Article on Mathematical Realism

Discussion in 'Physics & Math' started by Nyr, Aug 14, 2009.

  1. Nyr Registered Senior Member

    Messages:
    102
    First of all, I wasn't sure of where to put this - in the physics and mathematics section, or in the human sciences section. But since it is essentially about math, I decided on here.

    Anyway, this is an article I came across from an unknown author regarding mathematics, in which (s)he disagrees with the notion of mathematical realism; in other words, the commonly held belief that mathematics is the accurate picture of the physical world. Instead, (s)he regards that mathematics nothing more than an abstraction of the human mind to conceptualize experiences. Here's how it goes:

    There is a certain romance to mathematics,
    undoubtedly the result of the belief that
    mathematics is a concept that transcends humanity.
    A great many philosophers and mathematicians
    view mathematics as the last bastion of objective
    truth in an otherwise subjective world. According to
    these thinkers, mathematics embodies not only the
    natural processes of the world, but also the very
    nature of reason itself. Learn mathematics, so the
    thinking goes, and you have learned the nature of
    the world.
    This mythology, while certainly appealing,
    is unfortunately false. Mathematics does not exist in
    some ideal Platonic world; without the human mind
    to conceive of mathematics, there would be no
    mathematics. Mathematical relationships are not
    “discovered” by humans in the sense that the
    relationships exist independent of the discoverer.
    Rather, these relationships are created by humans to
    fit existing experiences of the world. The Greeks,
    for example, did not discover pi, the relationship
    between the circumference of a circle and its radius.
    Instead, they invented a relationship between two
    arbitrary concepts. Without the human designations
    of “radius” and “circle,” there would be no pi.
    Skeptics will argue that mathematics must
    be objectively true; how else could it so accurately
    describe the world? The answer to this question lies
    in the realm of cognitive science, the study of the
    relationship between the brain and the mind. Recent
    studies in this field demonstrate that mathematics
    arises from the human brain’s experience of the
    world around it. Using information gathered from
    the senses, the brain is able to move from a concrete
    concept to an abstract one. For example, the basis of
    arithmetic is the concept of object collection. The
    human brain has the capacity to recognize the
    general size of a collection and classify it as “big”
    or “small.” This idea can be translated into the
    concept of a number, an abstract representation of a
    collection. If an object is removed from a collection,
    the brain will note that the collection has become
    smaller. Add an object, and the collection becomes
    larger. Thus, abstract addition and subtraction are
    derived from concrete experiential evidence.
    In fact, mathematics is just another way of
    experiencing the world. Mathematical equations fit
    natural relationships because it is the natural
    relationships that spawned our understanding of
    math in the first place.​

    IMO, this view tends to the question of scientific realism, but doesn't quite get there, because here the author is unconcerned with the ultimate nature of the universe, but is merely making a point that human perception doesn't necessarily equate to objective truth.

    Anyway, you people convinced? For me, this certainly created doubts on how I regarded math as the ultimate viewing glass; broke the romance surrounding it, as the author puts it. Yet, how far would you go in giving importance to mathematics; how perfect an indicator of reality would you hold it to be?
     
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  3. CptBork Valued Senior Member

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    From a philosophical point of view, I would agree that mathematics is nothing more than a construction of the mind. Or rather, it's a logical system whose basic assumptions are chosen so as to fit with the computational structures in our brains. Historically we started off gaining an instinct through numbers by counting, trading, measuring etc., but once we set up a bunch of abstract axioms to describe and underpin these notions, the ideas no longer depend on human intuition, but rather they become arbitary logical constructions, whether or not they have any use or application in the real world.

    On the other hand, regarding the explanation as to why our reality is so well-described by mathematics, at this point it looks to me like the author is stepping outside their domain of expertise and making some very questionable assertions. If I understand their argument correctly, they're saying math works so well because our minds already gained an intuition of the universe, and math is just the language our minds use to interpret it. I simply don't see how this can rationally explain math's predictive power. It's one thing to claim we've seen the whole world already, and now our brains are sorting it into mathematical expressions. It's another thing altogether to claim our minds have previously created the mathematical structures which then correctly predicted things which had never been measured or tested before. That would imply our minds are somehow dictating what reality should be and thus making it conform to our mathematical notions, whereas it seems much more plausible that reality has always been that way, and math predicts things we haven't seen or thoroughly measured yet because, in fact, it actually does describe how things really work.
     
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  5. Nyr Registered Senior Member

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    102
    I pretty much agree with that view; the fact that math falls into place so perfectly in describing the universe makes it dubious that it were it not for the existence of our minds, such harmony wouldn't occur.

