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Math Science

Is Mathematics Discovered Or Invented? 798

An anonymous reader points out an article up at Science News on a question that, remarkably, is still being debated after a few thousand years: is mathematics discovered, or is it invented? Those who answer "discovered" are the intellectual descendants of Plato; their number includes Roger Penrose. The article notes that one difficulty with the Platonic view: if mathematical ideas exist in some way independent of humans or minds, then human minds engaged in doing mathematics must somehow be able to connect with this non-physical state. The European Mathematical Society recently devoted space to the debate. One of the papers, Let Platonism die, can be found on page 24 of this PDF. The author believes that Platonism "has more in common with mystical religions than with modern science."
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Is Mathematics Discovered Or Invented?

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  • by dreamchaser ( 49529 ) on Saturday April 26, 2008 @06:09PM (#23209322) Homepage Journal
    The language we use to describe mathematics is not the math itself. The math exists regardless of the symbolism used to describe it. Hence, you are incorrect. It is all discovered, but the means to describe it and put it to use is invented.
  • by nine-times ( 778537 ) <nine.times@gmail.com> on Saturday April 26, 2008 @06:22PM (#23209406) Homepage

    Yes, it's also amazing that the equation isn't 2.14332544988e=2.14332544988mc^2.

    Yes, sorry, I'm being a smart-ass and it's not polite. But c^2 is just a constant.

  • by SEMW ( 967629 ) on Saturday April 26, 2008 @06:39PM (#23209494)

    Actually its not quite e=mc^2, thats just the first term in a taylor series for the actual answer.
    No. For an object measured in its rest frame, the energy is possesses is exactly mc^2 (where m = m_0 = rest mass). The only situation where you're using a Taylor series approximation is when you approximate the energy of a moving object with speed v much less than c by mc^2 + (1/2)mv^2. But if you want the exact answer for a moving object it's easy enough to use E = \gamma mc^2 anyway.
  • by Beryllium Sphere(tm) ( 193358 ) on Saturday April 26, 2008 @06:45PM (#23209530) Homepage Journal
    The late mathematician Paul Erds used to say, perhaps metaphorically, that the most elegant proof of every mathematical theorem was written in a great book in God's library. When he came up with a beautiful proof, he would say it was one from the book.

    Feynman also felt like coming up with a proof was more discovery than invention. He said that the proof felt like it was already there all along, raising the question of where "there" is.
  • by John Hasler ( 414242 ) on Saturday April 26, 2008 @06:55PM (#23209592) Homepage
    > If it's discovered, we can patent it. If it's invented we can copyright it.

    No. If it is invented it can be patented. If it is created it can be copyrighted. If it is discovered it can be neither patented nor copyrighted.
  • by mmcuh ( 1088773 ) on Saturday April 26, 2008 @07:02PM (#23209646)
    You got it all wrong. You can not patent a discovery, and you have no copyright to an invention. You can, however, patent an invention.
  • by maxume ( 22995 ) on Saturday April 26, 2008 @07:28PM (#23209834)
    What happens when you define your system of measurement such that c=1?


    It's remarkable that the relationship between energy and mass is related to the speed of light, but c^2 really is just a constant.
  • by Estanislao Martínez ( 203477 ) on Saturday April 26, 2008 @07:30PM (#23209854) Homepage

    Suppose you had a definitive, 100% guaranteed answer to the "discovered vs invented" question. What would it allow you to do that you couldn't do before?

    That approach is not logical positivism. It's Pragmatism [wikipedia.org]. Two completely different schools. Logical positivists regard all statements as meaningless that do not have a truth value determined by either the logical system itself (tautologies, contradictions) or by contingent empirical facts ascertainable through observation. Pragmatists, on the other hand, don't believe in truth-conditional semantics; the meaning of a linguistic expression is a function of the practical consequences of its use.

  • One of my teachers in the physics department mused upon this a few years ago, and he said there was actually a paper proving from a logical/mathematical perspective that all units *had* to be integer combinations. Something to do with how we model dimensions. So, no m(1/2)s(-2) etc.

    I understand what you mean, but it's not something that hasn't been considered in scientific circles.
  • by Ralph Spoilsport ( 673134 ) * on Saturday April 26, 2008 @07:58PM (#23210024) Journal
    we have modelled it - it's called fractal dimensions. [wikipedia.org]

    Check it out. cool stuff.


  • by felipekk ( 1007591 ) on Saturday April 26, 2008 @07:58PM (#23210026) Journal
    I guess you haven't seen the golden ratio [wikipedia.org] yet then:

    g = 1.6180339887...
    g^2 = 2.6180339887...
    1/g = 0.6180339887...

