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

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4.3 Kolmogorov’s “5/3” Law 35

Therefore we come to Kolmo gorov’s “5/3” law:

E(k) ≈ Cǫ

2/3

−5/3

We have done it!

It is claimed in the specialist literature that the dimensionless

constant C can be determined from experiments and happens to be

pretty close to 1.

Fig. 4 .2. The vortex nature of the Universe. Vincent van Gogh, Starry

Night, 1889. Source: Wikimedia Commons. Public domain.

4.3.3 Discussion

The status of Kolmogorov’s celebrated result is quite remarkable.

In the words o f an expert o n turbulence, Alexander Chorin [11],

Nothing illustrates better the way in which turbulence is sus-

pended between ignorance and light tha n the Kolmogorov theory

of turbulence, which is both the cornerstone of what we know and

a mystery that ha s not been fathomed.

The same spectrum [. . . ] appears in the sun, in the oceans, and

in man-made machinery. The 5/3 law is well veriﬁed experimen-

tally and , by suggesting that not all scales must b e computed anew

in each problem, opens the door to practical modelling.

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36 4 History of Dimensional Analysis

Vladimir Ar nold [96] reminds us that the main premises o f Kol-

mogorov’s argument remain unproven—after m ore than 60 years!

Even worse, Chorin points to the rather disturbing fact that

Kolmogorov’s spectrum often appears in problems where his as-

sumptions clearly fail. [. . . ] The 5/3 law can now b e derived in

many ways, often under assumptions tha t are antithetical to Kol-

mogorov’s. Turbulence theory ﬁnds itself in the odd s ituation of

having to build on its main resul t while still struggli ng to under-

stand it.

4.4 Dimension o f Lagrange multipliers

By contrast with stories fr om my childhood, this one comes fr om my

very mature age. Only working over this bo ok, I ﬁnally managed

to get over my old incomprehension. It was about the nature of

Lagrange multipliers.

Recall that Lagrange multipliers are u se d in search of maxima

and minima of functions subject to a constraint. For example, we

may wish to maximize a function of two variables f (x, y), where x

and y cannot take arbitrary values but are constrained by condition

g(x, y) = c.

Fig. 4. 3 . Constrained optimization. Source: Wikipedia. Public domain.

Lagrange proposed to in troduce an extra v ariable λ and con-

sider a function

Λ(x, y, λ) = f(x, y) + λ · (g(x, y) − c).

At the point of constrained extremum, all three partial derivative

of Λ vanish:

∂Λ

∂x

∂Λ

∂y

∂Λ

∂λ

= 0.

It is obvious for

∂Λ

∂λ

= g(x, y) − c = 0

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4.4 Dimension of Lagrange multipliers 37

because of the imposed constraint. For the other two partial deriva-

tives, we have a simple geometric exp lanation: w hen w e look at the

contour lines (level curves) of the fun ction f (x, y), we notice that at

the point of c onstrained extremum it has to be tange nt to the curve

g(x, y) = c; compare Figures 4.3 and 4.4. We have a kind of duality

here: the curve g(x, y) = c is a level curve of the func tion g(x, y).

Since le vel curves of fu nctions are perpendicular to gradients, we

have that the two gradients

∂f (x,y)

∂x

∂f (x,y)

∂y

and

∂g(x,y)

∂x

∂g(x,y)

∂y

are collinear at the point whe re level curves are tange nt. Hence

∂f (x,y)

∂x

∂f (x,y)

∂y

+ λ

∂g(x,y)

∂x

∂g(x,y)

∂y





for some λ, which immediately translates into

∂Λ

∂x

∂Λ

∂y

= 0.

Fig. 4.4 . Contour lines are tangent at the point of constrained extremum.

Source: Wikipedia. Pub lic domain.

But what is the meaning of λ? What is dime nsion of the La-

grange multiplier, when we maximise, say, energy,

mass × length

time

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38 4 History of Dimensional Analysis

by vary in g co nstrained space coordinates x and y which have di-

mension of length, while the constraint function g(x, y) has dimen-

sion of time and measures how much time it takes to get our system

into the state (position) (x, y)? What is the meaning of this dimen-

sion?

