Understanding formulas, shapes, algorithms etc

This is a discussion on Understanding formulas, shapes, algorithms etc within the A Brief History of Cprogramming.com forums, part of the Community Boards category; It is sometimes hard to... understand. To look at a formula and not "understand" it is like a torture for ...

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    Understanding formulas, shapes, algorithms etc

    It is sometimes hard to... understand. To look at a formula and not "understand" it is like a torture for me. The questions that run around in my mind are: "Is it proven/Where does it come from/How can I imagine it, if possible" etc...
    About the proof of a formula I can bring an example from my own schoollife:
    Today, we solved an exercise about a cone and in the end, the teacher asked us: "Do you have any questions". I asked: "I'd like to see the proof for the formula ( V = 1/3*S*H) (I am in the 9-th grade)." But he said that the formula is too hard to prove at the moment - due to my knowledge about geometry.

    Believing a formula without a proof for its correctness is a real torture for me...
    How do you cope with formulas, that you have not seen a proof for?

    Now, what about imagining a formula?
    Let's take a "simple" one:

    S = a * b
    It's for calculating the area of a rectangle.
    We are all different personalities and we all think in our own unique way and our imaginations are different too...
    When I try "imagine" how the formula works, I think of 2 sides... Then I "multiply" one side with another and the other 2 sides of the rectangle form in front of my eyes.
    That was an easy one.

    But now, let's take a "harder" formula: A = I * I * R * t (Physics. The work that the electric field does).

    I mean, how do I imagine I * I? How do I imagine I * I * R? multiplied with time!?
    Even if you are able to imagine it, imagine formulas that you can't imagine !
    How do you deal with that?

    Do we even need to be able to "imagine" a formula?

    For me, it's a real torture to live without a proof and/or "imagination" - it goes for all the areas that i have mentioned in the topic of this non-programming-related post.

    And what is understanding afterall?

    thanks for your time...
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    Sometimes you just have to accept formulas as they are. You will understand a lot more when you get to calculus and linear algebra.

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    I know that in physics, we have no axioms on which to build up our formulas and thigs. Because our physics is still in its infancy - the are things we are not even aware of yet.

    The formula i posted above has been "proved" by experiments, and that's the heart of proof in physics imho. One day, the formula will be outdated, because we will find factors that influence A and of which we aren't even aware of(that's a mere speculation, hehe).
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    Quote Originally Posted by hardi View Post
    But now, let's take a "harder" formula: A = I * I * R * t (Physics. The work that the electric field does).

    I mean, how do I imagine I * I? How do I imagine I * I * R? multiplied with time!?
    How do you imagine 30 miles per hour multiplied by 3 hours? You don't "visualize" such a thing. It is a matter of units.

    Almost all physical formulae are derived from a small set of first principles, call them "laws" or "physical axioms" or whatever. If you understand those, you can synthesize the rest of physics.

    It is easy to see that U=I^2*R*t must be correct, or at least correct to within a dimensionless factor, simply on dimensional principles.

    It's fine to try to visualize the meaning of formulas, but some simply don't have any visualizable meaning. Take the quantum wavefunction for instance. The fundamental unit of the wave function is "inverse-square-root-of-meters." It is widely accepted that the wavefunction has no physically realizable form, and nobody tries to "imagine" what it actually is. The formulas have been worked out, proven countless times by experiment, and they are simply accepted.

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    Quote Originally Posted by hardi View Post
    I know that in physics, we have no axioms on which to build up our formulas and thigs. Because our physics is still in its infancy - the are things we are not even aware of yet.
    Clearly, those things we are not aware of cannot be described. So?

    The formula i posted above has been "proved" by experiments, and that's the heart of proof in physics imho. One day, the formula will be outdated, because we will find factors that influence A and of which we aren't even aware of(that's a mere speculation, hehe).
    It's a mistake to talk about the "heart of proof" in ANY scientific discipline. The best you can do is some up with a model that predicts all the observed behaviors of whatever phenomenon you are studying. Nobody would ever claim to have "proved" anything, no matter how much evidence was available.

    However, that doesn't mean that anything and everything is possible.

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    Quote Originally Posted by brewbuck View Post
    How do you imagine 30 miles per hour multiplied by 3 hours? You don't "visualize" such a thing. It is a matter of units.

    Almost all physical formulae are derived from a small set of first principles, call them "laws" or "physical axioms" or whatever. If you understand those, you can synthesize the rest of physics.

    It is easy to see that U=I^2*R*t must be correct, or at least correct to within a dimensionless factor, simply on dimensional principles.

    It's fine to try to visualize the meaning of formulas, but some simply don't have any visualizable meaning. Take the quantum wavefunction for instance. The fundamental unit of the wave function is "inverse-square-root-of-meters." It is widely accepted that the wavefunction has no physically realizable form, and nobody tries to "imagine" what it actually is. The formulas have been worked out, proven countless times by experiment, and they are simply accepted.

    Yup, they are accepted...
    But for me, it's just hard to accept things that don't seem "natural"/"proven"/"imaginable". Maybe I just need to educate myself and open my eyes, so as to see, that there are things we are not capable of. Well, maybe we are, but we just don't know it...

