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July 3

Does mathematics invent or discover?

When something new appears in math, is it a discovery or an invention?--Yppieyei (talk) 13:55, 3 July 2015 (UTC)[reply]

Yes. --Stephan Schulz (talk) 13:56, 3 July 2015 (UTC)[reply]

Some of each. There are basic laws of nature, like the value of pi. Those they discover. But there are also mathematical conventions, like there being 360 degrees in a circle, which mathematicians invent. (You can tell degrees are an invention because there are alternatives, like gradients and radians. You could use different approximations of pi, such as 3.14, depending on the accuracy required, but you can't just decide to use a completely different value, like 10, if you want to get correct answers.) StuRat (talk) 14:34, 3 July 2015 (UTC)[reply]

(ec) @Stu, the angle unit is a Gradian, not Gradient (although the gradient may be expressed in gradians, which in turn proves they are different notions). :) --CiaPan (talk) 15:22, 3 July 2015 (UTC)[reply]

Thanks for the clarification. I guess I misheard it, and it's only listed as "GRAD" on calculators, which doesn't help to clarify it. StuRat (talk) 15:33, 3 July 2015 (UTC) [reply]

See Philosophy of mathematics, particularly §Mathematical realism and §Mathematical anti-realism. -- ToE 15:18, 3 July 2015 (UTC)[reply]

...and for an extreme version of mathematical realism see Max Tegmark's mathematical universe hypothesis, which asserts that every conceivable mathematical object physically exists somewhere in some universe, and they are all that exists. Gandalf61 (talk) 15:49, 3 July 2015 (UTC)[reply]

Quite extreme indeed. I'd like to see the number 2. I can see two apples, and I can count to two, but the positing the physicality of the abstract concept is a bit much for me. I assume he has never produced such a number for us to consider... SemanticMantis (talk) 17:27, 3 July 2015 (UTC)[reply]

Tell me what the difference between invention and discovery is and I might have a chance at distinguishing between them. Dmcq (talk) 16:26, 3 July 2015 (UTC)[reply]

We have articles at invention and Discovery_(observation) that explain the common distinction. Usually we'd say something like the Nucleic_acid_double_helix was discovered, while the cotton gin was invented. It would be weird and basically incorrect to swap the terms in that case. Math is of course much murkier (and this question is indeed perennial), but OP has plenty of reading links above if they are interested. Some math examples that might be tolerable to many mathematicians - the infinitude of primes is better described as being discovered, while the Markov_chain_Monte_Carlo methods can be described as being invented (or not - please let's not bicker about my examples too much - I'm not making any bold categorical assertions about philosophy of math, just trying to demonstrate that in some cases both terms can be reasonably used in math). SemanticMantis (talk) 16:56, 3 July 2015 (UTC)[reply]

I say that math is discovered - it is all already "out there". The example of 360 degrees in a circle given above is just a convention. That isn't really mathematics. Bubba73 ^{You talkin' to me?} 06:44, 6 July 2015 (UTC)[reply]

Math invents some of its axioms, e.g. Axiom of infinity, being the base of whole Modern Mathematics.

Note that if the set S of mathematical universal axioms - had reflected a convention only - so that every mathematical theorem T should have only been interpreted as "T is provable from S", then there would have been no difference - between the weight of any mathematical axiom - and the weight of any hypothetical assumption like "The current French leader is a king" (and likewise), hence there would have been no difference - between the weight of any mathematical theorem - and the weight of any consequence deriving from any hypothetical assumption like "The current French leader is a king" (and likewise). HOOTmag (talk) 07:17, 6 July 2015 (UTC)[reply]

Without getting too philosophically committed, I would say that the English verb "invent" always seems rather odd to me when it refers to something in mathematics (although I supposed certain things in applied mathematics, like numerical schemes, might aptly be described as "inventions"). I think a better word is "introduce", as in "Galois cohomology was first introduced by Mordell". Also, "described" is another neutral word. Theorems are legitimately called discoveries. To call a theorem an invention seems like a semantic error. Sławomir Biały (talk) 18:28, 9 July 2015 (UTC)[reply]