"We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation!" (Henri Poincaré, "Science and Hypothesis", 1901)
"The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results." (Josiah W Gibbs, "Elementary Principles in Statistical Mechanics", 1902)
"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)
"An exceedingly small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation 'approximately'. If that enabled us to predict the succeeding situation with 'the same approximation', that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. (Jules H Poincaré, "Science and Method", 1908)
"The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to use a wide range of mathematical theory, the chemist is only just beginning to apply the first derivative, i.e. the rate of change at which certain processes are going on; for second derivatives he does not seem to have found any use as yet." (Felix Klein, "Lectures on Mathematics", 1911)
"Much of the skill of the true mathematical
physicist and of the mathematical astronomer consists in the power of adapting
methods and results carried out on an exact mathematical basis to obtain
approximations sufficient for the purposes of physical measurements. It might
perhaps be thought that a scheme of Mathematics on a frankly approximative
basis would be sufficient for all the practical purposes of application in
Physics, Engineering Science, and Astronomy, and no doubt it would be possible
to develop, to some extent at least, a species of Mathematics on these lines.
Such a system would, however, involve an intolerable awkwardness and prolixity
in the statements of results, especially in view of the fact that the degree of
approximation necessary for various purposes is very different, and thus that
unassigned grades of approximation would have to be provided for. Moreover, the
mathematician working on these lines would be cut off from the chief sources of
inspiration, the ideals of exactitude and logical rigour, as well as from one
of his most indispensable guides to discovery, symmetry, and permanence of
mathematical form. The history of the actual movements of mathematical thought
through the centuries shows that these ideals are the very life-blood of the
science, and warrants the conclusion that a constant striving toward their
attainment is an absolutely essential condition of vigorous growth. These
ideals have their roots in irresistible impulses and deep-seated needs of the
human mind, manifested in its efforts to introduce intelligibility in certain
great domains of the world of thought." (Ernest W Hobson, [address] 1910)
"The ordinary mathematical treatment of any applied
science substitutes exact axioms for the approximate results of experience, and
deduces from these axioms the rigid mathematical conclusions. In applying this
method it must not be forgotten that the mathematical developments transcending
the limits of exactness of the science are of no practical value. It follows
that a large portion of abstract mathematics remains without finding any
practical application, the amount of mathematics that can be usefully employed
in any science being in proportion to the degree of accuracy attained in the
science. Thus, while the astronomer can put to use a wide range of mathematical
theory, the chemist is only just beginning to apply the first derivative, i. e.
the rate of change at which certain processes are going on; for second
derivatives he does not seem to have found any use as yet." (Felix Klein,
"Lectures on Mathematics", 1911)
"The objects of abstract Geometry possess in
absolute precision properties which are only approximately realized in the
corresponding objects of physical Geometry." (Ernest W Hobson,
"Squaring the Circle", 1913)
"It is well to notice in this connection [the mutual relations between the results of counting and measuring] that a natural law, in the statement of which measurable magnitudes occur, can only be understood to hold in nature with a certain degree of approximation; indeed natural laws as a rule are not proof against sufficient refinement of the measuring tools." (Luitzen E J Brouwer, "Intuitionism and Formalism", Bulletin of the American Mathematical Society, Vol. 20, 1913)
"[…] as the sciences have developed further, the notion has gained ground that most, perhaps all, of our laws are only approximations." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1914)
"Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line [...]" (Vladimir I Lenin, "On the Question of Dialectics", 1915)
"It would be a mistake to suppose that a science
consists entirely of strictly proved theses, and it would be unjust to require
this. […] Science has only a few apodeictic propositions in its catechism: the
rest are assertions promoted by it to some particular degree of probability. It
is actually a sign of a scientific mode of thought to find satisfaction in
these approximations to certainty and to be able to pursue constructive work
further in spite of the absence of final confirmation." (Sigmund Freud,
"Introductory Lectures on Psycho-Analysis", 1916)
"It is the nature of a real thing to be
inexhaustible in content; we can get an ever deeper insight into this content
by the continual addition of new experiences, partly in apparent contradiction,
by bringing them into harmony with one another. In this interpretation, things
of the real world are approximate ideas. From this arises the empirical
character of all our knowledge of reality." (Hermann Weyl,
"Space-Time-Matter", 1918)
"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)
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