J. L. Austin and language

J. L. Austin (1911-1960) was an British philosopher philosopher best known for his analysis of ordinary language and speech acts. Guy Longworth said,

"[J. L. Austin] made a number of contributions in various areas of philosophy, including important work on knowledge, perception, action, freedom, truth, language and the use of language in speech acts... His work on truth has played an important role in recent discussions of the extent to which sentence meaning can be accounted for in terms of truth-conditions." (The Stanford Encyclopedia of Philosophy, 2013)

The rest of this post is some quotes from Austin.

Language and reality

"After all we speak of people 'taking refuge' in vagueness - the more precise you are, in general the more likely you are to be wrong, whereas you stand a good chance of not being wrong if you make it vague enough." (Sense and Sensibilia, 1962 posthumous)

"Words are not (except in their own little corner) facts or things: we need therefore to prise them off the world, to hold them apart from and against it, so that we can realize their inadequacies and arbitrariness, and can relook at the world without blinkers." (Philosophical Papers, 1979 posthumous)

Word meaning

"Faced with the nonsense question, 'What is the meaning of a word?' and perhaps dimly recognizing it to be nonsense, we are nevertheless not inclined to give it up." (Philosophical Papers, 1979 posthumous)

"Philosophers often seem to think that they can just 'assign' any meaning whatever to any word; and so no doubt, in an absolutely trivial sense, they can (like Humpty-Dumpty)." (Sense and Sensibilia, 1962 posthumous)

Word creation

"Our common stock of words embodies all the distinctions men have found worth drawing, the connexions they have found worth marketing, in the lifetimes of many generations; these surely are likely to be more numerous, more sound, since they have stood up to the long test of the survival of the fittest, and more subtle, at least in all ordinary and reasonably practical matter, than any that you or I are likely to think up in our arm-chairs of an afternoon..." (AZquotes.com)

"Going back into the history of a word, very often into Latin, we come back pretty commonly to pictures or models of how things happen or are done... but one of the commonest and most primitive types of model is one which is apt to baffle us through its very naturalness and simplicity." (A Plea for Excuses, 1956)

A. J. Ayer and verification

A. J. Ayer (1910-1989) was a British philosopher best known for his contributions to logical positivism and the verification principle. Ben Rogers said,

"In seeking to refound philosophy as an analytic discipline, Ayer was not just trying to separate philosophy from life but to liberate life from philosophy... To Ayer all this was not only unjustified - talk of supernatural reality, of beings existing outside space and time, or of the fundamental unity of things was literally senseless - but also reactionary." (A. J. Ayer: A Life, 1999)

Philosopher Julian Baggini said,

"Ayer's central mistake was to think that his principle could distinguish between the meaningful and meaningless. This error has two parts. First, a better distinction would be between the objective and subjective... The second part is to make the distinction binary where it should be spectral." (The Edge of Reason: A Rational Skeptic in an Irrational World, 2016)

The rest of this post is some quotes from Ayer.

What is philosophy?

"I see philosophy as a fairly abstract activity, as concerned mainly with the analysis of criticism and concepts, and of course most usefully of scientific concepts." (Quoted in A.J Ayers: A Life by Ben Rogers)

"In other words, the propositions of philosophy are not factual, but linguistic in character... Accordingly, we may say that philosophy is a department of logic. For we will see that the characteristic mark of a purely logical enquiry, is that it is concerned with the formal consequences of our definitions and not with questions of empirical fact." (AZquotes.com)

Verification

"The criterion which we use to test the genuineness of apparent statements of fact is the criterion of verifiability. We say that a sentence is factually significant to any given person, if, and only if, he knows how to verify the proposition which it purports to express - that is, if he knows what observations, would lead him, under certain conditions, to accept the proposition as being true, or reject it as being false." (Language, Truth and Logic, 1936)

"There never comes a point where a theory can be said to be true. The most that one can claim for any theory is that it has shared the successes of all its rivals and that it has passed at least one test which they have failed." (Philosophy in the Twentieth Century, 1982)

Transcendental reality

"Theism is so confused and the sentences in which God appears so incoherent and so incapable of verifiability or falsifiability that to speak of belief or unbelief, faith or unfaith, is logically impossible." (AZquotes.com)

"I do not believe in God. It seems to me that theists of all kinds have largely failed to make their concept of a deity intelligible; and to the extent that they have made it intelligible, they have given us no reason to think that anything answers to it." (Goodreads.com)

"The fact that people have religious experiences is interesting from the psychological point of view, but it does not in any way imply that there is knowledge... unless he can formulate this 'knowledge' in propositions that are empirically verifiable, we may be sure that he is deceiving himself." (AZquotes.com)

Metaphysics

"...It is possible to be a metaphysician without believing in a transcendent reality; for as we shall see that many metaphysical utterances are due to the commission of logical errors, rather than to a conscious desire on the part of their authors to go beyond the limits of experience." (Goodreads.com)

P. F. Strawson and the conceptual framework

P. F. Strawson (1919-2006) was a British philosopher best known for his contributions to the philosophy of language. Philosopher Jane O'Grady said,

"Strawson, first gained philosophical fame at the age of 29 in 1950, when he criticised Bertrand Russell' renowned theory of descriptions for failing to do justice to the richness of ordinary language." (Obituary Sir Peter Strawson, 2006)

Wikipedia says,

"In his book Individuals (1959), Strawson attempts to give a description of various concepts that form and interconnected web, representing (part of) our common, shared conceptual scheme. In particular, he examines our conceptions of basic particulars, and how they are variously brought under general spatio-temporal concepts." (Wikipedia: P. F. Strawson, 8.14.21 UTC 20:52)

The rest of this post is some quotes from Strawson.

Concepts

"There is a massive central core of human thinking which has no history - or none recorded in histories of thought; there are categories and concepts which, in their most fundamental character, change not at all... It is with these, their interconnexions, and the structure that they form, that descriptive metaphysics will be primarily be concerned." (Individuals, 1959)

"Part of my aim is to exhibit some general and structural features of the conceptual scheme in terms of which we think about particular things." (Individuals, 1959)

Language

"Neither Aristotelian nor Russellian rules give the exact logic of any expression of ordinary language; for ordinary language has no exact logic." (On Referring, 1950)

"It remains to mention some of the ways in which people have spoken misleadingly of logical form. One of the commonest of these is to talk of 'the logical form' of a statement; as if a statement could never have more than one kind of formal power; as if statements could, in respect of their formal powers, be grouped in mutually exclusive classes, like animals at a zoo in respect of their species." (Introduction to Logical Theory, 1952)

Rudolf Carnap and logical positivism

Rudolf Carnap (1891-1970) was a German philosopher best known for his contributions to logical positivism. Carnap was also a member of the Vienna Circle. Wikipedia says,

"From 1922 to 1925, Carnap worked on a book which became one of his major works... (translated as The Logical Structure of the World)... From 1928 to 1934, Carnap published papers... translated as Pseudoproblems in Philosophy) in which he appears overtly skeptical of the aims and methods of metaphysics." (Wikipedia: Rudolf Carnap, 7.27.21 UTC 20:53)

The rest of this post is some quotes from Carnap.

