Memorising formulas or deriving them?

In summary, the conversation discusses the feeling of awe and admiration towards those who have a deep understanding of mathematics and physics. The participants also touch on the history of these subjects and how they were developed by rare individuals, as well as the challenges of keeping up with the ever-expanding body of knowledge in these fields. They also mention the importance of learning from others and building upon existing ideas and theories.
  • #1
gocuriosity
3
0
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.
 
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  • #2
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.
 
  • #3
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. :)
 
  • #4
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.
 
  • #5
gocuriosity said:
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. :)

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.)
 
  • #6
gocuriosity said:
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.
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.

Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة‎, 'The Compendious Book on Calculation by Completion and Balancing') is a mathematical book written approximately 830 CE. The book was written with the encouragement of the Caliph al-Ma'mun as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance.[16] The term algebra is derived from the name of one of the basic operations with equations (al-jabr, meaning completion, or, subtracting a number from both sides of the equation) described in this book.

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.
 
  • #7
zoobyshoe said:
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.

Perhaps, but that how progress gets made.
 
  • #8
Drakkith said:
Perhaps, but that how progress gets made.
It's becoming unwieldy. It's becoming impossible to learn everything there is to know about smaller and smaller areas of investigation.
 
  • #9
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.
 
  • #10
gocuriosity said:
Hello,
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.

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.
 
  • #11
zoobyshoe said:
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.

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.
 
  • #12
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.
 
  • #13
zoobyshoe said:
It's becoming unwieldy. It's becoming impossible to learn everything there is to know about smaller and smaller areas of investigation.

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.
 
  • #14
gocuriosity said:
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.

Are you, in any way, trying to reconstruct/rediscover mathematics/science from scratch yourself?
 
  • #15
mathsciguy said:
Are you, in any way, trying to reconstruct/rediscover mathematics/science from scratch yourself?

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.
 
  • #16
micromass said:
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.
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 anyone 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.
 
  • #17
Drakkith said:
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.
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.
 
  • #18
zoobyshoe said:
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.

Yep.
 
  • #19
zoobyshoe said:
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 anyone 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.
Actually i think in that fashion SR should be called Lorentz/Fitzgerald/Poincare/Einstein theory :).
But I think you should give more credit to Euclid for Elements. Not only did he contribute with lot of proves (if I'm not mistaken), but he invented the first axiomatic system with it.
 
  • #20
xAxis said:
Actually i think in that fashion SR should be called Lorentz/Fitzgerald/Poincare/Einstein theory :).
But I think you should give more credit to Euclid for Elements. Not only did he contribute with lot of proves (if I'm not mistaken), but he invented the first axiomatic system with it.
I invite you and Micromass to rename all these things properly giving every contributor due credit. This new naming system will, of course, be called "The Zooby System".
 
  • #21
zoobyshoe said:
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.
That is a gross understating of his contributions.
 

1. How can I effectively memorize formulas?

Memorization of formulas can be achieved through practice and repetition. It is helpful to break down complex formulas into smaller parts and understand the underlying concepts. Visual aids, such as flashcards, can also aid in memorization.

2. Is it better to memorize formulas or derive them?

It is important to have a balance of both memorization and derivation when it comes to formulas. Memorization allows for quicker problem solving, but derivation helps with understanding the principles behind the formula and its applications.

3. How can I prevent mixing up formulas?

One way to prevent mixing up formulas is to understand the differences between them and their specific uses. It is also helpful to practice using the formulas in different scenarios to solidify their purpose and application.

4. How do I know which formula to use in a problem?

To determine which formula to use, it is important to carefully read and understand the problem. Identify the given information and what is being asked for, and then choose the appropriate formula that relates to those variables.

5. Can I use a calculator to help with memorizing formulas?

While calculators can be helpful in solving problems and checking answers, it is not recommended to rely on them for memorizing formulas. It is important to have a solid understanding of the concepts and principles behind the formulas rather than solely relying on a calculator.

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