"[…] although the symbol √-1 be beyond the power of arithmetical computation, the operations in which it is introduced are intelligible, and deserve, if any operations do, the name of reasoning." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"By use of the symbol √-1 and of the forms proved to obtain in the combination of real quantities, a mode of notation is obtained, by which we may express sines and cosines, relatively to their arc." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"It has been already observed, that demonstration ultimately depends on observations made on individual objects, and that a conclusion expressed by certain characters and signs, if general, must be true 'in each particular case that presents itself, on assigning specific values to the signs." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"It is to be desired, that the charges of paradox and mystery, said to be introduced into algebra by negative and impossible quantities, should be proposed distinctly, in a precise form, fit to be apprehended and made the subject of discussion." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"Mathematical science has been at times embarrassed with contradictions and paradoxes; yet they are not to be imputed to imaginary symbols, rather than to any other symbols invented for the purpose of rendering demonstration compendious, and expeditious. It may; however, be justly remarked, that mathematicians, neglecting to exercise mental superintendance, are too prone to trust to mechanical dexterity; and that some, instead of establishing the truth of conclusions on antecedent reasons, have endeavoured to prop it by imperfect analogies or mere algebraic forms. On the other hand, there are mathematicians, whose zeal for just reasoning has been alarmed at a verbal absurdity and, from a name improperly applied, or a definition incautiously given, l have been hurried to the precipitate conclusion, that operations with symbols of which the mind can form no idea, must necessarily be doubtful and unintelligible." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The application of imaginary quantities to the theory of equations, has perhaps been made more extensively than to any other part of analysis. To consider the propriety of this application on the grounds of perspicuity and conciseness, a long discussion would be necessary. I may, however, be here permited merely to state my opinion, that impossible quantities must be employed in the theory of equations, in order to obtain general rules and compendious methods." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The introduction of impossible quantities, is assigned as a great and primary cause of the evils under which mathematical science labours. During the operation of these quantities, it is said, all just reasoning is suspended, and the mind is bewildered by exhibitions that resemble the juggling tricks of mechanical dexterity." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The operations performed with imaginary characters, though destitute of meaning themselves, are yet notes of reference to others which are significant. They, point out indirectly a method of demonstrating a certain property of the hyperbola, and then leave us to conclude from analogy, that the same property belongs also to the circle. All that we are assured of by the imaginary investigation is, that its conclusion may, with all the strictness of mathematical reasoning, be proved of the hyperbola; but if from thence we would transfer that conclusion to the circle, it must be in consequence of the principle just now mentioned. The investigation therefore resolves itself ultimately into an argument from analogy; and, after the strictest examination, will be found without any other claim to the evidence of demonstration." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The symbol √-1 might arise from translating questions of which the statement involved a contradiction of ideas into algebraic language, and reasoning on them, as if they really admitted a solution." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"The use of a mathematical definition is, to deduce from it the properties of the thing defined […]" (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)
"Whether or not I have found a logic, by the role of which operations with imaginary quantities are conducted, is not now the question. but surely this is evident that since they lead to right conclusions they must have a logic! […] Till the doctrines of negative and imaginary quantities are better taught than they are at present taught in the University of Cambridge, I agree with you that they had better not be taught [...]" (Robert Woodhouse, [letter to Baron Meseres] 1801)
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