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Thread: Calculation without Understanding

  1. #1 Calculation without Understanding 
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    Calculation without Understanding

    Early in our institutional education system we learn arithmetic. We learn to add, subtract, multiply, and divide. We learn to calculate without understanding.

    This mode of education follows us throughout our formal education system. We learn to develop answers devoid of understanding. We do this because, in a society focused upon maximizing production and consumption, most citizens need only sufficient education to perform mechanical type operations; that is perhaps why our electronic gadgets fit so well within our culture.

    If we think about this situation we might well say that this form of education best serves our needs. It is efficient and quick. However, beyond the process of maximizing production and consumption we are ill prepared to deal with many of life’s problems because we have learned only how to develop answers that are “algorithmically friendly”.

    In grade school we are taught to manipulate numerals (symbols) not numbers (concepts). We are taught in grade school not ideas about numbers but automatic algorithmic processes that give consistent and stable results when dealing with symbols. With such capability we do not learn meaningful content about the nature of numbers but we do get results useful for a culture of production and consumption.

    We have a common metaphor Numbers are Things in the World, which has deep consequences. “The first is the wide spread view of mathematical Platonism…[it] leads to the metaphorical conclusion that numbers have an objective existence as real entities out there as a part of the universe…Given this metaphorical inference, other equally metaphorical inferences follow, shaping the intuitive core of the philosophy of mathematical Platonism.”

    Quotes from Where Mathematics Comes From by Lakoff and Nunez


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  3. #2  
    Universal Mind John Galt's Avatar
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    I'm pretty sure that Lakoff and Nunez are seriously better educated than I, and considerably smarter. That is likely why I find their perception to be badly flawed. I'm sure I'm wrong, but I'm just too much set in my ways to change.

    They seem to be complaining that our way of teaching leads to the metaphorical conclusion that numbers have an objective existence as real entities out there as a part of the universe. Well, I'm pretty glad it does lead to that conclusion.

    If I were a highly intelligent, mathematically gifted individual, I am sure I could appreciate the abstract quality of numbers in much the same way that I appreciate more substantial items, such as rocks (which interest me) or a wood carving (if I were an artist) or a precisely manufactured machine part (if I were an engineer).

    I learnt numbers by counting beads and apples and classmates. And I've since applied those skills, painfully upgraded, to count money on a balance sheet, to assess mechanical performance of industrial processes, and to determine the shape of a buried valley using seismic data.

    All of that is real world. All of that sees numbers as representing real things. And all of that was damnably useful to me and to my employers.

    So, from my intellectually stullified, mathematically inept, boringly materialistic stance, I have to say Lakoff and Nunez are talking bollocks. :wink:


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    Quote Originally Posted by John Galt
    I'm pretty sure that Lakoff and Nunez are seriously better educated than I, and considerably smarter. That is likely why I find their perception to be badly flawed. I'm sure I'm wrong, but I'm just too much set in my ways to change.
    :wink:
    I am sorry to "hear" that. I am 74 and learning new stuff makes my world go-round. You miss a great adventure by being "set in my ways".
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    Coberst,
    are you familiar with irony? Did you read the rest of my post? I'm sorry if my intent was obscure. I am mocking the conclusions of these purported experts and disagreeing wholeheartedly with their conclusions, which I find to be foolish, elitist and impractical. That's just an opinion, of course.
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    They do however, have a point (if I read that post correctly) that the average end-user of pretty much any product (excepting food) does not fully know how it works and would not be able to fix it or create a replica. To this end the human race is quite badly equipped to deal with isolation from the minority who have the knowledge and expertise to design and manufacture such products.
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    Quote Originally Posted by coberst
    I am 74 and learning new stuff makes my world go-round.
    I am seriously impressed by the width of your reading and the sheer number of different authors you quote in your many posts is a tribute to your determination to keep your mind active at an age when many of us would be out to pasture. However, there is room for a debate as to whether you are in fact learning new stuff. It seems to me that you are trapped in some kind of educational Ground Hog Day (movie reference) where the point of your existence is to continually re-learn that the American education system is seriously flawed. It is a source of wonder to me how much of your reading leads you to this conclusion, which you share with us on such a regular basis that I'm afraid it has now got to the point where, for me at least, you no longer sound like the astute Sage of the Ozarks but more like the sadly deluded Wild Man of the Adirondacks.

