thesis: Metaphor is not just a poetical way of speaking that can be ignored or paraphrased away if we so wish; it is deeply embedded in our language, culture, and the way we think, and hence affects how we experience and interact with the world and other people.
The book is crammed with claimed metaphors and example short phrases to back up the claims (a tiny selection is reproduced at the end here). The metaphors range from the obvious (ARGUMENT IS WAR), to ones that are so deeply embedded in our thought processes that it takes a little longer to appreciate the existence of a metaphor at all (MORE IS UP; LESS IS DOWN). The very number of examples provided goes a long way to supporting the thesis.
The authors argue that these truly are metaphors, and not just homonyms (words that coincidentally sound the same, such as bank for the side of a river and a place to keep money). They show that we can extend conventional use of a metaphor, usually poetically --- He prefers massive Gothic theories covered with gargoyles. If the metaphor were merely a coincidental homonym, we would not be able to do this. Also, when we use two metaphors simultaneously, we do so in a coherent manner, rather than arbitrarily.
Many of our deepest metaphors are based on our physical interaction with our environment: up/down, in-out, back-front. But even so, some of these deep metaphors are different in different cultures:
The first part of the book discusses metaphor in language; the second shows this has a profound impact on philosophy, which classically tends to ignore or downplay metaphor. A phrase like the fog bank is in front of the mountain can be understood only by assuming that
The latter part of the book is an explanation of why the new metaphorical experientialism is a better philosophy than either objectivism or subjectivism. Now we can realise that we are structuring our understanding by using just one of many possible metaphors, and maybe chose to use a different one (AN ARGUMENT IS A JOURNEY, rather than ARGUMENT IS WAR, say).
As well as detailed accounts of metaphor, there are some passing references to the way people carve up and classify the world: prototypical rather than hierarchical (robins and sparrows are prototypical birds, ostriches and penguins are not) and context-dependent (I may classify a bean-bag as a chair in some contexts but not in others). Lakoff goes into this more deeply in his fascinating later book Women, Fire, and Dangerous Things.
I originally read this book in 1993, and enjoyed enormously. I came at it from a background of Computer Science: it was recommended in Goldstein and Alger's Developing Object-oriented Software for the Macintosh, itself a refreshing antidote the 'natural categories' hype of object orientation. For quite a while now, I have been saying: 'An object-oriented hierarchy is just one possible way of modelling the world. We have been seduced into thinking a correct classification hierarchy must exist, because of biology.' And now, after reading the discussion of biological taxonomy in Chapter 8 here, I can strengthen my argument by adding, 'but it isn't even true for biology!': there are no such things as zebras, or fish.
Lakoff shows that the way we categorise things in language is more often radial and metaphorical than hierarchical. This means that certain 'prototypical' members can be thought to be 'good' or 'central' exemplars, while other, weirder members may be 'poorer' examples. A sparrow is a prototypical bird, an albatross less so, a penguin even less so. (And what about a duck-billed platypus? See the later section below.) Later in the book Lakoff provides a case study of the various meanings of 'over', and builds up a detailed, and amazingly complex, metaphorical map.
This is a thought-provoking and illuminating book. It should be recommended reading for all computer scientists who think hierarchical classification is 'the one true way' because 'that is the way our minds work'. It isn't, and they don't.
Here are some more detailed points I noted while reading:
I was fascinated by the account of classification in the Australian Aboriginal language Dyirbal, where all nouns are members of one of four classes. (Women, fire, and dangerous things are some of the nouns in class II)
Lakoff notes 'exceptions', such as the animals (dog, bandicoot, echidna and platypus) that are in class II, rather than class I where they 'should' be. However, the echidna and platypus both lay eggs, and both have a beak-like snout. Might they not 'really' be birds, and so properly in class II? (I am tormented by the thought that this language is dying out, and it might be impossible for anyone to do further research here. It's so easy to come up with glib explanations for strange observations; it's much more important to be able to check against reality.)
One example of his thesis that Lakoff gives, at the end of Chapter 9, is that "all other things begin equal, single digit numbers should be judged as better examples [of numbers] than double-digit numbers, which should be judged as better examples than larger numbers." Also, there are other prototypical effects, such as evenness, multiples-of-five, and multiples-of-10. [This kind of prototypical model of numbers, rather than detailed arithmetical knowledge, is built in to FARG's Numbo architecture.]
