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

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Visual Thinking vs Formal Logical

Thinking

LC:

Thanks for this email and the chance to tell a story that I have

never told anyone. In retrospect, this was an early sign of all my

mathematical curiosity.

I spent much of my childhood doing crochet with wool and mak-

ing things with fabric. I loved the colours and the regular patterns,

but also how the patterns could deviate and become a bit more

random.

This curiosity also extended to sketching things on paper, and

I remember one year (age 8-9) where I persistently tried to do a

certain type of drawing. I will explain it now usin g the language of

mathematics, but at the time, it was a simple childhood drawing.

I tried to discover and draw tessellations of the plane using dif-

ferent types of coloured regular polygons. That is, I tried to draw

aperiodic coloured tessellations of the Euclidean plane by regular

polygons. The existence of periodic regular polygons seemed obvi-

ous to me and I was exploring the boundaries of randomness in

this type of construction. After many attempts, I gave up, resolv-

ing that I d idn’t know what I was doing.

This question did not conti nue to intrigue me, but I remember

this playful exploration as a lot of fu n. I was born and went to

school in Australia, and my language of instructi on was the same

as my spoken la nguage, English.

It seems amusing now that this process is quite typical of a day

of research as a professional mathematician.

21.1

Michael Fried

It was not a mathematics teacher who awakened in me an interest

in mathematics, but an art teacher. In his 8th grade art clas s, Mr.

MF is male, Israeli, has a PhD in mathematics, teaches at a n university.

205

206 21 Visual Thinking vs Formal Logical Thinking

Spitz taught us perspective. I was completely enchanted with th is

and sp ent hours drawing perspective drawings on my own. I gave

myself various problems, for example, h ow to draw a spiral sta ir-

case, and I asked myself questi ons such as whether it is possible

to draw a tile ﬂoor in which the tiles were genuinely squares in

perspective, an d not just rectangles. Some of these problems and

questions had genuine mathematical content, s ome of which, on

hindsi ght, not entirely trivial. I had no id ea, though, that any of

this had a nything to do with mathematics; I thought I was just

doing art.

But it is not perspective drawing th at I really wanted to de-

scribe. It was something Mr. Spitz taught to me and only to me

(at least it was not s omething he s howed to the entire class). He

showed me how to draw what he called a “logarithmic spiral.” I

do not think he explained to me why it was called a logarithmic

spiral—and it was a long ti me before I learned why indeed it was

called that—but I was so taken with the shape and the “construc-

tion” he showed to me, th at I almost became obsessed with it.

The “constru ction” was as follows. Begin with a rectangle ABCD.

Draw a diagonal AC, and then draw a line DF from D to the side

BC, perpendicular to AC.

From F , draw F G to AC p arallel to AB, GH to F D pa rallel to

AD, HI to AC parallel to A B, and so on. Then with these lines in

place, one ca n sketch the spiral tangent to the lines BA, AD, DC,

CF , F G, and so on. Of cours e there was this fudging at the end: he

never explai ned, for instance, where the poin ts of tangency s hould

be, but this did not bother me.

A little later, I s aw another similar way of carry out the

construction—maybe Mr. Spitz suggested it to me, but I do not

think so. Instead of drawing DF perpendicular to AC, draw CF

from C to the extended side AD and perpendicular to AC. Then

draw F G to the extension of CD and perpendicular to CF , and

continue this construction over and over.

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21.1 207

Again, I thought I was just doing art. But in the 9 th grade, still

drawing spirals, I started studying geometry. By the early spring,

we were studying similar triangles. It was then that I realized,

looking at my second method of drawing the logarithmic spiral,

that each triangle I added was similar to the last. I think I also

noticed that if AD = a an d CD = b that DF = a(

)

, DG = a(

)

and so on, that is, I began to grasp that this was a spiral growing

(or rather shrinking) in a geometric sequence, though I did not

have such terms yet.

Certainly, the idea of a spiral similarity—turn and shrink, turn

and shrink—was in tuitively clear all at once, though, again, I did

not know the term “spiral similarity” then. I also noticed that the

original method of drawing the spiral was contained in the sec-

ond method! This thrilled me even more than the discovery of the

whirling simila r triangles. I could also ﬁnally see more clearly the

similar triangles spiraling in the ﬁrst method:

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208 21 Visual Thinking vs Formal Logical Thinking

All this happened in the evening, lying in my bed, with the

window open and the good ea rly spring air ﬁlling the room. I add

this because the experience making these little discoveries was a

powerfully aesthetic one, and the smell in the air added to it. More

importantly, I realized that its aesth eti c power had as much to do

with the mathematics i nvolved as it had with the a rt with which it

began. For the ﬁrst time, I began to understand that these thin gs

Mr. Spitz had shown me, these things which fascinated me and

which I loved, were, in fact, mathematics. And, conversely, no one

afterward would ever have to persuade me that mathematics could

be beautiful. T his, I think, was the most signiﬁcant discovery that

evening, and without a doubt, it turned me towards mathematics.