Language

"By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language - the systematic statement of the formal rules which govern it together with the development of consequences that follow from these rules." (Logical Syntax of Language, 1934)

"The task of making more exact a vague or not quite exact concept used in everyday life or in an earlier stage of scientific or logical development, or rather of replacing it by a newly constructed, more exact concept, belongs among the most important tasks of logical analysis and logical construction." (Meaning and Necessity, 1947)

"...it is the question of whether or not to accept the new linguistic forms. The acceptance cannot be judged as being either true or false because it is not an assertion. It can only be judged as being more or less expedient, fruitful, conducive to the aim for which the language is intended." (Empiricism, Semantics, and Ontology, 1950)

Logic

"Philosophy is to be replaced by the logic of science - that is to say, by the logical analysis of the concepts and sentences of the sciences, for the logic of science is nothing other than the logical syntax of the language of science." (Logical Syntax of Language, 1934)

"Logic itself does not  decide whether the first sentence is true, but surrenders that question to one or the other of the empirical sciences." (Quoted in The Language of Wisdom and Folly by Irving Lee)

Unified science

"Science is a system of statements based on direct experience, and controlled by experimental verification. Verification in science is not, however, of single statements but of the entire system or sub-system of such statements." (The Unity of Science, 1934)

"The function of logical analysis is to analyze all knowledge, all assertions of science and everyday life, in order to make sense of each such assertion and the connections between them." (Philosophy and Logical Syntax, 1935)

"If one is interested in the relations between fields which, according to customary academic divisions, belong to different departments, then he will not be welcomed as a builder of bridges, as he might have expected, but will rather be regarded by both sides as an outsider and troublesome intruder." (Goodreads.com)

Saul Kripke and names

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Saul Kripke (1940-now) is an American philosopher best known for his contributions to the  philosophy of language, modal logic and establishing Kripke semantics. Kripke is also known for his book Wittgenstein on Rules and Private Language (1982) which is an analysis of Ludwig Wittgenstein's book Philosophical Investigations (1953, posthumous). Wikipedia says,

"Since the 1960's Kripke has been a central figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and set theory." (Wikipedia: Saul Kripke, 8.20.21 UTC 21:30)

The rest of this post is some quotes from Kripke.

Names

"Let's call something a rigid designator if in every possible world it designates the same object..." (AZquotes.com)

"If I use the name 'Hesperus' to refer to a certain planetary body when seen in a certain celestial position in the evening, it will not therefore be a necessary truth that Hesperus is ever seen in that evening. That depends on various contingent facts about the people being there to see and things like that." (Naming and Necessity, 1980)

Logic

"Logical investigations can obviously be a useful tool for philosophy. They must, however, be informed by a sensitivity to the philosophical significance of the formalism and by a generous admixture of common sense, as well as understanding of both the basic concepts and of the technical details of the formal material used." (AZquotes.com)

Truth

"Any necessary truth, whether a priori or a posteriori, could nit have turned out otherwise." (AZquotes.com)

Gottlob Frege and formal language

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Gottlob Frege (1848-1925) was a German philosopher and mathematician best known for his contributions to logic and the philosophy of mathematics. Frege is also considered the father of analytic philosophy. Mathematician Michael Beeson said,

"Gottlob Frege created modern logic including 'for all', 'there exists' and rules of proof. Leibniz and Boole had dealt only with what we now call 'propositional logic' (that is, no 'for all' or 'there exists'). The also did not concern themselves with rules of proof, since their aim was to reach truth by pure calculation with symbols for the propositions. Frege took the opposite track: instead of trying to reduce logic to calculation, he tried to reduce mathematics to logic, including the concept of number." (Alan Turing: Life and Legacy of a Great Thinker, 2004)

The rest of this post is some quotes from Frege.

Formal language

"This ideography is a 'formula language', that is, a lingua characterica, a language written with special symbols, 'for pure thought', that is, free from rhetorical embellishments, 'modeled upon that of arithmetic', that is, constructed from specific symbols that are manipulated according to definite rules." (Begriffsschrift, 1879)

Mathematics

"Strictly speaking, it is really scandalous that science has not yet clarified the nature of number. It might be excusable that there is still no generally accepted definition of number if at least there were general agreement on the matter itself. However, science has not even decided on whether number is an assemblage of things, or a figure drawn on the blackboard by the hand of man; whether it is something psychical, about whose generation psychology must give information, or whether it is a logical structure; whether it is created and can vanish, or whether it is eternal." (Uber die Zahlen des Herrn H. Schubert, 1899)

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development  of logic, and every proposition of arithmetic a law of logic..." (The Foundations of Arithmetic, 1884)

"The novelty of this book does not lie in the theorems but in the development of the proofs and the foundations of which they are based." (Grundgesetze der Arithmetik, 1893)

Concepts

"Often it is only after immense intellectual effort, which may have continued over centuries, that humanity at last succeeds in achieving knowledge of a concept in its pure form, by stripping off the irrelevant accretions which veil it from the eye of the mind. (Grundgesetze der Arithmetik, 1893)

"We suppose, it would seem, that concepts grow in the individual mind like leaves on a tree, and we think to discover their nature by studying their growth; we seek to define them psychologically, in terms of the human mind. But this account makes everything subjective, and we follow it through to the end, does away with truth. (Grundgesetze der Arithmetik, 1893)

"If the task of philosophy is to break the domination of words over the human mind... then my concept notation, being developed for these purposes, can be a useful instrument for philosophers." (Begriffsschrift, 1879)

Science

"Facts, facts, facts' cries the scientist if he wants to emphasize the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. But the scientist will surely not recognize something which depends on men's varying states of mind to be the firm foundation of science." (The thought: A logical inquiry, 1956 posthumous)

Gilbert Ryle and the Ghost in the Machine

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Gilbert Ryle (1900-1976) was a British philosopher best known for his critique of Cartesian dualism. Scott Christensen and Dale Turner said,

"Ryle's book was a polemic against the Cartesian idea that mental states are states of an immaterial substance. This polemic, and the ensuing discussion, turned on the question of the reducibility of mental events to behavioral dispositions. Ryle's central argument was that we had misconceived the 'logic' of such words as 'belief', 'sensation', 'conscious', etc." (Folk Psychology and the Philosophy of Mind, 2013)

Wikipedia says,

"In The Concept of Mind, Ryle argues that dualism involves category mistakes and philosophical nonsense..." (Wikipedia: Gilbert Ryle, 8.12.21 UTC 07:00)

The rest of this post is some quotes from Ryle.

