Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics

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22.2 Pons Asinorum 215

side of the road) and in Britain (on the left hand side ). But I suspect

that I eventually beneﬁted from my slightly awkward relationship

with the left and the right when, at school, I was confronted with

the famous Pons AsinorumixPons Asinorum@Pons Asinorum theo-

rem of Euclidean geometry:

the base angles ∠B and ∠C of an isosceles triangle △ABC

are equal.

In the school tex tbooks the Pons AsinorumixPons Asinorum@Pons

Asinorum was prov en outside the axiomatic system, by a direct

application of the bilateral symmetry of the triangle viewed as a

cardboard cutout. This approach was promoted by Hadamard in his

highly inﬂuential Leçons de géométrie éleme ntaire [28] and adopted

by canonical Russian textbooks. I remember that I have instantly

seen a proof based on the formal symmetry of premises:

• AB = AC

• AC = AB

• ∠BAC = ∠CAB ( since the angle is equal to itself).

• △BAC = △CAB (by the Side-Angle-Side criterion of

congruence).

• Therefore, ∠B = ∠C.

A more detailed discussion of Pon s AsinorumixPons Asinorum@Pons

Asinorum (including its role in history of automated theorem prov -

ing) can be found in my book [102, Section 7.7].

Fig. 22.1 . Euclid proved Pons Asinorum by a sophisti cated auxiliary con-

struction, creating and us ing not just one, but two pairs of different but

congruent triangles: △DAC = △EAB and △CBD = △BCE.

I remember being puzzled why my teacher was referring to a

cardboard cutout when the theor em was obvious on its own. And

I also remember not being able to explain to my te acher what I

meant and why the theorem was obvious. Much later I happened

to te ach Euclidean geometry (in a Fo undation Year course in a Rus-

sian university), and I used this chance to try the “formal symme-

try” proof. The result was inte resting: there were students who ac-

cepted it instantly, as something self-evident, while other students

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216 22 Telling Left from Right

struggled. The reason for their difﬁculty is obvious: in the “formal

symmetry” proof, the same triangle plays two different roles em-

phasized by two different notations: △BAC and △CAB. In effect,

the symmetry o f the premises—but not the physical, real life sym-

metry of the card board cutout!—forces the triangle to split, like

an amoeba, into two twins. It is a subtle play with the concept of

identity of a mathematical object. Euclid in his El e ments gave a

complicated proof of Pons Asinorum and used construction lines to

make the two triangles to which the Side-Angle-Side criterion of

congruence (the key point of the proof) is applied to triangles that

are actually diffe rent; see Figure 22.1.

In mathematics, the identity of objects is not set in black and

white; it is more like a scale of grey. Some children are more sensi-

tive than their peers and are better at recognizing shades of iden-

tity. I would not be surp r ised at all if this sensitivity sometimes

affects a child’s ability to distinguish betw een the le ft boot and the

right boot, too.

22.3 TB

Age: 10 or so. I think. Gender: Male. Language: Finnish.

I am an undergraduate mathemati cs student right now and

will continue studying further.

I had trouble remembering if a : b means a divided by b or b

divided by a. There’s no way to tell. It is arbitrary. Likewise, some

years earlier, remembering which side is left and which is right.

I should also mention the notation a | b, as in a divides b. Or

maybe it was b divides a. Probably the ﬁrst one. For exampl e: 2 | 4

but not 2 | 3.

I learn ed a reliable mnemonic for left and right when learning

to drive at the age of 18 (about). Before that, it was guesswork. I’ve

never had much problem with it, aside from difﬁculties when com-

municating directions to somewhere. I still have to conﬁrm that

right is in deed where it is. I need to do the same thing with East

and West (Sweden is to the West of Finland, Russia to the East).

I’ll h ave to mention that I don’t remember dates or years, ei -

ther. I generally remember what year it currently is and what year

I was born and calculate my age from them, as an example. I’ve as -

sumed this comes from ignoring what I consider irrelevant or from

being lazy and not memorising a bunch of nigh-meaningless da ta.

22.4 Mar ia Zaturska

MZ is femail, Polish, studied in England, has a PhD in mathematics,

teaches in a British Univesity.

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22.6 Digression into ethn ography 217

I think m y experience may be more a matter of language than

mathematics.

I was brought up in the U K speaking Polish until I went to

school aged 5. I picked up English as I wen t along - there were

no Polish speaking children in my schools.

