Mathematics: Discovered or Invented?

Discussion in 'Math' started by PG1995, Sep 26, 2011.

  1. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hi :)

    Some people are of the position that math is discovered while others say it's invented. I think in a way both parties are at extreme ends. I think math is discovered as well as invented. There are some fixed mathematical relations in nature which exist on their own such as ratio of circumference over diameter, golden ration, ratio of sides of triangle, etc. These 'natural' relations constitute part of discovered mathematics. Then, humans use these relations and invent some of their own systems to create new math which is amalgamation of 'discovered' and 'invented' mathematics. This also implies that mathematics as a whole is not a universal language. I believe the part which includes natural relations (such as ration of circumference and diameter) is universal and will be true and applicable everywhere. But the part which contains human innovations and inventions is not universal. Recently someone told me that Godel's incompleteness theorem proves this point. What is your opinion on this? Please let me know. Thank you.

    Regards
    PG
     
  2. Wendy

    Moderator

    Mar 24, 2008
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    I already had my say, this is the same thread restated as far as I can tell.
     
  3. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Yes, Bill. But that thread was on imaginary numbers. I did'nt give it full attention there and I didn't put forward my case there. Here I have devoted a separate thread on this topic and have also stated my own opinion. As you have also given your opinion and I have read it, you are free to skip it if you like! ;)

    Best wishes
    PG
     
  4. steveb

    Senior Member

    Jul 3, 2008
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    I'll be a devil and argue that mathematics is abstraction, and hence an invention of man. The aspects of math that we feel are discoveries are simply abstractions that closely relate to the physical world. Those physical things are discoveries of science, not mathematics.

    For example, the fact that the ratio of the circumference of a circle to the diameter of a circle is always pi to very high accuracy is a scientific discovery. The mathematical description that pi is a transendental number with particular digits that is always exactly the same for any circle in any place, is an invented abstraction. We invent starting assumptions and create an abstract structure (called Euclidian space) and deduce these facts. If you've discovered anything, it is only a discovery in an artificial landscape you first had to invent, not a true discovery in the real world. Real discoveries like this are called scientific discoveries.

    In other words, the fact that mathematics is useful as an abstract language of physics and other sciences, makes it appear to be discovered truth, but the real world never exactly conforms to our mathematical models. Hence, the pure mathematics is an invention which creates an abstract false-reality that is invented by us, and nothing in that false reality can be discovered because it results from an invention, and is hence an invention itself. At best, it's an invented discovery.
     
  5. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Thank you, Steve.

    I think you are referring to pure mathematics. But pure mathematics is a later stage and highly abstract field, the stage which comes first and has driven almost all the math in the past is applied math which is closely intertwined with physical science. That's my two cents!

    Regards
    PG
     
  6. debjit625

    Well-Known Member

    Apr 17, 2010
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    I feel any invention is based on some discovery.Mathematics is a universal language, it was their all over the universe all the time, its just we humans discovered it on earth at some moment of time and still we are discovering it.

    I like what Cantor said

    In accadimics we don't get this freedom at least I feel so.
     
  7. steveb

    Senior Member

    Jul 3, 2008
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    You are welcome PG.

    I don't understand the distinction of "pure mathematics". Is there a pure and impure mathematics? :rolleyes: People seem to feel that "pure mathematics" is that which does not get applied to something, yet often the invented pure mathematics is later applied to something useful. Does the invention then become a discovery? Anyway, this term is not very useful to our discussion.

    But, I do understand the term "abstract mathematics", and my argument is that ALL mathematics is abstract mathematics. Just because the invention of the mathematics was motivated by a real world problem, and just because we apply mathematics to some real world situation does not change that.

    If you disagree, I would challenge you to find any mathematic model, that is applied to the real world, which is an EXACT description of reality. One example is all it takes for you to prove your point. Unfortunately, I have the more difficult problem of "trying to prove a negative". I can't prove that there is a reality that can not be exactly and perfectly described by mathematics. :p I can say that in all my studies, I have never found an example of it. As you may know, physicists are trying very hard to invent/discover a mathematical model of the universe that is exact, and they have, as yet, not even come close to achieving that goal.
     
    Last edited: Sep 26, 2011
  8. Wendy

    Moderator

    Mar 24, 2008
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    It could be argued math (all math) is a language, and is therefore an invention.

    I may have been talking in the imaginary numbers thread, but my reply's were specifically answered this thread. As a matter of fact imaginary numbers are but a subset of math, which is to say there is no difference.
     