    The 6180339887... thing is exactly the same.
    I used g to represent the golden ration here, although the correct entity is the greek letter phi. I couldn't get phi to show up correctly here.
  • by LaskoVortex ( 1153471 ) on Saturday April 26, 2008 @08:15PM (#23210118)

    I have mod points and I'd really love to mod you down, but I figure I'd educate you instead. (Of course you are probably wondering why I would mod you down and that you think your suggestions are intellectual, but they make about as much sense as racial supremacy arguments, which should be modded down as well.)

    Lots of the real neat roundedness of physics (different from math, though many get them confused) comes from how we define (don't forget that word, "define") properties that we require to explain phenomena or make predictions. For example, lets imagine that it is many years ago and we are the first to notice that it is hard to stop a bowling ball. Perhaps in our past we have already came up with a concept called velocity and one called mass. Clearly, the difficulty to stop the bowling ball is related to both, and we make two observations:

    • The more velocity it has, the harder it is to stop.
    • The heavier it weighs, the harder it is to stop.

    So we can take these observations into account and define a quantity which describes "hardness to stop" and give it a one-word name, like "momentum". The simplest formula that combines its component properties of mass and velocity is multiplication of these values. Or, to put it in mathematical terms p=mv.

    Now, someone who has studied the bible more than he has studied physics will look at the simplicity and elegance of the formula and call it proof that god exists. However, in reality its a matter of a simple and self-consistent method of accounting invented (or discovered if you like that word better) by people. So now please move along and convince yourself that some other area of science is proof of god. Hopefully someone else will correct you there as well.

  • by pclminion ( 145572 ) on Saturday April 26, 2008 @08:30PM (#23210226)

    What does this have to do with units?

    Absolutely everything. Many fundamental equations of physics can be correctly arrived at simply by manipulating units. The dimensions of energy are kg*m^2*s^2. A combination of physical quantities which does not have precisely this dimension cannot possibly be a quantity of energy.

    Dimensional analysis is an extremely powerful technique, and something which is learned in basic physics.

  • by knowsalot ( 810875 ) on Saturday April 26, 2008 @09:32PM (#23210636)
    I also have mod points and would love to mod you down, because education at this point is probably futile. There is a subtlety to understanding the nature of the universe that is difficult if not impossible to explain to the layman. But I will try.

    Your reasoning is subtly but fundamentally flawed. Yet as with all subtlties, pinpointing the exact nature of the flaw is difficult without having a back-and-forth conversation.

    You are right on target with respect to Ohm's law and Hooke's law -- but quite off base with your general assertion. The deep laws of physics *are* eerily symmetric, independent of our need to describe them so.

    For example, the inverse-square law of gravity or electromagnetism can be derived as a consequence of living in a 3-dimensional universe. (Integrate your favorite conserved quantity over concentric spherical surfaces and you get something that must "fan out" as 1/r^2). Nothing very suprising there. Nevertheless the deeper into exploration of physical laws you get, the more surprising interconnections pop up independent of our need to observe them.

    Your assertion that "momentum" is simply a convenient and observed quantity is both false and misleading. "Momentum" is a fundamental quantity that relates directly and ... well, fundamentally to the nature of energy, space, time, et cetera. It is particularly noteworthy that the nature of space and momentum should relate to our perception of time -- a property/dimension/quality which is quite distinct from all others in its one-way observable nature. The laws of "physics" seem to be time-invariant, yet the laws of "thermodynamics" which are equally fundamental seem to recognize that time is somehow special.

    Thus, it is misleading to imply that our physical laws are simple and elegant because we have simple and elegant requirements to describe the universe. An accurate description of the universe need not be simple -- and often it is not. For instance, I understand (although lack the mathematical sophistication to prove) that the electron spin g-factor has a theoretical value of exactly 2. Yet it is observed to be approximately 2.00232 and is one of the most precisely measured physical constants. So it is not always simple truth and beauty. Which makes it all the more surprising when the simplicity is there nevertheless.

    And while it is true that the inverse-square law breaks down at relativistic energies, even that corrective factor of "gamma" remains mathematically simple, and in fact geometrically constructable via a pythagorean triangle analysis of a certain thought-experiment.

    My point is that the easy examples are easily explained away by laymen, yet the surprisingly simple nature of the fundamental laws of the universe continue to pop up where you wouldn't expect. That is why expert scientists, true geniuses, of the sort that don't come along every day, routinely make comments about the "beauty" of physics. They have a deep understanding and "feeling" about the way the universe fits together that isn't captured by your example about momentum.

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