The Ghost in the Machine

"The dogma of the Ghost in the Machine... maintains that there exist both bodies and minds; that there occur physical processes and mental processes; that there are mechanical causes of corporeal movements and mental causes of corporeal movements. I shall argue that these and other analogous conjunctions are absurd." (The Concept of Mind, 1949)

"I shall often speak of it, with deliberate abusiveness, as 'the dogma of the Ghost in the Machine'. I hope to prove that it is entirely false, and false not in detail but in principle... It is, namely, a category mistake. It represents the facts of mental life as if they belonged to one logical type or category (or range of types or categories) when they actually belong to another." (The Concept of Mind, 1949)

Williard van Orman Quine and the web of belief

Williard von Orman Quine (1908-2000) was an American philosopher best known for his contributions to epistemology, logic and the philosophy of science. Quine was awarded the Schock Prize in Logic and Philosophy in 1993 for,

"...his systematical and penetrating discussions of how learning of language and communication are based on socially available evidence and of the consequences of this for theories on knowledge and linguistic meaning." (Royal Swedish Academy of Sciences)

The rest of this post is some quotes from Quine.

Belief

"The totality of our so-called knowledge or beliefs, from the most casual matter of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges." (AZquotes.com)

"Implication is thus the very texture of our web of belief, and logic is the theory that traces it." (The Web of Belief, 1970)

"Believing is a disposition. We could tire ourselves out thinking, if we put our minds to it, but believing takes no toll." (Goodreads.com)

Ontology

"A curious thing about the ontological problem is its simplicity. It can be put into three Anglo-Saxon monosyllables: 'What is there?' It can be answered, moreover, in a word - 'Everything' - and everyone will accept this answer as true." (On What There Is, 1948)

"Our acceptance of an ontology is, I think similar in principle to our acceptance of a scientific theory, say a system of physics; we adopt, at least insofar as we are reasonable, the simplest conceptual scheme into which the disordered fragments of raw experience can be fitted and arranged." (AZquotes.com)

Language

"Linguistically, and hence conceptually, the things in sharpest focus are the things that are public enough to be talked of publicity, common and conspicuous enough to be talked of often, and near enough to sense to be quickly identified and learned by name..." (AZquotes.com)

"Language is a social art." (Goodreads.com)

"No two of us learn our language alike, nor, in a sense, does any finish learning it while he lives." (Goodreads.com)

"We cannot stem linguistic change, but we can drag our feet." (Goodreads.com)

Science

"It is within science itself, and not some prior philosophy, that reality is to be identified and described." (Theories and Things, 1981)

"The scientist is indistinguishable from the common man in his sense of evidence, except that the scientist is more careful." (AZquotes.com)

"My position is a naturalistic one; I see philosophy not as a priori propaedeutic or groundwork for science, but as continuous with science." (AZquotes.com)

Giles Deleuze and creation

Giles Deleuze (1925-1995) was a French philosopher best known for his analysis of literature, metaphysics and art. Philosopher Jon Roffe said,

"Deleuze is a key figure in postmodern French philosophy. Considering himself an empiricist and a vitalist, his body of work, which rests upon concepts such as multiplicity, constructivism, post-structuralism, difference and desire, stands at a substantial remove from the main traditions of 20th century Continental thought." (Gilles Deleuze, 2002)

The rest of this post is some quotes from Deleuze.

Philosophy

"Something in the world forces us to think. This something is an object not of recognition but of a fundamental encounter." (Goodreads.com)

"Paradox is the pathos or the passion of philosophy." (Goodreads.com)

"What counts is the question, of what is a body capable? And thereby he sets out on of the most fundamental questions in his whole philosophy..." (Expressionism in Philosophy: Spinoza, 1990)

Creation

"Creation takes place in bottlenecks... it's by banging your head on the wall that you find a way through... You have to open up words, break things open, to free earth's vectors." (Goodreads.com)

"So it's not a problem of getting people to express themselves but of providing little gaps of solitude and silence in which they might eventually find something to say." (Negotiations, 1990)

"To write is to struggle and resist; to write is to become; to write is to draw a map: I am a cartographer." (Foucault, 1986)

"To affirm is not to bear, carry, or harness oneself to that which exists, but on the contrary to unburden, unharness, and set free that which lives." (Nietzsche and Philosophy, 1962)

"Every time someone puts an objection to me, I want to say, 'OK, OK, let's go on to something else'. Objections have never contributed anything." (AZquotes.com)

Intuition and reason

"Intuition is neither a feeling, an inspiration nor a disorderly sympathy but a fully developed method." (AZquotes.com)

"It is not the slumber of reason that engenders monsters, but vigilant and insomniac rationality." (Anti-Oedipus: Capitalism and Schizophrenia, 1972)

"Underneath all reason lies delirium and drift." (AZquotes.com)

Martin Heidegger and Being

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Martin Heidegger (1889-1976) was a German philosopher best known for his analysis of Being. Thomas Lange said,

"Heidegger's thought will fit no category; no more than Hegel can be presented as a Platonist or Kant as a follower of Leibniz and Hume. It is equally foolish to say that Heidegger is just Kierkegaard de-Christianized, or Nietzsche systematized, or Hegel de-absolutized. I consider Heidegger's philosophy one of those extraordinary manifestations of man's forging of a knowledge of Being which the ages cannot but recon with." (Meaning of Heidegger: A Critical Study of an Existentialist Phenomenology, 1959)

The rest of this post is some quotes from Heidegger.

Being

"We ourselves are the entities to be analyzed." (Being and Time, 1927)

"Everyone is the other, and no one is himself. The they, which supplies the answer to the who of everyday Dasein, is the nobody to whom every Dasein has always surrendered itself, in its being-among-one-another." (Being and Time, 1927)

"Death is the possibility of the absolute impossibility of Dasein." (Being and Time, 1927)

"Why are there beings at all, and why not rather nothing? That is the question... And yet, we are each touched once, maybe even every now and then, by the concealed power of this question, without properly grasping what is happening to us." (What is Metaphysics? 1929)

Time

"Being and time determine each other reciprocally, but in such a manner that neither can the former be addressed as something temporal nor can the latter be addressed as a being." (Brainyquote.com)

"Since time itself is not movement, it must somehow how have to do with movement. Time is initially encountered in those entities which are changeable; change is time." (The Concept of Time)

Language

"Nevertheless, the ultimate business of philosophy is to preserve the force of the most elemental words in which Dasein expresses itself, and to keep the common understanding from leveling them off to that unintelligibilty which functions in turn as a source of pseudo-problems." (Being and Time, 1927)

"Language is the house of the truth of Being." (Letter on Humanism, 1947)

"All the poems of the poet who has entered into his poethood are poems of homecoming." (Existence and Being, 1949)

Unknown

"Making itself intelligible is suicide for philosophy. Those who idolize 'facts' never notice that their idols only shine in a borrowed light." (Contributions to Philosophy, 1989 posthumous)

"In everything well known something worthy of thought still lurks." (Nietzsche, 1961)

Edmund Husserl and phenomenology

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Edmund Husserl (1859-1938) was a German philosopher best known for his contributions to phenomenology. Philosopher Leo Strauss said,

"No one in our century has raised the call for philosopher as a rigorous science with such clarity, purity, vigor and breadth as Husserl." (Philosophy as Rigorous Science and Political Philosophy, 1971)

Wikipedia says,

"In his early work, he elaborated critiques of historicism and of psychologism in logic based on analyses of intentionality. In his mature work, he sought to develop a systematic foundational science based on the so-called phenomenological reduction." (Wikipedia: Edmund Husserl, 8.7.21 UTC 15:10)

The rest of this post is some quotes from Husserl.