21.2 EH: Visualisation

Maths came easily to me until A-level, so I have no recollection of

any issues. After GCSE a teacher d id tell my parents that I should

rethink my ambition to do a maths degree as I woul d soon hi t my

ceiling. He was partly ri ght. I stru ggled for ages with the l anguage

of differentiation until I got the visual picture. Since then maths

has been ha rd work to understand, though I usually get through

with enough work. Making visu alisa tions and images reall y helps

me.

Around the same time (GCSE) I came across the Penrose tiling

for the ﬁrst time and was fascinated. I remember working for ages

trying to ﬁnd a version with seven fold rotational symmetry rather

than ﬁve. I drew lots of pi ctu res, but did not get anything to work.

I am now a mathematician, having done my PhD on the Pen-

rose and related tilings. And the problem? I eventually solved it

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21.3 Lego 209

(and the general ca se) and it became my second pub lished paper.

[30].

Jerry Swan:

My recollection is that a lot of early mathematics was taught with

reference to ‘scenes’, e.g. multiplication tables taught via the iso-

morphism with the corresponding rows and colu mns of numbers. I

believe that early emphasis on this kind of visual/geometric na ture

de-emphasises more abstract pattern-matching that must then be

re-introduced alongside these ‘sceni c’ perspectives as one gains

more mathematical maturity.

The mul tiplication table as a matrix of objects example is the

most concrete I can think of. My geometric intuition has always

been rubbish (or at least I’ve intrinsically distrusted it) relative to

my symbolic intuition.

My guess is that thi s is the opposite of the case for the ma-

jority of the people who are likely to teach early maths, hence the

geometric emphasis. I don’t really know what else to tell you on

this—if you can prompt me a bit, I might be able to say more.

On the “visual vs logical” issue, Roger Penrose uses an illus-

tration of thi s general kind (some kin d of hexagonal packing, if I

recall correctly) as an example of something that hu ma n mathe-

maticians can do that he claims is noncomputable (though as a

lapsed Platonist, I personally believe that he’s confusing a lack of

imagination with a n insight into necessity). I think it appears in

his book “Shadows of the Mind” (though it might possibly be “Em-

perors New Mind”). I strongly recommend that you give it a look.

21.3 Lego

Fig. 21.1. STICKLE BRICKS: A photograph showing how stickle bricks con-

nect. Photographer: Malcolm Tyrrell. Wikimedia Commons, licensed under

the Creative Commons Attribu tion-Share Alike 2.5 Generic license.

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210 21 Visual Thinking vs Formal Logical Thinking

Dean Wyles

My earliest mathematical thinking, when I must have been

just four or ﬁve years old, involved LEGO bricks. This was long be-

fore I became aware of formal mathematical th inking or language—

my parents certai nly didn ’t speak any mathematics to me at all,

not even simple arithmetic so I am not sure what prompted me to

think in this way.

I can remember wondering about possible combinations of

colour sequences, wh en several different coloured bricks were

aligned. I even remember including two reds and one blue, white

and green; coming up with the answer sixty. This I later recognised

as represented with factorials. Even n ow, when I teach to sixth

form students, I cannot get combinations and permutations across

to many students without reference to colour combinations of the

kind I did myself, mentally, some forty years ago. I am always

amazed when students appear to solve such problems directly

without any obvious physical analogy and simply write down s ome

factorial expression ins tinctively.

My appreciation of space—volume and area—was enhanced

with consideration of how many bricks could ﬁt in as small a box

as possible. I remember trying to esti mate how more efﬁcient the

packing was if the bricks were ti ghtly conn ected compared to be-

ing i ndividually stored in a box. Not just noticing th e efﬁciency but

calculating it!

