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Is Mathematics Discovered Or Invented?

Posted by kdawson on from the plato-says-yeah-but dept.

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."

16 of 798 comments (clear)

  1. Logical positivism to the rescue... by 26199 · · Score: 5, Insightful

    When faced with an awkward question, logical positivism asks: what would the answer tell me about the future?

    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? What could you predict? What would you gain?

    Nothing, nothing and nothing.

    It's meaningless; merely a matter of perception, wordplay and people having too much time on their hands.

    Oh, and the correct answer is "discovered".

    1. Re:Logical positivism to the rescue... by Anonymous Coward · · Score: 5, Insightful

      Oh, and the correct answer is "discovered".

      No, the correct answer is "both."

      The relationships and observations that we use mathematics to model are discovered. They are out there, we discover them, and then we model them. That should be obvious to all but the most die-hard of idealists.

      The language that we use to do this modeling is invented. It is also refined (i.e. slightly reinvented) over time to better fit our discoveries. That, too, should be obvious to all but the most die-hard of determinists.

      I know, this answer isn't very deep, but in my opinion the question isn't nearly as deep as it is being made out to be.

    2. Re:Logical positivism to the rescue... by Anonymous Coward · · Score: 5, Insightful

      Because squared gives you the right units.

    3. Re:Logical positivism to the rescue... by goombah99 · · Score: 5, Funny


      Oh, and the correct answer is "discovered"

      No, I think the correct answer is "Why are you asking the question?" There might be a more interesting (and perhaps answerable) question that underlies it. And how does that make you feel?

      --
      Some drink at the fountain of knowledge. Others just gargle.
    4. Re:Logical positivism to the rescue... by nine-times · · Score: 5, Informative

      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.

    5. Re:Logical positivism to the rescue... by nine-times · · Score: 5, Insightful

      No, the correct answer is "both."

      No, I think the correct answer is, "What are you asking?"

      The problem with questions like this is that it isn't clear what's in the mind of the person asking the question. What do you mean by "invented" and what do you mean by "discovered"? What difference do you see between the two?

      For example, some people will think that "invented" means "made up". So in that person's mind, if math is "invented", then it's based only on human thought, and not on real principles of the universe itself. Of course, this line of thought makes me want to ask what it would mean to be a "real principle", and what is the "universe itself" when detached from human conception, but I'll leave that aside.

      The problem I see immediately with this concept of "invented" is that real inventions don't exist independently of the universe. For example, was the wheel "invented", or did someone discover that rolling a circularly shaped object requires less energy than dragging an equally massive object? Was gunpowder "invented", or did someone discover than mixing certain chemicals together and setting fire to them caused an explosion? Was the telephone "invented", or did someone discover that you could convert sounds into electrical signals and back again by using magnets?

      All inventions are a discovery of sorts, which makes this whole question a bit nonsensical.

    6. Re:Logical positivism to the rescue... by MrNaz · · Score: 5, Insightful

      the reason that is it not (some value here)mc^2 is because c is a natural constant with a non-integer value, and all the "non-roundness" that seems to amaze you is contained in this constants. Another example of a fundamental constant is pi. Is it really so amazing that the ratio of circumference to diameter is exactly pi and not 2.143243*pi ? These numbers and constants are discovered, as they clearly exist whether or not we know what they are.

      Other parts of math do resemble invention more than discovery. E.g., the definition of mole being the number of atoms of carbon 12 needed to make exactly 12g and the Coulomb, both of which are numbers that are arbitrarily assigned to fit in with the system of measurements that has been devised over the years. All of these constants could easily be multiplied by any non-integer value and the whole system would still work.

      To answer the article's original question however, my answer would be: Who gives a toss? Math is useful. Whatever semantic definition we apply to the process by which we expand our mathematical capabilities has absolutely zero impact upon that expansion.