Phenomenology

"To begin with, we put the proposition: pure phenomenology is the science of pure consciousness." (Brainyquote.com)

"A new fundamental science, pure phenomenology, has developed within philosophy: this is a science of a thoroughly new type and endless scope. It is inferior in methodological rigor to none of the modern sciences. All philosophical disciplines are rooted in pure phenomenology..." (Pure Phenomenology, 1917)

"This concept of the phenomenon, which was developed under various names as early as the 18th century without being clarified, is what we shall have to deal with first of all." (Brainyquote.com)

"Within this widest concept of object, and specifically within the concept of individual object, objects and phenomena stand in contrast with each other." (Brainyquote.com)

"Philosophers, as things now stand, are all too fond of offering criticism from on high instead of studying and understanding things from within." (Brainyquote.com)

Adolphe Quetelet and social physics

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Adolphe Quetelet (1796-1874) was a Belgian sociologist and sociologist best known for proposing that society operate according to social laws. Sociologist Frank Hamilton Hankins said,

"The most important of Quetelet's statistical principles... include the conception of the Average Man as a type, the significance for social science of the regularities found in the moral actions of man, and the theoretical basis of the distribution of group phenomena about their type." (Adolphe Quetelet as Statistician, 1908)

The rest of this post is some quotes from Quetelet.

Statistics

"It has seemed to me that the theory (calcul) of probability ought to serve as the basis for the study of all the sciences, and particularly of the sciences of observation." (Instructions populaires sur le calcul des probability, 1825)

"It may be seen, in my work, that the course which I have adopted is that followed by the natural philosopher, in order to grasp the laws that regulate the material world. By the seizure of facts, I seek to rise to an appreciation of the causes whence the spring." (A Treatise on Man and the Development of his Faculties, 1842)

Social physics

"The collection of the laws, which exist independently of time and of the caprices of man, form a separate science which I have considered myself entitled to name social physics." (Quoted in Adolphe Quetelet Statistician by Frank Hamilton Hankins)

"I believe that I have achieved to some extent what I have achieved to some extent what I have long said about the possibility of founding a social mechanics on the model established by celestial mechanics... to find there again the same properties and laws of conservation." (Astronomie elementaire? 1834)

"We are struck with the inflexible constancy of the laws which regulate the march of worlds and which preside over the succession of human generations. (Letter to H. R. H. the Grand Duke of Saxe Coburg and Gotha)

"Moral phenomena, when observed o a great scale, are found to resemble physical phenomena." (A Treatise on Man and the development of His Faculties, 1842)

Human nature

"The time is come for studying the moral anatomy of also, and for uncovering its most afflicting aspects with the view of providing remedies." (A Treatise on Man and the development of His Faculties, 1842)

"...we are under the domination of our habitudes, our wants, our social relations and a host of causes which, all of them, draw us in a hundred different ways." (A Treatise on Man and the development of His Faculties, 1842)

Carl Friedrich Gauss and the darkness

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Carl Friedrich Gauss (1777-1855) was a German mathematician and physicist best known for his contributions to algebra, non-Euclidean geometry, statistics and magnetism. Mathematician Morris Kline said,

"The man who was to change the course of mathematics was but six years old when Euler and D'Alembert died in 1783... Gauss is commonly ranked with Archimedes and Newton... all three of these men were as much devoted to physical research as to mathematics." (Mathematics and the Physical World, 1959)

Physicist James Clerk Maxwell said,

"It is to Gauss, to the Magnetic Union and to magnetic observation in general, that we owe our deliverance from that absurd method of estimating forces by a variable standard which prevailed so long even among men of science." (The Scientific Papers of James Clerk Maxwell, 1890 posthumous)

Mathematician Florian Cajori,

"Some of the discoveries of Abel and Jacobi were anticipated by Gauss... Jacobi concluded that Gauss had thirty years earlier considered the nature and properties of elliptic functions and had discovered their double periodicity." (A History of Mathematics, 1893)

The rest of this post is some quotes from Gauss.

Discovery

"Dark are the paths which a higher hand allows us to traverse here... let us hold fast to the faith that a finer, more sublime solution of enigmas of earthly life will be present, will become part of us." (Quoted in Carl Friedrich Gauss: Titan of Science by Guy Waldo Dunnington)

"When I have clarified and exhausted a subject, then I turn away from it, in order to go into the darkness again." (Brainyquote.com)

"If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.' (Quoted in The World of Mathematics J. R. Newman)

"Finally, two days ago, I succeeded - not on account of my hard efforts but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible." (Quoted in Mathematical Circles Squared by Howard Eves)

Metaphysics

"There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science." (Quoted in The World of Mathematics J. R. Newman)

"I scarcely believe that psychology data are present which can be mathematically evaluated. But one cannot know this with certainty, without having made the experiment. God alone is in possession of the mathematical basis of psychic phenomena." (Quoted in Carl Friedrich Gauss: Titan of Science by Guy Waldo Dunnington)

Explanation

"You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length." (Brainyquote.com)

"By explanation the scientist understands nothing except the reduction of the least and simplest basic laws possible, beyond which he cannot go, but must plainly demand them; from them however he deduces the phenomena absolutely completely as necessary." (Quoted in Carl Friedrich Gauss: Titan of Science by Guy Waldo Dunnington)

Bernhard Riemann and mind-masses

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Bernhard Riemann (1826-1866) was a German mathematician and philosopher best known for his contributions to integrals and differential geometry. He is also known for the Riemann hypothesis. Science historian A. D'Abro said,

"Einstein's theory, more especially the second part (the general theory), is intimately connected with the discoveries of the non-Euclidean geometricians, Riemann in particular." (The Evolution of Scientific Thought from Newton to Einstein, 1927)

 Mathematician Harold Edwards said,

"Riemann's style is extremely difficult. His tragically brief life was too occupied with mathematical creativity for him to devote himself to elegant exposition or to... polished presentation... Riemann was so far ahead of his time that it was 30 years before anyone could begin to really grasp his ideas." (Riemann's Zeta Function, 1974)

The rest of this post is some quotes from Riemann.

