The Education of T.C. MITS --What Modern Mathematics Means to You
By Lillian Lieber, Illustrations by Hugh Gray Lieber
The following is excerpted, with permission of the publisher, from The Education of T.C. MITS --What Modern Mathematics Means to You.
V. Our Totem Pole
Let us symbolize our Totem Pole
by a column made up of
the five well-known regular solids,
as shown in the
drawing on the opposite page.
And let us think of the solids as
separate rooms, in each of which
a certain aspect of Science
is presented.
We shall now take you
on a guided tour of these rooms.
The First Floor, the CUBE,
contains all the scientific gadgets
with which we are all so familiar:
automobiles and refrigerators
and radios and airplanes,
and what seems like googols* of others.
This is of course also the room
in which you will find
tanks and bombers and
all the paraphernalia of war.
And this is why some people say
that Science is amoral,
since it produces,
with equal indifference,
the peaceful toys we enjoy so much
as well as the instruments of
destruction.
But these people have probably
never climbed the magic staircase
which leads from the gadget room
up into the other rooms of
our Totem Pole,
and are entirely unfamiliar with
their contents.
Let us therefore go up to
the Second Floor, the ICOSAHEDRON.
Here we find
a great industrial laboratory-
this is where the gadgets are
invented, tried out, manufactured;
the men working on this level
are not advertising and selling,
they are inventing.
They are told by
the people who hire them:
“We want a brighter light,
a cheaper light,
a more smoothly running car,
an effective defroster for
airplanes"—
and a thousand and one other things.
These research men do not
let their minds roam around
looking for interesting problems,
Their problems are handed to them,
and they must solve them within
a very reasonable length of time,
“or else.”
Only “practical” men are
wanted here,
and not oversentimental ones.
For they may be told at any moment
to find effective ways of
killing people-
they must make
the best long-range guns,
the best poison gases,
the bombs which can
destroy the most people.
But you thought we promised
to get away from all this
as we climbed upward:
and yet this floor seems to have
even more diabolical possibilities
than the first one.
Perhaps if this second Floor
were destroyed,
the war paraphernalia of
the first floor might
become obsolete and die out
of its own accord.
Are not these scientific inventors
the real devils after all?
But let us climb another flight
and see what goes on in
the OCTAHEDRON.
These men are doing research in .
“Pure" Science.
They are not employed by
manufacturers or governments;
they are usually
professors at universities who
select their own problems because
they are interested in them.
They are not concerned with
any practical applications of
their ideas.
They are the theoretical men-
they ask the most “useless" questions.
For instance;
“What happens when you mix
sugar and water and lemon?"
They call it "Sugar Hydrolysis"
instead of “Lemonade."
They study it in different solutions,
carefully varying the
relative amounts of
the substances involved,
and examine them with a polariscope
for days and days, and years and years,
keeping careful records
and publishing the results in
scientific journals.
Will these investigations
make them rich?
Or fat?
Or benefit them in any “practical” way?
Not at all.
Then why do they do it?
The answer is that
they are just driven by .
Curiosity.
Once in a while they are consulted by _
the men on the second floor,
but not so very often.
Usually they just
write up their results and
die without knowing whether
these will ever have any
practical use.
But the fact is that
their results ARE very often,
in the long run,
used by some second-floor scientist.
Indeed,
these second-story men find that
they must study the work of
the “pure” scientists
constantly.
But usually it is the work of
the ”pure" ones of the past-
work which has already found its way
into the textbooks which
they have studied at the
institutes of technology,
rather than the current work
published in the journals of
Pure Science.
In fact,
the gentlemen working at
any given time on the
second and third floors
seem to have very little in common:
the second-floor men consider
the upper-floor men to be
“wild-eyed, absent-minded
college professors.
Some of them are perhaps just
crackpots,
who knows?
It is safer to go to the theory of
the established past,
which has been duly
tried and tested."
And, on the other hand,
the inhabitants of the third floor
look down upon the second-story men,
considering them to be mere
“hirelings and ignoramuses,"
and prefer to leave their results to
the second-story men of the FUTURE,
“who will be in a better position to
appreciate them."
But, granting even that
this will be so,
what guarantee have we that
the uses that their ideas will find
WILL be decent, moral uses?
How do we know that
they are not storing up just
a lot of additional trouble for the
unfortunate future generations?
No,
let us climb up further,
and look at the Fourth Floor,
the DODECAHEDRON.
Here we find
the Mathematicians—
not the “Pure" Mathematicians,
for they live on the Fifth Floor,
in the TETRAHEDRON garret,
with the Modern Artists.
The fourth-floor mathematicians are
the ones who know the
Classical Mathematics of the past
and apply it to
the scientific findings of the
“Pure" Scientists of the third floor.
They take the scientific data
and organize it
and study it with
all the mathematical machinery
at their command.
If a second-story man ever
happens in on the fourth floor,
which is very rare,
he can hardly control his laughter.
These men seem to him to be
even more wild-eyed than those
on the floor below,
but the guide tells him:
“You ain't seen nothin' yet,
wait till you see the Top Floor,
the TETRAHEDRON.”
At least on this fourth floor
you hear them mention
Geometry and Algebra and Calculus,
subjects you have heard about
in high school or in college.
But on that top floor,
they draw geometric figures on
doughnuts and pretzels
(no fooling!)
and on rubber sheets.
And they have up there
Algebras and Arithmetics in which
twice two is NOT four!
ln which 3+2 does NOT give
the same answer as 2+3,
nor is 5 X 6 equal to 6 X 5!
They are indeed lit companions for
the Modern Artists who
share the garret with them!
They are lucky if they can even
get a job!
And yet the connoisseurs say that
their work is
tremendously important for
the future.
