"I believe that we need another analysis properly geometric or linear, which treats PLACE directly the way that algebra treats MAGNITUDE." (Gottfried W Leibniz, 1670s)
“Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality.” (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)
"Of late the speculations about Infinities have run so high, and grown to such strange notions, as have occasioned no small scruples and disputes among the geometers of the present age. Some there are of great note who, not contented with holding that finite lines may be divided into an infinite number of parts, do yet further maintain that each of these infinitesimals is itself subdivisible into an infinity of other parts or infinitesimals of a second order, and so on ad infinitum. These I say assert there are infinitesimals of infinitesimals, etc., without ever coming to an end; so that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts." (George Berkeley, "A Treatise Concerning the Principles of Human Knowledge ", 1710)
“There are two famous labyrinths where our reason very often goes astray. One concerns the great question of the free and the necessary, above all in the production and the origin of Evil. The other consists in the discussion of continuity, and of the indivisibles which appear to be the elements thereof, and where the consideration of the infinite must enter in.” (Gottfried W Leibniz, "Theodicy: Essays on the Goodness of God and Freedom of Man and the Origin of Evil", 1710)
"For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear." (Leonhard Euler, "De Curvis Elasticis", 1744)
"[...] these increments must be conceived to become continuously smaller, and in this way, their ratio is represented as continuously approaching a certain limit, which is finally attained when the increment becomes absolutely nothing. This limit, which is, as it were, the final ratio of those increments, is the true object of differential calculus." (Leonhard Euler, Institutiones Calculi Differentialis”, 1755)
"The algebraic analysis soon makes us forget the main object [of our researches] by focusing our attention on abstract combinations and it is only at the end that we return to the original objective. But in abandoning oneself to the operations of analysis, one is led to the generality of this method and the inestimable advantage of transforming the reasoning by mechanical procedures to results often inaccessible by geometry. Such is the fecundity of the analysis that it suffices to translate into this universal language particular truths In order to see emerge from their very expression a multitude of new and unexpected truths. No other language has the capacity for the elegance that arises from a long sequence of expressions linked one to the other and all stemming from one fundamental idea. Therefore the geometers [mathematicians] of this century convinced of its superiority have applied themselves primarily to extending Its and pushing back its bounds." (Pierre-Simon Laplace, "Exposition du system du monde" ["Explanation on the solar system"], 1796)
"The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which of times they are unable to account." (Pierre-Simon Laplace, "Analytical Theory of Probability, 1812)
“The differential calculus has all the exactitude of other algebraic operations.” (Pierre-Simon Laplace, "Analytical Theory of Probability, 1812)
“Mathematical analysis […] in the study of all phenomena, interprets them by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes. […] There cannot be a language more universal and more simple, more free from errors and obscurities, … more worthy to express the invariable relations of all natural things.” (Baron Jean-Baptiste-Joseph Fourier, “Théorie Analytique de la Chaleur”, 1822)
“Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.” (Baron Jean-Baptiste-Joseph Fourier, “Théorie Analytique de la Chaleur”, 1822)
“The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.” (Auguste Comte, “Course of Positive Philosophy”, 1830)
“A limit is a peculiar and fundamental conception, the use of which in proving the propositions of Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. The axiom just noted that what is true up to the limit is true at the limit, is involved in the very conception of a limit: and this principle, with its consequences, leads to all the results which form the subject of the higher mathematics, whether proved by the consideration of evanescent triangles, by the processes of the Differential Calculus, or in any other way.” (William Whewell, “The Philosophy of the Inductive Sciences”, 1840)
"[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values." (Auguste Comte, "Philosophy of Mathematics", 1851)
"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)
“A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion.” (Count Leo Tolstoy, “War and Peace”, 1869)
“Absolute continuity of motion is not comprehensible to the human mind. Laws of motion of any kind become comprehensible to man only when he examines arbitrarily selected elements of that motion; but at the same time, a large proportion of human error comes from the arbitrary division of continuous motion into discontinuous elements.” (Count Leo Tolstoy, “War and Peace”, 1869)
"The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits." (George H. Lewes, "Problems of Life and Mind", 1873)
"For an understanding of Nature, questions about the infinitely large are idle questions. It is different, however, with questions about the infinitely small. Our knowledge of their causal relations depends essentially on the precision with which we succeed in tracing phenomena on the infinitesimal level." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)
