Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being by George Lakoff
My rating: 5 of 5 stars
Cognitive linguistics has at its underlying aesthetic the very literal understanding that how we think of things is what they are. This follows post-structural rhetoricians like Paul Ricoeur who argue that the connective tissue of language is metaphor — where metaphor is the substantiation of the naked copula form is through content. We forget the form of the copula in metaphors and thus experience the content as a variation of the copula form instead of being the actual connection. In other words we understand our world through representations, never understanding that an ontologically reified point of view is only possible because metaphors position the copula through its latent content so that the form of the copula becomes seen as the “ding as such”. In other words, representations only appear to be representations because one of the formal representations comes to represent nothing but the pure presence of its own linguistic connectivity.
Having said this, I was surprised (but also not surprised) by the comments below. Many people were confused by this book, blaming either the psychologists for not living up to their expectations (of not being neurologists), or blaming the thickness of the mathematical concepts presented. We often think of the pure formalism of math as being objectively isometric (as one reviewer said) to the proposition that reality is always present beneath our representations. One key connection that Lakoff and Nunez being up repeatedly is that many mathematical formalisms (such as zero, negative numbers, complex numbers, limits, and so on) were not accepted even long after their calculatory prowess was proven effectual… what made these concepts acceptable wasn’t their caculatory significance, but rather their introduction to the cannon of mathematical concepts via metaphoric agency. For instance, we take zero for granted as being “real” even though we understand it to not be a true number. It only was after a new metaphoric concept was presented for zero to be sensible (numbers as containers and origin on a path) was then zero incorporated into the cannon of what was acceptable. This understanding proves to be the very “twist” needed for Lakoff and Nunez to write this book. While many of the concepts are perhaps difficult for some of us non-mathematicians to grasp, I found their presentation to be concise and illuminating. Their tabulatory presentation of metaphors side by side allow us to grasp the mapping of logically independent factors from one domain into another. This basic movement is in fact a methodology they may have picked up from analytic geometry as invented by Rene Descartes: the translation of continuums into discrete points.
While it is understandable that they trace the building of conceptual metaphors via simple to the more complex, I did find their delay of speaking of analytic geometric to be confusing. When a topic is presented I want it to be explained, rather than having to wait half a book to read on it again. This is really my only possible complaint.
Overall, this book helped me connect the observation of formalism being prevalent as an organizing feature of pretty much all procedure and knowledge formation today with the root of that formalization, being the atomization of discrete epistemes of knowledge, whether that knowledge is granular or point or vector, or some other kind of rigor. We can also thus understand mathematics as being synthetic, contrary to what most philosophers in the west (excluding the great Immanuel Kant, Alain Badiou and Gilles Deleuze) understood.
Today, through our rockstar mathematicians and physicists we revisit the old Platonic hat that math is somehow natural, only apparent in our minds and yet more real than anything else this world has to offer. This is a troubling and definitely cold and etymologically naive sentiment. It’s mysterious that anything in this world is the way that is, let alone consistent as though following laws, but that isn’t any reason to be hypnotized by our own intellectual conceptions. As Lakoff and Nunez point out, while some math is applicable in the physical world most conceptual math remains beyond application of the physical world, as there is no physical correlation with those domains. Such application may be possible in alternate universes, but such universes remain the sole conception of our mind.
In other words, how we think of something is what we understand it to be, that is true, but it’s also how we experience what we understand to be to be what it is. To get into that deeper thought requires an unpacking of the most erudite philosophical concept of all — that of the number One, arguably the only number there ever has been and in fact the only thing there has ever been. Understandably this is beyond the scope of mathematics itself, or at least beyond the tenants of what most mathematicians are willing to go. I don’t want to belabor the point here, but I will state that the case study at the back of the book is quite compelling. If Euler’s equation may work in formal procedure alone, but as Lakoff and Nunez point out, the construction of that equation is only possible through the discrete projections of layered metaphors to understand equivalence of conception regardless of the different construction domains these metaphors originate from (logarithms vs trigonometry, vs Cartesian rotation vs complex numbers)… ultimately a unity is made possible because such closure is driven by the singular domain of our minds. In our minds, with their ornate metaphors, their clearly trained disciplines and their innate mechanisms of spacial orientation, we are able to combine complex concepts into the most brilliant of abstractions.
As such this book may be too difficult for most of us to read, because it requires we re-orient our thinking along different parameters, different assumptions about who we are and what we are doing when we study and create math. This probably won’t jive with most people, as it seems for most people, knowledge is less about reworking what they already know into a new arrangement, and more about filling in gaps in the arrangements they already have.
I’m not saying that this cognitive linguistic approach is equivocally true, I’m saying that truth is more than how we arrange something, but the entire range of what we can conceive of to be a relation that brings to light new connections. In the end, I think for most of us, the only legitimatizer of reason remains one’s singular emotions, of what feels to be acceptable. To get around this, requires the most stern of discipline and the most unabashed eagerness to learn something new. This is also a reminder that math is not formal procedure as we learned long division in our elementary grades. Rather, math is the unabashed conceptualization of formal arrangements in their absolute complexity. In this way, even understanding how highly educated mathematicians think of math is illuminating to how you and I can understand something (ourselves and the universe) in new light. That alone is worth reading this book.
So do read this book because it’s beautiful, but also read this book because it’s another way of considering something you already think you know. After all, learning isn’t a matter of facts. Facts are boring; the world is full of facts we can never memorize (such as where your car was on such and such date and time. Kind of useless, except in special cases, such as in the immediate). Learning is the mastery of how to conceptualize, how to arrange information and how to further that arrangement through metaphor of what is.
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