The correct answer: price. It is the sh adow price of mathemat-

ical economics: if everything is expressed in money (which are di-

mensionless because of their universality), and , with energy priced

at the current market rate, λ gets the dimension

time

of the price of unit of time. And this is indeed price. Assume that

you have to hire an hourly paid w orkers, and that it takes him

g(x, y) man-hours to tune up your energy-producing installation up

to the state (x, y) at which point the installation will pr oduce your

a certain amount of energy which y ou sell for f (x, y) dollars. Yo u

have limited funds c, hence you are co nstrained by equation

g(x, y) = c.

But assume you got an extra dollar to invest; then the shadow price

λ tells you how much you have to be pr epared to pay for extra work-

ers. If you hire new workers at hour ly rate below λ, you will in-

crease your proﬁt; if you have to pay more than λ, e xpansion of the

work force lead s to decline in your proﬁt.

Disturbingly, when I started to type in into GOOGLE:

money is the measure . . .

GOOGLE automatically suggested to continue the phrase as

money is the measure of success.

Well, there is some mathematical justiﬁcation for this bland tru-

ism. Alas, the greatest ﬂaw of mathematics as a c ultural system is

that it is morally neutral.

Exercises

Pythagorean Theore m

Exercise 4.1 Alexander Givental [186] poin ts to a relatively unknown

fact:

It turns out that Book VI of the Elements contains a generaliza-

tion of the Pythagorean theorem that seems much less famous.

Namely, Euclid VI.31 asserts (see Figure 4.5) that A + B = C for

the areas A, B and C of similar ﬁgures of any sh ape built on the

sides of a right tria ngle. The Pythagorean theorem is clearly the

special case where the shape is the sq uare.

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4.4 Dimension of Lagrange multipliers 39

Fig. 4.5. Euclid VI.31: Generalization of Pyth agoras’ Theorem. After A.

Givental [186].

Prove this generalization of the Pythagorean Theorem.

Givental’s wonderful paper [186] shows that Euclid was in fact using

some form of scaling a rguments.

Froude’s Law of Steamshi p Comparisons

Froude’s Law of Steamship Comparisons was used to great effect in D’Arcy

Thompson’s book On Growth and Form [84 0, p. 24] to analyze the speed of

animals. D’Arcy Thompson applied the same deriva tion as i n Section 4.2.2

to ﬁsh becau se of the validity of the principal assumption: the energy is

produced internally. This leads us to a natural question:

Exercise 4.2 How would Froude’s Law look for solar powered ships? (Use

Froude’s drag.)

Exercise 4.3 Try to relate th e previous exercise to the fact th at all known

living organisms able to move in water and capable of photosynthesis (and

thus able to use solar power directly) consist of a single cell, see Figure 4.6.

Exercise 4.4 Prove the following corollary of Froude’s Law:

The relative speed of a ﬁsh (that is, speed measured in numbers of

its lengths covered by the ﬁsh p er unit of time) is inversely propor-

tional to the square root of its length.

This explains a well-known phenomenon: little ﬁsh in a stream appear to

be very quick.

Exercise 4.5 Estimate, which is relatively faster: an ant or a ra cehorse?

Exercise 4.6 Compare the expected maximal speeds of photosynth esising

(that is, solar powered) and ph agocytosising (that is, fuel powered) organ-

isms of comparable size. Use Stokes’ drag.

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40 4 History of Dimensional Analysis

Fig. 4.6. Euglena, one of many single cell solar powered living organ-

isms able to move in water, it propels itself us ing its tail (ﬂagellum).

http://biodidac.bio.uottawa.ca/info/regles.htm,

 BIODI-

DAC.

Fig. 4. 7 . Walking on stilts increases speed (for Exercise 4 .9).

Exercise 4.7 Why does a mouse have (relatively) a slimmer body build

than an elephant?

I refer the reader to [839] for more on this and other biological problems

from this section.

Exercise 4.8 Deduce a version of Froude’s Law for high speeds and quadratic

drag.

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4.4 Dimension of Lagrange multipliers 41

Legs and paces

Exercise 4.9 Analyze the following quote from [25, p. 233]:

Thus treating a man’s leg as swinging like a pendulum through

a ﬁxed deﬁnite angle, the length of a pace is proportional to his

height, but the number of paces a minute is inversely as the square

root of the height; so that his pace of walking and gettin g over the

ground will vary as the square root of his height, as in Froude’s

Law again.