    Yet, I have more of a middle stance on the uniqueness of math. Agreed, math is a wonderful way to represent the world; a very efficient logical system to abstractize (if the word exists) our empirical observations; but this doesn't make it the only possible way of describing it. I'm not an expert here, but I do think that if some of the axioms initially conceived of by the Greeks, that set the road for what is now science, were different, i.e. used a different approach, the outcome would be the same. Mathematics as we know it isn't the only way of describing reality. One example that I think might be valid to substantiate my point is that of Vedic Mathematics, on which most of today's highly commercialized Speed-Math systems and all are based. Though leading to the same outcomes, for a very basic example, solving a set of linear equations, the approaches used are completely different from conventional maths. But my point is that such systems do not have to be mathematical at all, any such sound logic-based system should be good for describing reality; mathematics isn't the only tool. After all, for explaining physical phenomena, we are boiling all such explanatory-system's worth down to their instrumentalist effectiveness; and here, IMHO, math doesn't have to be the only possibility.
     
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  7. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    Nyr---please post a link to your article.
     
  8. Nyr Registered Senior Member

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    102
    It actually came from an educational C.D. - the article was a critical reading comprehension passage.
     
  9. temur man of no words Registered Senior Member

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    1,330
    How do you know the world existed before human?

    No, they did not invent it. They discovered it.

    This is a straw man. The question is not so much about objective truth of mathematics, as it is about objectivity of mathematics.

    This is not so obvious. Math can accurately describe phenomena like quantum effects and relativity, which are not really part of the everyday reality that gives rise to mathematical intuition.

    If there are laws and patterns in the real world then math is the only way to go since math defines what we regard as laws and patterns.
     
  10. CptBork Valued Senior Member

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    6,460
    Well I've never heard of Vedic math before, so I did a wee bit of research on it. Looks like a bunch of strategies for doing certain numerical calculations, but it wasn't hard to understand from a mathematical POV why the ones I was looking at work. It's still math, it's just a specific methodology for approaching calculations that can really be done infinitely many different ways (using mathematical operations). You can represent it by diagrams, mantras, symbols or whatever, it's still a method that can be built from the basic tools and axioms of mathematics.

    Mathematics is, after all, the study of logical structure. Numbers are just one example of such a structure- quantum physics wouldn't even be possible if it were restricted to just thinking about numbers and numeric equations. Any logical system you develop to describe the universe, if it gives the same predictions and calculations as conventional math, would be logically equivalent to ordinary math in everything but notation.
     
  11. Nyr Registered Senior Member

    Messages:
    102
    Paleontology, archeology, evolution, etc.


    That is true, if we in the first place assume that the things mathematics describes have an independent existence outside of the human mind.
    Your point being?

    Yes, I agree with that..
     
  12. kurros Registered Senior Member

    Messages:
    793
    As the article says, all math eventually boils down to tricky ways of counting things, in some sense. So as long as the world itself is an objective phenomena (which I generally feel would be hugely self-centred of us to deny), and it is possible to count things in that world, then to me that says that math itself is an objective part of that world.
     
  13. Pete It's not rocket surgery Registered Senior Member

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    10,167
    I think this is fundamentally incorrect. I think that the author is confusing designations with concepts with real entities.

    "Circle" is a designation, a human invention. It corresponds to a concept, the concept of a round shape, and the concept of the set of points in a plane that are a particular distance from a fixed point.
    But the designation and the concept also correspond to real things, actual shapes. The shape of a circle is unambiguous. It has been realized independently many times, by different cultures, by other organisms consciously (e.g. dolphins making bubble rings) instinctively (spider webs, bird's nests) accidentally (ant hills, worm holes, fairy rings) and inherently (your eye, a daisy), and by non-living things such as bubbles, craters, weather patterns, stars, galaxies, and planets. Clearly, the concept of a circle is not a human invention. It is a discovery that has been given an invented designation.

    The properties of a circle (radius and circumference) are part of the discovery. These properties are properties of actual circles that really exist. They may be given arbitrary designations, but the concepts themselves are not arbitrary, and nor is the relationship between them.

    If pi were a human invention, if pi depended solely on its inventor, then it would be unlikely to be given the same value by two independent inventors. The fact that it has been discovered independently with the same value indicates that it relies on something beyond those inventors, something with its own independent reality.
     
  14. Nyr Registered Senior Member

    Messages:
    102
    I see your point. I think the author chose the wrong word in 'invented'; 'discovered' is the only way the sentence would be correct. But then, that would contradict the whole gist of his/her argument.
     
  15. temur man of no words Registered Senior Member

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    1,330
    My point being that it does not make sense to talk about objective truth of math, because it is trivially true by definition. What I mean by objectivity is that (the essence of) math does not depend on who is working on math. Here I partly rely on Pete's answer.
     
  16. phyti Registered Senior Member

    Messages:
    732
    It seems the straight line/diameter, the circle/circumference are simple idealized mental patterns used to approximate what is perceived in the external world.
    Pi is then the relation between the two concepts. You used that pattern when referring to the above examples.
    To prove my point, find me one real circle. I'll wait.
     
  17. Pete It's not rocket surgery Registered Senior Member

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    10,167

    Please Register or Log in to view the hidden image!


    Find me an approximate circle that doesn't have a ratio of approximately pi.
    I'll wait.
     

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