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A very theoretical question I came across with friends some time ago. I am not a native English speaker, so I may be misunderstanding some details, but as I understand:

1) Discovery relates to a concept which existed prior to ... well "discovery" but was unknown before the act.

2) Invention relates to a creation of a concept which never existed before the act

Following from this, are mathematical concepts such as sinus "discovered" or "invented"? In one sense, being a fundamental property hence could only be discovered. On the other hand, the logical metaphor was created by humans, hence invented.

Which one is right and which one is wrong and why?

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Are you sure this doesn't belong on a different site? – SamB Mar 18 '11 at 0:51
Ghm, my first post here, sorry if that ended up being off topic. It is mostly a "would want to know coz its fun" kind of question. – alex Mar 18 '11 at 0:52
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We have a Math site, but I'm not sure whether or not this particular question would be welcomed there these days. Edit: they do have a soft-questions tag, populated with questions such as "Do complex numbers exist?" and "Who invented vectors and why do we need them?" – RegDwighт♦ Mar 18 '11 at 1:08
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This is not an English language question; it's a math or a philosophy question. As for the English language component, both terms are used by mathematicians, but possibly with slightly different meanings. – Peter Shor Mar 18 '11 at 2:02
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@Billare: I guess you can interpret the question "Which one is right and which is wrong, and why?" in two ways. Linguistically, both are right (since both are used, possibly depending on the philosophical leanings of the mathematician using them). Philosophically, it's been disputed for a long time, and will probably be under discussion as long as there are mathematicians. – Peter Shor Mar 18 '11 at 2:11
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6 Answers

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This is a very old question, even among mathematicians who are accomplished native English speakers. There are definitely aspects of discovery as well as invention in math, and some things lend themselves more to one than the other. For example, it is more appropriate to say that notation is invented, rather than discovered. It's the other way for proofs.

In general, if you're not sure which one, it's a little safer to say discover, as it is a more "humble" term; but in casual use, either one is usually acceptable.

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Discover is typically used among mathematicians. Most mathematicians are typically Platonists internally, even though the Platonist view has no real credibility among philosophers or those that study the philosophy of mathematics.

Since most mathematicians are atheists (based on my experience in grad school and in working as a math professor), it would be more consistent for them to speak of inventing theorems, but they just don't. I suspect that the reason for this is that in the process of doing mathematics it is expedient to think as if this stuff is simply out there.

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"Since most mathematicians are atheists": uh, [citation-needed]? – Marthaª Mar 18 '11 at 17:09
Personal experience, I have a Ph.D. in mathematics and worked as a professor. – Eric Wilson Mar 18 '11 at 17:33
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@FarmBoy: one example does not a pattern make. (I have a B.A. in math and am definitely not an atheist.) – Marthaª Mar 18 '11 at 17:36
@Martha I'm basing my statement on the fact that I know about 100 mathematicians, and the vast majority of them are atheists. (Trust me, there is a pattern here.) I worship the Lord Jesus Christ myself, and am glad to hear you are not the fool that says 'there is no god.' But if you had gone on in your mathematical studies, you would have found yourself increasingly in the minority. – Eric Wilson Mar 18 '11 at 18:03
@Martha There are plenty of studies which show that people with graduate-level educations and those in academia are far more likely to be atheist on average. See this page analyzing the GSS: halfsigma.com/2006/06/religious_peopl.html Average IQ of Ph. D mathematician: ~130, so the result follows that many likely to be atheist. whyevolutionistrue.wordpress.com/2010/10/08/… Here Jerry Coyne analyzes a survey to show that professors 3x as likely to be atheist as the American population. FarmBoy has ample support for his assertions. – Billare Mar 18 '11 at 18:06
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It depends on what you believe.

A. Mathematical concepts exist. (we find them out)
B. Mathematical concepts don't exist. (we build them)

If you believe A then you can use "discover" else you can use "invent". Even if you believe A you can still use "invent" but that won't be consistent with your belief.

If you believe that A is true then B is false for you and if you believe B is true then A is false for you.

There is no way to decided which one is actually true. That is why this debate can go on forever.

In real life, people don't really care which one (A,B) is true. So they are comfortable with either word.

Such questions are bread and butter of philosophers.

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1) Discovery relates to a concept which existed prior to ... well "discovery" but was unknown before the act.

2) Invention relates to a creation of a concept which never existed before the act

That is exactly correct.

Both terms are used. In popular writing, they seem to be used interchangeably. My gut feeling is that no mathematician would say that either Pierre de Fermat or Andrew Wiles invented Fermat's Last Theorem. Generally, I think a mathematician would seldom if ever use the word invention to refer to a specific recent mathematical result (a theorem proved in a new paper).

But I think mathematicians do use the adjective inventive to describe specific people, approaches, techniques, papers, etc. that they find really impressive and unexpected; and I think some mathematicians use the noun invention for those things. So you might hear a mathematician call Galois theory an amazing invention.

I am not a mathematician. All this is speculation.

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up vote 1 down vote

An English speaker probably won't mind hearing discover (or even invent) in the context of mathmatics. However, if you want to be technical you could always use the mathematic term proof.

Pythagoras is credited with the Pythagorean Theorem proof.

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"...credited with the proof of the Pythagorean Theorem" sounds better to me, though I would probably say "Pythagoras is credited with the proof of the theorem that bears his name" to avoid the repetition :-) – psmears Mar 18 '11 at 7:35
up vote 2 down vote

I think neither verb is entirely appropriate for mathematical concepts in ordinary English. You can't "discover" a square root, nor can you "invent" a radius: they just are, I'd say. I suppose you could use these verbs, depending on context, but I suspect that it would nearly always be better to avoid the situation.

If I said "Pythagoras discovered the Pythagorean theorem", that would be possible; but it'd sound a tiny bit childish, as though I weren't aware of the very problem you present here. The same applies to "invent".

(If you like Kant, you will probably agree that mathematical concepts are best considered models created by the mind in which we shape our sensory impressions. We have always used these models subconsciously in various ways: discovering or inventing them means just becoming aware of patterns in the ideas we have about reality.)

You might use to conceive of:

Surprisingly it was Pythagoras who first conceived of the Pythagorean theorem.

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Doesn't discover also apply to newly acquired knowledge? I can discover something I didn't know before the discovery. Why would Math be situated outside of this classification? – MrHen Mar 18 '11 at 18:51
@MrHen: Because "discovering" mathematics might imply that it was a thing that existed outside the human mind, like a physical object. You don't ordinarily discover knowledge, you gain it. Hmm the more I think about, the less sure I am of any of this. I suppose you could discover a mathematical concept if it were presented as a solution to a problem: you then discover the fact that it is a solution, not the concept itself. My head hurts while I thought it was all sort of clear. But perhaps this is all too vague and/or philosophical. – Cerberus Mar 18 '11 at 19:03
Yeah, I agree it is a little vague. I often claim I have "discovered" a solution to a problem or puzzle and figure its use in that context will transfer to Mathematics. Honestly, my hunch is that any objection to either discover or invent is an issue of pride and/or hurt feelings by the Mathematicians themselves. The meaning is certainly obvious in both cases. – MrHen Mar 18 '11 at 19:10

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