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Is mathematics a discovery or invention?

  1. Jan 10, 2010 #1
    As I read, this seems to be a tough question, even for Philosophers.

    Other animals too do some maths concept. A prey understands concepts like bigger/larger/smaller concepts. If the number of predators are higher, it runs away. If it's smaller, it's tries to defend. It's possible that an animal knows to count, may be up to 3.

    To me zero, positive numbers and all the arithmetic using positive numbers and zero look as a discovery. Negative numbers, subtraction that results in a negative number looks like man-made.

    Another way to look at is modifying the question a bit.

    Mathematics is applied logic.

    Is logic man's (or anything with a brain) invention?

  2. jcsd
  3. Jan 10, 2010 #2
    Kronecker once said "God made the integers, all else is the work of man" (Although I've always been more of a Cantor guy)
  4. Jan 10, 2010 #3
    I think of mathematics at just a set of theories and proofs, which are invented by mankind. If man didn't exist then there would be no mathematics. The same can't be said for things like gravity.
  5. Jan 10, 2010 #4
    By "man", you mean any life form that is capable of thinking, rt? Theorems and proofs are higher mathematics. Animals seem to have concepts like zero, small positive integers and some relationship logic among integers (as bigger/smaller).
  6. Jan 10, 2010 #5


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    It's probably a bit of both actually.

    One had to discover or invent counting, or quantifying things.

    After a while, it evolved into more discovery of relationships of variables.

    I suppose calculus and abstract algebras were inventions, but I think there's discovery in that as well.
  7. Jan 10, 2010 #6
    As it turns out, every mathematician (professional or amateur) has an opinion. It's an interesting discussion, but it's hardly possible to convince somebody out of their opinion.

    Just in terms of logic rules, for me (personal opinion), mathematics are awaiting to be discovered. We do not add anything, our imagination just guides us towards the most fruitful path. The logic construction does not need us, and would be discovered equally as well on the other side of the Universe, or in another time, by conscious beings supported by completely different life forms. That applies to the entire body of mathematics. That also works for Universes were physical laws would be completely different, life forms would be completely different, and mathematics would still be the same.
  8. Jan 10, 2010 #7
    Mathematics is nothing more than a construct of a set of postulates. It cannot be 'discovered' because it does not exist outside its own framework. There is no such thing as a circle, a square, or a rectangle, in the real world. These are all idealizations, and because of them we have a set of rules to the game that lead to 'discoveries'.

    If I start with new assumptions (eliminate the circle, for instance), then you don't have pi, area of a circle, sectors - all our 'rules' go out the window. We may get something that works, and it may give equally elegant answers and proofs. But they wont be the same as our system of mathematics.

    Some ancient cultures used different number base systems than what we use today, as an example.

    I think this is a cute question, with good answers from both sides.
  9. Jan 11, 2010 #8


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    I see math as a set of inventions in which we discover relationships. For example, we invent the concept of 1 and the concept of 2, and the concept of addition, and then we discover the relationship that 1+1=2. Or something like that...
  10. Jan 11, 2010 #9
    That doesnt really matter. It is the rules that matter. Redefining symbology doesn't change anything.
  11. Jan 11, 2010 #10


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    I'm not talking about the symbols 1 2 and + I'm talking about the concepts 1 2 and +

    I suppose a better word than inventing would be defining. We define 1 2 and + then go around discovering stuff to do with it.
  12. Jan 11, 2010 #11
    Oh, I skimmed your over your post too quickly. My bad.
  13. Jan 11, 2010 #12
    I doubt that there could be any intelligent life anywhere that doesn't have some of the the same concepts as ourselves. I put rocks into a container and I count them with positive numbers, but sometimes I also take rock out of the container and I need negative numbers to keep track of those. If I account for going uphill with positive numbers, then I must also account for going downhill with negative numbers. That much must be in common to any life in the universe that has made the leap to abstract thinking that was made when our tool-making species appeared. As for the original question, I'll say "discovery" and not "invention", because what happened could not have happened otherwise.
  14. Jan 11, 2010 #13


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    IMO it's more a question of lexicography than philosophy -- "invent" and "discover" aren't really disjoint terms.

    e.g. does one invent a wheel, or discover round things roll?
  15. Jan 11, 2010 #14


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    I like to think 1 + 1 = 2 (or * + * = **) even if there is nobody around to think so.

    Last edited: Jan 11, 2010
  16. Jan 11, 2010 #15
    I would argue that this is true but not for the reason you state. As you have written this, it is simply arithmetic bookkeeping for the sake of bookkeeping.

    In contrast, we do know that conservation of mass is a property of the universe. This therefore implies that the property of addition is a fundamental property of the universe. It no longer exists for the 'sake of' existing: it is now physical.

    Heh, but then again, the critic can rightly argue that 'energy' is simply a construct!

    So, looks like it isn't a property of the universe. Its a property of the model we adopted to understand it.
  17. Jan 11, 2010 #16
    Mathematics is the invention of man used to describe and help discern the meaning behind the discoveries of the world around him.

    I think there was some "pure math," too, which predated some of the discoveries...
  18. Jan 11, 2010 #17
    I think math is just like any other science. The science is an invention that explains how the universe works. We invented biology, but biology has been happening for a lot longer. We invented math, but things have been adding and subtracting for a lot longer.
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