I have noticed that for people who are well-acquainted with numbers, the 'best' (or at least, a 'very good') example of a prime number seems to be 17. So why not a single-digit, or smaller double-digit, prime? Maybe because 17 is the first prime number that is not special in some other, 'stronger' way. 2 is the 'best' example of an even number (and it is a 'bad' example of a prime, being the only even prime, which shows how important 'evenness' is to us); 3 is the 'best' example of an odd number (since 1 is too special); 5 is special because of the 'multiples-of-five' model; 7 has mystical overtones; 11 does not 'look' prime (since all repeated-digit numbers are divisible by the repeated digit); 13 is strongly associated with 'bad luck'. To numerate people 17 is so quintessentially prime that, if they are asked the question (which was once actually set in a school maths exam) "what is the difference between 12 and 17" they often reply "17 is prime, 12 is composite", rather than the 'correct' answer "12 is even, 17 is odd". It seems that, to numerate people at least, 17 is a better example of a prime number than of an odd number. (What this says about the people who set the question, I hesitate to think.)
I make a prediction: where 'mother' is being used as a genetically-motivated analogy (with no overtones of birthing or nurturing, just relationship) it should be possible to use 'father' in the same way. In computing, we do see this. Disk backups use the 'grandfather, father, son' scheme; tree structures, looking like genealogical charts, can have 'father', 'son' and 'brother' nodes. Non-sexist language is overtaking this usage, and the terms 'parent', 'sibling' and 'child' are heard more often. But only in genetic analogies; I have yet to hear 'parent-board' instead of 'mother-board', for example (though maybe I've just been lucky there).
Lakoff’s thesis is that metaphor is deeply embedded in the way we think about everything. So what happens when this mindset is turned to poetry, which is supposed to be metaphorical? Well, we get that it is even more metaphorical: it takes everyday metaphors, extends them in novel ways, and combines them into a dense interwoven structure that implies more than can the individual metaphors alone. It is the combination and overlapping of metaphors that gives poetry its added richness and depth. [Allegedly. I will admit right up front that I know nothing about poetry, and care somewhat less. But I am interested in Lakoff’s ideas about cognition. And it seems worth studying those ideas where they occur in extreme situations – like poetry.]
So Lakoff and Turner identify a bunch of metaphors used in everyday life (LIFE IS A JOURNEY gets a lot of mileage), and illustrate how they are used, extended and combined in certain poems. The point is that these metaphors are everyday concepts. They aren’t just “pure poetry”, in contrast to “literal” prose. Our understanding of how they work in the everyday context is necessary for our understanding of them in the poetical context.
Having dissected individual lines of various poems, the authors go on to analyse an entire (short) poem, “The Jasmine Lightness of the Moon”, in terms of its layered metaphorical imagery. [If poetry had been analysed like this when I was at school, I might have been more interested. But then again, I’m interested in the analysis – not the poem!]
But their key message is in the final chapter, on proverbs as poems, where they identify THE GREAT CHAIN OF BEING metaphor, deeply embedded in our culture and cognition, and deeply antithetical to ideas of equality and conservation.
It has taken me three months, off and on, to read this book. Not because it is dull or boring, I hasten to add. Rather the opposite: it was recasting fundamental concepts in my head, and I can take that kind of action only so quickly.
The authors are interested in why mathematics is the way it is, and why certain ideas are true. (This emphasis on explanation, here and elsewhere, is to be applauded; it contrasts well with the rather less helpful "shut up and calculate" schools of thought.) They explain that structure of mathematics is built from various metaphors, ultimately grounded in our embodied reality. As I understand it, the metaphors are cognitive descriptions, that describe the way we embodied creatures actually think and understand; the mathematics is a construct that makes use of the metaphors (usually implicitly).
