Friday 18 March 2011

The Difficulties of Math

File:De Morgan Augustus.jpg
It was on this day, March 18, in 1871 that Agustus de Morgan died.  The student and friend of George Peacock, the Algebra Reformer, de Morgan went on to shape the math program of a new college in London, (the present University College of London)  founded on principles of religious freedom in direct opposition to the exclusion of Jews and other non-Anglican religions in Cambridge and Oxford. 

One of his popular books, "On the Study and Difficulties of Mathematics",  written essentially for the student studying "without a tutor", points out the difference between math and other disciplines in his first chapter. 

No person commences the study of mathematics
without soon discovering that it is of a very different
nature from those to which he has been accustomed.
The pursuits to which the mind is usually directed be-
fore entering on the sciences of algebra and geometry,
are such as languages and history, etc. Of these,
neither appears to have any affinity with mathemat-
ics ; yet, in order to see the difference which exists be-
tween these studies, — for instance, history and geom-
etr)', — it will be useful to ask how we come by knowl-
edge in each. Suppose, for example, we feel certain
of a fact related in history, such as the murder of
Caesar, whence did we derive the certainty? how came
we to feel sure of the general truth of the circum-
stances of the narrative? The ready answer to this
question will be, that we have not absolute certainty
upon this point; but that we have the relation of his-
torians, men of credit, who lived and published their
accounts in the very time of which they write ; that
succeeding ages have leceived those accounts as true,
and that succeeding historians have backed them with
a mass of circumstantial evidence which makes it the
most improbable thing in the world that the account,
or any material part of it, should be false. This is
perfectly correct, nor can there be the slightest ob-
jection to believing the whole narration upon such
grounds; nay, our minds are so constituted, that,
upon our knowledge of these arguments, we cannot
help believing, in spite of ourselves. But this brings
us to the point to which we wish to come ; we believe
that Caesar was assassinated by Brutus and his friends,
not because there is any absurdity in supposing the
contrary, since every one must allow that there is just
a possibility that the event never happened : not be-
cause we can show that it must necessarily have been
that, at a particular day, at a particular place, a c-
cessful adventurer must have been murdered in the
manner described, but because our evidence of the
fact is such, that, if we apply the notions of evidence
which every-day experience justifies us in entertain-
ing, we feel that the improbability of the contrary
compels us to take refuge in the belief of the fact ;
and, if we allow that there is still a possibility of its
falsehood, it is because this supposition does not in-
volve absolute absurdity, but only extreme improb-

In mathematics the case is wholly different. It is
true that the facts asserted in these sciences are of a
nature totally distinct from those of history ; so much
so, that a comparison of the evidence of the two may
almost excite a smile. But if it be remembered that
acute reasoners, in every branch of learning, have
acknowledged the use, we might almost say the neces-
sity, of a mathematical education, it must be admitted
that the points of connexion between these pursuits
and others are worth attending to. They are the more
so, because there is a mistake into which several have
fallen, and have deceived others, and perhaps them-
selves, by clothing some false reasoning in what they
called a mathematical dress, imagining that, by the
application of mathematical symbols to their subject,
they secured mathematical argument. This could not
have happened if they had possessed a knowledge of
the bounds within which the empire of mathematics
is contained. That empire is sufiiciently wide, and
might have been better known, had the time which
has been wasted in aggressions upon the domains of
others, been spent in exploring the immense tracts
which are yet untrodden.

We have said that the nature of mathematical dem-
onstration is totally different from all other, and the
difference consists in this — that, instead of showing
the contrary of the proposition asserted to be only im-
probable, it proves it at once to be absurd and impos-
sible. This is done by showing that the contrary of
the proposition which is asserted is in direct contra-
diction to some extremely evident fact, of the truth of
which our eyes and hands convince us. In geometry,
of the principles alluded to, those which are most
commonly used are —

I. If a magnitude be divided into parts, the whole
is greater than either of those parts.

II. Two straight lines cannot inclose a space.

III. Through one point only one straight line can
be drawn, which never meets another straight line, or
which is parallel to it.

It is on such principles as these that the whole of
geometry is founded, and the demonstration of every
proposition consists in proving the contrary of it to be
inconsistent with one of these. Thus, in Euclid, Book
I., Prop. 4, it is shown that two triangles which have
two sides and the included angle respectively equal
are equal in all respects, by proving that, if they are
not equal, two straight lines will inclose a space, which
is impossible.
In other treatises on geometry, the
same thing is proved in the same way, only the self-
evident truth asserted sometimes differs in form from
that of Euclid, but may be deduced from it, thus —

Two straight lines which pass through the same
two points must either inclose a space, or coincide
and be one and the same line, but they cannot inclose
a space, therefore they must coincide. Either of these
propositions being granted, the other follows imme-
diately ; it is, therefore, immaterial which of them we
use. We shall return to this subject in treating
specially of the first principles of geometry.

Such being the nature of mathematical demonstra-
tion, what we have before asserted is evident, that
our assurance of a geometrical truth is of a nature
wholly distinct from that which we can by any means
obtain of a fact in history or an asserted truth of meta-
physics. In reality, our senses are our first mathe-
matical instructors; they furnish us with notions
which we cannot trace any further or represent in any
other way than by using single words, which every
one understands. Of this nature are the ideas to
which we attach the terms number, one, two, three,
etc., point, straight line, surface; all of which, let
them be ever so much explained, can never be made
any clearer than they are already to a child of ten
years old.


I wonder if one of the problems with mathematics education today is that student's want to treat it as if it was a study like other disciplines, and perhaps if teachers want to teach it like it was any other discipline.


Sue VanHattum said...

Not just today, but throughout history, perhaps?

I got a chuckle out of his assertion that: Two straight lines which pass through the same two points must either inclose a space, or coincide and be one and the same line, but they cannot inclose a space, therefore they must coincide.

I wonder what he'd think of non-Euclidean geometry. The foundations of math have changed quite a bit since deMorgan's time, haven't they?

Pat's Blog said...

You are correct... De Morgan was well aware in his life of the "parallel" question, and even comments in his "budget of Paradoxes"... but the big questions would come after this book was written... Lobachevsky, for instance, was little known before about 1840..