Explanation

"The completion or amelioration of the concept-system forms the 'explanation' of the unexpected observation. By this process our comprehension of nature becomes gradually always more complete and assured but at the same time recedes even farther behind the surface of phenomena." (Gesammelte Mathematische Werke, 1876)

Mind-masses

"With every simple act of thinking, something permanent, substantial, enters the soul. This substantial somewhat appears to us as a unit but (insofar so it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it 'mind-mass'. All thinking is, accordingly, formation of new mind-masses." (Gesammelte Mathematische Werke, 1876)

"Forming mind-masses amalgamate, combine or compound themselves in definite degree, partly with eachother, partly with older mind-masses." (Gesammelte Mathematische Werke, 1876)

"Every entering mind-mass excites all related mind-masses..." (Gesammelte Mathematische Werke, 1876)

"Mind-masses, once formed, are imperishable, their combinations are indissoluble; only the relative strength of these combinations is altered by the incoming of new mind-masses." (Gesammelte Mathematische Werke, 1876)

"Every mind-mass strives to produce a like formed mind-mass..." (Gesammelte Mathematische Werke, 1876)

"The souls of perished creatures shall... form the elements of the soul-life of earth." (Gesammelte Mathematische Werke, 1876)

Georg Cantor and infinity

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Georg Cantor (1845-1918) was a German mathematician best known for pioneering work on set theory and transfinite numbers. Mathematics historian Eli Maor said,

"In 1874 the German mathematician Georg Cantor made the startling discovery that there are more irrational numbers than rational ones, and more transcendental numbers than algebraic ones." (The Story of a Number, 1994)

The rest of this post is some quotes from Cantor.

Infinity

"The actual infinite arises in 3 contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the absolute infinity or simply absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type." (Quoted in Mind Tools by Rudy Rucker)

"Thus I believe that there is no part of matter which is not - I do not say divisible - but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures." (Quoted in Out of the Mouths of Mathematicians by Rosemary Schmalz)

"This view [of the infinite] which I to be the sole correct one, is held by only a few. While possibly I am the very first in history to take the position so explicitly, with all its logical consequences, I know for sure that I shall not be the last!" (Quoted in Journey Through Genius by William Dunham)

Mathematical freedom

"Mathematics is in its development entirely free and is only bound in the self evident respect that its concepts must both be consistent with eachother and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already established." (Quoted in From Kant to Hilbert by William Bragg Ewald)

"The essence of mathematics lies entirely in its freedom." (Quoted in From Kant to Hilbert by William Bragg Ewald)

Advancement of knowledge

"Great innovation only happens when people aren't afraid to do things differently." (AZquotes.com)

"A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held." (AZquotes.com)

Benjamin Peirce and geometry

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Benjamin Peirce (1809-1880) was an American mathematician best known for his contributions to number theory and statistics. He is also the father of philosopher Charles Sanders Peirce.

Geometry

"Geometry, to which I have devoted my life, is honored with the title of the key of the sciences." (Ben Yamen's Song of Geometry, 1853)

"There is proof enough furnished by every science, but by none more than geometry, that the world to which we have been allotted is peculiarly adapted to our minds, and admirably fitted to promote our intellectual progress." (Ben Yamen's Song of Geometry, 1853)

"I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry." (Linear Associative Algebra, 1982)

Logic and mathematics

"Mathematics is the science which draws necessary conclusions." (Linear Associative Algebra,1882)

Metaphysics

"What is man? ... What a strange union of matter and mind! A machine for converting material into spiritual force." (AZQuotes.com)

"Gentlemen, as we study the universe we see everywhere the most tremendous manifestations of force. In our experience we know of but one source of force, namely will. How then can we help regarding the forces we see in nature as due to the will of some omnipresent, omnipotent being? Gentlemen, there must be a God." (AZQuotes.com)

Edward Witten and string theory

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Edward Witten (1951-now) is an American physicist and mathematician best known for his contributions to string theory. Physicist Brian Greene said,

"In the spring of 1995... Edward Witten - who for decades has been the world's most renowned string theorist - uncovered a hidden unity that tied all five string theories together. Witten showed that rather than being distinct, the five theories are actually just five different ways of mathematically analyzing a single theory... The unifying master theory has tentatively been called M-theory." (The Fabric of the Cosmos, 2003)

The rest of this post is some quotes from Witten.

Concepts

"Most people who haven't been trained in physics probably think of what physicists do as a question of incredibly complicated calculations, but that's not really the essence of it. The essence of it is that physics is about concepts, wanting to understand the concepts, the principles by which the world works." (Interview with P. W. C. Davies and Julian Brown, 1992)

"I would expect that a proper elucidation of what string theory really is all about would involve a revolution in our concepts of the basic laws of physics - similar in scope to any that occurred in the past." (Interview with P. W. C. Davies and Julian Brown, 1992)

String Theory

"String theory is an attempt at a deeper description of nature by thinking of an elementary particle not as a little point but as a little loop of vibrating string." (Goodreads.com)

"Generally speaking, all the really great ideas of physics are really spin-offs of string theory... some of them were discovered first, but I consider that a mere accident of the development of planet earth." (Quoted in The End of Science by John Horgan)

"Spreading out the particle into a string is a step in the direction of making everything we're familiar with fuzzy. You enter a completely new world where things aren't at all what you're used to." (Brainyquote.com)

Dimensions

"As of now, string theorists have no explanation of why there are three large dimensions as well as time, and the other dimensions are are microscopic. Proposals about that have been all over the map." (Brainyquote.com)

"If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't know how big they are yet, but we hope to explore that in various ways." (Brainyquote.com)

"Vibrating strings in 10 dimensions is just a weird fact... An explanation of that weird fact would tell you why there are 10 dimensions in the first place." (Quoted in A Theory of Everything by K. C. Cole)

G. H. Hardy and mathematical reality

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G. H. Hardy (1877-1947) was a British mathematician best known for his contributions to number theory and population genetics. Philosopher Brand Blanshard said,

"I do not think that G. H. Hardy was talking nonsense when he insisted that the mathematician was discovering rather than creating, nor was it wholly nonsense for Kepler to exult that he was thinking God's thoughts after him." (The Philosophy of Brand Blanshard, 1980)

Mathematician Bruce Berndt said,

"Paul Erdos has passed on to us Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100. Hardy gave himself a score of 25, Littlewood 30, Hilbert 80 and Ramanujan 100." (Ramanujan's Notebooks, 1994)

The rest of this post is some quotes from Hardy.

