Augustus De Morgan Quotes

The number of mathematical students, increased as it has been of late years, would be much augmented if those who hold the highest rank in science would condescend to give more effective assistance in clearing the elements of the difficulties which they present. If any one claiming that title should think my attempt obscure or enormus, he must share the blame with me, since it is through his neglect that I have been enabled to avail myself of an opportunity to perform a task which I would gladly have seen confided to more skilful hands.

Find a fraction which, multiplied by itself, shall give 6, or... find the square root of 6. This can be shown to be an impossible problem; for it can be shown that no fraction whatsoever multiplied by itself, can give a whole number, unless it be itself a whole number disguised in a fractional form, such as 4⁄2 or 21⁄3. To this problem, then, there is but one answer, that it is self-contradictory. But if we propose the following problem,—to find a fraction which, multiplied by itself, shall give a product lying between 6 and 6 + a; we find that this problem admits of solution in every case. It therefore admits of solution however small a may be... as small as you please. ...there is such a thing as the square root of 6, and it is denoted by √6. But we do not say we actually find this, but that we approximate to it.

During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical.

When... we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And if the series of values increase in succession, so that name any quantity we may, however great, all after a certain point will be greater, then the series is said to increase without limit. It is also frequently said, when a quantity diminishes without limit, that it has nothing, zero or 0, for its limit: and that when it increases without limit it has infinity or ∞ or 1⁄0 for its limit.

Although there is no study which presents so simple a beginning as that of geometry, there is none in which difficulties grow more rapidly as we proceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very commencement, for it is evident that every student will feel a claim to have his objections answered, not by authority, but by argument, and that the intelligent student will perceive more readily than another the force of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise.

The moving power of mathematical invention is not reasoning, but imagination.

There is a mistake into which several have fallen, and have deceived others, and perhaps themselves, 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 sufficiently 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.

Take a unit, halve it, halve the result, and so on continually. This gives—1 1⁄2 1⁄4 1⁄8 1⁄16 1⁄32 1⁄64 1⁄128 &c.Add these together, beginning from the first, namely, add the first two, the first three, the first four, &c... We see then a continual approach to 2, which is not reached, nor ever will be, for the deficit from 2 is always equal to the last term added.
...We say that—1, 1 + 1⁄2, 1 + 1⁄2 + 1⁄4, 1 + 1⁄2 + 1⁄4 + 1⁄8, &c. &c.is a series of quantities which continually approximate to the limit 2. Now the truth is, these several quantities are fixed, and do not approximate to 2. ...it is we ourselves who approximate to 2, by passing from one to another. Similarly when we say, "let x be a quantity which continually approximates to the limit 2," we mean, let us assign different values to x, each nearer to 2 than the preceding, and following such a law that we shall, by continuing our steps sufficiently far, actually find a value for x which shall be as near to 2 as we please.

The student of the Differential Calculus may... be brought to think it possible that the terms and ideas which that science requires may exist in his own mind in the same rude form as that of a straight line in the conceptions of a beginner in geometry. ...he must be prepared to stop his course until he can form exact notions, acquire precise ideas, both of resemblance between those things which have appeared most distinct, and of distinction between those which have appeared most alike. To do this... formal definitions would be useless; for he cannot be supposed to have one single notion in that precise form which would make it worth while to attach it to a word. One reason of the great difficulty which is found in treatises on this subject... the tacit assumption that nothing is necessary previously to actually embodying the terms and rules of the science, as if mere statement of definitions could give instantaneous power of using terms rightly. We shall here attempt... a wider degree of verbal explanation than is usual with the view of enabling the student to come to the definitions in some state of previous preparation.

I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. ...Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold...

I am far from saying that this Treatise will be easy; the subject is a difficult one, as all know who have tried it.

It is not true, out of geometry, that the mathematical sciences are, in all their parts those models of finished accuracy which many suppose. The extreme boundaries of analysis have always been as imperfectly understood as the tract beyond the boundaries was absolutely unknown. But the way to enlarge the settled country has not been by keeping within it, but by making voyages of discovery, and I am perfectly convinced that the student should be exercised in this manner; that is, that he should be taught how to examine the boundary, as well as how to cultivate the interior. ...allowing all students whose capacity will let them read on the higher branches of applied mathematics, to have each his chance of being led to the cultivation of those parts of analysis on which rather depends its future progress than its present use in the sciences of matter.

We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it can see better with one eye than with two.

There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid.

The following Treatise... has been endeavoured to make the theory of limits, or ultimate ratios... the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit...