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Mathematics proof vs Physicists proof

  1. Jan 28, 2010 #1
    Firstly, a little background: I'm a compsci major with a new found love of mathematics and particularly formal logic. To put it bluntly, I like symbols. Reality doesn't really give me a warm fuzzy feeling inside, but when I see that two algebraic equations are identical, I feel like god is smiling on me (and, as you would guess, when I make a mistake, it's like I'm being tortured by demons)

    Anyways.. How do physicists prove theoretical / abstract concepts? I understand that all principles must be held up to the light of physical evidence and measurement, but say that you don't have that. Say you're having a conversation with a friend, a thought experiment or what-have-you. You're just trying to convince them that what you say is true, but without the benefit of experiment. How do you do this without the use of formal logic like in mathematics?

    Is convincing another physicist more about experience with a long list of principles, gathered by old dudes over centuries, and just becoming familiar with them, or is there a way to derive secondary laws from the primary ones?
     
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  3. Jan 28, 2010 #2
    The problem is that there is no "proof" in science. Proof is, as you seem to suspect, a mathematical concept. So if you are having a conversation with a friend and are trying to convince them of something, you will need either experimental and observational evidence, as you pointed out, or math. Preferably both. And the more you have of these than the closer you can get to this "proof"...although, you can never truly attain it.

    If you go and try to convince someone of some theory without either, then you are probably a crackpot. How much of a crackpot is directly proportional to how extravagant your theory is, and how little evidence you have for it.

    However, you can definitely derive new theories from existing ones. For example, I can use the theory of gravity to derive the theory that if I drop a bowling ball on someone's head, they will get hurt/ die. I don't need to make a direct experiment to be fairly certain of this.
     
  4. Jan 28, 2010 #3
    you can obviously mathematically or logically derive certain things that must hold if your original findings were true. For example, if you know that if the velocity of an object is porportional to its motion with respect to time, then its motion cannot follow the equation ct^2. But the foundations of physics will always be observational and experimental.

    An interesting thought is what physicists are actually looking for, I wouldn't say that a physicist actually wants to find what is "true" about the universe, a physicist only wants to come up with a useful system with which they may view the world. "An apple doesn't fall down because of gravity, it falls down because it falls". On the other hand, finding "absolute truths" in the universe is a very romantic and directionless goal. Many things are true, you can say something that must be true - and this is how arbitrary it can be.
    I originally wanted to study math - and while doing math is fun, physics is just so much more exciting.
     
    Last edited: Jan 28, 2010
  5. Jan 28, 2010 #4

    nicksauce

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    There is still lots of mathematical proofs in physics. Look at a GR book like Wald, or Hawking&Ellis. In there you will see lots of Theorem/Proof blocks.
     
  6. Jan 28, 2010 #5
    Hmm.. Turns out I have a pdf of Wald.. Looking at it now.. Seems like there are a couple of proofs.. I'm in Calc III now, so I recognize but am not familiar with the operators they use, but that's only a matter of time.. I have no real understanding of what a tensor other than that it was described as an n-dimensional matrix. At the very least I don't feel like I'm being talked down to and asked to use my intuition for everything like my undergrad physics book does. I hate that crap... I'll add it to my wishlist.. Maybe I'll pick up a copy when I have the money.
     
  7. Jan 28, 2010 #6

    nicksauce

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    Look at Chapter 8 of Wald especially. It reads like a math textbook.
     
  8. Jan 28, 2010 #7
    Ahh... okay, I see what you mean.. This is probably the kind of physics that people fall in love with :) Not the "A train travels east at 100 km/h while a car travels north at..." kind of crap that I was exposed to previously...
     
  9. Jan 28, 2010 #8
    The proof is in the pudding.

    We have a few sets of fundamental functions, postulates, constants, and operators from which we can derive many equations which we use to calculate many things...and it works.