      --
      I hate printers.
    7. Re:Logical positivism to the rescue... by Anonymous Coward · · Score: 5, Insightful

      math is truth
      truth is discovered
      truthiness is invented

    8. Re:Logical positivism to the rescue... by Ralph+Spoilsport · · Score: 5, Informative
      we have modelled it - it's called fractal dimensions.

      Check it out. cool stuff.

      RS

      --
      Shoes for Industry. Shoes for the Dead.
    9. Re:Logical positivism to the rescue... by ZombieWomble · · Score: 5, Insightful
      You say the "simplest" formula which combines the properties of mass and velocity is a multiplication of these values - but it also happens to be the only correct one to describe this new property of matter (barring tomfoolery with constants and so forth).

      Momentum scales linearly with both mass and velocity, fields fall off with inverse square relations, and so on. You cannot change the equations describing them away from these truths in any meaningful fashion without making the equations wrong - this is not human convention or definition, it is how the universe works.

    10. Re:Logical positivism to the rescue... by knowsalot · · Score: 5, Informative
      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.

  2. I know this! by ForumTroll · · Score: 5, Funny

    It's intelligently designed.

    --
    "A Lisp programmer knows the value of everything, but the cost of nothing." - Alan Perlis
  3. Is Mathematics Discovered Or Invented? by SamP2 · · Score: 5, Interesting

    Is Mathematics Discovered Or Invented?
     
    Neither. It is defined.

  4. Parallel by blaster151 · · Score: 5, Interesting

    Are songs discovered or written?

  5. Re:It's neither by SEMW · · Score: 5, Insightful

    You can go a lot more basic than 1+1=2. Go back to the Peano axioms and you'll find that all you have to assume is the existance of "0", a "successor" function, induction, and a few trivial things like the properties of equality and addition, and you get the whole of arithmetic -- including 1+1=2.

    So you invent/assume your choice of axioms, and everything else follows from them and can be discovered at leisure.

    --
    What's purple and commutes? An Abelian grape.
  6. Just reading about this... by underworld · · Score: 5, Interesting

    It is coincidental that I was just reading about this in Paul Davies' book "The Mind of God". My opinion on the matter is fairly simple. Mathematics are invented. Period. The reason is simple... all of mathematics is an abstraction. There is no "real" thing called 1 or 2 or 3. In fact, the "integers" we use for counting things is only allowed because of the way we abstract the thing which we count. If we really defined whatever we were counting (say, coins for instance), then we could not count more than one of them.

    Here's a thought problem for you.

    You have the following in your hand:

    A one-cent piece from 1978
    A one-cent piece from 1986
    A one-cent piece from 2004

    I could have said you have 3 cents. But there is no such thing as 3 cents. 3 cents is an idea, an abstraction. It is not a concrete thing in the real world.

    So, despite all that we appear to discover about the world through mathematics, we cannot really say that math is "out there" somewhere waiting for our discovery. Rather, mathematics is our projection onto the universe. It it because of the shortcomings of our abstractions and models that our science must be continuously revised.

    For example, Newton did not discover anything about the universe. He made observations and rationalized (projected?) an abstract model which works very similarly to the observations. It's repeatable and consistent, so we call it a theory.

    But then along comes Einstein. He makes some new observations, some new hypothesis, and voila, a new theory. Even if you argue that Einstein, or anyone else for that matter, has made such discoveries through mathematical observation, that doesn't discount the fact that the observation in that case is made upon the abstraction of the universe, not the universe itself.

    In summary, mathematics is a simulation of the universe. It's an abstraction. One we humans invent. The fact that our model is observable, predictable, and so on in no way justifies the position that we are discovering some thing which pre-existed. Here's a final analogy - a computer model can be created to simulate the design of a car. We can study, observe, made predictions, corrections, and so on with the model. Yet, despite how relevant those observations, predictions, corrections, and so on are to the real car, they are still NOT the real car. The model is our interpretation, our abstraction of the car. We invent it. We make it. We project our ideas about the car into it. We do not "discover" it. The model does not exist without us.