Mathematical discovery

"The function of a mathematician then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps..." (The Theory of Numbers, 1922)

"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way (A Mathematician's Apology, 1941)

"I believe that mathematical reality lies outside us, that our function is yo discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations." (Goodreads.com)

"We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas." (Goodreads.com)

"The mathematician's patterns, like the painter's of the poet's must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way." (Goodreads.com)

George Boole and the logical mind

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George Boole (1815-1864) was an English mathematician best known for the development of Boolean algebra. Astrophysicist Mario Livio said,

"Boole literally transformed logic into a type of algebra (which came to be known as Boolean algebra) and extended the analysis of logic even to probabilistic reasoning... Boole managed to mathematically tame the logical connectives 'and', 'or', 'if... then', and 'not', which are currently at the very core of computer operations and various switching circuits." (Is God a Mathematician? 2009)

The rest of this post is some quotes from Boole.

Logic and the mind

"There is not only a close analogy between the operations of the mind in general reasoning and its operation int he particular science of algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted." (An Investigation into the Laws of Thought, 1854)

"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 symbolical language of a calculus, and upon this foundation to establish the science of logic and construct its method..." (An Investigation into the Laws of Thought, 1854)

"A successful attempt to express logical propositions by symbols, the laws of whose combinations should be found upon the laws of the mental processes which they represent, would, so far, be a step towards a philosophical language." (The Mathematical Analysis of Logic, 1847)

History of mathematics

"I presume that few who have paid any attention to the history of the mathematical analysis, will doubt that it has been developed in a certain order... by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived." (A Treatise on Differential Equations, 1859)

John Sowa and conceptual graphs

John Sowa (1940-now) is an American computer scientist best known for his contributions to conceptual graphs. Tawan Banchuen said,

"Sowa (1992) observed that various kinds of semantics networks had been developed for multiple purposes, ranging from modeling human cognitive mechanisms to optimizing computational efficiency. He commented that computational motivations had occasionally produced the same network as psychological purposes." (The Geographical Analog Engine, 2008)

Computer scientist Jeffrey Schiffel said,

"Among the formal graphical methods are Frege's (1879) Begriffsschrift, Pierce's (1909) existential graphs, and Sowa's (1984) conceptual graphs. These three are based in first-order predicate logic." (Improving Knowledge Management Programs Using Marginal Utility in a Metric Space Generated by Conceptual Graphs, 2008)

The rest of this post is a some quotes from Sowa.

Conceptual graphs

"A conceptual graph is a finite connected bipartite graph which consists of concepts and conceptual relations. Every conceptual relation has one or more arcs each of which is linked to a concept." (Conceptual Structures, 1984)

"We define a semantic network as 'the collection of all the relationships that concepts have to other concepts, to percepts, to procedures, and to the motor mechanisms' of the knowledge." (Conceptual Structures, 1984)

"The purpose of [conceptual graphs] is to express meaning in a form that is logically precise, humanly readable and computationally tractable. With their direct mapping to language, conceptual graphs can serve as an intermediate language for translating computer oriented formalism to and from natural languages." (Conceptual graphs for knowledge representation, 1993)

Giuseppe Peano and zero

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Giuseppe Peano (1858-1932) was an Italian mathematician best known for his contributions to set theory and the axiomatization of natural numbers. Mathematician Ugo Cassina said,

"Peano - whether in Logic or in Mathematics - never worked with pure symbolism - he always required that the primitive symbols introduced represent intuitive ideas to be explained with ordinary language." (Quoted in The Mathematical Philosophy of Giuseppe Peano by Hubert Kennedy, 1963)

Mathematician Gregory H. Moore said,

"Bertrand Russell never wavered in acknowledging his intellectual debt to Giuseppe Peano. In many ways the contribution that Russell made to the foundations of mathematics, culminating in Principia Mathematica, strongly bears Peano's mark." (1980)

The rest of this post is some quotes from Peano.

Numbers

"No number before zero. The numbers may go on forever, but like the cosmos, they have a beginning." (AZquotes.com)

"1. Zero is a number. 2. The immediate successor of any number is also a number. 3. Zero is not the immediate successor of any number. 4. No two numbers have the same immediate successor. 5. Any property belonging to 0 and to the immediate successor of any number that also has that property belongs to all numbers." (Quoted in The Mathematical Philosophy of Giuseppe Peano by Hubert Kennedy, 1963)

"In every science, after having analyzed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction." (Notations de Logique Mathematique, 1894)

David Hilbert and mathematical problems

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David Hilbert (1862-1943) was a German Mathematician best known for his contributions to the axiomatization of geometry and Hilbert spaces. Hilbert is also known for compiling a list of 23 unsolved mathematical problems in 1900 for the International Congress of Mathematicians which became influential for the development of mathematics in the 20th century. Science historian A. D'Abro said,

"A more thorough study of Euclid's axioms and postulates proved them to be inadequate for the deduction of Euclid's geometry... Hilbert and others succeeded in filling the gap by stating explicitly a complete system of postulates for Euclidean and non-Euclidean geometries alike." (The Evolution of Scientific Thought from Newton to Einstein, 1927)

This post is a list of quotes from Hilbert.

Mathematics is indivisible

"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts." (Brainyquote.com)

"The further a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separated branches of the science." (Brainyquote.com)

Mathematical problems

"A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be a guide post on the mazy paths to hidden truths..." (Mathematical Problems, 1900)

"As long as a branch of science offers an abundance of problems, so long it is alive; a lack of problems foreshadows extinction or the cessation of independent development." (AZquotes.com)

"If I were to awaken after having slept for a thousand years, my first question would be: has the the Riemann hypothesis been proven? (Quoted in Mathematical Mysteries by Calvin Clawson)

"We can measure the importance of a scientific work by the number of earlier publications rendered superfluous by it." (Quoted in Mathematical Circles Revisited by Howard Eves, 1971)

Formalism

"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 more easily comprehend. The very effort for rigour forces us to find out simpler methods of proof." (AZquotes.com)

"...it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper and simpler methods..." (Mathematical Problems, 1900)

Ernst Mach and the organization of thoughts

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Ernst Mach (1838-1916) was an Austrian physicist and philosopher best known his study of shock waves and his criticism Newton's theory of space and time. Mach also known for influencing logical positivism. Mathematician Henri Poincare said,

"Scientists believe there is a hierarchy of facts and that among them may be made a judicious choice. They are right, since otherwise there would be no science... As Mach says, these devotees have spared their successors the trouble of thinking." (The Value of Science, 1907)

Political scientist Alan Ebenstein said,

"There was a loud echo of Hume in Mach's work, as both emphasized the tangibility of all knowledge - ultimately, all knowledge is based in the senses. Mach also emphasized the internal nature of all knowledge, in that it is experienced in mind." (The Mind of Friedrich Hayek, 2003)

This post is some quotes from Mach.