I always (probably from age 7 up) had problems with subtrac-

tion when it was worded ‘what is x from y?’ as I thought symboli-

cally, i.e. y − x, and so the order was confusing. (In later life I also

had proble ms with ‘x divides y’.) At the same sort of age I remem-

ber people saying it was ’‘ﬁve and twenty past six o’clock’ and again

I couldn’t get my head around ‘ﬁve and twenty’ - why weren ’ t they

using twenty ﬁve?

22.5 MP

Your question does bring to mind two speciﬁc, somewhat mathe-

matical, intellectual difﬁculties from (some time at) primary school:

one was the issue of including/excluding endpoints (eg in comput-

ing the number of days between, say, 7th and 23rd of June: so

whether to use 23−7 or 23−7+1; I think my usual strategy was to

compute a si mi lar example which could be counted on my (mental)

ﬁngers in order to determine whether or not it was correct (which

would depend on the nature of the exact questi on being asked) to

add 1).

The other was the problem of reconciling the personal right

/ left frame of reference with the view-from-“above” frame which

is necessary for north / south . . . (th e latter won and I still have

problems in qu ickly telling right from l eft).

22.6 Digression into ethnography

[328, p. 68]

Speakers of English and other Indo-European languages favor th e

use of an egocentric ( relative) system to represent the location of

objects—that is, relative to the self (e.g., “the man is on the right

side of the ﬂagpole”). In contrast, many if not most langua ges favor

an allocentric frame, which comes in two ﬂavors. Some allocentric

languages such as Guugu Yimithirr (an Australian language) and

Tzeltal (a Mayan l anguage) f avor a geocentric system in which ab-

solute reference is based on cardinal directions (“the man is west

of the house”). T he other allocentric fra me is an object-centered

(intrinsic) approach that locates objects in space, relative to some

coordinate system anchored to the ob ject (“the man is behind the

house”).

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218 22 Telling Left from Right

One can add that Tenejapan Tzeltal people of Central America

have no words or concepts for the left and the right—they live in

square houses, with a door (made of two equal halves) positioned

in the middle of the south wall [463, 462]; in their language, an

egocentric refe r ence system is virtually impossible.

The role of cardinal directions [508] in the geocentric re ference

system of Guugu Yimithirr can be explained by the harsh environ-

ment in which they live —in the wilderness, it pays to remember

where the north is; if you are lost, you are dead. Lera Boroditsky re-

marks [458] that, in tradition of anothe r Abo r iginal community in

Queensland, Pormpuraaw,, cardinal directions do minate even the

simplest for mulae of polite small talk:

to say hello, one a sks, “Where are you going?”, and an app ropriate

response might be, “A long way to the south-southwest. How about

you?” If you don’t know which way is wh ich, you literally can’t get

past hello. [458]

Lera Boroditsky continues:

People rely on their spatial knowledge to buil d many other more

complex or a bstract representations including time, number, mu-

sical pitch, kinship relations, morality and emotions. So if Porm-

puraawans think d ifferently about space, do they also think differ-

ently about other things, like time?

To ﬁnd out, my colleague Alice Gaby and I traveled to Australia

and gave Pormpuraawans sets of pictures that showed temporal

progressions (for example, pictures of a man at different ages, or

a crocodile growing, or a banana bei ng eaten). Their job was to ar-

range the shufﬂed photos on the ground to show the correct tem-

poral order. We tested each person in two separate sittings, each

time facing in a different cardinal direction. When asked to do this,

English speakers arrange time from left to right. Hebrew speak-

ers do i t from right to left (because Hebrew is written from right

to left).

Pormpuraawans, we found, arranged time from east to west.

That is, seated facing south, time went left to right. When facing

north, right to left. When facing east, toward the body, and so on.

Of course, we never told any of our participants which direction

they faced. The Pormpuraawans not only knew that already, but

they also spontaneously used this spatial orientation to construct

their representations of time.

A detailed account of this research is pu blished in [459].

For a mathematician, the very existence of world o utlook build

around cardinal directions is a revelation: the ap proach to geom-

etry in our (Western, that is) mathematics is strictly local! Of

course, the value jud gements—which of the two philosophies, local

or global is best—but the “local” approach of traditional Euclidean

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22.8 PD 219

geometry was perhaps conducive for the development of differen-

tial calculus and differential geometry.

