Books : reviews

George Lakoff, Rafael E. Nunez.
Where Mathematics Comes From.
Basic Books. 2000

rating : 2 : great stuff
review : 15 August 2002

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).

  • First, there are grounding metaphors --- metaphors that ground our understanding of mathematical ideas in terms of everyday experience. ...
  • Second, there are redefinitional metaphors --- metaphors that impose a technical understanding replacing ordinary concepts. ...
  • Third, there are linking metaphors --- metaphors within mathematics itself that allow us to conceptualize one mathematical domain in terms of another mathematical domain. ...

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.

Minor observations:

I have a few niggles with some of the metaphors.

There is a rather nice distinction made amongst:

  • The number (e.g., thirteen)
  • The conceptual representation of the number: the sum of products of powers adding up to that number (e.g., one times ten to the first power plus three times ten to the zeroth power)
  • The numeral that symbolizes the number by, in turn, symbolizing the sum of products of powers (e.g., 13)

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:

the axiomatic method is the manifestation in Western mathematics of the folk theory of essences inherited from the Greeks.

and, possibly more problematically, on science:

it is not true that the theory of essences, in either its folk or expert version, fits the physical world. ... A species cannot be defined by necessary and sufficient conditions ... Indeed, in biology, the folk theory of essences has interfered with the practice of 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?

Other quotes:

Everything we perceive or think of as an action or event is conceptualized as having [aspect] structure.

... that is, Readiness; Starting up; The main process; Possible interruption and resumption; Iteration or continuing; Check purpose achieved; Completion; Final state

it is no accident that 1 is used for true and 0 for false and not the reverse.

It is important to contrast our everyday concept of Same Number As with ... Cantor's concept of pairability --- that is, capable of being put into one-to-one correspondence. ... the ideas are different in a significant way, but the happen to correlate precisely for finite sets. The same is not true for infinite sets. ... This distinction has never before been stated explicitly using the idea of conceptual metaphor. ... [pairability] is a metaphorical rather than a literal extension of our everyday concept. The failure to teach the difference between Cantor's technical metaphorical concept and out ordinary concept confuses generation after generation of introductory students.

On the formalist view of the axiomatic method, a "set" is any mathematical structure that "satisfies" the axioms of set theory ... Many writers speak of sets as "containing" their members ... Even the choice of the word "member" suggests such a reading, as do the Venn diagrams used to introduce the subject. But if you look carefully through those axioms, you will find nothing in them that characterizes a container. The terms "set" and "member of" are both taken as undefined primitives. ... most of us do conceptualise sets in terms of Containment schemas, and that is perfectly consistent with the axioms ... However ... a constraint follows automatically:Sets cannot be members of themselves ... this constraint does not follow from the axioms ... [So the axiom of Foundation] was proposed [to] rule out this possibility. ... our ordinary grounding metaphor that Classes are Containers gets in the way of modeling [recursive] phenomena. ... Set theorists have realized that a new noncontainer metaphor is is needed for thinking about sets, and they have explicitly constructed one: hyperset theory ... The idea is to use graphs, not containers, for characterizing sets.

Outside mathematics, a process is seen as infinite if it continues (or iterates) indefinitely without stopping. That is, it has imperfective aspect (it continues indefinitely) without an endpoint. This is the literal concept of infinity outside mathematics.

Processes are commonly conceptualized as if they were static things --- often containers, or paths of motion, or physical objects. ... We speak of the parts of a process, as if it were an object with parts and with a size. ... one of the most important cognitive mechanisms for linking processes is ... "fictive motion", cases in which an elongated path ... can be conceptualized metaphorically as a process tracing the length of that path ... in mathematics, processes can be conceptualized as atemporal. ... The [Fibonacci] sequence can be conceptualized either as an ongoing infinite process of producing ever more terms or as a thing, an infinite sequence that is atemporal. This dual conceptualization ... is part of everyday cognition.

On Hardy's warning not thinking of infinity, infinity, as a number:

when there are explicit culturally sanctioned warnings not to do something, you can be sure that people are doing it. Otherwise there would be no point to the warnings. ... there are, cognitively, different uses for numbers --- enumeration, comparison, and calculation. ... mathematicians have devised notions ... in which infinity is a number with respect to enumeration, though not calculation. For Hardy, an entity either was a number or it wasn't, since he believed that numbers were objectively existing entities. The idea of a "number" that had one of the function of a number (enumeration) but not other functions (e.g., calculation) was an impossibility for him. But it is not an impossibility from a cognitive perspective

The BMI [Basic Metaphor of Infinity] ... is often the conceptual equivalent of some axiom that guarantees the existence of some kind of infinite entity (e.g., a least upper bound). And just as axioms do, the special cases of the BMI determines the right set of inference required.

