Was mathematics invented or discovered?

  • What would it mean to say that mathematics was invented and how would this be different from saying mathematics was discovered?

    Is this even a serious philosophical question or just a meaningless/tautological linguistic ambiguity?

    Here is a headline from 2008: http://news.bbc.co.uk/2/hi/americas/7640183.stm *Huge new prime number discovered* I wonder who would say that this prime number was invented in 2008?

    @GEdgar Intuitionists do not claim that individual numbers are invented - in fact an infinite number series is a principle of intuitionism. They would say that it *became true* that the number is question was prime. Prior to that, it was neither true nor false that it was prime.

    My first reaction was "man, don't ask me a question like that", so great question...lol! But I would say it's BOTH! Certainly, there was imagination involved in the discovery and the unfolding of those processes relates to the spontaneity of genius in the establishment of reaching great mathematical solutions.

    Just to make it clear: discovery = finding something that existed before (e.g., a frog, black holes) invention = intellectual creation (e.g., of a system)

    What we invent is called application. And what we find already existing is called discovery. Mathematics is the logic of understanding the facts through measures. Here measure is only a medium of understanding. So, I feel mathematics a discovery..........!!!

    Mathematics is a language. Just as we didn't invent a tree, we can describe a tree using English, or French, or Mathematics.

    @ProfessorFluffy But did we *invent* English, or *discover* it?

    @nocomprende Well to the point -- We converged upon it. If Mathematics is a structure made of shared intuitions, they *become shared* by some process. I would blame evolution and a little bit of culture. In that case, the answer is *neither*: it was gifted to us by an outside force.

    @jobermark I would say the answer is neither also: no two people can ever know the same thing, there are no objects or truths and language, time and causality are delusions. Mathematics goes "poof!" Except for all the people riding in elevators.

    If maths were invented, it wouldn't be possible to draw a circle without knowing the value of Pi ... but it is.

    what if `invented` = `discovered`?

    It's a valid historical question. Philosophically the question poses a false dichotomy. It is likely that we started a system of counting in our head that gradually evolved into the abstract system of symbol manipulation that we have now, so it was probably a continual interaction between the two until the formal system emerged.

    @Ooker, indeed; no one discusses that possibility. There is indeed a philosophy which poses that, and it even has axioms (though not in the sense that modern mathematicians uses them—more like Euclid's definition).

    @Ooker I wouldn't say so. As discovering means finding out something that exists naturally, like gravity. You cannot invent gravity, at most you can invent a machine which can mock it. They are really different. Since maths existed even before humans started to make calculations, it means that it is discovered. The electrons didn't start to spin around protons and neutrons after we found out that they do, in an orbit(which has a ellipsis-like form)

    @Redfx are dinosaurs invented or discovered? Surely we discovered the fossil, but for an imaginary human leaves at the birth of the Earth, they are invented, right?

    @Ooker as you said yourself, it is neither invention nor discovery, it is just imagination. It has nothing to do with these concepts. But lets say they really really believed in this and then they figured out that the statement was really wrong, then it would be the discovery of the truth. Because you perceive 'something' that was there or not there. In very simple words, you discover something that is already there. But you invent something unusual. Like you cannot invent 'dark matter', at most you can discover it is there or it is not there.

    @Ooker dinosaurs are so discovered in the sense: we discovered that there were these kind of living creatures before us. We didn't create them, or we didn't put them in the past ourselves. [my prev. comment on imagination was about the human birth with leaves if you meant that]

    As of December 21, 2018, *The World Has A New Largest-Known Prime Number*: There's a new behemoth in the ongoing search for ever-larger prime numbers — and it's nearly 25 million digits long. https://www.npr.org/2018/12/21/679207604/the-world-has-a-new-largest-known-prime-number?utm_campaign=storyshare&utm_source=facebook.com&utm_medium=social&fbclid=IwAR3JPtG_toEtR1XiZsZU_OvEOkD2vH17uGZmDTPM1PughJLSMsVRnZr4uec

    Were Bachelors (the people) discovered or invented? It's kind of a moot question.

    Math is invented through consensus on naming, axioms, logic, etc. The implications within these systems can appear to be discoveries but they're also an inherent result of the invention.

  • user16

    user16 Correct answer

    10 years ago

    "Intuitionists" believe that mathematics is just a creation of the human mind. In that sense you can argue that mathematics is invented by humans. Any mathematical object exists only in our mind and doesn't as such have an existence.

    "Platonists", on the other hand, argue that any mathematical object exists and we can only "see" them through our mind. Hence in some sense Platonists would vote that mathematics was discovered.

    @eMansipater: I am not an expert on this topic. All I can refer to is probably wiki which I have now added.

    Good enough for my upvote--I just want readers to realise these are specific schools of thought with defined characteristics.

    True Platonists would argue that anything we learn is in fact remembered. This is the point of Socrates walking Meno's slave through a simple Euclidean proof about squares -- i.e., that through dialogue and introspection, our 'innate' knowledge (memory!) of mathematical Reality can be recovered in some partial way.

    I don't think this is a good answer. It reminds me too much of academic philosophers' evasive ism-dodges, i.e. when you ask them 'Does you argument fail because of X?' and they respond: 'Well if you're a Y-ist then no and if you're a Z-ist then maybe yes, but I'm not sure which one I am yet etc etc.' - The schools themselves mean nothing and people who haven't explored the issue independently should not, I believe, get told to look up a certain school of philosophy as their point of first contact

    @Joe: That's quite correct. Plato (via Socrates) held that we knew everything from previous 'afterlives' and that we remembered them, but thought we were learning something new. The passage with the slave boy sets this out very clearly.

    Ok, good definitions; but what is your **actual** answer?

    Mathematical platonism isn't intended to be the same as Plato's views but just similar in certain respects. That's why its often spelt with a lowercase P.

    @GeoffroyCALA Why does it matter whether any one individual is an intuitionist or a platonist?

    @boehj To which passage do you refer?

    @Chuck: suppose somebody, whose vision is limited to 2-D objects, asks you: "Are Egyptian pyramids triangles or squares?" Would it be acceptable for you to answer: "Well, according to Plato, who lives in a spaceship and looks down on Earth, they are squares; according to Aristotle, who stands on the ground, they are triangles; but IMHO they are a little bit of both."

    This is hardly an answer. Completely a comment. I wish I had enough reputation to downvote.

    What do you mean exactly when you say that - for the Platonist - we can only *see* mathematical objects through our *mind*? Do you have in mind some kind of Godelian faculty of intuition? If so, I fear this isn't really the acknowledged understanding of the Platonist's position. If not, can you make it clearer?

    I think it's fair to say that the number of professional mathematicians who apply _instuitionist logic_ are in a small minority, and that the vast majority have no qualms about, for example, using the law of excluded middle and non-constructive existence proofs.

    This is one of my favorite philosophical debates.

    Interestingly and somewhat ironically, pragmatism would say this is the only important answer.

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Content dated before 7/24/2021 11:53 AM