"Generality of points of view and of methods, precision and elegance in presentation, have become, since Lagrange, the common property of all who would lay claim to the rank of scientific mathematicians. And, even if this generality leads at times to abstruseness at the expense of intuition and applicability, so that general theorems are formulated which fail to apply to a single special case, if furthermore precision at times degenerates into a studied brevity which makes it more difficult to read an article than it was to write it; if, finally, elegance of form has well-nigh become in our day the criterion of the worth or worthlessness of a proposition, - yet are these conditions of the highest importance to a wholesome development, in that they keep the scientific material within the limits which are necessary both intrinsically and extrinsically if mathematics is not to spend itself in trivialities or smother in profusion." (Hermann Hankel, "Theorie der Complexen Zahlensysteme", 1867)
"Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blow-pipe flame of Gauss’s transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; or Legendre’s law of reciprocity; or Sturm’s theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen’s method of continued fractions, characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; [...]" (James J Sylvester,"A Plea for the Mathematician", 1869)
"Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems." (Florian Cajori, "A History of Mathematics", 1893)
"Strange as it may sound, the power of mathematics rests upon its evasion of all unnecessary thought and on its wonderful saving of mental operation. Even those arrangement-signs which we call numbers are a system of marvelous simplicity and economy. When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations; when we resolve new integral expressions into familiar old integrals; we see in this simply a feeble reflexion of the intellectual activity of a Lagrange or a Cauchy, who, with the keen discernment of a great military commander, substituted new operations for whole hosts of old ones. No one will dispute me when I say that the most elementary as well as the highest mathematics are economically-ordered experiences of counting, put in forms ready for use." (Ernst Mach, "Popular Scientific Lectures", 1895)
"In the dynamics of a rotating heavy body, the only substantial extension of our knowledge since the time of Lagrange has accrued from associating the general properties of functions with the discussion of the equations of motion. Further, under the title of conjugate functions, the theory has been applied to various questions in electrostatics, particularly in connection with condensers and electrometers. And, lastly, in the domain of physical astronomy, some of the most conspicuous advances made in the last few years have been achieved by introducing into the discussion the ideas, the principles, the methods, and the results of the theory of functions. […] the refined and extremely difficult work of Poincare and others in physical astronomy has been possible only by the use of the most elaborate developments of some purely mathematical subjects, developments which were made without a thought of such applications." (Andrew Forsyth, Presidential Address British Association for the Advancement of Science - Section A, Nature 56, 1897)
"To emphasize this opinion that mathematicians would be unwise to accept practical issues as the sole guide or the chief guide in the current of their investigations, [...] let me take one more instance, by choosing a subject in which the purely mathematical interest is deemed supreme, the theory of functions of a complex variable. That at least is a theory in pure mathematics, initiated in that region, and developed in that region; it is built up in scores of papers, and its plan certainly has not been, and is not now, dominated or guided by considerations of applicability to natural phenomena. Yet what has turned out to be its relation to practical issues? The investigations of Lagrange and others upon the construction of maps appear as a portion of the general property of conformal representation; which is merely the general geometrical method of regarding functional relations in that theory. Again, the interesting and important investigations upon discontinuous two-dimensional fluid motion in hydrodynamics, made in the last twenty years, can all be, and now are all, I believe, deduced from similar considerations by interpreting functional relations between complex variables." (Andrew Forsyth, Presidential Address British Association for the Advancement of Science - Section A, Nature 56, 1897)
"What is mass? According to Newton, it is the product of the volume by the density. According to Thomson and Tait, it would be better to say that density is the quotient of the mass by the volume. What is force? It, is replies Lagrange, that which moves or tends to move a body. It is, Kirchhoff will say, the product of the mass by the acceleration. But then, why not say the mass is the quotient of the force by the acceleration? These difficulties are inextricable. When we say force is the cause of motion, we talk metaphysics, and this definition, if one were content with it, would be absolutely sterile. For a definition to be of any use, it must teach us to measure force; moreover that suffices; it is not at all necessary that it teach us what force is in itself, nor whether it is the cause or the effect of motion." (Henri Poincaré, "Science and Hypothesis", 1901)
"Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar, - we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one." (Ernst Mach,"Populär-wissenschafliche Vorlesungen", 1908)
"Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, in conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination." (Ernest W Hobson, "Presidential Address British Association for the Advancement of Science", Nature, 1910)
"The unfortunate controversy over the discovery of the calculus between Newton and Leibniz and their protagonists occupies more space in the literature than its real importance justifies, but in this case (perhaps the most striking one of this type) the additional space redounds about equally to the advantage of both men. If it were eliminated entirely, Newton's position would still be preeminent, but Leibniz would probably be placed below Lagrange and Euler, rather than above them. As a matter of fact, however, all that we are justified in concluding, considering the probable errors of their positions, is that all three of these men are of practically equal rank [...]" (Walter C Eells, "One hundred eminent mathematicians", The Mathematics Teacher, 1962)
"The two most outstanding mathematicians of the eighteenth century were Euler and Lagrange, and which of the two is to be accorded first place is a matter of debate that often reflects the varying sensitivities of the debaters. Euler certainly published far more than Lagrange, and worked in many more diverse fields of mathematics than Lagrange, but he was largely a formalist or manipulator of formulas. Lagrange, on the other hand, may be considered the first true analyst and, though his collection of publications is a molehill compared with the Vesuvius of Euler's output, his work has a rare perfection, elegance, and exactness about it. Whereas Euler wrote with a profusion of detail and a free employment of intuition, Lagrange wrote concisely and with attempted rigor." (Howard W Eves, "In Mathematical Circles: Quadrants III and IV", 1969)
"In most courses on group theory, students usually study cosets and Lagrange's Theorem in one part of the course and group actions in another. Pedagogically, this makes sense, but it is also important to remember that historically, things are often more complicated. In considering resolvent polynomials, Lagrange had to deal with many issues all at once. It is a testament to his power as a mathematician that Lagrange could see what was important and thereby enable his successors to sort out the details of what he did." (David A Cox, "Galois Theory" 2nd Ed., 2012)
"On reading Lagrange's work, one is struck by his feeling for the general. [...] His extreme love of generality was unusual for this time and contrasts With the emphasis of many of his contemporaries on solving specific problems. His algebraic foundation for the calculus was consistent WIth his generalizing tendency." (Judith Grabiner, "The Origins of Cauchy's Rigorous Calculus"
No comments:
Post a Comment