Augustus De Morgan citáty

August de Morgan bol britský logik a matematik, ktorý sa zaoberal algebraickým vyjadrením problémov formálnej logiky.

De Morganov zápis ovplyvnila algebraická logika Georgea Boolea, ktorý sa neskôr prihlásil o časť zásluh na týchto logických zákonoch. Hoci podobné pozorovania zapísal už Aristoteles a boli známy už v starom Grécku a v stredoveku, De Morgan im dal klasickú formu a zaviedol tým matematickú reč do matematickej logiky. De Morganove zákony sú ľahko dokázateľné a môžu sa zdať triviálne, avšak sú užitočné pre dôkazy platnosti záverov a pre deduktívne úlohy. Wikipedia  

✵ 27. jún 1806 – 18. marec 1871
Augustus De Morgan fotka
Augustus De Morgan: 41   citátov 0   Páči sa

Augustus De Morgan: Citáty v angličtine

“In order to see the difference which exists between… studies,—for instance, history and geometry, it will be useful to ask how we come by knowledge in each. Suppose, for example, we feel certain of a fact related in history… if we apply the notions of evidence which every-day experience justifies us in entertaining, 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 involve absolute absurdity, but only extreme improbability.
In mathematics the case is wholly different… 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. In geometry, of the principles alluded to, those which are most commonly used are—
I. If a magnitude is 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.”

Zdroj: On the Study and Difficulties of Mathematics (1831), Ch. I.

“The lowest steps of the ladder are as useful as the highest.”

Zdroj: On the Study and Difficulties of Mathematics (1831), Ch. I.

“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.”

"Short Supplementary Remarks on the First Six Books of Euclid's Elements" (Oct, 1848) Companion to the Almanac for 1849 as quoted by Sir Thomas Little Heath, The Thirteen Books of Euclid's Elements Vol.1 https://books.google.com/books?id=UhgPAAAAIAAJ, Introduction and Books I, II. Preface, p. v.

“A large quantity of examples is indispensable.”

Preface, pp. viii
The Differential and Integral Calculus (1836)

“A finished or even a competent reasoner is not the work of nature alone… education develops faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to learn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something must be reasoned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history may be chosen for this purpose. Now, of all these, it is desirable to choose the one… in which we can find out by other means, such as measurement and ocular demonstration of all sorts, whether the results are true or not.
.. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds:—
1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing.
2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general.
3. The demonstration is strictly logical, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion.
4. When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if… reason is not to be the instructor, but the pupil.
5. There are no words whose meanings are so much alike that the ideas which they stand for may be confounded.
…These are the principal grounds on which… the utility of mathematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these studies have a tendency to form are valuable in the highest degree. The most important of all is the power of concentrating the ideas which a successful study of them increases where it did exist, and creates where it did not. A difficult position or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exertions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life.”

Zdroj: On the Study and Difficulties of Mathematics (1831), Ch. I.

“If much difficulty should be experienced in the elementary chapters, I know of no work which I can so confidently recommend to be used with the present one, as that of M. Duhamel.”

Duhamel, Cours d'Analyse de l'Ecole Polytechnique. Paris, Bachelier. vol i 1841 vol. ii. 1840.
The Differential and Integral Calculus (1836)

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