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Derivations in textbooks

  1. Aug 12, 2018 at 1:48 PM #1
    G'day,

    Would anybody else agree that derivations of key results in physics texts - school or college level - proceed in a manner that betrays no compunctions in the mind of the author about fore-knowledge of the pre-known form of the final expression?

    I have a strong conviction that the great ones of yore, those responsible for determining the underpinnings of modern science, didn't proceed by thinking in such terms as, for example..."Next, dividing everything by j*w, I get:..."

    Would appreciate a discussion.

    Best regards,
    wirefree
     
  2. jcsd
  3. Aug 12, 2018 at 3:23 PM #2
    I think it doesn't really matter how you prove something unless your proof is wrong.

    For example:

    Let's say I know A for some reason.
    From A I show we can derive B, and from B I discover C, D and F.
    Now let's say I find a way to prove A given D and F.

    Ok, now imagine I forget about A. What am I going to do? Well, I could use D and F to get A, but - according to your reasoning - this would be cheating because I used A to derive D and F :wideeyed:. If you say so then you are implying that A is "more important", "more fundamental" than B, C, D and F, but I don't think this would be a valid assumption - I mean, it is totally arbitrary.

    Hope I explained myself clear enough

    PS: if you find a very nice and smooth proof why would you ignore it just because who solved the problem first used a different approach? It doesn't make much sense. Also if you find an other proof for an argument maybe you will see it under a different point of view and you will be able to discover new things, even if the proof seems artificial.
     
  4. Aug 12, 2018 at 4:23 PM #3
    .

    What gives you such a strong conviction? Take the Pythagorean Theorem. There are many ways to prove the result. Have you seen Euclid's proof (proposition 1.47 of the Elements)? Why do you think he chose to prove it that way? Hint: read 'alternate methods of proof' in the comments below the proof on the linked page.

    Have you read Newton's Principia? What do you think of his proof of the shell theorem (propositions 30 and 31 in the link). Do you think that Newton had already had an inkling of how gravity behaved in (and near) spherical bodies before he worked this out?

    Newton still worked in the spirit of axiomatic geometry. Have a look at Lagrange's Analytic Mechanics and you'll find classic results (once stated geometrically) re-stated (and derived) with the new tool of analysis.

    I'm not sure I understand the issue.
     
  5. Aug 13, 2018 at 12:48 PM #4
    I used to feel this way, like it was cheating to always know the answer that you were working to derive. But after practicing physics a lot more, I realize that this is kind of how it works. Much of the time you know or have a feeling of what the answer should look like, or how it should behave. So proceeding with a derivation has a lot to do with your intuition about what form the answer should take.
     
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