  9. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hi Bill

    Yes, your reply was specifically about this topic and was helpful (perhaps, your reply was one of the reasons which made me think about the foundation of mathematics specifically). That's the reason I said that you can skip it if you like.

    Peace
    PG
     
  10. tgotwalt1158

    Member

    Feb 28, 2011
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    The manifestation of the whole universe is based upon a divine algorithm. The existing known mathematics to humans or by humans may be a tiny sub set of that algorithm.
     
  11. BillO

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    Nov 24, 2008
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    Yeah, right. Divine....
     
  12. BillO

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    Nov 24, 2008
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    Although....

    Mathematics is a wholly human construct inspired by the nature of our universe.

    Just an opinion now, not the gospel.
     
  13. Adjuster

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    Dec 26, 2010
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    I wonder if this is a question we can ever expect to answer, and even whether it is worth trying. One difficulty I see is that some things may seem "obvious" to us which rest on assumptions we are not conscious of, but take for granted.

    Trying to imagine for instance how a non-human intelligence would view the Universe, we might imagine that it would use a system of mathematics as we do. However, if the basis of this other intelligence was something very different from the human brain, it might have totally different ways of processing information. There appear to be structures in our brain that facilitate calculation: if these are damaged, then that ability is lost. Do we assume that we process information this way because mathematics is universal, or do we assume mathematics is universal because our brain structure makes it seem that way?
     
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  14. count_volta

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    Feb 4, 2009
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    I like what you said Adjuster. I agree.

    But lets look at the simple definition of math. You have a tiger in the jungle. Then you also have another tiger in the same jungle. Now we say that there are two tigers.

    2 is a human invention to represent the fact that there are multiple tigers in the jungle. We did not create the first tiger, it was just born and there it is. Nor did we create the second tiger. So even if human beings did not exist there would be 2 tigers in the universe.

    So the thing that we call 1 2 3 4 5 6....... exists independently of humans. Or hell just take the billions of stars in the universe. And they follow a certain universal law that Newton just happened to call gravity. So gravity is independant of humans also. This can be extended to all natural phenomena.
     
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  15. THE_RB

    AAC Fanatic!

    Feb 11, 2008
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    I think it's necessary to separate the meaning of the terms.

    To me, "discovered" implies that which already existed before the "discovery" like "the island of Hawaii was discovered".

    And "invented" implies that it did not exist before being invented, as in "Dubai's Palm Island was invented".

    So when referring to math, the correct word would depend on whether that particular part of math already clearly existed (like Pi) or was "created" when no obvious examples were in existence.

    I would say "infinity" was invented. Zero was discovered.
     
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  16. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Thank you, RB.

    Actually this was my take in my original post. I don't think we can say math is entirely human invention.

    Regards
    PG
     
  17. steveb

    Senior Member

    Jul 3, 2008
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    It's ok to say zero and pi are discoverd and it's ok to believe it . I'm not sure one can really prove numbers exists in reality, and as I said above, I can't prove a negative and claim that you can't find a place where these numbers have real existence. But, what is your argument to convince others that zero and pi are discovered?

    What you said is confusing to me. You are saying that infinity was invented, but "pi already clearly existed" Huh? Clearly? Where does pi exist in the real world? We think of it as an infinite series of numbers. How can something made of infinity, which is invented, be something real to be discovered.

    It seems to me pi is an abstraction that does not exist anywhere in the real world. It doesn't even exist as the ratio of circumference to diameter of a circle because circles in the real world don't obey this formula. Even if you could make a circle perfectly (which you can't), the warping of spacetime prevents the ratio from being perfectly accurate. And even if we lived in Euclidean space and could make a perfect circle, how could you do the measurement that proves with certainty that the space is Euclidean and the circle is perfect? The concept of pi in this context is an abstraction.

    The concept of zero also is not straightforward. Where does zero exist in the real world? We can make an abstraction that there are zero tigers in the room, and the fact that there are no tigers there could even be termed as "a discovery". But the concept of zero is an abstraction that simplifies the reality and is not that reality. If the room has people in it, then all the basic ingredients of tigers are in that room, since maybe 4 people could have all the atoms and molecules needed to make one tiger. But, we abstract the idea of tiger and classify it, and then we count the number of things that fit the abstract concept in our minds. Yes, tigers exist and can be discovered once we define what a tiger is, but zero tigers can't exist. You can't discover zero of anything, but you can invent the abstract idea of the absence of something. Further, zero does not exist in the real world as in independent thing, as far as we know. Even a vacuum has stuff in it.
     