Simplicity

"I know of nothing more terrible than the poor creatures who have learned too much... What they have acquired is a spider's web of thoughts too weak to furnish sure supports, but complicated enough to provide confusion." (Popular Scientific Lectures, 1898)

"Science is the most complete presentment of facts with the least expenditure of thought." (AZquotes.com)

"Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations." (AZquotes.com)

Concepts

"Nature consists of the elements given by the senses. Primitive man first takes out of them certain complexes of these elements that present themselves with a certain stability and are most important to him. The first and oldest words are names for things." (The Analysis of Sensations, 1902)

"Of all the concepts which the natural inquirer employs, the simplest are the concepts of space and time." (Space and Geometry in the Light of Physiological, Psychological and Physical Inquiry, 1906)

Laws of reality

"In reality, the law always contains less than the fact itself, because it does not reproduce the fact as a whole but only in that aspect of it which is important for us, the rest being intentionally or from necessity omitted." (The Economical Nature of Physical Inquiry, 1898)

"The aim of research is the discovery of the equations which subsist between the elements of phenomena." (Popular Scientific Lectures, 1898)

Frank Ramsey and the philosophy of logic

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Frank Ramsey (1903-1930) was a British mathematician and economist best known for his contributions to logic and decision theory. He also known for translating Ludwig Wittgenstein's Tractatus Logico-Philosophicus into English. Philosopher G. E. Moore said,

"He was an extraordinarily clear thinker: no-one could avoid more easily... the sort of confusions of thought to which even the best philosophers are liable, and he was capable of apprehending clearly... the subtlest distinctions. He had... an exceptional power of drawing conclusions from a complicated set of facts. (Preface to The Foundations of Mathematics and Other Logical Essays by Frank Ramsey, 1931)

This post is a some quotes from Ramsey.

Logic

"It is worth pausing for a moment to consider how far our conclusions are affected by considerations which our simplifying assumptions have forced us to neglect." (A Mathematical Theory of Saving, 1928)

"Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organizing thought." (The Foundations of Mathematics and Other Logical Essays, 1931 posthumous)

"The chief danger to our philosophy, apart from laziness and woolliness is scholasticism, the essence of which is treating what is vague as if it were precise and trying to fit it into an exact logical category." (Goodreads.com)

"...I hold that mathematics is part of logic, and so belong to what may be called the logical school as opposed to the formalist and intuitionist schools." (The Foundations of Mathematics, 1925)

Oskar Morgenstern and qualitative economics

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Oskar Morgenstern (1902-1977) was a German economist best known for pioneering game theory and its application to economics with John von Neumann. Economist Leonid Hurwicz said,

"It would be doing [Morgenstern and von Neumann] an injustice to say that their is a contribution to economics only. The scope of the book is much broader. The techniques applied by the authors in tackling economic problems are of sufficient generality to be valid in political science, sociology or even military strategy." (The Theory of Economic Behavior, 1945)

The rest of this post is some quotes from Morgenstern.

Philosophy of economics

"All economic decisions, whether private or business, as well as those involving economic policy, have the characteristic that quantitative and non-quantitative information must be combined into one act of decision. It would be desirable to understand how these two classes of information can best be combined." (Quoted in industrial Planning in France by John McArthur and Bruce Scott)

"...the facts of economic life cannot be comprehensively described in terms of statistics." (Limits of the Use of Mathematics in Economics, 1963)

"At the present time there is an unmistakable tendency to consider only a contribution to mathematical economics as a contributions of value. Hence a certain craving to press everything into some mathematical form... This preoccupation appears to me to be unnecessary and possibly dangerous." (Limits of the Use of Mathematics in Economics, 1963)

"As far as the use of mathematics in economics is concerned, there is an abundance of formulas where such are not needed. They are frequently introduced, one fears, in order to show off. The more difficult the mathematical theorem, the more esoteric the name of the mathematician quoted, the better." (Limits of the Use of Mathematics in Economics, 1963)

Richard von Mises and applied statistics

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Richard von Mises (1883-1953) was an Austrian statistician and physicist best known for his contributions to solid mechanics and probability theory. He is also the brother of economist Ludwig von Mises. Wikipedia says,

"Although best known for his mathematical work, von Mises also contributed to the philosophy of science as a neo-positivist, following in the line of Ernst Mach." (Wikipedia: Richard von Mises, 7.9.21 UTC 04:28)

Wikipedia also says

"In aerodynamics, von Mises made notable advances in boundary-layer-flow theory and  airfoil design. He developed the distortion energy theory of stress which is one of the most important concepts used by engineers in material strength calculations." (Wikipedia: Richard von Mises, 7.9.21 UTC 04:28)

The rest of this post is some quotes from von Mises.

Probability

"If the concept of probability and the formulae of the theory of probability are used without clear understanding of the collectives involved, one may arrive at entirely misleading results." (Probability, Statistics and Truth, 1928)

"In games of chance, in the problems of insurance, and in the molecular processes we find events repeating themselves again and again." (Probability, Statistics and Truth, 1928)

"It has been asserted - and this is not overstatement - that whereas other sciences draw their conclusions from what we know, the science of probability derives its most important results form what we don't know." (Probability, Statistics and Truth, 1928)

Philosophy of mathematics

"I am prepared to concede without further argument that all the theoretical constructions, including geometry, which are used in the various branches of physics are only imperfect instruments to enable the world of empirical fact to be reconstructed in our minds." (Probability, Statistics and Truth, 1928)

Reason

"No contradiction exists, if the events are correctly interpreted." (Probability, Statistics and Truth, 1957)

Plato and the dialogues

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Plato (427-347 BC) was a Greek philosopher best known as the founder of the Academy in Athens and regarded by some as the most influential philosopher in history. Physicist Wolfgang Pauli said,

"Modern man, seeking a middle position in the evaluation of sense impression and thought, can follow Plato, interpret the process of understanding nature as a correspondence, that is, a coming into congruence of pre-existing images of the human psyche with external objects and their behavior." (Writings on Physics and Philosophy, 1994 posthumous)

The rest of this post is some quotes from Plato.

Knowledge

"Rhetoric, it seems, is a producer of persuasion for belief, not for instruction in the matter of right and wrong... and so the rhetorician's business is not to instruct a law court or a public meeting in matters of right and wrong but only to make them believe." (Menexenus)

"How can you prove whether at this moment we are sleeping, and all our thoughts are a dream; or whether we are awake, and talking to one another in the waking state?" (Brainyquote.com)

Language

"A poet, you see, is a light thing, and winged and holy, and cannot compose before he gets inspiration and loses control of his senses and his reason has deserted him." (Goodreads.com)

"All that is said by us can only be imitation and representation." (Critias)

"Calligraphy is a geometry of the soul which manifests itself physically." (Goodreads.com)

Aesthetics

"Beauty of style, harmony, grace and good rhythm depend on simplicity - mean the true simplicity of a rightly and nobly ordered mind and character..." (The Republic)

"Musical training is a more potent instrument than any other, because rhythm and harmony find their way into the inward places of the soul." (The Republic)

Socrates and skepticism

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Socrates (470-399 BC) was a Greek philosopher best known for the Socratic method of debate. Philosopher John M. Cooper said,

"Socrates was a totally new kind of Greek philosopher. He denied that he had discovered some new wisdom, indeed that he possessed any wisdom at all, and he refused to hand anything down to anyone as his personal 'truth', his claim to fame."

