"I look upon it [mechanical notation] as one of the most important additions I have made to human knowledge. It has placed the construction of machinery in the rank of a demonstrative science. The day will arrive when no school of mechanical drawing will be thought complete without teaching it." Charles Babbage, "Passages From the Life of a Philosopher", 1864)
"If the task of philosophy is to break the domination of words over the human mind […], then my concept notation, being developed for these purposes, can be a useful instrument for philosophers […] I believe the cause of logic has been advanced already by the invention of this concept notation." (Gottlob Frege, "Begriffsschrift", 1879)
"The origin of a science is usually to be sought for not in any systematic treatise, but in the investigation and solution of some particular problem. This is especially the case in the ordinary history of the great improvements in any department of mathematical science. Some problem, mathematical or physical, is proposed, which is found to be insoluble by known methods. This condition of insolubility may arise from one of two causes: Either there exists no machinery powerful enough to effect the required reduction, or the workmen are not sufficiently expert to employ their tools in the performance of an entirely new piece of work. The problem proposed is, however, finally solved, and in its solution some new principle, or new application of old principles, is necessarily introduced. If a principle is brought to light it is soon found that in its application it is not necessarily limited to the particular question which occasioned its discovery, and it is then stated in an abstract form and applied to problems of gradually increasing generality. [/] Other principles, similar in their nature, are added, and the original principle itself receives such modifications and extensions as are from time to time deemed necessary. The same is true of new applications of old principles; the application is first thought to be merely confined to a particular problem, but it is soon recognized that this problem is but one, and generally a very simple one, out of a large class, to which the same process of investigation and solution are applicable. The result in both of these cases is the same. A time comes when these several problems, solutions, and principles are grouped together and found to produce an entirely new and consistent method; a nomenclature and uniform system of notation is adopted, and the principles of the new method become entitled to rank as a distinct science." (Thomas Craig, "A Treatise on Projections", 1880)
"Boole's work is not so much an attempt (as used to be commonly said) to 'reduce logic to mathematics', as the employment of symbolic language and notation in a wide generalisation of purely logical processes. His fundamental process is really that of continued dichotomy, or subdivision, in respect of all the class terms which enter into the system of propositions in question. [...] This process in its priori form furnishes us with a complete set of possibilities, which, however, the conditions involved in the statement of the assigned propositions necessary necessarily?) reduce to a limited number of actualities: Boole's system being essentially one for displaying the solution of the problem in the form of a complete enumeration of these actualities." (John Venn, [in "Dictionary of National Biography"], 1886)
"I believe, however, that the increasing extent of the territory of mathematics will always be counteracted by increased facilities in the means of communication. Additional knowledge opens to us new principles and methods which may conduct us with the greatest ease to results which previously were most difficult of access; and improvements in notation may exercise the most powerful effects both in the simplification and accessibility of a subject. It rests with the worker in mathematics not only to explore new truths, but to devise the language by which they may be discovered and expressed; and the genius of a great mathematician displays itself no less in the notation he invents for deciphering his subject than in the results attained." (James W L Glaisher, British Association for the Advancement of Science Nature, Section A Vol. 42, [presidential address], 1089)
"I have great faith in the power of well-chosen notation to simplify complicated theories and to bring remote ones near and I think it is safe to predict that the increased knowledge of principles and the resulting improvements in the symbolic language of mathematics will always enable us to grapple satisfactorily with the difficulties arising from the mere extent of the subject." (James W L Glaisher, British Association for the Advancement of Science Nature, Section A Vol. 42, [presidential address] 1089)
"The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms." (Florian Cajori, "A History of Mathematics", 1894)
"A mathematical argument is, after all, only organized common sense, and it is well that men of science should not always expound their work to the few behind a veil of technical language, but should from time to time explain to a larger public the reasoning which lies behind their mathematical notation." (George Darwin, "The Tides and Kindred Phenomena in the Solar System", 1898)
"There are now two systems of notations, giving the same formal results, one of which gives them with self-evident force and meaning, the other by dark and symbolic processes. The burden of proof is shifted, and it must be for the author or the supporter of the dark system to show that it is in some way superior to the evident system." (William S Jevons)
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