    It is, of course, entirely possible that there is something wrong with the education system in America, but seeing as you can't personally do anything about it can I suggest that there might be other things worth learning, things that fill you with joy and wonder rather than cynical disdain?
    Everything the laws of the universe do not prohibit must finally happen.
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    I have a great deal of sympathy with the Lakoff etc point of view here, because I have never been able to quite grasp mathematical platonism.

    What does it mean to say that is real? Or that it is universal and absolute?

    Mathematical Platonists tend to say that the value of is the same everywhere and anywhere, but they haven't checked. Instead, it is true by definition . But that simply leads to the circular argument: how can something that is only true by definition also be true of the real world? Unless you decide that your definitions change the real world to match...

    Anybody else get where I'm coming from here?

    For what it's worth, every study I've read, or read of, suggests that humans (and animals) are born with an ability to differentiate number up to 4 or 5, after which they tend to think in terms of amount and proportion. As a result, I suggest, for the mathematicians to say (as they have): "God gave us the Natural numbers, all ther est is the work of mathematicians", is perhaps overstating the case. From all we can tell, 'God' may have given us only 1, 2, 3, 4 and 5! Everything else in mathematics is only true by definition.

    And there's little point speaking to a mathematician about this because he/she is usually so submerged in the subject, he/she cannot even appreciate the question, let alone any validity it may have.

    It's hard, I tell you, hard!
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    Sunshine

    What does it mean to say that pi, love, god, home, 3, law and order, fairness, freedom, etc are real. All are abstract ideas (concepts). What are abstract concepts? Abstract concepts are constructed from concepts created by various experiences.

    Grounding metaphors and linking metaphors are two of our most important metaphor types, which preserve inference structure gained from our common experiences by the neural process of transporting these structures from the source domain to the target domain.

    These metaphors are conceptual metaphors because the concepts, the neural structures, can be mapped, unconsciously and automatically, from the source domain, a common experience, to a target domain, an abstract or another concrete concept.

    “Grounding metaphors yield basic, directly grounded ideas.” A child at play discovers that placing one object with two other objects yields three objects. The same child discovers that removing one object from three objects yields two objects. These capabilities are common to all infants and likewise with many other types of non human animals. The first grade arithmetic teacher relies on this inherent understanding to facilitate the next step in teaching arithmetic.

    “Linking metaphors yield sophisticated ideas, sometimes called abstract ideas. Examples: numbers as points on a line, geometrical figures as algebraic equations, operations on classes as algebraic operations.” All of which require careful teacher guidance.

    “One of the major ways in which metaphor preserves inference is via the preservation of image-schema structure.” The child collecting a group of objects as her very own is conceptualizing the objects as being in a container totally separate from her playmates container of objects. She also knows when her playmate has entered her container without approval. “Mine” becomes a very important “container” and “container” being a very important image-schema.

    Quotes from “Where Mathematics Comes From” Lakoff and Nunez

    Quickie from Wiki

    Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato's belief in a "World of Ideas" (typified by Plato's cave): the everyday world can only imperfectly approximate of an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers.

    The major problem of mathematical Platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be Ultimate ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.
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    Quote Originally Posted by sunshinewarrior
    Anybody else get where I'm coming from here?
    Huh?

    Somehow I think that if we built a spaceship on the assumption that pi is 22/7 and flew it to the end of the Universe, our astronaut would have no fear that it might enter a region where pi is 3.0 and promptly fall apart. That same astronaut would never assume that all swans are white just because she had never seen a black swan. The belief that pi is always the same is somehow different from the belief that all swans are white, isn’t it?

    It's hard, I tell you, hard!
    For philosophers, not for engineers!
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  11. #10  
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    Quote Originally Posted by Bunbury
    Quote Originally Posted by sunshinewarrior
    Anybody else get where I'm coming from here?
    Huh?