22.7 BB

And reluctance to distinguish between things that are undistin-

guishable from the formal logical point of view goes deeper and

deepe r. Here is what BB

wrote to me:

From the time I learned matrices (age 16 or so) I cannot remember

which are the columns and which are the rows. Given that the ar-

rangement of coefﬁcients in a linear transformation can be written

equally well in a matrix in two ways, it is something that always

takes me 10–15 seconds to recall even now.

Perhaps a few words of (mathematical) explanations are needed.

Most m athematicians would agree that BB has reasons to be con-

fused: for a mathematician, a matrix is an element in the tensor

product V ⊗ V

∗

of a ﬁnite d im ensional vector space V and its dual

∗

. Since the dual of the dual of a ﬁnite dimensional space is the

same as the original space, that is,

∗

)

∗

≃ V

canonically, there is no intrinsic reason to distinguish between V

and V

∗

, that is, betw een the rows and the columns of a square ma-

trix. (Notice that PD also mentions psychological difﬁcu lties arising

from duality.)

22.8 PD

I had a sh ort moment of bewilderment (around 14 or 15 or 16?)

when I learned that the graphs of f (x − a) and f(nx) when com-

pared to those of f (x) behave “in the opposite way” to one’s ex-

pectation. I got it very quickly and easily, but I remember being

surprised.

And this ﬂows into similar adult experiences:

Does the “transition matrix” transform the basis or the coordi-

nates? (Actually, many books hide the appearance of the i nverse

of the transpose b y suitably deﬁning the transition matrix. ) Given

a matrix of a linear map, am I writing the map between the vec-

tor spaces or between their duals? Do the transition functions of

a vector bundle transform the f rame or the sections? A m I looking

BB is male, Russian, has a PhD from an American university and holds

a research position in a British university.

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220 22 Telling Left from Right

at the sheaf O(D) or O(−D) (D is a divisor on a variety), and do

its sections have a pole or zero at D?

This is all one and the same question and one learns to recog-

nize it, but it is amazing how persistent it is.

What I am going to say now probab ly won’t make sense. It

seems that often there is a “ﬁxed part” and a “moving part” in a

problem, and I have the feeling that my mind often gets confused

by looking at the “moving part” and forgetting that i t “moves” with

respect to the “ﬁxed part”.

This refers both to mathematical and worldly experiences. In

math this is often a question about inclusions vs natural inclu-

sions (equal ities). In daily life one example of a similar problem

is learning the streets of a city. I have the feeling that often my

brain stores some images separately, say, the direction in which I

traverse a street for the ﬁrst time gets stored wi th higher priority.

Or there may be several images of the same pl ace (from differ-

ent viewpoints) which don’t exactly glue. As if the brain prefers to

work with the moduli stack and ha s a problem when passing to

the moduli space.

In that slightly metaphorical phrase I meant exactly what you

say.

That it feels as if my brain often keeps track of the i ndividual

representatives of an equivalence class, or at least has a preferred

one, instead of dealing with the class itself. For example, if there

are several roads meeting at an in tersection, it may h appen that

I perceive each of them slightly differently, they carry a different

emotional charge, if you want. Of course, if I make a conscious

effort, I can destroy th is asymmetry.

(I don’t think I am describing a pathology, I don’t get lost on

campus, it’s a more subtle effect. Though I am pretty bad at ori-

enting, and when I do it, it is more like a ﬂow, rather than a log-

ical sequential process: I prefer the right-brain orienting to the

left-brain orienting.)

I consider this to be a weakness of mine, of course. Mathemat-

ically, this has the effect that I can spend a lot of time trying to

understand d ifferent equivalent formulations of the same result,

then dash along a route, then b ack up and agai n look for eq uiva-

lent formulations. . . This often leads to very slow progress. And at

least when it comes to science, I have the feeling that this effect

gets stronger with time.

I deﬁni tely prefer writing a < b < c, as opposed to c > b > a,

and my suspicion i s that in the ﬁrst case the numbers appear in

the same order as on the real line—if we take the positive direction

of the real line to be to the right of 0.

I have a small empirical conjecture. I think a wi d e class of er-

rors and confusions appear when there is some discrete (usually,

Z/2Z) ambiguity/choice/convention in the problem, and one has to

ﬁx the choice and stick to it while solving. The issue is, however,

if your brain decides to ﬂi p the choice halfway through the compu-

tation. For instance, if you thin k too much about conventions you

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22.9 Digression into Estonian language 221

may notice that certain choice is more convenient in some situa-

tions and less convenient in others, and you may decide to change

your convention halfway through.

