Augustus de Morgan - Collected Quotes

"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." (Augustus De Morgan, "On the Study and Difficulties of Mathematics", 1830)

"The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connection with one another. This cannot be attained by a mere reading of the book, however great the attention which may be given. It is impossible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have completely mastered it. Many results must be given of which the details are suppressed, such are the additions, multiplications, extractions of the square root, etc., with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand, while engaged in any algebraical process." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1830)

"The nature of mathematical demonstration 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 improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely evident fact, of the truth of which our eyes and hands convince us." (Augustus De Morgan, "On the Study and Difficulties of Mathematics", 1830)

"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." (Augustus De Morgan, "On the Study and Difficulties of Mathematics", 1830)

"If we then compare the position in which we stand with respect to divergent series, with that in which we stood a few years ago with respect to impossible quantities [that is, complex numbers], we shall find a perfect similarity […] It became notorious that such use [of complex numbers] generally led to true results, with now and then an apparent exception. […] But at last came the complete explanation of the impossible quantity, showing that all the difficulty had arisen from too great limitation of definitions." (Augustus de Morgan, Penny Cyclopaedia, cca. 1833-1843) 

"Every mathematical method has its inverse, as truly, and for the same reason, as it is impossible to make a road from one town to another, without at the same time making one from the second to the first. The combinatorial analysis is analysis by means of combinations; the calculus of generating functions is combination by means of analysis." (Augustus de Morgan, "The Differential and Integral Calculus", 1836)

"I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical."(Augustus De Morgan, "The Differential and Integral Calculus", 1836)

"The greatest writers on mathematical subjects have a genius which saves them from their own slips, and guides them to true results through inaccurate expression, and sometimes through absolute error." (Augustus De Morgan, "The Differential and Integral Calculus", 1836)

"The science of algebra, independently of any of its uses, has all the advantages which belong to mathematics in general as an object of study, and which it is not necessary to enumerate. Viewed either as a science of quantity, or as a language of symbols, it may be made of the greatest service to those who are sufficiently acquainted with arithmetic, and who have sufficient power of comprehension to enter fairly upon its difficulties." (Augustus de Morgan, "Elements of Algebra", 1837)

"I consider the world probability as meaning the state of mind with respect to an assertion, a coming event, or any other matter on which absolute knowledge does not exist." (Augustus De Morgan, "Essay on Probability", 1838)

"My own impression [...] is that the mathematical results have outrun their interpretation and that some simple explanation of the force and meaning of the celebrated interval [...] will one day be found [...] which will at once render useless all the works hitherto written." (Augustus de Morgan, 1838)

"Common integration is only the memory of differentiation, the different artifices by which integration is effected are changes, not from the known to the unknown, but from forms in which memory will not serve us to those in which it will." (Augustus De Morgan, Transactions of the Cambridge Philosophical Society Vol. 8, 1844)

"All existing things upon this earth, which have knowledge of their own existence, possess, some in one degree and some in another, the power of thought, accompanied by perception, which is the awakening of thought by the effects of external objects upon the senses." (Augustus De Morgan, "Formal Logic: Or, The Calculus of Inference, Necessary and Probable", 1847)

"By degree of probability we really mean, or ought to mean, degree  of belief [...] Probability then, refers to and implies belief, more or  less, and belief is but another name for imperfect knowledge, or it  may be, expresses the mind in a state of imperfect knowledge." (Augustus De Morgan, "Formal Logic: Or, The Calculus of Inference, Necessary and Probable", 1847)

"Trigonometry contains the science of continually undulating magnitude: meaning magnitude which becomes alternately greater and less, without any termination to succession of increase and decrease." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)

"Metaphysics. The science to which ignorance goes to learn its knowledge, and knowledge to learn its ignorance. On which all men agree that it is the key, but no two upon how it is to be put into the lock." (Augustus De Morgan, [letter to Dr. Whewell] 1850)

"Every kind of human knowledge has unanswerable questions at its end: and until the insoluble problems are revealed, we plume ourselves upon our progress." (Augustus De Morgan, "On Infinity: and on the Sign of Equality", Transactions of the Cambridge Philosophical Society, 1871)

"Great discoveries are always laughed at: but it is very often not the laugh of incredulity; it is a mode of distorting the sense of inferiority into a sense of superiority, or a mimicry of superiority interposed between the laughter and his feeling of inferiority." (Augustus De Morgan, "On Infinity: and on the Sign of Equality", Transactions of the Cambridge Philosophical Society, 1871)

"If you wish to oppose a mathematician, confront him on some question in the parts of his subject which are supposed to be perfectly demonstrated." (Augustus De Morgan, "On Infinity: and on the Sign of Equality", Transactions of the Cambridge Philosophical Society, 1871)

"In all matters we have learnt to say that we do not know what things are, we only know something about them: that is, we have subjects with attributes, and therefore propositions which can be affirmed." (Augustus De Morgan, "On Infinity: and on the Sign of Equality", Transactions of the Cambridge Philosophical Society, 1871)

"Infinity is a pertinacious meddler, who will not be turned out: we must find out what he wants, and give it him." (Augustus De Morgan, "On Infinity: and on the Sign of Equality", Transactions of the Cambridge Philosophical Society, 1871)

"My opinion of mankind is founded upon the mournful fact that, so far as I can see, they find within themselves the means of believing in a thousand times as much as there is to believe in." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"Proof requires a person who can give and a person who can receive." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"The gambling reasoner is incorrigible; if he would but take to the squaring of the circle, what a load of misery would be saved." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself."  (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"There are no limits in mathematics, and those that assert there are, are infinite ruffians, ignorant, lying blackguards. There is no differential calculus, no Taylor's theorem, no calculus of variations, [...] in mathematics. There is no quackery whatever in mathematics." (Augustus De Morgan, "A Budget of Paradoxes", 1872)

"[…] wrong hypotheses, rightly worked from, have produced more useful results than unguided observation." (Augustus de Morgan, "A Budget of Paradoxes", 1872)

"[…] there are terms which cannot be defined, such as number and quantity. Any attempt at a definition would only throw difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatise on that subject." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"It is generally true, that wherever an imaginary expression occurs the same results will follow from the application of these expressions in any process as would have followed had the proposed problem been possible and its solution real." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"Many theorems are obvious upon looking at a moderately-sized figure; but the reasoning must be such as to convince the mind of their truth when, from excessive increase or diminution of the scale, the figures themselves have past the boundary even of imagination." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols." (Augustus De Morgan, "On the syllogism: and other logical writings", 1966)

"Among the mere talkers, so far as mathematics are concerned, are to be ranked three out of four of those who apply mathematics to physics, who, wanting a tool only, are very impatient of everything which is not of direct aid to the actual methods which are in their hands." (Augustus De Morgan) 

"The moving power of mathematical invention is not reasoning but imagination." (Augustus De Morgan)

"We have shown the symbol √-1 to be void of meaning, or rather self-contradictory and absurd. Nevertheless, by means of such symbols, a part of algebra is established which is of great utility. It depends upon the fact, which must be verified by experience, that the common rules of algebra may be applied to these expressions without leading to any false results." (Augustus De Morgan)

"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 that with two." (Augustus De Morgan)