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Have we discovered or invented maths?

  1. Mar 5, 2005 #1
    Hi everybody,
    This must have been discussed before but I would like to hear your opinions about the above question. Are mathematics just an invention, a creation of humans that helps them in their everyday life, or are they actually connected to nature, and are part of it that we just happened to discover? I personally think that we have created maths, and certain needs made humans do so, and during the years we have changed and defined them in such a way, that we can apply them in many different aspects of nature and our lives. And as for theoritical mathematics, theorems and properties are,in my opinion, just consequences of the definitions we have created, some of which have been found and proved ,while some others not.
    What do you think?

  2. jcsd
  3. Mar 5, 2005 #2
    Invented or Discovered.

    Hopefully it does not deviate us from reallity. I mean if maths is wrong then its like a piraymid; One wrong block at the bottom and the entire structures looks stranger than facts.
  4. Mar 5, 2005 #3


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    There have been similar threads here. I always throw into the mix the idea that I will expect any space faring race of extraterrestrials that we may run into "out there" to have a plane geometry (for example) that will have a great deal of overlap with our own plane geometry. If they can design and build spacecraft, they will most definitely not claim that for a planar right triangle residing in a flat space, the cube of the hypotenuse is equal to the sum of the cubes of the sides.

    So at least to a degree, I believe that the math we come up with is indeed connected to nature. Remember that the very word "geometry" means "measuring the earth."
  5. Mar 5, 2005 #4


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    Relationships between things exist in the universe. Selecting and relating them to each other is the work of minds. But minds can also invent relationships that are not between outer things, but between thoughts. So the answer is both: people may discover whether the geometry of space is euclidean or not, but they may also discover there is a principle bundle over spacetime with such-and-such a group, not something the intrepid cosmo/astro-naut will ever encounter.
  6. Mar 5, 2005 #5


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    selfAdjoint said it very nicely: mathematics is about the relationships in the abstract- what are all the possible ways that things can be related to one another.

    As to whether we invent or discover things in mathematics, the answer is BOTH!
    Mathematical theorems are "invented" when we first construct the "mathematical structure": develop the axioms, undefined terms, etc., then "discovered" when we prove them. Yes, that requires that we invent things we don't know about! The term "emergent properties" is very evocative here: properties that were inherent in the structure but that we did not know about.
  7. Mar 5, 2005 #6


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    We invent Mathematics.

    Numbers don't have a separate existence from the mind: The earth and sun were here long before us but the equations which describe the orbit of the earth were not.

    Von Neumann considers the natural numbers as a play on empty sets by the mind: The mind ponders the empty set. Conceiving the empty set generates the number 1. Conceiving the set containing the empty set and the empty set generates the number 2 and so on. Thus the natural numbers from his perspective are created independently from any physical existence.

    Mathematics fits nature so well because the geometry of Mathematics itself and that of nature are very similar: The real numbers system is dense: Between any two real numbers lies another. Nature appears dense as well: no smallest small nor largest large. Nature is "nested" (plans within plans). So too is math: Think of the chain rule. Nature is non-linear, so too is the geometry of math: Make one small mistake in solving a problem and it's not just a small error in the results.

    It makes perfect Darwinian sense for a human mind to conceive of a math which fits nature from the perspective of survival: When in New York, act like a New Yorker. When the human brain finds itself in a non-linear world, a successful survival strategy it seems to me would be to devise a system of metaphors (mathematics) which fit that non-linearity.
  8. Mar 6, 2005 #7
    Math is both a discovery and an invention. We invented a universally accepted code to express the discovery of mathematical principles in terms others will understand. Math is the language we use to encode those principles.
  9. Mar 6, 2005 #8
    Thanks for all your answers.
  10. Mar 10, 2005 #9
    Math has always existed, we have just invented the language in order to speak it.
  11. Mar 10, 2005 #10
    Humans invented math, just as humans invented god, or just as humans invented internet.

    Mathematics is the way we use to express the languange of nature. It is not, I repeat, it is not the languange of nature, it is the way we describe it.
  12. Mar 14, 2005 #11
    Book of numbers

    The astrophysicist Paul Davies asked this same question in his book, "About Time". It is an question that I cannot answer for certain but I am in the camp of those who believe that mathematics is an invention of humans.
  13. Mar 15, 2005 #12
    I am trying to learn the math of modern physics by reading Penrose's Road to Reality, and the author is clearly a Platonist.

