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?


closed as off-topic by agha, beep-boop, Andrés E. Caicedo, quid, Milo Brandt Jan 6 '15 at 23:36

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – agha, beep-boop, Andrés E. Caicedo, Milo Brandt
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Philosophical questions about math are far more interesting to philosophers than to mathematicians. $\endgroup$ – Matt Samuel Jan 6 '15 at 23:28
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    $\begingroup$ I would say just as in physics, it is an interplay of both. Mathematicians invent definitions and axiom sets, and then discover consequences. $\endgroup$ – David H Jan 6 '15 at 23:30
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    $\begingroup$ @MattSamuel Only if they’re also mathematicians, I’d say. $\endgroup$ – k.stm Jan 6 '15 at 23:32
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    $\begingroup$ I think this question is correctly tagged, and that philosophical questions are close to the surface in mathematics. I would't close it. $\endgroup$ – Mark Bennet Jan 6 '15 at 23:34
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    $\begingroup$ @MattSamuel Actually this has already been asked on the philosophy SE. For anyone interested, it has some interesting points of view: philosophy.stackexchange.com/questions/1/… $\endgroup$ – andrepd Jan 6 '15 at 23:47

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.

  • $\begingroup$ I quite like this view. Numbers are invented, their properties and relations are discovered. $\endgroup$ – andrepd Jan 6 '15 at 23:33

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.

  • $\begingroup$ Does this mean that numbers exist in nature outside the human mind, like electrons and quarks? $\endgroup$ – andrepd Jan 6 '15 at 23:38
  • $\begingroup$ 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. $\endgroup$ – Konstantinos Gaitanas Jan 6 '15 at 23:47
  • $\begingroup$ 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". $\endgroup$ – andrepd Jan 7 '15 at 0:22
  • $\begingroup$ 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. $\endgroup$ – Konstantinos Gaitanas Jan 7 '15 at 7:30
  • $\begingroup$ You may destroy a radio, but you can not destroy the idea behind the radio, which is what the invention is $\endgroup$ – TheQuantumMan Mar 6 '16 at 19:51

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