On Truth (1650-1699)

“Men are apt to prefer a prosperous error before an afflicted truth.” (Jeremy Taylor, “The Rule and Exercises of Holy Living”, 1650)

 “All things being double-handed, and having the appearances both of truth and falsehood, where our affections have engaged us we attend only to the former.” (Joseph Glanvill, “Scepsis”, 1665)

“Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.” (Blaise Pascal, “Pensées”, 1670)

“We see neither justice nor injustice which does not change its nature with change in climate. Three degrees of latitude reverse all jurisprudence: a meridian decides the truth.” (Blaise Pascal, “Pensées”, 1670)

“In logic, they teach that contraries laid together more evidently appear: it follows, then, that all controversy being permitted, falsehood will appear more false, and truth the more true; which must needs conduce much to the general confirmation of an implicit truth.” (John Milton, “True Religion, Heresy, Schism, Toleration, and what best means may be used against the Growth of Popery”, 1673)

“In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow.” (Baruch Spinoza, [letter to Hugo Boxel], 1674)

“Truth is always consistent with itself, and needs nothing to help it out; it is always near at hand, and sits upon our lips, and is ready to drop out before we are aware; whereas a lie is troublesome, and sets a man’s invention upon the rack, and one trick needs a great many more to make it good.” (John Tillotson, “Sermons”, 1682)

"And thus many are ignorant of mathematical truths, not out of any imperfection of their faculties, or uncertainty in the things themselves, but for want of application in acquiring, examining, and by due ways comparing those ideas." (John Locke, "An Essay Concerning Human Understanding", 1689)

On Truth (1600-1649)

“The first and most ancient inquirers into truth were wont to throw their knowledge into aphorisms, or short, scattered, unmethodical sentences.” (Lord Bacon, “Novum Organum”, 1620)

"There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." (Francis Bacon, "Novum Organum", 1620)

“[…] thus each truth discovered was a rule available in the discovery of subsequent ones.” (René Descartes, “Discourse on Method”, 1637)

 “Every man is not a proper champion for truth, nor fit to take up the gauntlet in the cause of verity: many from the ignorance of these maxims, and an inconsiderate zeal for truth, have too rashly charged the troops of error, and remain as trophies unto the enemies of truth. A man may be in as just possession of truth as of a city, and yet be forced to surrender: ’tis therefore far better to enjoy her with peace than to hazard her on a battle: if therefore there rise any doubts in my way, I do forget them, or at least defer them, till my better settled judgment and more manly reason be able to resolve them.” (Sir Thomas Browne, ”Religio Medici”, 1643)

“In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.” (René Descartes, “Principles of Philosophy”, 1644)

“Knowledge is made by oblivion, and to purchase a clear and warrantable body of truth, we must forget and part with much we know.” (Sir Thomas Browne, “Pseudodoxia Epidemica”, 1646)

On Truth (until 1599)

“As being is to become, so is truth to belief” (Plato, “Timaeus”, cca. 360 BC)

“The first duty of man is the seeking after and the investigation of truth.” (Marcus Tullius Cicero, “De Officiis”, [“On Duties”], cca. 44 BC)

“The exact kind of language we employ in philosophical analyses of abstract truth is one thing, and the language used in attempts to popularize the subject is another.” (Marcus Tullius Cicero, “De officiis” [“On Duties”], cca.44 BC)

“Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.” (Marcus Aurelius, "Meditations", cca. 2nd century)

“We are meant to take them [the words ‘increase and multiply’] in a figurative sense. […] It is only in the case of signs outwardly given that we find increase and multiplication in the sense that a single truth can be expressed by several different means […] that a single expression can be interpreted in several different ways.” (St. Augustine, “Confessions”, 397- 400)

"Truth is sought for itself, but the truths are immersed in uncertainties." (Abu Ali al-Hasan ibn al-Haytham [Alhazen], "Aporias against Ptolemy", 1025-1028)

"In creation, on the other hand, truth is one thing, reason another. For in creation, truth is an image of the divinity, which is sought and found by reason in created things. Reason is a virtue or activity of the mind, whose object is to discern truth. Truth, like reason, does not have any contrary, and this for the same cause that was given and explained above in regard to reason." (John of Salisbury, "Metalogicon", 1159)

