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Was mathematics invented or discovered?

To be more precise, does mathematics describe the physical world or does it describe a mental representation of the physical world? If the latter is true, then an empirical science, physics namely, tries to describe mental phenomena via what is believed to be the physical world and its behavior?

Thanks for the suggestions. Actually, my question refers to reality or the physical world, and not to mathematics. Whether mathematics was invented or not is a different question, I believe. Anyway, I think that my question is most likely unanswerable. ... That mathematics is real and that it is capable of describing our understanding of things in a systematic and very efficient way, for me is out of question. But, consider the following situation: there is a huge sculpture in the middle of a room. There is also a wall before the sculpture. Johnny has never seen a sculpture and he has been told that sculptures are solid and white; that's all he knows about sculptures. So, Johnny stands in front of the wall and thinks that it is a sculpture, as the wall is solid and white. For the sake of this stupid argument, let's assume that Johnny is a prolific mathematician. As he contemplates the wall which he believes is a sculpture, he starts to generate a number of very accurate mathematical formulations that indeed describe the reality of the wall. Then, Johnny is describing a reality (the mathematics he produces describe a real thing), but they fail to describe what is being intended in this case, the sculpture. In this way, if one substitutes the wall for mental representation and the sculpture for the physical world, then (at this point I am feeling quite ashamed) does the conclusion stated above hold? I really hope this was easy to understand; I surely tried.

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Gabriella
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    This question largely reduces to this one: Was mathematics invented or discovered? – Michael Dorfman Aug 19 '11 at 07:13
  • Gabriella, is there any chance you could indicate here what might not be covered by the answers to the other question? – Joseph Weissman Aug 24 '11 at 01:46
  • @Joseph: Really, that's a duplicate? The question is about representations, descriptions and reality, not provenance. The question might be better worded in that regard but it is not about discovery vs invention. Which is to say, don't close it on that account. – Mitch Aug 25 '11 at 12:51
  • @Mitch, it's been significantly reformulated since yesterday. I'm happy to reopen. I'm a little worried this still isn't as clear as it might be. – Joseph Weissman Aug 25 '11 at 13:07
  • @Gabriela: You gave an example (a sculpture, and Johnny's mathematical description of him mistakingly describing/representing a similar object (the wall)) and then said "does the above conclusion hold?" presumably for the example. But What conclusion? can you clarify? – Mitch Aug 25 '11 at 13:59
  • Just out of curiousity, what would be the difference between mathematics "describing the physical world" and "describing a mental representation of the physical world?" Surely it's possible for something to do both. – eMansipater Aug 25 '11 at 14:00
  • @Michael: can you explain how the reduction to invention/discovery works? – Mitch Aug 25 '11 at 14:00
  • @Mitch: If one believes that mathematical objects actually exist in and of themselves, it follows that mathematics is discovered. If, on the other hand, one believes that mathematical objects are merely invented abstractions, abstracted (presumably) from empirical objects. – Michael Dorfman Aug 25 '11 at 14:05
  • @Michael: I don't get that. A machine exists in and of itself. Is it discovered? Also, I couldn't follow the second sentence. – Mitch Aug 25 '11 at 14:25
  • @Gabriela: how are the wall and the sculpture connected? Is it by coincidence that they are near each other and also both solid and white? or is there a picture of the sculpture on the wall? – Mitch Aug 25 '11 at 14:27
  • @Gabriela: more questions!! Are you asking: "Is math (or description) in the real world or in our heads?" or "Does math describe the real world or does it describe our mental processes about the real world?" – Mitch Aug 25 '11 at 14:30
  • @Mitch: A machine is invented. A rock is discovered. The question is: are mathematical objects already existing out there, ready to be discovered, or are they invented in our minds, non-existent until somebody invents them? – Michael Dorfman Aug 25 '11 at 14:54
  • @Michael: You said: "If one believes that X objects actually exist in and of themselves , it follows that X is discovered." You substituted X=math, and I substituted X=machine. I agree with you that machines are usually considered to be invented, but using your inference, it seems that machines are discovered. So which is it or am I not inferring or substituting correctly? – Mitch Aug 25 '11 at 17:07
  • @Mitch: the key concept you are missing is "in and of themselves". A machine does not exist in and of itself; it is constructed from parts (i.e., invented.) I assume you are aware of the debate between mathematical Platonists and Intuitionists; the distinction I am referring to is well documented in the literature. – Michael Dorfman Aug 25 '11 at 18:04
  • @Michael: I don't interpret 'in and of themselves' that way. In fact I don't know what that could possibly mean. If you stipulate to mean anything, then the argument will be circular. Can you give a non-stipulative meaning for 'in and of itself' that supports the inference to 'discovered'? A math object can be constructed from parts..so that makes it discovered? – Mitch Aug 25 '11 at 20:31
  • @Michael: yes, I am aware of those two philosophical attitudes, and I thought of those as the more relevant ideas for the OP, rather than the invention/discovery dichotomy which I find to be related but not the same thing and too simplistic anyway (just an emotional appeal to show importance of provenance: discovering a mountain is one thing but -building- a mountain is so much more impressive). – Mitch Aug 25 '11 at 20:35
  • Any chance I could persuade you guys to move this to chat? – Joseph Weissman Aug 26 '11 at 02:08
  • Sure, Joseph. Is anyone else available? – Gabriella Aug 26 '11 at 03:58
  • @Gabriella, I think Mitch has messaged you in the primary chat (Schrodinger's Discussion). – Joseph Weissman Aug 27 '11 at 00:55
  • What about a Turing machine?