    Maybe he'll have me converted by the time I'm finished with the book, but right now I'm incredulous, and believe that math is a construction. If math existed in a timeless and an independent manner, how come we don't have timeless and independent meanings for our symbology? Or to rephrase it, why is not easier for us to have timeless and independent meanings in our symbology?
  14. Mar 15, 2005 #13
    Is it possible that Math will fall short to answer all questions ?

    If we have invented Maths then will there be any other invention that will be more significant ?
  15. Mar 15, 2005 #14

    matt grime

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    I think you miss the point: the objects exist in this platonic realm, but the symbols are merely ways of referring to the objects, the language of referral is not itself part of the platonic realm. Language is a human invention, so it will change, the things it refers to may not.

    As it happens I do not adhere to the platonic theory. But, as someone else has observed, it doesn't matter whether or not you are a platonist or a formalist, the maths is still the same. However, as the book is about physics, platonism is not surprising.
    Last edited: Mar 15, 2005
  16. Mar 15, 2005 #15
    matt, Penrose does make it clear that "Platonism" is really just a way of agreeing to the "objectiveness" of mathematical objects.

    I know that I am missing the point. lol, it's just that I feel I understand and can manipulate mathematical objects quite well, but the language gets in my way. I sort of just feel the objects interact with one another and in my mind outputs a solution. I have heard that autistic savants do something similar - but of course I'm no where near that developed.

    So, I really should be a Platonist because I experience math as independent from its language, but then because I haven't accepted the way much of math is written, I've effectively closed myself from expressing it. And, if I can't express it - do I know it? No, not according to pedagogy.

    So, is it possible for one to know math but at the same time not know how to express it?

    It seems that if you answer "yes," you're a Platonist, and if you answer, "no," you're a formalist.

    Maybe it matters for pedagogical reasons? Which are also related to communicative reasons?
    Last edited: Mar 15, 2005
  17. Mar 16, 2005 #16

    matt grime

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    No, it doesn't matter whether or not you think that mathematical objects have an independent existence in some other reality (and "in your head" isn't what that means). All that matters is, for example, that you understand that a 2-sphere has vanishing fundamental group. It doesn't matter whether you think that there is in some other universe an object that *is* a 2-sphere, and this comes down to understanding the definitions and being able to express yourself clearly. Many people con themselves into thinking they "understand" mathematics but can't explain it. No, the standard is that unless you can explain it you can't say you understand it. Some people have better intuition than others and see the results faster, and don't need to verify details since they can see an argument as to why its true. Witten for instance is widely credited with having fanatastic intuition and little time to verify his conjectures personally - they predomintantly are true, it appears.

    The idiot savants you are talking about aren't mathematicians, they are arithmeticists - knowing how to take square roots in your head isn't mathematics.

    I think yo'ud have to explain what you meant by "objects" and "seeing things in your head". To go back to the sphere example, there is abig difference between intuitively understanding something about the sphere by visualizing it and it actually having a platonic existence.
    Last edited: Mar 16, 2005
  18. Mar 16, 2005 #17
    Thank you once again, matt. I think my relating Platonic existence with an experience "in my head" may have something to do with my forays into lucid dream studies, and the new age perception that many of these so-called lucid dreams actually take place in separate realities (or as they term it "wider reality"). I don't completely subscribe to those notions, but I can't ignore the experiences I've had that cause me to be curious of them.

    You've helped me see it differently, and therefore have helped opened doors to mathematics for me. I'll repay you by understanding as much math as I can.

    It makes you wonder how many Ed Wittens are out there who don't learn math because they think too intuitively, doesn't it?
    Last edited: Mar 16, 2005
  19. Mar 17, 2005 #18
    Are mathematics just an invention, a creation of humans that helps them in their everyday life, or are they actually connected to nature, and are part of it that we just happened to discover?

    We discover the consequences of our inventions.

    That logic/mathematics can be applied to the world is a result of the generality of logic, i.e. it applies to all existent things.

    There are no existent things that are excluded from logic.
  20. Mar 17, 2005 #19
    I don't agree.

    Logic and mathematics are mental phenomena, i.e. there is no logic or mathematics without mind. Indeed, there are no languages either.

    There cannot be any timeless things!

    Concepts are dependent on mind.
    Mind is dependent on brain.
    Brain is dependent on physical things.
    Physical things are dependent on time.
    Concepts are dependent on time!
    Last edited: Mar 17, 2005
  21. Mar 22, 2005 #20


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    Read Steven Pinker's book, Language, and discover that there are strong reasons to believe that we are wired for language, hence mathematics as well. But this does not indicate there's no invention -- calculus, Fourier Series, Bessel Functions, Banach Spaces and on and on were certainly invented. It's just that we have a genetic propensity to be able to invent such things.
    Reilly Atkinson
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