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.

In other treatises on geometry, the

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-

ability.

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.

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.

## 2 comments:

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?

Sue,

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..

Post a Comment