On Mathematical Reasoning

"The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning; and when they cannot it’s a sign our knowledge of them is very small and confus’d; and when a Mathematical Reasoning can be had it’s as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you." (John Arbuthnot, "Of the Laws of Chance", 1692)

"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)

"There is nothing physical to be learned a priori. We have no right whatever to ascertain a single physical truth without seeking for it physically, unless it be a necessary consequence of other truths already acquired by experiment, in which case mathematical reasoning is alone requisite." (Peter G Tait, "Lectures on Some Recent Advances in Physical Science, With a Special, Lecture on Force", 1876)

"[…] in the Law of Errors we are concerned only with the objective quantities about which mathematical reasoning is ordinarily exercised; whereas in the Method of Least Squares, as in the moral sciences, we are concerned with a psychical quantity - the greatest possible quantity of advantage." (Francis Y Edgeworth, "The method of least squares", 1883)

"The type of reasoning found in mathematics seems thus not only available but essentially interwoven with every inference in non-mathematical reasoning, being always used in one of its two steps ; facility in making the other step, the more difficult one, must be attained through other than purely mathematical training." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"What is the nature of mathematical reasoning? Is is really deductive, as is commonly supposed? A deeper analysis shows us that it is not, that it partakes in a certain measure of the nature of inductive reasoning, and just because of this is it so fruitful. None the less does it retain its character of rigor absolute; this is the first thing that had to be shown." (Henri Poincaré, "Science and Hypothesis" [in "The Foundations of Science", 1913])

“Philosophy in its old form could exist only in the absence of engineering, but with engineering in existence and daily more active and far reaching, the old verbalistic philosophy and metaphysics have lost their reason to exist. They were no more able to understand the ‘production’ of the universe and life than they are now able to understand or grapple with 'production' as a means to provide a happier existence for humanity. They failed because their venerated method of ‘speculation’ can not produce, and its place must be taken by mathematical thinking. Mathematical reasoning is displacing metaphysical reasoning. Engineering is driving verbalistic philosophy out of existence and humanity gains decidedly thereby.” (Alfred Korzybski,  “Manhood of Humanity”, 1921)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning. [...] The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45 (2), 1939)

"[...] nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop." (Chen-Ning Yang, "The Law of Parity Conservation and Other Symmetry Laws of Physics", [Nobel lecture] 1957)

"Mathematical Reasoning is not only exact; it has its own criteria of reality." (Paul K Feyerabend, “Science in a Free Society”, 1978)

"For most problems found in mathematics textbooks, mathematical reasoning is quite useful. But how often do people find textbook problems in real life? At work or in daily life, factors other than strict reasoning are often more important. Sometimes intuition and instinct provide better guides; sometimes computer simulations are more convenient or more reliable; sometimes rules of thumb or back-of-the-envelope estimates are all that is needed." (Lynn A Steen,"Twenty Questions about Mathematical Reasoning", 1999)

"In chaos theory this 'butterfly effect' highlights the extreme sensitivity of nonlinear systems at their bifurcation points. There the slightest perturbation can push them into chaos, or into some quite different form of ordered behavior. Because we can never have total information or work to an infinite number of decimal places, there will always be a tiny level of uncertainty that can magnify to the point where it begins to dominate the system. It is for this reason that chaos theory reminds us that uncertainty can always subvert our attempts to encompass the cosmos with our schemes and mathematical reasoning." (F David Peat, "From Certainty to Uncertainty", 2002)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity [...] (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The first things I found out were that all mathematical reasoning is diagrammatic and that all necessary reasoning is mathematical reasoning, no matter how simple it may be. By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, assures itself that similar experiments performed upon any diagram constructed according to the same precept would have the same results, and expresses this in general terms. This was a discovery of no little importance, showing, as it does, that all knowledge without exception comes from observation." (Charles S Peirce)