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In physics for example, and in science in general, facts are "discovered" in the sense that they arise from observing nature. A particle is discovered if we can measure its existence in nature. A law is discovered if the predictions it makes are observed in nature.

Is this the case with mathematics? Is mathematics "invented", in the sense that we think up concepts and ways to relate them and all mathematical derivations are just toying with those base elements we made up, or "discovered", in the sense that there is some underlying fundamental "natural" framework we shed light on?

On one hand, one could think that mathematics is an invention, as it usually derived from a set of axioms and all we do is take those axioms as given and creatively work from there, but on the other hand, there seem to be truths akin to physical laws or constants. Take $\pi$ for example. Mathematicians in India computed the same number as Archimedes. If mathematics was just an invention by man, how could there be an agreement on this as there is with any other fundamental physical law? Same with the binomial theorem, for example. This suggests that mathematical truths are "discovered", not merely "invented".

In fact, does it even make sense to ask this? Is this an open philosophical question?

andrepd
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    Philosophical questions about math are far more interesting to philosophers than to mathematicians. – Matt Samuel Jan 06 '15 at 23:28
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    I would say just as in physics, it is an interplay of both. Mathematicians invent definitions and axiom sets, and then discover consequences. – David H Jan 06 '15 at 23:30
  • @Arthur Fermat's last theorem is true.... – Matt Samuel Jan 06 '15 at 23:32
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    @MattSamuel Only if they’re also mathematicians, I’d say. – k.stm Jan 06 '15 at 23:32
  • @Arthur I don't think saying theorems are false until proven true is reasonable. – andrepd Jan 06 '15 at 23:34
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    I think this question is correctly tagged, and that philosophical questions are close to the surface in mathematics. I would't close it. – Mark Bennet Jan 06 '15 at 23:34
  • @MattSamuel Maybe I should post on philosophy.SE then? :P I only posted here because I saw there was a philosophy tag. – andrepd Jan 06 '15 at 23:35
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    Compare Is there any difference between a math invention and a math discovery?, closed for being not constructive. – MJD Jan 06 '15 at 23:36
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    @andrepd unless it's against some sort of policy it may be worthwhile to post on both. – Matt Samuel Jan 06 '15 at 23:36
  • In technology (which is applied science) TVs, computers or smartphones were invented instead of discovered – janmarqz Jan 06 '15 at 23:39
  • I was posting an answer to this question when it was closed. Mathematics is abstract until it is used as a model of reality. Abstract mathematics is, on the whole invented, though there may be underlying motivations. Euclid had a good model of reality before real numbers, Riemannian Geometry and Special and General Relativity - I want to propose that good models of reality are discovered. – Mark Bennet Jan 06 '15 at 23:39
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    Consider reading http://andrewlias.blogspot.com.br/2004/08/is-mathematics-science.html by Arthuro Magidin. – Moishe Kohan Jan 06 '15 at 23:43
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    @MattSamuel Actually this has already been asked on the philosophy SE. For anyone interested, it has some interesting points of view: http://philosophy.stackexchange.com/questions/1/was-mathematics-invented-or-discovered – andrepd Jan 06 '15 at 23:47
  • @andrepd A tag's existence is not a particularly good heuristic - tags exist merely because someone once tagged a question with them. If you look at the other questions, they're quite specific - they are dealing with particular mathematical concepts - and their answers are mathematical, or at least expository of particular mathematical experience. This question is just not as "sharp" as the other ones - its statement is fuzzy, and its answer most definitely lies both outside of mathematics and, critically for this site, the expertise of mathematicians. – Milo Brandt Jan 06 '15 at 23:47
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    To come to think of it... did Ramanujan discover or invent his results? He built upon a certain foundation, so he invented it we may say. In the process of such invention he discovered great order in maths that was revealed to him and he expressed it as discovery.Same with Newton or Euler. Math is practically inventing and discovering new items on the way.Perhaps it is so for all of Science. – Narasimham Jan 06 '15 at 23:56

2 Answers2

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I would say, personally, that axioms are invented, and theorems are discovered (as a result of those axioms). Different people will probably have different opinions.

An analogy may be that by inventing axioms, we are planting the seed of a tree. The resulting tree that grows from that seed is the mathematical framework that is a result of our planting of the tree, but we didn't have a part in shaping each branch.

Jonathan Hebert
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America was discovered.It was there when people lived in Europe but they didn't know about it's existence.When I was 6 years old I could certainly count up to 20.I surely counted 8 and 9.I did not know of course that 8 and 9 are the only consecutive powers but this was and is actually true.I know this is now a fact (thanks to Preda Mihailescu, 2004).It is the same thing with America.Mathematics is discovered.

One more thing:You may destroy an "invention" such as the radio but you cannot destroy the fact that 8 and 9 are consecutive powers.Even if you destroy the radio the ability to construct one from the beginning will remain alive.
Certainly Mathematics is discovered.

Konstantinos Gaitanas
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  • Does this mean that numbers exist in nature outside the human mind, like electrons and quarks? – andrepd Jan 06 '15 at 23:38
  • A thing does not exist in nature outside the human mind if it exists ONLY in human mind.How can be proved that there exists at least one thing which belongs not only in my mind?I am not sure.But anyway this has nothing to do with what I gave as an answer. – Konstantinos Gaitanas Jan 06 '15 at 23:47
  • Using your analogy, America existed before it was discovered. If you say "Certainly Mathematics is discovered." then you are stating that natural numbers, for example, also exist a priori. If you say you can't prove anything exists outside your mind, then maybe "Mathematics is discovered" is not so "Certain". – andrepd Jan 07 '15 at 00:22
  • You cannot prove that something "exists" in general.I can't prove that an elephant isn't behind me right now but this doesn't mean that there is one.We can ask continuously questions about how we know this thing or the other thing and never reach a point.But this exactly is philosophy. – Konstantinos Gaitanas Jan 07 '15 at 07:30
  • You may destroy a radio, but you can not destroy the idea behind the radio, which is what the invention is – TheQuantumMan Mar 06 '16 at 19:51