"[Intuitive] Understanding is consequent upon deliberation, and firmly embraces the better part. For [intuitive] understanding concerns itself with divine truths, and the relish, love, and observance of the latter constitutes true wisdom. Rather than being the [mere] product of nature, these successive steps are the result of grace. The latter, according to its own free determination, derives the various rivulets of the sciences and wisdom from the fountainhead of sense perception. Grace reveals hidden divine truths by means of those things which have been made, and by that unity which belongs to love, communicates what it has made manifest, thus uniting man to God." (John of Salisbury, "Metalogicon", 1159)

"Truth is both the light of the mind and the subject matter of reason." (John of Salisbury, "Metalogicon", 1159)

“If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.” (Roger Bacon, “Opus Majus” Book 1, 1267)

"Reasoning draws a conclusion and makes us grant the conclusion, but does not make the conclusion certain, nor does it remove doubt so that the mind may rest on the intuition of truth, unless the mind discovers it by the path of experience."(Roger Bacon, "Opus Majus", cca. 1267) 

“The truth of voice perishes with the sound; truth latent in the mind is hidden wisdom and invisible treasure; but the truth which illuminates books desires to manifest itself to every disciplinable sense. Let us consider how great a commodity of doctrine exists in books, - how easily, how secretly, how safely, they expose the nakedness of human ignorance without putting it to shame. These are the masters that instruct us without rods and ferules, without hard words and anger, without clothes or money. If you approach them, they are not asleep; if, investigating, you interrogate them, they conceal nothing; if you mistake them, they never grumble; if you are ignorant, they cannot laugh at you.” (Richard de Burry, “Philobiblon”, 1344)

"Man's mind is so formed that it is far more susceptible to falsehood than to truth." (Desiderius Erasmus, "Praise of Folly", 1509)

“[…] no pleasure is comparable to the standing upon the vantage ground of truth […]” (Sir Francis Bacon, “Essays”, 1597) 

On Proofs (1900 - 1924)

"[…] it is an error to believe that rigor in the proof is the enemy of simplicity." (David Hilbert, Paris International Congress, 1900)

“Besides it is an error to believe that rigour is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof.” (David Hilbert, “Mathematical Problems”, Bulletin of the American Mathematical Society, 1902)

"It is one of the chief merits of proofs that they instill a certain skepticism as to the result proved." (Bertrand Russell, "The Principles of Mathematics", 1903)

"It is by logic that we prove, but by intuition that we discover. [...] Every definition implies an axiom, since it asserts the existence of the object defined. The definition then will not be justified, from the purely logical point of view, until we have proved that it involves no contradiction either in its terms or with the truths previously admitted." (Henri Poincaré, "Science and Method", 1908)

"Banishing fundamental facts or problems from science merely because they cannot be dealt with by means of certain prescribed principles would be like forbidding the further extension of the theory of parallels in geometry because the axiom upon which this theory rests has been shown to be unprovable. Actually, principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)

"Now even in mathematics unprovability, as is well known, is in no way equivalent to nonvalidity, since, after all, not everything can be proved, but every proof in turn presupposes unproved principles. Thus, in order to reject such a fundamental principle, one would have to ascertain that in some particular case it did not hold or to derive contradictory consequences from it; but none of my opponents has made any attempt to do this." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)

"To reach our goal [of proving consistency], we must make the proofs as such the object of our investigation; we are thus compelled to a sort of proof theory which studies operations with the proofs themselves." (David Hilbert, 1922)

"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." (Charles P Steinmetz, 1923)

Beyond the History of Mathematics III

“The history of mathematics is important […] as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development.” (Florian Cajori, “A History of Mathematics”, 1893)

 “This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time.” (Florian Cajori, “A History of Mathematical Notations”, 1928)

“There are no absolutes [...] in mathematics or in its history.” (Eric T Bell, The Development of Mathematics, 1940)

“Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution.” (Richard Curand [forward to Carl B Boyer’s “The History of the Calculus and Its Conceptual Development”, 1949])

"All followers of the axiomatic method and most mathematicians think that there is some such thing as an absolute ‘mathematical rigor’ which has to be satisfied by any deduction if it is to be valid. The history of mathematics shows that this is not the case, that, on the contrary, every generation is surpassed in rigor again and again by its successors.” (Richard von Mises, “Positivism: A Study in Human Understanding”, 1951)

“It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands.” (Philip J Davis, “The Mathematics of Matrices”, 1965)

“Students enjoy […] and gain in their understanding of today's mathematics through analyzing older and alternative approaches.” (Lucas N H Bunt, Phillip S Jones & Jack D Bedient, “The Historical Roots of Elementary Mathematics”, 1976)