Or, it may become purely i nvoluntarily. One elementary exam-

ple is subtraction: I’ve seen many students with weak arithmetic

background getting the signs wrong in linear algebra, where they

start with the intent of subtracting one row from another . . . but

when they reach a column where there is (2 − 5) they decide to

compute it as −(5 − 2), and then forget where they were going . . .

A similar example is multiplication (a + b)(c + d), where one

has to make the choice about enumerating the terms: either a s

a(c + d) + b(c + d) or as (a + b)c + (a + b)d. After computing 2 of the

4 terms you may decide to change the choice . . .

Of course, a similar problem appears in homological algebra:

when you talk about a left/right exact functor/cofunctor, you know

that the chirality i s determined in the ta rget category, but that

works if you write all complexes from right to left.

This somewhat ties with my feeling that there is often a “ﬁxed”

and a “moving” part of the problem . . .

By the way, here is one other “chiral ” kind of asymmetry—this

time related to the multiplication table.

Despite the commutativity of multiplication, I think certain

products are computed faster than others: e.g., 6 × 7, 8 × 7, 7 × 9,

7 × 5 are faster to compute tha n 7 × 6, 7 × 8, 9 × 7, 5 × 7 etc. (Not

all examples of such pairs include 7.) Of course, the dif ference in

“computational time” is very small, but I’ve found several p eople

who have noti ced such a phenomenon (with mostly the same pairs

of numbers, and with different native languages). Probably this

has a fairly simple and prosaic explanation, and I ca n give a few

myself, but I thought it is still worth mentioning.

22.9 Digression into Estonian language

This quote from Urmas Sutrop [542] was brough t to my attention

by Yagmur Denizhan.

Even the most ordinary everyday Estonian language conta ins nu-

merous ancient expressions, possibly going back as far as the Ice

Age.

The Estonians say kü lma käes, päikese, tuule kä es ‘in the hand

of the cold, rain, sun , wind’, or ta sai koerte käest hammustada ‘he

was bitten by the hand of dogs’, i.e. ‘he was bitten by dogs’, or ta

sai nõgeste käest kõrvetada ‘h e was stung from the hand of nettles ’.

In all the above Estonian expressions ‘hand’ occurs in singu-

lar. This is ass ociated with the integral concept of the world of our

ancestors. Everything formed a whole, a totality, also the paired

parts of the body which were used only in singular. If one wanted

to speak about one hand, one had to say pool kätt ‘half a hand’.

Hence the division of the holis tic world into the right and left

halves, right an d left sides.

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222 22 Telling Left from Right

22.10 Standing arches, hanging chains

The symmetry between the mechanical behavior of a standing arch

and a hanging chain is not obvious at all, and a genius on the scale

of Robert Hoo ke and Antoni Gaudi was needed for its d isco very

and a systematic exploitation. A mathematical justiﬁcation of the

surprising connection is very simple but subtle.

I quote fro m Robert Osserman’s paper [67]:

It was Rob ert Hooke who in 1675 made the connection b etween

the ideal shape of an arch and that of a hanging chain in an apho-

rism that says, in abbreviated form, “As hangs the chain, so stands

the arch.” In other words, the geometry of a standing arch should

mirror th at of a ha nging chain. The horizontal and vertical forces

in a hanging chain must add to a force directed along the chain ,

since any component perpendicular to the chain would cause it to

move in that direction to gain equilibri um. Similarly, one wants

the combined forces at each point of an arch to add up to a vector

tangent to the arch. In both cases, the horizontal component of the

force i s constant and simply transmitted along the arch or chain,

while the vertical forces are mirror images.

Wikipedia provided a full quote from Hooke:

Ut pendet continuum ﬂexile, sic stabit contiguum rigidum inver-

sum,

meaning

“As hangs a ﬂexible cable so, inverted, stand the touching pieces

of an arch.”

Incidentally, the language of Tenejapan Tzeltal (which has no

words for “left” and “right”) has special grammatical forms of verbs

for expression of moveme nt uphill or d ownhill [464] . . .

22.11 Orientation o f surfaces

Let S be a regular surface in R

. Explain what it means to say that

S is orientable. Show that the sphere S

is orientable.

A Ga uss map on a regular surface S is a smooth map

N : S → S

such that N(p) is normal to S at p for all p ∈ S. The surface S is

orientable if there is a Gauss map on S. Let N : S

→ S

be the

identity map on S

. Then N is a Gauss map on S

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