"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire)
“As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).” (Bernhard Riemann, “Die partiellen Differentialgleichungen der mathematischen Physik”, 1882)
“If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, American Journal of Physics 12, 1890)
“Now, in the development of our knowledge of the workings of Nature out of the tremendously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in some respects a very limited, in others a very important part. As regards the limitations, it is merely necessary to refer to the sciences connected with living matter, and to the ologies generally, to see that the facts and their connections are too indistinctly known to render mathematical analysis practicable, to say nothing of the complexity.” (Oliver Heaviside, “Electro-magnetic Theory II”, The Electrician, 1891)
"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)
“Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression.” (Antoine-Augustin Cournot, “Mathematical Theory of the Principles of Wealth", 1897)
“The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.” (Alfred N Whitehead, “Universal Algebra”, 1898)
“Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.” (Bertrand Russell, “Recent Work on the Principles of Mathematics” The International Monthly 4 (1), 1901)
“If we must confine ourselves to one system of notation then there can be no doubt that that which was invented by Leibnitz is better fitted for most of the purposes to which the infinitesimal calculus is applied than that of fluxions, and for some (such as the calculus of variations) it is indeed almost essential.” (Rouse W W Ball, “A Short Account of the History of Mathematics”, 1901)
“The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.” (Herbert G Wells, “Mankind in the Making”, 1903)
“If we turn to the problems to which the calculus owes its origin, we find that not merely, not even primarily, geometry, but every other branch of mathematical physics - astronomy, mechanics, hydrodynamics, elasticity, gravitation, and later electricity and magnetism - in its fundamental concepts and basal laws contributed to its development and that the new science became the direct product of these influences. [...] The calculus is the greatest aid we have to the appreciation of physical truth in the broadest sense of the word.” (William Osgood, “The Calculus in Colleges in Colleges and Technical Schools”, Bulletin of the American Mathematical Society, 1907)
“The invention of the differential calculus marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level.” (Alfred N Whitehead, “An Introduction to Mathematics”, 1911)
"In Continuity, it is impossible to distinguish phenomena at their merging-points, so we look for them at their extremes." (Charles Fort, "The Book of the Damned", 1919)
“The notion of continuity depends upon that of order, since continuity is merely a particular type of order.” (Bertrand Russell, “Mysticism and Logic and Other Essays”, 1925)
“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.” (John von Neumann, “Works of the Mind”, 1947)
"Analysis […] would lose immensely in beauty and balance and would be forced to add very hampering restrictions to truths which would hold generally otherwise, if […] imaginary quantities were to be neglected." (Garrett Birkhoff, 1973)
“Natura non facit saltum or, Nature does not make leaps […] If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).” (Benoit B Mandelbrot and Richard Hudson, “The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward”, 2004)
" […] nothing takes place in the world whose meaning is not that of some maximum or minimum." (Leonhard Euler)
"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented." (Leonhard Euler)
"The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which of times they are unable to account." (Pierre-Simon Laplace, "Analytical Theory of Probability, 1812)
“The differential calculus has all the exactitude of other algebraic operations.” (Pierre-Simon Laplace, "Analytical Theory of Probability, 1812)
“Mathematical analysis […] in the study of all phenomena, interprets them by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes. […] There cannot be a language more universal and more simple, more free from errors and obscurities, … more worthy to express the invariable relations of all natural things.” (Baron Jean-Baptiste-Joseph Fourier, “Théorie Analytique de la Chaleur”, 1822)
“Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind.” (Baron Jean-Baptiste-Joseph Fourier, “Théorie Analytique de la Chaleur”, 1822)
“The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.” (Auguste Comte, “Course of Positive Philosophy”, 1830)
“A limit is a peculiar and fundamental conception, the use of which in proving the propositions of Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. The axiom just noted that what is true up to the limit is true at the limit, is involved in the very conception of a limit: and this principle, with its consequences, leads to all the results which form the subject of the higher mathematics, whether proved by the consideration of evanescent triangles, by the processes of the Differential Calculus, or in any other way.” (William Whewell, “The Philosophy of the Inductive Sciences”, 1840)
"[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values." (Auguste Comte, "Philosophy of Mathematics", 1851)
"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)