Once I had exhausted my curiosity with LEGO I was delighted

with the huge possibilities with STICKLE BRICKS; they can been

linked in man y more ways. Each brick has a rectangular array of

stickles and so two m × n bricks can be stuck together in

(m − 1) × (n − 1)

ways, in addition to different alignments. I am afraid to say I was

more interested in the possible constructions of a small number

of bricks than trying to gather many many bricks to build some-

thing like a bridge or tower! This meant that I soon appreciated

isomorphism and clearly many arrangements would be the same

once you started to look at the pieces from different views.

Once formal schooling kicked in and I became exposed to hours

and hours of ‘sums’ I think I was quite put off by what most people

call mathematics. I think I only started to excel when an Asian lad

joined my class and a competitive streak began to come to me—I

must have been about seven by this time; we would race each other

to complete the hundred or so arithmetic or algebraic tasks on a

page (large Beta books?).

I suspect that my ‘approach’ to mathematical thinking has

lead me to think that when I teach students mathematics, it is

DW is male, British, a teacher of mathematics in an academically selec-

tive college.

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21.3 Lego 211

their mathematics they are discovering, even with simple devel-

opments, and hope that I try to avoid the cramming approach

so many of my school teachers adopted. Everyones’ mathemati-

cal world, mainly visual I gu es s, is a very personal thi ng. Surely

that is the beautiful and wonderful thing about mathematics.

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Telling Left from Right

The mirror is the master of painters.

Leonardo Da Vinci

22.1 Why does the mirro r change left and right

but does not change up and down?

The answer to this old chestnut appears to be well known. Here are

two classical explanations.

Blaise Pascal:

Our notion of symmetry is derived from the human face. Hence, we

demand symmetry horizontally a nd in breadth only, not vertically

nor in depth.

Immanuel Kant [128]:

In physical space, on account of its three dimensions, we can con-

ceive three planes which intersect one another at right angles.

Since through the senses we know that which is outside us only

in so far a s it stands in relation to ourselves, it is not s urprising

that in the relationship of these intersecting planes to our body we

ﬁnd the ﬁrst ground from which to derive the concept of regions in

space . . .

One of these vertical planes divides the body into two out-

wardly similar parts and su pplies the ground for the distinction

between right and left; the other, which is perpendicula r to it,

makes it poss ible for us to have the concept before and behind.

Quoted in W. H. Auden and L. Kronenberger, The Viking Book of Aph o-

risms, New York, 1966.

213

214 22 Telling Left from Right

I believe that the actual explanation is both simpler and deeper:

the left-right symmetry is built in the human neurophysiology at

the level m ore primitive and elementary in relation to even most

basic structures of h uman cognition.

It is claimed that 1 in 6500 people possesses ability for mirror

writing, when the non-dom inant hand writes in direction oppo site

to the natural direction f or a given language, producing the result

that is a mirror image of the normal writing; intriguingly, this trait,

although o f no evolutionary signiﬁcance, appears to be genetically

predeter mined [344]. Many mo re people can do a weaker version of

mirror writing, when both hands write simultaneously, but in op-

posite directions. Figures 22.1 and 22.1 show my own experiment:

I wrote the word “mirror” without any preparation or training. The

sample produced by my left (non-dominant) hand bears all the spe-

ciﬁc features of my handwriting—but in mirro r reﬂection.

I have no time to discuss the huge literature on mirror writ-

ing; see, for example, bibliography in [319]. What matters for the

purposes of this paper is the existence of built-in symmetry of our

mind that, as children’s stories show, c ould be a mixed blessing in

mastering mathematics.

First of all, I was surpr ised to discover that a number of my

correspondents told me about their difﬁculties of learning the left

from the right. I could be biased on this point because I myself

remember, an d in surprisingly clear detail, how, in a kindergarten,

my frien d taught me to distinguish between the left shoe and the

right shoe, by placing them on the ﬂoor in such a way that they

“face” each other. I could not see the real reason why it should be

that way. The two bo ots, for me, were logically equivalent.

From my kindergarten times, I hardly remember anything else.

Perhaps, the smell o f burned milk from the kitchen . . .

Anna, my wife, who is a mathematician, tells similar stories.

Even now she gives me directions, when I drive, in the f orm “turn in

your dire ction” instead of “turn right”. She explains that it takes too

much time for her to ﬁgure out that “driver’s side ” is the “right hand

side”. Anna told me recently that when, aged four, she was eating

sweets, she developed a custom of moving the candy from one side

of the mouth to another, with a rationalization for this rule that

both cheeks were equal and no ne of them had to be disadvantaged.

22.2 Pons Asinorum

I eventually learned to tell a left boot from the right one, and was

able to dr ive, without accidents, both in the USA (on the right hand

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