    For instance, newton's equations can be used to derive the ideal gas equation with a thought experiment of how ideal gasses work (spheres of mass bumping into each other in free space). The ideal gas equation is experimentally found to work with near ideal gasses with in ideal ranges. We have corrections for non ideal behavior that comes from the electromagnetic interactions and the fact that atoms take up space. These electromagnetic interactions are, fundamentally, calculated from quantum mechanics and electromagnetic force equations. When we apply these corrections the math nearly identically describes measured values. Quantum and electromagnetic force equations were both developed by different people of different fields at different times and are independently verified experimentally.

    If you looked at my Physical Chemistry book (basically a class on the molecular approach to quantum, thermo, and stat mech) you'd think it were a math book with real world examples. In fact, between each chapter is an appendix on things like operators and complex numbers and what not.

    Edit:
    And for that matter, many contributions to physics have come from mathematicians. Newton developed "modern" calculus.
     
    Last edited: Jan 28, 2010
  10. Jan 28, 2010 #9

    Andy Resnick

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    Physicists (in general) are notorious for the lack of mathematical rigor. What constitutes a 'theorem' in physics is usually a half-baked derivation that mathematicians laugh at. Which is fine: Physics deals with the real world, not an abstract human invention.

    If you would like to see a couple of mathematicians present an axiomatized branch of physics, check out "Mathematical Foundations of Elasticity" by Marsden and Hughes (Dover). I guarantee you that it looks like no physics book you have ever seen.
     
  11. Jan 28, 2010 #10
    Hmm.. https://www.amazon.com/reader/0486678652?_encoding=UTF8&ref_=sib_dp_ptu#reader-link" talk like that? Geez.. It's like, how does it make you feel when someone tries to explain centripetal acceleration by talking about what it feels like to be on a ferris wheel, something I probably haven't done for years, instead of just giving me the god damn theorem in R^2 like I'm an adult?
     
    Last edited by a moderator: Apr 24, 2017
  12. Jan 28, 2010 #11
    I suggest you never ever read books on evolution. Science sometimes has a very practical but messy side. We make models that work well, but dont conform to great mathematical rigor because you cant use math, the phenomena you are studying is way too complex. Does not mean its a waste.

    When you come up with a mathmatical formula that tells us exactly what time in a persons life a cancer cell will go ballistic I will bow to you. There are very concrete differences between science and math. And I love math, but am also skeptical of some basic foundations. Kurt Goedel is quite interesting imo.

    There are mathematical philosophers that tear up quite a few assumptions made by mathematics. We are only human... You think a hominid evolved over 100,000 years that somehow came up with a symbolic language is going to reach some sort of math nervana?

    stay humble...
     
    Last edited by a moderator: Apr 24, 2017
  13. Jan 28, 2010 #12
    Erg.. must resist urge to start debate.. erg..

    Well, much of evolution has not been formalized but that doesn't mean it cannot be. Gregory Chaitin proposed that a field of mathematics called metabiology be developed and studied that would do exactly that. No such field exists but it would be a place where Darwin's theory of evolution could be "proved", given sufficient conditions.

    And incidentally, Kurt Godel was probably the first lisp programmer :)

    http://xkcd.com/435/" [Broken] is my general feeling about the topic. I hope I don't anger too many with it :)
     
    Last edited by a moderator: May 4, 2017
  14. Jan 28, 2010 #13
    Once we fully understand the mechanism of cancer we will be able to.

    Mathematics is used to describe physics which dictates chemistry which dictates biology.
     
  15. Jan 28, 2010 #14
    So now I really want to know.. Since I'm browsing Amazon for the topic: does there exist any books that can do this level of axiomatization for undergrad physics?
     