“A modern branch of mathematics, having achieved the art of dealing with the infinitely small, can now yield solutions in other more complex problems of motion, which used to appear insoluble. This modern branch of mathematics, unknown to the ancients, when dealing with problems of motion, admits the conception of the infinitely small, and so conforms to the chief condition of motion (absolute continuity) and thereby corrects the inevitable error which the human mind cannot avoid when dealing with separate elements of motion instead of examining continuous motion.” (Count Leo Tolstoy, “War and Peace”, 1869)
“Absolute continuity of motion is not comprehensible to the human mind. Laws of motion of any kind become comprehensible to man only when he examines arbitrarily selected elements of that motion; but at the same time, a large proportion of human error comes from the arbitrary division of continuous motion into discontinuous elements.” (Count Leo Tolstoy, “War and Peace”, 1869)
"The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits." (George H. Lewes, "Problems of Life and Mind", 1873)
"For an understanding of Nature, questions about the infinitely large are idle questions. It is different, however, with questions about the infinitely small. Our knowledge of their causal relations depends essentially on the precision with which we succeed in tracing phenomena on the infinitesimal level." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)
"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire)
“As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation (can be compared with experience).” (Bernhard Riemann, “Die partiellen Differentialgleichungen der mathematischen Physik”, 1882)
“If one looks at the different problems of the integral calculus which arise naturally when one wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics or electrodynamics, the propagation of heat, optics, elasticity, or hydrodynamics, we are led always to differential equations of the same family." (Henri Poincaré, American Journal of Physics 12, 1890)
“Now, in the development of our knowledge of the workings of Nature out of the tremendously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in some respects a very limited, in others a very important part. As regards the limitations, it is merely necessary to refer to the sciences connected with living matter, and to the ologies generally, to see that the facts and their connections are too indistinctly known to render mathematical analysis practicable, to say nothing of the complexity.” (Oliver Heaviside, “Electro-magnetic Theory II”, The Electrician, 1891)
"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)
“Those skilled in mathematical analysis know that its object is not simply to calculate numbers, but that it is also employed to find the relations between magnitudes which cannot be expressed in numbers and between functions whose law is not capable of algebraic expression.” (Antoine-Augustin Cournot, “Mathematical Theory of the Principles of Wealth", 1897)
“The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.” (Alfred N Whitehead, “Universal Algebra”, 1898)
“Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be.” (Bertrand Russell, “Recent Work on the Principles of Mathematics” The International Monthly 4 (1), 1901)
“If we must confine ourselves to one system of notation then there can be no doubt that that which was invented by Leibnitz is better fitted for most of the purposes to which the infinitesimal calculus is applied than that of fluxions, and for some (such as the calculus of variations) it is indeed almost essential.” (Rouse W W Ball, “A Short Account of the History of Mathematics”, 1901)
“The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write.” (Herbert G Wells, “Mankind in the Making”, 1903)
“If we turn to the problems to which the calculus owes its origin, we find that not merely, not even primarily, geometry, but every other branch of mathematical physics - astronomy, mechanics, hydrodynamics, elasticity, gravitation, and later electricity and magnetism - in its fundamental concepts and basal laws contributed to its development and that the new science became the direct product of these influences. [...] The calculus is the greatest aid we have to the appreciation of physical truth in the broadest sense of the word.” (William Osgood, “The Calculus in Colleges in Colleges and Technical Schools”, Bulletin of the American Mathematical Society, 1907)
“The invention of the differential calculus marks a crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level.” (Alfred N Whitehead, “An Introduction to Mathematics”, 1911)
"In Continuity, it is impossible to distinguish phenomena at their merging-points, so we look for them at their extremes." (Charles Fort, "The Book of the Damned", 1919)
“The notion of continuity depends upon that of order, since continuity is merely a particular type of order.” (Bertrand Russell, “Mysticism and Logic and Other Essays”, 1925)
“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.” (John von Neumann, “Works of the Mind”, 1947)
"Analysis […] would lose immensely in beauty and balance and would be forced to add very hampering restrictions to truths which would hold generally otherwise, if […] imaginary quantities were to be neglected." (Garrett Birkhoff, 1973)
“Natura non facit saltum or, Nature does not make leaps […] If you assume continuity, you can open the well-stocked mathematical toolkit of continuous functions and differential equations, the saws and hammers of engineering and physics for the past two centuries (and the foreseeable future).” (Benoit B Mandelbrot and Richard Hudson, “The (Mis)Behaviour of Markets: A Fractal View of Risk, Ruin and Reward”, 2004)
" […] nothing takes place in the world whose meaning is not that of some maximum or minimum." (Leonhard Euler)
"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented." (Leonhard Euler)
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