“[…] how completely inadequate it is to limit the history of mathematics to the history of what has been formalized and made rigorous. The unrigorous and the contradictory play important parts in this history.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

"We who are heirs to three recent centuries of scientific development can hardly imagine a state of mind in which many mathematical objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, a fraction of the history of mathematics is incomprehensible.” (Philip J Davis & Rueben Hersh, “The Mathematical Experience”, 1985)

“Like anything else, mathematics is created within the context of history […]” (William Dunham, “Journey Through Genius”, 1990)

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From Facts to Theory

“[…] ideas may be both novel and important, and yet, if they are incorrect – if they lack the very essential support of incontrovertible fact, they are unworthy of credence. Without this, a theory may be both beautiful and grand, but must be as evanescent as it is beautiful, and as unsubstantial as it is grand.” (George Brewster, “A New Philosophy of Matter”, 1858)

“Perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind. But it, would be a mistake to suppose that this candour has anything akin to fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining an hypothesis as long as its falsity is not actually apparent.” (William S Jevons, “The Principles of Science”, 1887)

”Scientific facts accumulate rapidly, and give rise to theories with almost equal rapidity. These theories are often wonderfully enticing, and one is apt to pass from one to another, from theory to theory, without taking care to establish each before passing on to the next, without assuring oneself that the foundation on which one is building is secure. Then comes the crash; the last theory
breaks down utterly, and on attempting to retrace our steps to firm ground and start anew, we may find too late that one of the cards, possibly at the very foundation of the pagoda, is either faultily placed or in itself defective, and that this blemish easily remedied if detected in time has, neglected, caused the collapse of the whole structure on whose erection so much skill and perseverance have been spent.” (Arthur M Marshall, 1894)

”Facts are carpet-tacks under the pneumatic tires of theory.” (Austin O’Malley, “Keystones of Thought”, 1918)

“Nothing is more interesting to the true theorist than a fact which directly contradicts a theory generally accepted up to that time, for this is his particular work.” (Max Planck, “A Survey of Physics”, 1925)

“[…] the mere collection of facts, without some basis of theory for guidance and elucidation, is foolish and profitless.” (Gamaliel Bradford, “Darwin”, 1926)

“[…] facts are too bulky to be lugged about conveniently except on the wheels of theory.” (Julian Huxley, “Essays of a Biologist”, 1929)

”[while] the traditional way is to regard the facts of science as something like the parts of a jig-saw puzzle, which can be fitted together in one and only one way, I regard them rather as the tiny pieces of a mosaic, which can be fitted together in many ways. A new theory in an old subject is, for me, a new mosaic pattern made with the pieces taken from an older pattern.” (William H George, “The Scientist in Action”, 1936)

”We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts.” (James B Conant, “On Understanding Science”, 1947)

”Without facts we have no science. Facts are to the scientist what words are to the poet. The scientist has a love of facts, even isolated facts, similar to a poet’s love of words. But a collection of facts is not a science any more than a dictionary is poetry. Around his facts the scientist weaves a logical pattern or theory which gives the facts meaning, order and significance.” (Isidor Isaac Rabi, “Faith in Science”, Atlantic Monthly , Vol. 187, 1951)

“The true aim of science is to discover a simple theory which is necessary and sufficient to cover the facts, when they have been purified of traditional prejudices.” (Lancelot L Whyte, “Accent on Form”, 1954)

“Science does not begin with facts; one of its tasks is to uncover the facts by removing misconceptions.” (Lancelot L Whyte, “Accent on Form”, 1954) 

“When we meet a fact which contradicts a prevailing theory, we must accept the fact and abandon the theory, even when the theory is supported by great names and generally accepted.” (Claude Bernard, “An Introduction to the Study of Experimental Medicine”, 1957)

”Facts do not ‘speak for themselves’; they are read in the light of theory. Creative thought, in science as much as in the arts, is the motor of changing opinion. Science is a quintessentially human activity, not a mechanized, robot-like accumulation of objective information, leading by laws of logic to inescapable interpretation.” (Stephen J Gould, “Ever Since Darwin”, 1977)

”No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress.” (Paul K Feyerabend, “Against Method”, 1978)

[Maier’s Law:] ”If the facts do not conform to the theory, they must be disposed of.” (Norman R F Maier)

“First accumulate a mass of Facts: and then construct a Theory.” (Lewis Carroll)