Memorising formulas or deriving them?

gocuriosity

Hello,

I'm new here and this is the first thing I ever write on this forum, hence the question if this is the wrong place to post anything remains unanswered on my part.

I came across this forum doing something on the internet, I can't remember what. I'm certain though that it was related to science, most likely got stuck with homework.

Anyway, this question might seem idiotic and I don't really know where or how to start. However, many things are intuitive to me, why the definition of derivative looks like what it looks like, why 2+2=4, why you can add a 5x to the side of the equation that has -5x to make that side equal to 0, and add the 5x to the other side.

What my point is that I don't believe that I would ever have come up with it, discovered it, that I had to learn it from someone else. Eventhough I understand it as well as anything else, I couldn't as a caveman have discovered it.

Now, I'm alright at maths, I love learning maths and learn it rather quickly. What I want to know is how people on your level of intelligence and ability experience subjects such as maths or physics. Do you read books, go to classes and struggle to comprehend a new concept or have you already in your day dreams come up with a similar idea and just had it confirmed from your teachers? A maths Ph.D. for example, has he gone to classes to learn from others or did he already, eventhough he wasn't sure or had created sufficient proof, understand it from just thinking about it?

I apologise for this question, yet I really wish to know how it feels for those who are at your level.

Pythagorean

I've had that feeling many many times (how the hell is some genius come up with this?)

Not sure what your post has to do with your title.

gocuriosity

Well, mainly what I meant with the title was that how do you "do math" without knowing the formulas... I guess the naming was off, sorry. :)

Vorde

I'm very young, at least for this forum, and I already know that feeling. Today in my Multivariable Calc class the professor asked us a question about a function. We all looked at it quietly for about 30 seconds until one person answered it in about 3 words.

Did the answer make complete sense to me? Yes. Was there a chance in a billion I would have answered that question correctly in comparative time? No.

Pythagorean

Well, before the theory of relativity there was the retarded potential. It's not as surprising when you see how one leads into the other.

Some guy came up with the d'Alembertian and turned maxwell's four equations into one that allowed for gauge transforms.

They were all "standing on the shoulders of giants" as Newton would say. That's why it's important to learn the field you want to contribute to, because a lot of work has already been done and lot of your intuitions may already be wrong. But if you go to the edge of research, you don't have to waste time testing all your hypotheses and you can focus your questions better and find the next step in understanding some obscure branch that nobody else cares about (unless you want to be a celebrity scientist... then you better be good at presenting. Might even try minoring in drama.)

zoobyshoe

It took about two thousand years for the physics misconceptions disseminated by Aristotle to be finally, irrevocably corrected by Newton's compilation of the three laws of motion. In between only rare, scattered individuals conceived of and wrote down partial, tentative foreshadowings of the mechanics Newton laid out and which we accept today. I don't think there's anyone at PF who could have derived even one the three laws from scratch, and there may not be anyone alive who could have done it.

Algebra, which you admire but don't think you could have invented, likewise, seems to have been the creation of a rare individual, Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī.

Just about anyone can follow someones instructions to get from here to there, but that's a vastly easier thing than finding the way from here to there from scratch with only a vague intuition there might be something there to get to.

But there's another problem. Today I think science and math are bogged down in so many useful tools that a lot of people's creative energies are sapped just trying to familiarize themselves with those tools so they can get a job exploring some unbelievably specialized vein, or more likely capillary, of science. If there's a Galileo out there today he's probably anonymously making amazing headway in some minute sub-specialty no one outside his workplace will ever hear about.

Drakkith

Perhaps, but that how progress gets made.

zoobyshoe

It's becoming unwieldy. It's becoming impossible to learn everything there is to know about smaller and smaller areas of investigation.

arildno

It is not a question of either memorization or derivation.
If you should derive everything everytime, then you have barred yourself from doing any progress at all.

mathsciguy

Every time a very neat theorem is presented in front of me, I'm having similar thoughts. It's just a bit of a shame though, a lot of my peers, or even my instructors seem to see theorems only as a very useful tool, and wouldn't really care how they are proved or expound on them.

About memorizing or deriving, I have some rule of thumb about it. I 'memorize and not derive' if I'm in an unusual hurry, but would still insist that I have some intuition about what the theorem/formula says. Otherwise, I usually re-derive and re-prove pretty much most of what I am trying to learn.

micromass

He just invented the term algebra (or rather, he wrote a book that contained "algebra" and people after him kept using the same name), but he didn't invent algebra itself. Algebra (like in: solving equations) was already known in some form to Egyptians and Babylonians. But perhaps the major first book on algebra was written by the Greek Diophantos. The term "Diophantine equation" is still named after him.
Al Khwarizmi certainly did make significant contributions to mathematics. As an example, he solved equations using geometric shapes. But it wouldn't be right to credit him with the entire invention of algebra.

Needless to say that algebra back then and algebra now are significantly different. For example, the concept of 0 and the concept of negative numbers did not exist yet.

gocuriosity

Thank you so much for your answers. I'm really feeling more confident in myself now, seeing that you aren't so different from me afterall, hehe.

So in plain words, it is NOT shameful to go to class and read books or browse the internet to learn new scientific or mathematical concepts... I guess.

Drakkith

Of course. People can only learn so much at a time, and as our body of knowledge grows this will get worse. I don't see it as a good or bad thing, it's just the way it is.

mathsciguy

Are you, in any way, trying to reconstruct/rediscover mathematics/science from scratch yourself?

gocuriosity

I wish, but no, lol. I'm just trying to figure out how things are invented and whether the great minds, let's say John Nash, really ever struggled to learn anything or if everything came naturally. Of course things such as calculus and trig existed before his time and no human being has time to discover everything by himself in a lifetime but I'm wondering if he actually had a hard time learning anything or was the only difficult thing he ever did in mathematics discovering what he actually discovered and became renowned for.

zoobyshoe

Good point, but in some circumstances (like when I feel lazy) it's just easier to pin responsibility onto a specific individual for things that were actually the result of various important contributors. Newton did not personally discover any one of the three laws, and some people think SR should be called The Einstein/Lorentz Theory. "The Galileo/Royal Society/Euler Laws of Motion" is a bit unwieldy. Euclid, likewise, was merely a collector and editor of centuries of prior knowledge from all over the ancient world, but "Euclid's Elements" has a nice, compact ring to it.

zoobyshoe

The problem is that people are born just as dumb as they have always been but the bar keeps getting raised higher and higher on how much they have to learn to keep their heads above water.