    Somehow I think that if we built a spaceship on the assumption that pi is 22/7 and flew it to the end of the Universe, our astronaut would have no fear that it might enter a region where pi is 3.0 and promptly fall apart. That same astronaut would never assume that all swans are white just because she had never seen a black swan. The belief that pi is always the same is somehow different from the belief that all swans are white, isn’t it?
    We operate under the assumption that the physical structure of our universe holds in all directions - but near a Black Hole, for instance, the value of pi will change, won't it, because the geometry of space will be non-Euclidean? The tearing forces associated with the gravity of the big beast will make this consideration seem small and irrelevant, perhaps, but I thought it was a 'real' issue?

    Quote Originally Posted by Bunbury
    Quote Originally Posted by shanksie!
    It's hard, I tell you, hard!
    For philosophers, not for engineers!
    And not hard for mathematicians either, as I said - both youse guys are sure of what you're doing, black swans to the hindmost, but yes, us philo dilettantes have trouble with these things. After all, is pi is 'real' or absolute or whatever, where is it? Who set it up? Why? etc etc.

    Simply stating it as fiat, which is sufficient for maths, does not help the philosopher at all - it merely raises further, and worse, questions.

    Whoops, I should simply have quoted coberst quoting wiki on this:

    Quote Originally Posted by coberst quoting wiki
    The major problem of mathematical Platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be Ultimate ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.
    On this one, interestingly, coberst and I seem to be of the same mind (albeit I do not go as far as to state positively that mathematical entities are simply abstractions from our concrete concepts, I simply suspect that this is the likeliest answer).
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  12. #11  
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    Quote Originally Posted by sunshinewarrior
    near a Black Hole, for instance, the value of pi will change, won't it, because the geometry of space will be non-Euclidean? The tearing forces associated with the gravity of the big beast will make this consideration seem small and irrelevant, perhaps, but I thought it was a 'real' issue?
    I don't know. Does a black hole have a circumference and a diameter? How are they related if not by pi?
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  13. #12  
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    Quote Originally Posted by Bunbury
    Quote Originally Posted by sunshinewarrior
    near a Black Hole, for instance, the value of pi will change, won't it, because the geometry of space will be non-Euclidean? The tearing forces associated with the gravity of the big beast will make this consideration seem small and irrelevant, perhaps, but I thought it was a 'real' issue?
    I don't know. Does a black hole have a circumference and a diameter? How are they related if not by pi?
    Now here's one for the physics forum innit?

    My understanding was that, according to the General Theory of Relativity, space itself becomes non-Euclidean in the presence of large masses, and it is this non-Euclidean space that affects the path of light beams passing these massive objects and hence gives the appearance of their having been affected by 'gravity'.

    And, of course, a BNlack Hole is actually a point - mass but no volume and hence of theoretically infinite density - a consideration of its shape is therefore not really possible (unless it's rotating, in which case, see Stephen Hawking...)
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    And, of course, a BNlack Hole is actually a point
    The singularity at the center is a point but the black hole is considered to begin at the event horizon, which could be inches or miles in diameter (AFAIK).
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    Quote Originally Posted by Bunbury
    And, of course, a BNlack Hole is actually a point
    The singularity at the center is a point but the black hole is considered to begin at the event horizon, which could be inches or miles in diameter (AFAIK).
    ah, the infamous Event Horizon. Perhaps we ought to take this to the Astronomy section and ask what they think: "What is the shape of the Event Horizon, and is it non-Euclidean due to the distortion of space that the Black Hole itself creates?"

    What do you think?
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  16. #15  
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    I agree - and perhaps we need to decide if our black hole is spinning or not.

    I am in danger of missing my own event horizon, in the form of an annoying project deadline, so will leave it to you to post the question in Astronomy, if you don't mind, and will come back this evening.
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  17. #16  
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  18. #17  
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    And responded to:

    http://www.thescienceforum.com/Circl...ole-16616t.php

    Bunbury

    Where are you?
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  19. #18  
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    I'm here.

    Where are you?

    And if it's established that the ratio between circumference and diameter need not be pi, then can we still call it a circle?

    What would Euclid say?
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  20. #19  
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    Well, a mathematician would say that pi is not equal to C/D, and therefore a circle is not dependent upon it.

    I'm just a philosophy dilettante so I say, fair enough, but then want to dig a little deeper...
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