    I think that our senses play a very important role in "understanding". I mean, imagine how hard it would be to teach geometry to a blind man... The more senses we have, the more ways we can link things with each other and the more we understand...
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    Quote Originally Posted by brewbuck View Post
    Clearly, those things we are not aware of cannot be described. So?



    It's a mistake to talk about the "heart of proof" in ANY scientific discipline. The best you can do is some up with a model that predicts all the observed behaviors of whatever phenomenon you are studying. Nobody would ever claim to have "proved" anything, no matter how much evidence was available.

    However, that doesn't mean that anything and everything is possible.
    I just meant that the formula A = I * I * R * t might be proven wrong in the future, and for now it's good enough, beacause in practice, it works good enough.
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    Quote Originally Posted by hardi View Post
    Yup, they are accepted...
    But for me, it's just hard to accept things that don't seem "natural"/"proven"/"imaginable". Maybe I just need to educate myself and open my eyes, so as to see, that there are things we are not capable of. Well, maybe we are, but we just don't know it...
    For me, all of physics is quite easy to accept, since the unprovable axioms it is founded on are so beautiful. But even if we threw away the axioms, the formulae themselves would still predict reality quite well. So if I was forced to choose between a universe where the formulae worked "just because, for no reason," or a universe where they worked because of fundamental truths relating to symmetry, I would choose the latter understanding.

    I think that our senses play a very important role in "understanding". I mean, imagine how hard it would be to teach geometry to a blind man... The more senses we have, the more ways we can link things with each other and the more we understand...
    That's a very dangerous assumption to make. Beethoven was deaf. And what is a sense, anyway? It's just a facet of reality presenting itself to our awareness. Closing your eyes and grasping a sphere in your hand is just as much a demonstration of geometry as looking at a sphere could be.

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    Quote Originally Posted by hardi View Post
    I just meant that the formula A = I * I * R * t might be proven wrong in the future, and for now it's good enough, beacause in practice, it works good enough.
    Just because some better description is found doesn't prove the older one "wrong." Even a simple law like F = ma is "wrong" in the sense that it doesn't predict things properly when the velocity is extremely high (relativity and all that). That doesn't mean that we do day-to-day physical calculations taking relativity into account.

    It could be found that under certain circumstances, A=I*I*R*t is not a correct description of what happens. But that would mean a new law would be found for this new set of circumstances. We wouldn't just throw the old one out the window because it isn't the most general theory.

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    I just meant that the formula A = I * I * R * t might be proven wrong in the future, and for now it's good enough, beacause in practice, it works good enough.
    That is true, though like Newton's laws of motion, it may not so much be wrong, but just not apply under certain circumstances that we currently think it should.
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    I agree with you both... Thanks.
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    Quote Originally Posted by hardi View Post
    I agree with you both... Thanks.
    Well don't just stop arguing! ;-)

    What proof is enough for you? The 1/3*s*h formula can be proved by calculus -- although you have to believe calculus first! So we could get to the fundamental ideas of calculus and build it up from scratch.

    But consider an alternate experiment. Make a hollow cylinder, cross section s, height h. Made a solid cone, base area s, height h. Stick the cone in the cylinder. Pour water into the cylinder. Now remove the cone. The cylinder is exactly 2/3rds full! Is this a "proof?"

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    Quote Originally Posted by brewbuck View Post
    Well don't just stop arguing! ;-)

    What proof is enough for you? The 1/3*s*h formula can be proved by calculus -- although you have to believe calculus first! So we could get to the fundamental ideas of calculus and build it up from scratch.

    But consider an alternate experiment. Make a hollow cylinder, cross section s, height h. Made a solid cone, base area s, height h. Stick the cone in the cylinder. Pour water into the cylinder. Now remove the cone. The cylinder is exactly 2/3rds full! Is this a "proof?"
    First of all, I consider this a discussion, not arguing. I am not trying to convince anyone in anything - just trying to share my experiences and views.

    And secondly, it would not be a proof, because to know that the cylinder is 2/3rds full, we have to measure it somehow, and measuring is not accurate(I don't know of any case where we could say that measuring is accurate). Even if the measurements would be accurate, making 1000 different experiments would still not be enough for me - I'd like to see a mathematical proof
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    Quote Originally Posted by hardi View Post
    First of all, I consider this a discussion, not arguing.
    I didn't intend any negative connotation. I view pretty much any exchange of thoughts as an argument. I'm just trying to keep you talking.

    And secondly, it would not be a proof, because to know that the cylinder is 2/3rds full, we have to measure it somehow, and measuring is not accurate(I don't know of any case where we could say that measuring is accurate). Even if the measurements would be accurate, making 1000 different experiments would still not be enough for me - I'd like to see a mathematical proof
    Consider this: if there is no way to accurately measure a thing, is there even any quantity to be measured at all? Think about what it means for some property to have a particular quantity in the first place.

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    I think that by "I'd like to see a mathematical proof", hardi is trying to say that an example does not prove a universal statement.
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