    Last edited: Oct 5, 2011
  18. THE_RB

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    Well let's say there is an alien civilisation out there on another planet somewhere...

    At some early point in their science history the early mathematicians would have realised that there is a simple relationship (which exists even before discovery) between the diameter of a circle and the distance around its circumference. They would have the same discovery of Pi that we had, as the Pi relationship is real and always exists. The circle is a real thing, and both it's diameter and circumference are real characteristics, simple and empirically measurable.

    I think that would always be the basis for "discovering" math principles, the fundamental principles that are real, can be measured and demonstrated with real models, and compared to a pure math model.

    Likewise zero is a real model. A box can hold 3,2,1, or 0 apples. All 4 conditions are equally demonstratable. An object may have a speed of zero, or 1 or 2 or 3 MPH. A geometric object drawn on a grid can have one side with a length of 3cm, 2cm, 1cm or 0cm. All demonstratable and real. I have to conclude zero is a real thing and would be discovered quite early within the scientific development of a civilisation.

    So any aliens we bump into will understand Pi, and have the same value for Pi we have, likewise other relationships for triangles, squares and square roots, multiplication and division etc. It is generally accepted that communication with an alien race (if one existed) would start with these math concepts as they MUST be universal and clearly understood by any advanced civilisation.

    Now infinity on the other hand, to me seems "invented". It can't be modelled or demonstrated in any real way and only really exists conceptually as the reciprocal of zero. Or not the reciprocal of zero, as math nerds will argue. :)
     
  19. steveb

    Senior Member

    Jul 3, 2008
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    Interesting discussion. I like the fact that you bring simple numbers into the discussion because it seems to me that it is easier to claim that counting numbers have real existence than numbers like 0, infinity, transcendental numbers and i=sqrt(-1). Many things seem to quantize to discrete levels in the real world. Atoms have discrete energy states and they also have a discrete number of particles. The hydrogen atom is like the number 1 if we count protons. A stray neutron is also like the number one. But, a neutron and a proton is also like the number 3 because they are made of 3 quarks, and this is where the abstraction reveals itself. It all depends on how you look at reality.

    Anyway, a person who wants to prove that math has any aspects that are discovered, should start with the simplest example, since he only needs to prove one example, and natural counting numbers are a good place to look.

    But, it's hard to prove anything here on either side.

    I also like your argument that another alien civilization would come up with the same mathematics. If every independent intelligent being always arrives at the same mathematical language, it would tend to suggest that there is something "out there" tangible that we are all "discovering". But, I guess we can't know that for sure, unless we meet many other civilizations.

    I don't like the argument about pi. I've mentioned this a couple of times above and nobody seems to want to address my points about the fact that we do not live in Euclidean space. Real circles with the property that ratio of circumference to diameter is a constant and that that constant is exactly equal to the transcendental number we call pi is a complete fairy story. It's just not true physics. It's approximately true physics for sure, but "approximately true" is infinitely far from "perfectly true" when it comes to mathematics. Perhaps this point is too subtle for anyone to notice, but I believe it is the crux of the entire question.

    Personally, I feel that numbers are abstractions, but counting numbers come the closest to having real existence, in my mind. I also feel that infinity is invented, but I think it is the one candidate that had the potential to be a real discovery. If physics had discovered that the universe is infinite and unbounded (which doesn't seem to be the current view), then that would be an example of the math discovering something real. But it seems there is no place where you can find infinity of anything (distance, mass, charge, energy ... etc). But, if we find an example, then PG's point is proved conclusively.

    So, my view is that mathematics has the potential to really discover something, but I don't know a good example of it, and perhaps it hasn't happened yet. But, that's why I'm asking for examples. I don't know everything and there may be a good example I'm not thinking of. It seems, so far, math has been abstraction that can model interesting aspects of the real world, but it never does so perfectly. Physics uses math more than any other science, and physics discoveries often take the form of ... "hey look, I have discovered that this particular aspect of nature behaves very similarly to this mathematical model I've developed". If only that model were exact, then a mathematical derivation might have discovered something real. But, it is usually experiments that reveal the discovery, and the model is made to fit the data.

    One thing is certain however. Mathematics is a great tool for discovery. Whether we are inventing that tool, discovering that tool or doing a combination of both doesn't change that fact.
     
  20. t_n_k

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    Mar 6, 2009
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    King Solomon apparently had an object [the "Sea"] made for his temple which points to an "estimate" for the value of what we call Pi - see Second Chronicles Chapter 4 verse 2.
     
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