    http://philosophy.stackexchange.com/questions/11175/not-turing-reducible-non-physical

     – yters May 14 '14 at 03:28

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As he contemplates the wall which he believes is a sculpture, he starts to generate a number of very accurate mathematical formulations that indeed describe the reality of the wall

At this point you've answered your own question. The mathematical formulations approximate the the reality of the wall in some register, but there is still a significant distinction to be drawn between the (abstract, intelligible) mathematical object and the (empirical, sensible) physical object it describes.

No empirically found circular object is actually a circle.

Michael Dorfman
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  • Yeah, you are right. I think I got confused with 'asking a question' and "wanting an answer." – Gabriella Aug 26 '11 at 03:46
  • Mitch. Yeah, of course I can clarify. By conclusion I mean 'producing an accurate description of an object and this object being different from the one that was intended to be described.' – Gabriella Aug 26 '11 at 03:52
  • @Gabriella: I'll start something over in chat..it's pretty sparse now so easy to see who is responding to what (even if we're not there at the same time). – Mitch Aug 26 '11 at 15:06
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Mathmatics is a set of definitions. Many things in the world can be represented mathmatically but that does not mean that the formala that represents the object is the object. Mathmatics can be used to describe all of those things that have a measureable quality. However it requires our understanding to be meaningful. E=mc2 is only meaningful if you understand how to apply it. So it is possible that to define equations that define how things that we can not currently measure work. But until we understand those measurements those equations are meaningless.

So while it may be possible to describe through a set of equations all of the physical characteristics of an object they are meaningless without understanding. However I do have an answer for you and it is 42.

Chad
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    Chad, Thanks for taking the time to comment and try to answer my question. For the most part, I agree with what you say. But, it is not very clear to me what type of insight are trying to provide me with. Anyway, as Michael Dorfman suggested, I am basically answering my own question by giving a practical example of the concept I am questioning... – Gabriella Aug 26 '11 at 03:43
  • The point is that I could have a simple equation in front of you that did represent everything that exists mathmatically but that equation would be meaningless to us because we do not understand it. So while it potentially could be represented mathmatically because of the immensity of the scope this description would be beyond our comprehension. Thus 42 is the representation of that equation. – Chad Sep 08 '11 at 20:46
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Max Tegmark has written a wonderful paper on this actually: http://arxiv.org/pdf/0704.0646.pdf

Please give it a chance, I think you'll find it wonderful. So let me try to describe what his paper is on, Max takes the approach that the universe itself is a mathematical structure within which exist smaller structures. These structures can be defined nicely and are computable which explains in a sense why the physical laws that govern the world appear rather simple. He describes this in a more detailed way and more importantly he handles the implications of proposing that the universe itself is a mathematical structure in the paper.

dhillonv10
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