Philosopher Bertrand Russell said,

"Socrates was the chief saint of the Stoics throughout their history; his attitude at the time of his trial, his refusal to escape, his calmness in the face of death, and his contention that the perpetrator of injustice injures himself more than his victim, all fitted in perfectly with Stoic teaching." (A History of Western Philosophy, 1945)

The rest of this post is some quotes from Socrates.

Skepticism

"I know nothing except the fact of my ignorance." (Goodreads.com)

"True wisdom comes to each of us when we realize how little we understand about life, ourselves, and the world around us." (Brainyquote.com)

Education

"Employ your time in improving yourself by other men's writings, so that you shall gain easily what others have labored hard for." (Goodreads.com)

"Smart people learn from everything and everyone, average people from their experiences, stupid people already have all the answers." (Goodreads.com)

Language

"The beginning of wisdom is a definition of terms." (Goodreads.com)

"Everything is plainer when spoken than when unspoken." (Goodreads.com)

"I realized that it was not by wisdom that poets write their poetry, but by a kind of nature or inspiration, such as you find in seers and prophets..." (Quoted in The Apology by Plato)

Metaphysics

"Suppose that there are two sorts of existence, one seen, and the other unseen... the seen is the changing and the unseen is the unchanging... and further, is not one part of us body and the rest of us soul?" (Quoted in Phaedo by Plato)

Augustus De Morgan and exploration

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Augustus De Morgan (1806-1871) was a British logician and mathematician best known for his contributions to mathematical induction. Cecil J. Monro said,

"The fact is known that having very thoroughly worked at the generalisations of mathematics in theory and practice, Mr. De Morgan was enabled to establish with perfect precision the most highly generalised conception of logic, perhaps, which it is possible to entertain." (Quoted in Memoir of Augustus De Morgan by Sophia Elizabeth De Morgan)

The rest of this post is some quotes from De Morgan.

Discovery

"The way to enlarge the settled country has not been by keeping within it, but by making voyages of discovery, and I am perfectly convinced that the student should be exercised in this manner; that is, that they should be taught how to examine the boundary, as well as how to cultivate the interior." (The Differential and Integral Calculus, 1836)

"New knowledge... must come by contemplation of old knowledge." (A Budget of Paradoxes, 1872)

"All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors and learned in what had been before them. There is not one exception." (A Budget of Paradoxes, 1872)

Philosophy of mathematics

"This mysterious 3.141592... which comes in at every door and window, and down every chimney." (Todayinsci.com)

"The moving power of mathematical invention is not reasoning but imagination." (Quoted in The Life of Sir William Rowan Hamilton by Robert Perceval Graves)

"Geometrical reasoning, and arithmetical process, have its own office: to mix the two in elementary instruction, is injurious to the proper acquisition of both." (Trigonometry and Double Algebra, 1849)

History of knowledge

"The thirteen books of Euclid must have been a tremendous advance, probably even greater than that contained in the Principia of Newton." (Quoted in Dictionary of Greek and Roman Biology and Mythology by William Smith)

"The genius of Laplace was a sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results." (In Review of Theorie Analytique des Probabilities, 1837)

"Bacon himself was very ignorant of all that had been done by mathematics; and strange to say, he especially objected to astronomy being handed over to the mathematicians." (A Budget of Paradoxes, 1872)

Karl Mannheim and arguments

Karl Mannheim (1893-1947) was a Hungarian sociologist and philosopher best known for his contributions to the sociology of knowledge. Lloyd Spencer said,

"Of all the classical sociologists, Mannheim is the one whose biography and mode of questioning connects him most directly to the problems of our own time... the questions he posed in the diagnosis of conflict, on the role of the intelligentsia, on education and on democratic planning remain as pertinent as ever." (Quoted in The A-Z Guide to Modern Social and Political Theorists by Noel Parker and Stuart Sim)

The rest of this post is some quotes form Mannheim.

Analysis of arguments

"As long as one does not call his own position into question but regards it as absolute, while interpreting his opponents' ideas as a mere function of the social positions they occupy, the decisive step forward has not yet been taken." (Ideology and Utopia, 1929)

"In attempting to expose the views of another, one is forced to make one's own view appear infallible and absolute, which is a procedure altogether to be avoided if one is making a specifically non-evaluative investigation." (Ideology and Utopia, 1929)

"Conflicting intellectual positions may actually come to supplement one another." (Ideology and Utopia, 1929)

George Henry Lewes and imagination

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George Henry Lewes (1817-1878) was a British literary critic best known for his philosophical approach to literature. The Encyclopedia Britannica says,

"A versatile writer and thinker in many fields, Lewes contributed significantly to the development of empirical metaphysics; his treatment of mental phenomena as related to social and historical conditions was a major advance in psychological thought."

The rest of this post is some quotes from Lewes.

Simplicity

"There are occasions when the simplest and fewest words surpass in effect all the wealth of rhetorical amplification." (The Principles of Success in Literature, 1865)

"...it is in the selection of the characteristic details that the artistic power is manifested." (The Principles of Success in Literature, 1865)

"Although the mind instinctively rejects all needless complexity, we shall greatly err if we fail to recognize the fact, that what the mind recoils from is not the complexity, but the needlessness." (The Principles of Success in Literature, 1865)

Literature and thought

"[The great writer] knows how to blend vividness with vagueness, knows where images are needed and where by their vivacity they would be obstacles to the rapid appreciation of thought." (The Principles of Success in Literature, 1865)

"The magic of the pen lies in the concentration of your thoughts upon one object." (AZquotes.com)

"An energetic crudity, even a riotous absurdity, has more promise in it than a clever and elegant mediocrity because it shows that the young man is speaking out of his own heart, and struggling to express himself in his own way rather than in the way he finds other men's books." (The Principles of Success in Literature, 1865)

Imagination and discovery

"Everyone who has seriously investigated a novel question, who has really interrogated nature with a view to a distinct answer, will bear me out in saying that it requires intense and sustained effort of imagination." (The Principles of Success in Literature, 1865)

"Where sense observes two isolated objects, imagination discloses two related objects. We had not see it before; it is apparent now." (The Principles of Success in Literature, 1865)

"Artists brood over the chaos of their suggestions, and thus shape them into creations." (The Principles of Success in Literature, 1865)