pi) of an inch, regardless of the radius.

A sixth of an inch is close to six inches, I guess, when compared to the radius of the earth.

If you meant, pinching it up at a single point and pulling it taut there, the problem is a bit more complicated. It's taken me a while (I used usoft excel) but I've come up with about 26 feet*.
Don't worry about it
This post might've been able to be broken into two, for readability and taxonomic purposes.

The line of sight distance is approx sqrt(13h) kilometers, where h is in meters. 26 feet is about 9 meters, so 13*9 is about 121, and the square root is 11 kilometers. The radius of the Earth is about 6400 kilometers, so the tangent of the angle is 11/6400, or 0.00171875. Since that is approx. the same as theta, we can find the sine of that, which is 0.001718749. Multiply the difference 0.000000001 by the radius of the earth and I get half a centimeter. Hmmm, I should have got one centimeter, about half an inch.

It's more than 26 feet??

It's not a matter of choosing between mathematical symbols and language. Mathematical symbols ARE a part of the language. It might be a specialized part that only some people use, but other examples of that aren't separated from the language and treated as not a part of it; they're just called "jargon" or something like that.