Indeed,
if you trace back some of the
most practical and useful gadgets,
you will find that
if it had not been for a series of
"wild-eyed,” “impractical" men,
these gadgets could not exist today.
As you will see in the next chapter.
VI. THE TOTEM POLE (Cont.)
Take the radio for example,
with all its variety of
concerts and important
broadcasts of all kinds.
Trace it to the second floor
and you will find that
many men on that floor have been
improving reception by inventing
better tubes and aerials, etc.
But all this could not have happened
had it not been for
a man named Marconi,
a second-story man,
who sent the first
crude radio messages.
And even his work
would have been impossible
had it not been for
another man, named Hertz.
who worked on the third floor,
and who proved that the very idea
of sending a wireless message
was actually possible,
since he demonstrated the existence of
electromagnetic waves.
But where did he get the idea of
even looking for these waves?
Why, from a fourth-floor man,
of course,
a man named Clerk Maxwell,
who first conceived the idea of waves in
an “electromagnetic field" and
applied the Calculus to it, obtaining
a set of differential equations
from which he declared
the consequence followed that
there MUST be
electromagnetic waves.
And, as we have already said,
Hertz subsequently proved
that he was right.
And, obviously,
Maxwell could not have done his job
had not Newton invented the Calculus.
And so it goes.
Take any gadget you like
and trace it back,
and you will find that
invariably you will have to
go up into
all the five floors
before you can have its complete story.
“But,” you will say,
“you have not proved
your initial point
at all,
since the same is true of
tanks and bombers also,
and therefore
Science IS indifferent to
Good and Evil,
and IS amoral after all."
You will soon admit, however,
if you read again
the story we have told you,
more attentively this time,
that Science is trying to
tell you something else,
if you will but listen.
For instance,
go back and you will see that
in the little story of the radio,
there are
Americans,
Italians,
Germans,
Englishmen.
If you take the airplane,
you will also find
Russians,
Frenchmen,
and others.
In short you will be very much
impressed by the fact that
SCIENCE IS INTERNATIONAL,
that it is trying to tell us that
Hitler’s racial theories are
utterly false.
It is also trying to tell us—
if we would only listen—
that co-operation is essential
for accomplishing things,
that it is really absurd
for the first- and second-story men
to laugh at those who live upstairs,
or for the latter
to look down upon the others.
For they are all needed
to do the job.
Is not this DEMOCRACY?
Thus we see that
Science is NOT AMORAL,
but has a PHILOSOPHY
to offer us,
provided that we do not
merely identify Science with
first-floor gadgets,
and thus
cut oft its HEAD
(the upper Floors!)
and stop its
BLOOD STREAM
(the interrelationship
between ALL the floors)!
And as we tell you more about
those strange
algebras and geometries we mentioned,
you will see that
Mathematics has many important
messages for us-
that it is trying to tell us that
VARIOUS mathematical systems
are possible,
that they are all man-made
and controllable by man,
and that
if you apply this idea
to the social world,
you will realize that
it is up to you
to build a good world if you want one-
that man has a great deal more
freedom and creative ability than
he is sometimes aware of.
The idea of a fixed “human nature"
that has us by the throat
is just a Fiction,
for
the activities on the top floor
are trying to tell us that
HUMAN NATURE HAS
INF INITE POSSIBILITIES.
In short,
it is not the guns and tanks
which are the real evils—
for a gun may be a great “good”
under certain circumstances.
But rather
such false IDEAS as
"Nationalism,"
"Dictatorship,"
narrow views of .
“Human Nature,"
etc.,
are the real DEVILS.
Thus
FALSE IDEAS ARE MORE DANGEROUS
THAN GUNS!!
Guns and tanks are mere tools,
they may be used for good or evil.
But they are only
first-floor gadgets,
whereas the philosophy of science,
which comes from
a contemplation of all the floors,
and of their relationship
to each other,
has for us
unmistakable messages:
we must rise up above
the first and second floors .
and realize that
these alone are
NOT SUFFICIENT
for the human race,
whose nature is so
beautifully revealed by
a study of Science as a whole,
which, as we have seen, has
Internationalism and Democracy
at its very heart.
We therefore advocate:
(1) A broader view of Science which
enables us to appreciate the
philosophy in it-
Science as a whole,
as seen in the Totem Pole,
can really protect us from evil.
(2) A more appreciative attitude
toward the top-floor men.
By knowing how much we owe
to the top-floor men of the past,
we should stop treating them with
the brutality with which
they have been treated
in the past-
just because they do not use
their energy to make themselves
physically comfortable.
And we should stop heckling them
by asking them:
“What is the practical use of
what you are doing?"
or
“What does this mean for
The Average Man?"
Since the truth is that
they themselves do not know.
Their work is as much
a “Natural Phenomenon” as
a natural oil well or
natural gas or
mountains or
rivers.
Let us give them the
freedom they need
to do what is in them to do.
Let us turn their garret into a
penthouse,
and marvel at their
strange products.
Perhaps some day we shall find
a “practical" use for them,
as has so often happened
in the past.
And besides,
the philosophical implications
of their work
already make them
invaluable to us NOW-
as we shall see.
The Moral: Oh, listen to the
Totem Pole!
* A “googol” is a very large number,
namely, 10^100, or 1 with a
hundred zeros after it.
The term “googol" was
invented in fun by
a nephew of one of our
great American mathematicians,
Edward Kasner.
Lillian Lieber was the head of the Department of Mathematics at Long Island University. She wrote a series of lighthearted (and well-respected) math books in the 1940s. Her Husband, Hugh Gray Lieber, was the head of the Department of Fine Arts at Long Island University. He illustrated many books written by his wife Lillian.
