lakoff and nunez’s 4 G’s

15 09 2010

Very interesting reading, this Where Mathematics Comes From, but there is a whiff of smoke and mirrors, as I shall draw attention to towards the end. But first, what are their 4 G’s. They are the metaphoric mappings they have constructed to explain arithmetic, namely:

  • object collection – how piles of things behave
  • object construction – how things are composed of parts
  • measuring stick – a comparative standard
  • motion along a path – how things, including ourselves, move in space
  • Brilliant, I have to say. Very useful I would have thought in terms of teaching. Simple explanations as to how to make sense of the magic that is numbers and arithmetic.

    I have six/seven observations about the first section which takes up the first quarter of the book, and mostly regarding the all important third chapter.

    First, the language of “grounding metaphors”. There is some confusion as to the “direction” of this metaphor. It sounds like the abstract arithmetic is grounded metaphorically in physical systems. But this is clarified on page 101: the physical experience is the basis from which the metaphor projects into arithmetic. That is, from “sensory-motor” they call it to abstraction of number system. This may appear pedantic, but it reveals something about the active intent of lakoff and Nunez; think Reflexive Imposition here: their thinking is influencing the thing they are thinking about, rather forcefully.

    Second, for us to be playing around with piles and measuring sticks and the rest of it, there must be a mind to make sense of it. L&N don’t examine this, for this is the direction of Buddhism. They present mind’s rather splendid ability to shift piles and manipulate wholes and parts a priori, and use this to explain the jump to maths. Nowhere in this is an understanding, or even an explanation as to how we do these things in the first place. Academically, at least, this is an interesting if not peculiar position — not that their exercise is thus invalidated, for actually, I think this represents the self-referential and self-enclosed nature of explanation that this realm entails. Indeed, I think it supports the methodology I have set out with XQ.

    Third, as an explanation of arithmetic, including the metaphoric interpretations to 0 and 1, it fulfills the operational standard of science: it explains the mysteries of the mind with our understanding of things. Definitely a lot less crude than slapping electrical nodes onto the skulls of buddhists to attempt to divine the effects of meditation, but nevertheless, still a little clumsy in attempting to push our objective thinking into the subtleties of our conscious condition. Neat as it is, and useful to, it is the same as the instructions you’d find on a tin of paint.

    (This is coming across as being more scathing than when first these thoughts came to me. I suspect it is to do with the medium of writing, and the association of exaggeration that written polemic induces. I think L&N are brilliant in their exposition. I need to show the limitations of their western methodology. And the more convincing it appears, and indeed useful it is, the more we must be vigilant that we do not get seduced by this way of thinking; after all, as westerners steeped in scientific methodology, we are the most susceptible to it.)

    Fourth, L&N talk of the grounding domain as “forming collections”, “putting objects together”, “using measuring sticks” and “moving through space”. These are all processes. Again, requiring mind to perform them. It sounded like it was grounded in things, in motor-sensory experience, but really, it is all process. Thus, the metaphoric mapping they are producing is rather sophisticated, equivalent IMHO to the complex mapping of functions that is calculus.

    Which leads me to the fifth observation: their list of simple cognitive capacities on page 51 equate to arithmetic, but their additional cognitive capacities on page 52 equate to algebra; that is, the ability to use symbols or representations, that is, words. It would be wise for us to be aware that L&N may appear to start simple and then go into more complex fields, as is the want of a standard scientific text book, but they have actually begun at rather sophisticated levels to start with. After all, these are not high school teachers, but university academics at the bleeding edge of cognitive psychology.

    Which leads further to my sixth observation — what do you think of their final sentence to their first section on page 103?

  • We are now in a position to move from basic arithmetic to more sophisticated mathematics.
  • XQ may start with arithmetic too, but the direction of greater… subtlety… is towards counting. That is, it is not about creating a foundation of understanding, as L&N put forth so enticingly, but rather to intuit interpretations of internal processing in one’s own mind. And the simpler it is, the deeper it’s operation in mind, and the calmer the mind of the subjectivity needs to be. The direction of genuine inquiry in this direction, in terms of examining the mental processes involved in mathematics, is characterized by simplicity and sensitivity.

    It all makes for interesting reading, for sure. And I suspect that their explanations will appear in my mind while I continue with my XQ exploration. Indeed, they can’t fail to because they are now in my mind because I’ve their their book! Nevertheless, there is a striking validity to their contributions; I’d just like to approach them from the inside, as it were.

    Finally, almost like their commentary on the metaphors for 0 and 1, a few small points:

  • motion on a path refers to time — a topic blaringly omitted from a serious academic text that purports to examine the processing of mind, tut
  • the measuring stick is the minimal artificial mechanism, actually probably after the tally sticks; the first virtualizations of our ability to notice patterns in space-time
  • object construction is related to wholeness, to completeness, something which is central to a way of thnking that comes naturally to us human beings, thus giving rise to part and fractions
  • object collection relates to our granular way of comprehending actuality, as sums of wholes likes apples, say
  • Thus, the 4G’s are not like the three laws of subjective reality, something which we can depend upon no matter what thoughts and ideas we navigate in our intellectual endeavours, but more a useful constellation of terms and concepts to help us explain, or even describe, how to do arithmetic well. After all, all kids have these experiences, and what better way than make the shift of abstraction to number and arithmetic than through their direct experience?

    The book is shaping up to show more about how Lakoff and Nunez think about the world, or rather reveal about their thinking processes, than it might about mind, consciousness and mathematics. Interesting not only for L&N, but as an indicator of current scientific progress in this realm.




    One response

    16 09 2010
    Leon Conrad

    David –
    L&N cover more of the stuff you (rightly, I think) think is missing in the book I’m reading of theirs – The Embodied Mind, particularly notions of time as metaphor, and the topic of levels of looking at thought – I like the inclusionist attitude they take. I also think the emphasis on the role of metaphor is sound – important – seminal.
    There is a danger with their work that the symbol gets confused with the thing symbolised.
    Their take on metaphor is novel – yes, it’s metaphor, but it’s grounded metaphor – based on neural networks formed through our experiences of the world – of taking in cycles, moving through space, observing motion, gathering and counting.
    Your interest in maths enables you to come to the equation they choose to analyse in Where Mathematics Comes From with some informed knowledge – it means little to me. I’m not as evolved in terms of mathematical thought. Perhaps that is why I’m looking at their work from a different point of view.
    What struck me in reading their work is that they take many things for granted – to what extent does gravity influence the formation of grounding metaphor – to what extent are our grounding metaphors influenced by the movement of the sun (and moon) and our observation of them?

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