  16. Jan 28, 2010 #15
    Physicists can and do find exact solutions to basic problems without doing an experiment. Physicists would never have found the exact fastest path of a frictionless bead sliding down a curved wire from point A to point B by experimentation. One physicist, Isaac Newton, calculated the correct curve in 1697. See brachistochrone solution and animation in

    http://curvebank.calstatela.edu/brach/brach.htm

    http://mathworld.wolfram.com/BrachistochroneProblem.html

    Bob S
     
  17. Jan 28, 2010 #16
    When you go up a level to chemistry you increase the complexity enormously. And then take a step up to Biology... its nuts. The math becomes less rigorous because the complexity in modeling is crazy hard.

    What breakthroughs do you see in biology and physics that will EVER allow us to predict when a person will die to the very second right when he is born? The complexity piles up on the probability to make this way too difficult.
    When will we see a unified field theory? Since we start with the most basic science.
     
  18. Jan 28, 2010 #17
    maybe we should just try and get the weather right...

    I watched the Jetsons, no rocket cars yet. Very dissappointing.
     
  19. Jan 29, 2010 #18
    I would not expect evolution to be proved. Its the mechanisms of evolution that we attempt to quantify with things like population genetics. I would expect that we might be able to predict conditions at a later time. This is basically what most physics does. And I dont personally see this happening in this area. I see attempts, but it will be jelly because the variables are out of control.
     
  20. Jan 29, 2010 #19

    dx

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    It's always good to be precise and think clearly, but the kind of axiomatic approach used by mathematicians is rarely useful (or even possible) in physics.
     
    Last edited by a moderator: Apr 24, 2017
  21. Jan 29, 2010 #20
    You wouldn't be able to predict much of anything. That's not quite the point. The point would simply be to have a paper-and-pencil simulation of evolution that would show that given certain properties of an organism (or whatever mathematical formalism we use instead) it would "mutate" and "evolve" towards a set of fitness constraints. What would be the immediate benefit of this? Well, you may be already aware of this, but the field of genetic algorithms would benefit most certainly. Perhaps bioinformatics or quantitative biology? Who knows.. But a mathematical proof would be nice to finally kill of any notion of intelligent design.
     
  22. Jan 29, 2010 #21
    It's useful because it helps me to understand it clearly :) A good general / abstract proof is not just an axiomatization but a very thorough and complete explaination. It carries out the very special cases. It does not bother giving actual examples (although they may be illustrative, they are not the proof itself). And most important, it does not rely on intuition. If I'm taking a test and I give some crazy answer, and I tell the teacher "but it's right because I used my intuition!", he'll probably fail me on the spot. But that's what I'm expected to use to understand certain physics principles in my undergrad book. Screw that. If it cannot be proved, take it as an axiom. But if it can, I'd like to see it.
     
    Last edited: Jan 29, 2010
  23. Jan 29, 2010 #22
    A mathematical proof would do nothing to stop these ideas. You dont utilize reasoning with people that did not use reason to establish their position in the first place.

    The ultimate reason for using math in science is to be able to predict future conditions. Or look back into the past for a possible glimpse of how we got to present status. My take...
     
  24. Jan 29, 2010 #23
    That's right.

    I don't really understand the OP's issue. Physics is about the real world, a.k.a. "our universe." How can you "prove" anything relevant to that? After all, this (all that you see & feel) could all be a dream of some butterfly. To make progress, we choose to discount such notions. But we don't delude ourselves that we can construct 'proofs' regarding how the world works. And even if you do, someone will come along and find the faults in your proof. Didn't Kant & company 'prove' that the world has three dimensions?
     
  25. Jan 30, 2010 #24
    IMO, be careful of mathematical proofs. Mathematics is a symbolic representation of reality. Not reality itself. It is a form of language and can be used to communicate, but never expresses the actual reality.

    EG. Zero is not a number.
     
  26. Jan 30, 2010 #25
    I can let go of the idea that certain things cannot be mathematically proved but are still evident by observation. My real problem is when a teacher asks me to solve a problem and I cannot give him a rigorous explanation of why the result goes such-and-such a way due to the lack of rigor of the principles of physics.
     
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