an ordinary natural number ... can have a cardinal use; that is, it can be used to indicate how many elements there are in some collection. It can also have an ordinal use; that it, it can be use to indicate a position in a sequence. .... These are two very different uses of numbers. ... the arithmetic of the natural numbers is the same for cardinal and ordinal uses. But this is not true for transfinite numbers. Cantor's metaphor determines "size" for an infinite collection by pairing, not counting in a sequence. Cantor's metaphor, therefore, is only about cardinality (i.e., "size") not about ordinality (i.e., sequence). ... there are no transfinite numbers that can have both cardinal and ordinal uses. Rather, two different types of numbers are needed, each with its own properties and its own arithmetic. ... You can get different results by "counting" the members of a fixed collection in different orders! The reason for this has to do with the different metaphors needed to extend the concept of number to the transfinite domain for ordinal as opposed to cardinal uses. ... Recall that "addition" for transfinite cardinals is defined in terms of set union. ... it follows that ... ℵ0+1=ℵ0 ... But the situation is very different with ordinal numbers. ... we can go on forming longer sequences by appending a further sequence "after" the ω position. ... ω≠ω+1. However, 1+ω=ω.

ignoring certain differences is absolutely vital to mathematics! ... calculus is defined by ignoring infinitely small differences.

Conceptually, [Cantor's transfinites, and the hyperreals] are two utterly different structures, leading to two utterly different notions of "infinite numbers". ... How can there be two different conceptions of "infinite number", both valid in mathematics? By the use of different conceptual metaphors, of course

any technical discipline develops metaphors that are not in the everyday conceptual system. In order to teach mathematics, one must teach the difference between everyday concepts and technical concepts, making clear the metaphorical nature of the technical concepts.

Given the metaphor A Space is a Set of Points, "points" are not necessarily spatial in nature, but can be any kind of mathematical entities at all. ... one must learn which kinds of mathematical problems require which metaphors.

In modern mathematics, the lack of a feature is conceptualized metaphorically as the presence of that feature with value zero.

The rational numbers, by themselves, have no gaps. In the set of rational numbers, rational numbers are all there is. ... The "gap" is in the space domain, where there are points on a naturally continuous line that are not paired with rational numbers. This "gap" makes sense only in the metaphorical conceptual blend that Dedekind was constructing and we have inherited.

Of the "bumpy curve" paradox:

this metaphorical image ignored the derivatives (the tangents) of the functions, which are crucial to the question of length [because the measuring rods lie along the tangents]

the claim that transcendent mathematics exists appears to be untenable. One important reason is that mathematical entities such as numbers are characterized in mathematics in ontologically inconsistent ways. ... since transcendent mathematics takes each branch of mathematics to be literally and objectively true, it inherently claims that it is literally true of the number line that numbers are points, literally true of set theory that numbers are sets, and literally true of combinatorial game theory that numbers are values of positions. ... But again, according to transcendent mathematics, there should be a single kind of thing that numbers are; that is, there should be a unique ontology of numbers.

... 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 the only access that human beings have to any mathematics at all, either transcendent or otherwise, is through concepts in our minds that are shaped by our bodies and brains and realized physically in our neural systems. For human beings --- or any other embodied beings --- mathematics is embodied mathematics. The only mathematics we can know is the mathematics that our bodies and brains allow us to know. For this reason, the theory of embodied mathematics ... [is] a theory of the only mathematics we know or can know, it is a theory of what mathematics is --- what it really is! ... But there are excellent reasons why so many people, including professional mathematicians, think that mathematics does have an independent, objective, external existence. The properties of mathematics are, in many ways, properties that one would expect from our folk theories of external objects.

... 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".

a significant part of mathematics itself is a product of historical moments, peculiarities of history, culture, and economics. This is simply a fact. In recognizing the facts for what they are, we are not adopting a postmodernist philosophy that says mathematics is merely a cultural artifact. We have gone to great lengths to argue against such a view. ... In recognizing all the ways that mathematics makes use of cognitive universals and universal aspects of experience, the theory of embodied mathematics explicitly rejects any possible claim that mathematics is arbitrarily shaped by history and culture alone.

Even an idea as apparently simple as equality involves considerable cognitive complexity. From a cognitive perspective, there is no single meaning of "=" that covers all these cases ["yields", "gives", "produces", "can be decomposed into", "can be factored into", "results in", ...].

The authors give away their ages:

Think for a moment of how a slide rule works.

Why do e, π, i, 1, and 0 come up all the time when we do mathematics, while most numbers ... do not? The reason is that these numbers express common and important concepts via arithmetization metaphors. Those concepts, like recurrence, rotation, change and self-regulation are important in our everyday life.