Using dimensional anal ysis, prove:

• the nu mber of swings of a pend ulum per minute varies inversely as the

square root of the length of the pendulum;

and use this fact to derive Greenhill’s conclusion:

• his pace of walking and getting over the ground wi ll vary as the s quare

root of his height (or of length of legs, see Figure 4.7).

Exercise 4.10 Investigate further:

• How do the frequency of a pendulum’s swings and the pace of a man

walking depend on the acceleration due to gravity? (Read more on that

in [228].)

And even further:

• Read a n article [843] which proves that desert a nts count their paces:

Ants walked on normal legs from their nest to a feeder,

where they were placed on stilts (made of pig bristles glued on

to their legs) an d released. (See Figure 4.8.)

On their back home ants on stilts misjudged the 10-meter

distance back and overshot the nest entrance.

When the experiment was repeated with the sh ort-legged

ants (their legs cruell y amputated), they stopped and were look-

ing for nest entrance too early.

In this experiment, it is crucial to dis tinguish between counting steps

and measuring time—how would you handle this issue if you were an

experimenter?

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Fig. 4. 8 . An ant walks with stilts glu ed on to its legs.

Adding One by O n e

5.1 Adding one by one

It appears that addition comes from counting. In words of Iain Cur-

rie

My Mother told me the following story. When I was about two and

a half a small ﬂock of b irds ﬂew overhead. I sai d: “Look, there

are two and three birds”. I didn’t yet know the nu mber ﬁve but I

understood simple counting.

So far so good; but challenges arise fairly so on.

My colleague Elizabeth Kimber

told me about a dif ﬁculty she

experienced in her ﬁrst encounter with arithmetic, aged 6. She

could easily solve “put a number in the box” problems of the type

7 +  = 12,

by counting how many 1’s she had to add to 7 in order to get 12 but

struggled with

 + 6 = 11,

because she did not know where to start. Worse, she felt that she

could not communicate her difﬁculty to adults. Her teacher forgot

to explain to her that addition was commutative.

Another one of my colleagues, AB

, told me how afraid she was

of subtraction. She could easily visualize subtraction of 4 from 100,

say, as a stack of 100 objects; after removing 4 objects from the top

IC is male, British, has a PhD in mathematics, teaches statistics and

actuarial science at an university.

For the record: EHK is English , female, ha s a PhD in Mathematics,

teaches mathematics at a hi ghly selective secondary school.

AB is Turkish, f emale, has a PhD in Mathemati cs, teaches mathematics

in a research-led university.

44 5 Adding One by One

(by reverse counting: 100, 99, 98, 97), 96 are left. But what will hap-

pen if you remove 4 objects from the bottom of the stack?

This is a phenome non well kn own and well studied in the main-

stream pedagogical research. James Hiebert reports from a ﬁeld

study [667]:

Fifty-ﬁve percent of the responses to the verbal problem

a + b = 

and to the verbal problem

a − b = 

included modeling the sets with cubes. This percentage drops to

about 40% for the

a +  = c

problems and to a bout 18% for the

 + b = c

problems.

Statistics is always instructive, but I wo uld rather understand

the intrinsic logic of individual personal stories. I ﬁnd an ally in

the neurologist Vilayanur Ramachandran who said about statisti-

cal analysis [370, pp. xi–xii]:

There is also a tension in the ﬁeld of neurology between the ‘single

case study’ approach, the intensive stu d y of just one or two pa-

tients with a syndrome, and sifting through a large number of pa-

tients and doi ng a statistical analysis. The criticism is sometimes

made that it’s easy to b e misled by single strange cases, but this

is nons ense. Most of the syndromes in neurology that have stood

the test of time [. . . ] were initially discovered by a careful study

of single case and I don’t know of even one that was discovered by

averaging results from a large sample.

My ap proach is different from the traditional pedagogical re-

search: instead of looking at pedagogical and psychological ex pla-

nation for EHK’s and AB’s problems, I am asking a question: what

were hidden mathematical reasons for their difﬁculties? For an an-

swer, I look at Peano arithmetic and some elementary category the-

ory.

I will be happy if my educationalist colleagues recognize that my

approach complements, not contradicts that one of more traditional

educational research. I am open to any criticism and comments; I

am prepared to accept that my project is far-fetched. However, the

remarkable level of support that I have already received from my

correspondents—mostly pro fessional research mathe maticians—

leads me to arg ue conﬁdently that my proje ct is worth trying.

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