“I have in this treatise followed the mathematical method, if not with all strictness, at least imitatively, not in order, by a display of profundity, to procure a better reception for it, but because I believe such a system to be quite capable of it, and that perfection may in time be obtained by a cleverer hand, if stimulated by this sketch, mathematical investigators of nature should find it not unimportant to treat the metaphysical portion, which anyway cannot be got rid of, as a special fundamental department of general physics, and to bring it into unison with the mathematical doctrine of motion.”

— Immanuel Kant, book Metaphysical Foundations of Natural Science

Preface, Tr. Bax (1883)
Metaphysical Foundations of Natural Science (1786)

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German philosopher 1724–1804

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“I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results.”

Hermann Grassmann (1809–1877) German polymath, linguist and mathematician

Ausdehnungslehre (1844)

“I maintain that in every special natural doctrine only so much science proper is to be met with as mathematics; for… science proper, especially of nature, requires a pure portion, lying at the foundation of the empirical, and based upon à priori knowledge of natural things. …the conception should be constructed. But the cognition of the reason through construction of conceptions is mathematical. A pure philosophy of nature in general, namely, one that only investigates what constitutes a nature in general, may thus be possible without mathematics; but a pure doctrine of nature respecting determinate natural things (corporeal doctrine and mental doctrine), is only possible by means of mathematics; and as in every natural doctrine only so much science proper is to be met with therein as there is cognition à priori, a doctrine of nature can only contain so much science proper as there is in it of applied mathematics.”

Immanuel Kant book Metaphysical Foundations of Natural Science

Preface, Tr. https://books.google.com/books?id=OCJLAAAAMAAJ Ernest Belfort Bax (1883) <br class="br">Metaphysical Foundations of Natural Science (1786)

“If you are receptive and humble, mathematics will lead you by the hand.”

Paul Dirac (1902–1984) theoretical physicist

As quoted in The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom (2009) by Graham Farmelo, p. 435
Context: If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics led me by the hand. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress.

“The following work is not a republication of a former treatise by the Author, entitled, "The Mathematical Analysis of Logic." Its earlier portion is indeed devoted to the same object, and it begins by establishing the same system of fundamental laws, but its methods are more general, and its range of applications far wider. It exhibits the results, matured by some years of study and reflection, of a principle of investigation relating to the intellectual operations, the previous exposition of which was written within a few weeks after its idea had been conceived.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. i; Preface, lead paragraph

“I am fully assured, that no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognize, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form.”

George Boole (1815–1864) English mathematician, philosopher and logician

George Boole, " Solution of a Question in the Theory of Probabilities http://books.google.nl/books?id=9xtDAQAAIAAJ&pg=PA32" (30 November 1853) published in The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science‬‎ (January 1854), p. 32 <br class="br">1850s

“Simulation is the use of models and/or the actual conditions of either the thing being modeled or the environment in which it operates, with the models or conditions in physical, mathematical, or some other form. The purpose of simulation is to explore the various results which might be obtained from the real system by subjecting the model to representative environments which are equivalent to, or in some way representative of, the situations it is desired to understand or investigate. Simulation may involve system hardware and the actual physical environment, or it may involve mathematical models subjected to mathematical forcing or disturbance functions representative of the systems conditions to be studied.”

Harold Chestnut (1917–2001) American engineer

Source: Systems Engineering Tools, (1965), p. 111 as cited in

“There is no problem in all mathematics that cannot be solved by direct counting. But with the present implements of mathematics many operations can be performed in a few minutes which without mathematical methods would take a lifetime.”

Ernst Mach (1838–1916) Austrian physicist and university educator

Source: 19th century, Popular Scientific Lectures [McCormack] (Chicago, 1898), p. 197; On mathematics and counting.

“The viewpoint of the formalist must lead to the conviction that if other symbolic formulas should be substituted for the ones that now represent the fundamental mathematical relations and the mathematical-logical laws, the absence of the sensation of delight, called "consciousness of legitimacy," which might be the result of such substitution would not in the least invalidate its mathematical exactness. To the philosopher or to the anthropologist, but not to the mathematician, belongs the task of investigating why certain systems of symbolic logic rather than others may be effectively projected upon nature. Not to the mathematician, but to the psychologist, belongs the task of explaining why we believe in certain systems of symbolic logic and not in others, in particular why we are averse to the so-called contradictory systems in which the negative as well as the positive of certain propositions are valid.”

L. E. J. Brouwer (1881–1966) Dutch mathematician and logician

as translated by Arnold Dresden from: Brouwer, L. E. J. (1913). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20(2), 81–96. (quote on p. 84)

“If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.”

George Dantzig (1914–2005) American mathematician

Source: Linear programming and extensions (1963), p. 2

“The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.”

George Boole (1815–1864) English mathematician, philosopher and logician

Source: 1850s, An Investigation of the Laws of Thought (1854), p. 1; Ch. 1. Nature And Design Of This Work, lead paragraph

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