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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."

8 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 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.

    4. 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.
    5. Re:Logical positivism to the rescue... by Anonymous Coward · · Score: 5, Insightful

      math is truth
      truth is discovered
      truthiness is invented

    6. 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.

  2. 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.