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B Would Mathematics be Considered a discovery or an Invention?

  1. Aug 30, 2017 #1
    Sorry for the wrong prefix, but it is the most appropriate one that I could have chosen.
    I have been for long wondering if mathematics would be considered a discovery about the universe or an invention of the human brain. If it is a discovery, then math is the universal language of the universe. If it is a discovery of the human brain, then we just invented it to accurately describe the universe and our surroundings.
    What do you think?
  2. jcsd
  3. Aug 30, 2017 #2
    I think that maths was discovery until Pythagoras discovered it but then people started inventing different ways of calculating maths and different theories about maths. So in total my answer would be that maths was both discovered and invented.
  4. Aug 30, 2017 #3
    I agree with your opinion, but how can math be both a discovery and an invention?
  5. Aug 30, 2017 #4


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    I would say that mathematicians discover the truths encompassed by the systems of mathematics that humankind invented.
  6. Aug 31, 2017 #5
    Wise words and language!
  7. Aug 31, 2017 #6


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    I'm a pure Platonist, "Platonism describes idea as prior to matter". I push the entire existence question into the heaven of ideas. Real world objects are discrete - nothing smooth is out there - at latest on the level of electron microscopes. And nobody has ever drawn a true circle. However, neither stop us from "knowing" what is meant and calculate with it. I even consider pieces of music as a real world manifestation of an idea of the perfect version of named piece. So all those things are discoveries to me, and the invention part of it is merely an insight, a short glimpse of the idea.

    I recently counted ways to consider a derivative and found ten different descriptions of the same single formula. Those observations are a kind of evidence to my point of view, and not really suited to raise doubts. Nevertheless, in the end it is a matter of taste and a question about a preferred philosophical school.
  8. Sep 4, 2017 #7
    Yes, as I said in my original thread comment, I think that mathematics is an invention, because we invented it to accurately describe the universe. It is an invention that just happens to confirm observations so well, that it is the most important invention of the human kind, in my opinion.
  9. Sep 4, 2017 #8


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    It's neither an invention nor a discovery . Just something that evolved over time .
  10. Sep 4, 2017 #9
    Yes, however what did it evolve from? Can you explain yourself a bit more, please?
  11. Sep 5, 2017 #10
    Mathematical facts are discovered or as yet undiscovered, and mathematical procedures by which mathematical facts are established, are primarily invented.

    However, if one is searching for a proof of something that he intuits to be or supposes or hypothesizes to be a mathematical fact, and he finds such a proof , he may regard the proof to have been by him discovered.

    Some may consider him to have invented the proof, or to have discovered it, or to have invented or discovered the method by which he arrived at it, or may take the view that discovery and invention, while not the same thing, are not necessarily mutually exclusive categories

    Some mathematical facts, e.g. those regarding complex numbers, are partly discoveries based partly on invention.
    Last edited: Sep 5, 2017
  12. Sep 11, 2017 #11


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    In my mind (and I am a mathematician) mathematics is a branch of philosophy. The difference from the standard philosophic systems is that mathematics is more open about its basic tenets (axioms) and rules of deduction.

    Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess.
    -- Robert A. Heinlein --
  13. Sep 11, 2017 #12


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    This is what I think about physics. The fact the apple fell from the tree doesn't prove it will do tomorrow, too. O.k. the likelihood is pretty high but it is still a likelihood. An example closer to reality might be the proton decay. Mathematics on the other hand actually proves statements in the clearest way possible to us: Given certain deduction rules and certain conditions, then a conclusion is true. If someone means by proof an absolute truth in what sense ever, then neither can, nor mathematics, nor physics, nor any other scientific branch. Absolute truth simply doesn't exist. To consider mathematics as a form of philosophy is actually what it had been through centuries. The view as a language in which physics and other sciences are written is a quite modern point of view which disguises the nature of mathematics in my opinion. Or to say it with another quotation:

    What the difference between pure and applied mathematics is? There is none. One has nothing to do with the other.
    -- David Hilbert --
  14. Sep 11, 2017 #13
    Rather than a discovery or invention, I would call mathematics a development coming from our interaction with the universe. Mathematics needs arithmetic, it needs theory and discrimination. My assumption is that mathematics originated from observing & relating arithmetic with physical situations. How many apples do I have? My 2 apples together with his 3 apples give enough that all 4 of us can have one each. The last apple can be divided...

    Empirical geometric observations led to reliable calculations and theorems, which allowed cross-application from hand-tools to buildings to the stars. At every point along the way, the human mind carried math away from its tether to physical objects & situations allowing for both purity & application.
  15. Sep 11, 2017 #14
    Yes, but I would rather say that mathematics is a development made by humankind to accurately describe our surroundings, as I have already mentioned. We have adapted mathematics and fit it with physics to dig deeper and deeper with our understanding of the universe.
  16. Sep 11, 2017 #15
    I agree-- just saying it allows for more... deep understanding isn't always unambiguous. Also there can be misinterpretation, and there's my personal favorite, castles in the air, which are beautiful without having any particular function. (I've been called a space cadet on more than one occasion.)
  17. Sep 11, 2017 #16

    jim mcnamara

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    What you are discussing is whether or not a Platonic view applies to mathematics. Platonism says:
    In other words we discover, not invent, mathematics - if you take the Platonic view. Most of the mathematicians I worked with at a National lab from my view as an outsider were:

    1. Platonists
    2. "Closet" Platonists - their language use betrayed their beliefs, but they never, AFAIK, said that objects existed independent of "discovery" or elucidation.

    I asked quite a few of them over the years.

    This is an old discussion, BTW. And it is, IMO, completely philosophical. Not really a good fit for PF. @fresh_42 is one of the keepers of the keys for this area.
  18. Sep 11, 2017 #17
    I do believe that it is the discretion of mods to close of course. As I understand though, this kind of debate might be more closely related (or interesting possibly) to a set theorist or logician(from area other than set theory) than perhaps a working mathematician from other areas.

    Even a logician such as late Solomon Fefferman has written at one point along the lines of "what is a real number anyway?" towards the end of an article, while also raising important concerns regarding more philosophically troubling issues such as CH. So I think logicians do tend to take these kind of topics more seriously (sometimes ofc, not always).

    I was going to add a response based upon some basic issues that I see (quite briefly and without taking any opinionated side). But indeed there have been few threads similar to this before (in the last year or so). So, in any case, if mods want to close this, I guess it is up to their discretion.

    So I guess I will first wait to see what fresh_42's opinion is about this thread (regarding a possible closure).
  19. Sep 11, 2017 #18
    Or it could get relocated to General Discussions section
  20. Sep 11, 2017 #19
    I do not think this belongs to "general discussion". Because a lot of the associated issues are closely related to strictly mathematical topics (where there is also subtelty involved). I believe one can get some sense of the issues (just as a general roadmap, after that details start to become more important) involved even without going to the level of detail all the way, since it gives some understanding of wider perspective.

    As an example, I wouldn't want to study constructivist stance in more detail before I have a sense of potential shortcomings and strengths associated with it (and same holds for many other topics of study). Ofc that's just my personal view and many might reasonably disagree. Admittedly sometimes I have becomes overly stubborn with selection of topics to study, which can be personally detrimental at times in terms of learning (but that's another thing).

    Anyway, lets wait for mods to decide.
  21. Sep 11, 2017 #20


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    I always just assumed the axioms were invented and the theorems were discovered.
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