This is spellbinding stuff. The authors illustrate their thesis with a host of deep and fascinating examples, teasing out the actual metaphors that seem to underlie many parts of mathematics. These metaphors are (for the most part) stunningly simple and utterly compelling. They start with the four grounding metaphors of arithmetic, then move on to a host of linking metaphors. Much of the book is taken up with applications of the Basic Metaphor of Infinity (BMI), showing how this single metaphor, when blended and combined with others, can explain the approach to infinity taken in many different branches of mathematics. They cover a wide range of mathematical concepts, and illuminate every single one.
I found the discussion of the role of infinitesimals, the reason for the differences between transfinite cardinals and ordinals, and the case study of the "bumpy line" paradox, particularly fine. (I did find the final big case study, of why exp iπ + 1 = 0, just a bit of a let-down, possibly because I has come across most of the explanations before, and the final explanation of just why exp iθ = cos θ + i sin θ felt a little bit of a cop-out. But these exceptionally clear explanations given would make an excellent addition to a course teaching this for the first time.)
The authors' argument that learning mathematics would be made simpler by making these metaphors explicit sounds very plausible. Their own explanations of usually difficult concepts are mostly examples of remarkable clarity. (There are some problems, admittedly. In particular, I found the explanation of sequence limits, involving so-called critical elements, to be incomprehensible; I think they are just wrong here.)
The authors claim that the existence of these metaphors disproves the existence of objective mathematical reality (but they hasten to add they are not advocating a postmodernist philosophy; embodiment means there really are universal human aspects to mathematics, if not transcendent, Platonic, ones). Whether or not you believe the disproof, and my other quibbles notwithstanding, this is well worth reading. Read it if you want to understand the metaphors underlying mathematics, or even just to deepen your understanding of (some) of the mathematics itself.
I have a few niggles with some of the metaphors.
There is a rather nice distinction made amongst:
This is a useful distinction to make: our base ten (sum of products of powers of ten) representation is so deeply ingrained it is difficult even to notice it. But base ten is not the only conceptual representation we have. We can choose a conceptual representation to make a particular calculation easier. As well as the familiar base ten, we could choose other bases (such two, eight and sixteen for computer arithmetic, or 360 for angles); products of prime factors for whole numbers; fractions or decimal expansions for rational numbers; continued fractions for irrational numbers; etc. Similarly, complex numbers can be represented as rectangular or polar coordinates. The "numerals" that symbolise these other representations are often less familiar than our base ten symbolisation, however.
The authors point out that the folk theory of essences, with its categories and necessary and sufficient conditions, has has a deep influence on (Western) mathematics:
and, possibly more problematically, on science:
This links in to the much more extensive discussion in Women, Fire, and Dangerous Things about its influence on the way we think we categorise things, in contrast to the way we actually categorise things. It has implications for the current enthusiasm for object-oriented modelling in computer systems (which is why I started reading Lakoff in the first place).
The Numbers are Points on a Line metaphor may be more grounded than the authors assume. Ramachandran, in Phantoms in the Brain, reports there is a line of neurons in the brain that represent numbers.
The authors argue that Weierstrass continuity is not a generalisation of natural continuity (as characterised by Euler), because there are some monster curves that are not naturally continuous, but are Weierstrass continuous, such as x sin(1/x) at the origin. But there is an implicit infinity in this curve at the origin, so maybe natural continuity as characterised by Euler first needs to be extended with the BMI (Basic Metaphor of Infinity) to capture the motion of a point at infinity, at which point the curve is "naturally" continuous?
... that is, Readiness; Starting up; The main process; Possible interruption and resumption; Iteration or continuing; Check purpose achieved; Completion; Final state
On Hardy's warning not thinking of , infinity, as a number:
Of the "bumpy curve" paradox:
... I can almost hear the mutters of "only up to an isomorphism", though. What I find fascinating is that although these areas have their own numbers, those numbers are all the "same" (obey the same properties). This sameness is used by some as an argument of their external objective existence, and by others as an argument of their single origin, our own minds.
... but surely this is true of anything: "The only X we can know is the X that our bodies and brains allow us to know". Since the authors admit there are external objects, why no external mathematical objects? I assume it is that they admit the existence of objects, but not categories of objects, and hence not of mathematical categories like number. Nevertheless, I feel a residual Johnson-esque need to kick a stone (one stone, two stones, ...) and mutter "I refute it thus".
The authors give away their ages: