1980, with Mark Johnson*Metaphors We Live By*.1987*Women, Fire, and Dangerous Things*.1989, with Mark Turner*More than Cool Reason*.*Philosophy in the Flesh*. 1999, with Mark Johnson2000, with Rafael E. Nunez*Where Mathematics Comes From*.*Moral Politics: 2nd edn*. 2002

- The Idea of an Idea. 2000. (In
*The Greatest Inventions of the Past 2000 Years*)

**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:

p161.
We project a front-back orientation in
context onto objects that have no intrinsic fronts or backs. Given a
medium-sized rock in our visual field and a ball between us and the
rock, say a foot from it, we would perceive the ball as being *in
front of* the rock. The Hausas make a different projection than we
do and would understand the ball as being *in back of* the rock.

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

- a
*fog bank*is an object [watching time-lapse films of clouds is a good way to disabuse yourself of the idea that a cloud is an object with a well-defined spatial*and temporal*extent] - a
*mountain*is an object - a mountain has a
*front* - the mountain's front is oriented towards the (implicit)
*observer*

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).

p232.
Communication theories based on the
CONDUIT metaphor turn from the pathetic to the
evil when they are applied indiscriminately on a large scale, say, in
government surveillance or computerized files. There, what is most
crucial for real understanding is almost never included, and it is
assumed that the words in the file have meaning in themselves ---
disembodied, objective, understandable meaning. When a society lives
by the CONDUIT metaphor on a large scale,
misunderstanding, persecution, and much worse are the likely products.

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*.

- ARGUMENT IS WAR
- He
*attacked*my argument -- I've never*won*an argument - ARGUMENTS/THEORIES ARE BUILDINGS
- The argument will
*fall apart*-- We need to*buttress*the theory with*solid*arguments - AN ARGUMENT IS A JOURNEY
- We have
*set out*to prove... -- We have*arrived*at a disturbing conclusion - TIME IS MONEY
- You're
*wasting*my time -- He's living on*borrowed*time - TIME IS A MOVING OBJECT
- The time for action
*has arrived*-- Time*flies* - LOVE IS A JOURNEY
- Look
*how far we've come*-- We can't*turn back*now - LOVE IS MADNESS
- He's gone
*mad*over her -- I'm just*wild*about Harry - LOVE IS WAR
- He is known for his
*conquests*-- He*won*her hand -- She is*besieged*by suitors - LABOUR IS A RESOURCE
- MORE IS UP; LESS IS DOWN
- My income
*rose*last year -- He is*under*age -- Turn the heating*down* - HAPPY IS UP; SAD IS DOWN

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.

p110.
As we see it, there are six fundamental
positions that we consider mistaken. The first, and biggest, mistake
concerns what has been called “literal meaning.” The second
is to fail to seek general principles and to focus instead on
individual metaphorical expressions as if each were unique. The third
is a confusion between presently existing conventional metaphors and
metaphors that once existed but no longer do, the so-called “dead”
metaphors. The fourth mistake is the claim that metaphors do not have
a source and a target domain, but are merely bidirectional linkages
across domains. The fifth mistake is the claim that metaphor resides
in linguistic expressions alone and not in conceptual structure.
Finally, the sixth mistake is the claim that everything in language
and thought is metaphorical, that there are no aspects of language or
thought that are not metaphorical.

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).

- 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 *i*π + 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 *i*θ
= 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.

- The "object collection" metaphor entailment, used to
explain why
*n** 1/*n*= 1, can't be that concrete. We all know if you cut something in half, then glue it back together again, you*don't*get quite what you started with: there's missing sawdust, and a joint. - The four metaphors identified for innate arithmetic are "object
collection", "object construction", "measuring
stick," and "motion along a path". Notice that these
all give the
*same*laws of arithmetic. The authors explain this is because there are*isomorphisms*across the metaphors. But why*should*there be these isomorphisms? - "Multiplication by -1 is rotation". Certainly, once you
have complex numbers, and the complex plane, this is an appropriate
metaphor. But the authors' argument seems to be that the new complex
numbers fit into neatly into this existing metaphorical structure.
But it seems to me when all you have is the 1D real number line, a
more natural metaphor is "Multiplication by -1 is
*reflection*" -- which doesn't then extend naturally to the 2D complex plane. I wonder if the rotation metaphor came later?

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?

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, 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
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 ...
ℵ_{0}+1=ℵ_{0}
... 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.

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.