🪷On Mind: On Reasoning (1875-1899)

"The very genius of the common geometry as a method of reasoning - a system of investigation - is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn" (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds." (Edward Olney, "Mathematics", The Cyclopedia of Education, 1877

"[…] it must be noticed that these diagrams do not naturally harmonize with the propositions of ordinary life or ordinary logic. […] The great bulk of the propositions which we commonly meet with are founded, and rightly founded, on an imperfect knowledge of the actual mutual relations of the implied classes to one another. […] one very marked characteristic about these circular diagrams is that they forbid the natural expression of such uncertainty, and are therefore only directly applicable to a very small number of such propositions as we commonly meet with." (John Venn, "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings", 1880)

"Symbolical reasoning may be said to have pretty much the same relation to ordinary reasoning that machine-labour has to manual labour. In the case of machine labour we see some ingeniously contrived arrangement of wheels, levers, &c., producing with speed and facility results which the hands of man without such aid could only accomplish slowly and with difficulty, or which they would be utterly powerless to accomplish at all. In the case of symbolical reasoning we find in an analogous manner some regular system of rules and formulae, easy to retain in the memory from their general symmetry and interdependence, economizing or superseding the labour of the brain, and enabling any ordinary mind to obtain by simple mechanical processes results which would be beyond the reach of the strongest intellect if left entirely to its own resources." (Hugh MacColl, Symbolical reasoning. Mind 5" (17), 1880)

"We endeavour to employ only symmetrical figures, such as should not only be an aid to reasoning, through the sense of sight, but should also be to some extent elegant in themselves." (John Venn, "Symbolic Logic", 1881)

"[…] deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts." (Charles S Peirce, 1885)

"I call a sign which stands for something merely because it resembles it, an icon. Icons are so completely substituted for their objects as hardly to be distinguished from them. Such are the diagrams of geometry. A diagram, indeed, so far as it has a general signification, is not a pure icon; but in the middle part of our reasonings we forget that abstractness in great measure, and the diagram is for us the very thing. So in contemplating a painting, there is a moment when we lose the consciousness that it is not the thing, the distinction of the real and the copy disappears, and it is for the moment a pure dream, - not any particular existence, and yet not general. At that moment we are contemplating an icon." (Charles S Peirce, "On The Algebra of Logic : A Contribution to the Philosophy of Notation" in The American Journal of Mathematics 7, 1885)

"There is no more remarkable feature in the mathematical theory of probability than the manner in which it has been found to harmonize with, and justify, the conclusions to which mankind have been led, not by reasoning, but by instinct and experience, both of the individual and of the race. At the same time it has corrected, extended, and invested them with a definiteness and precision of which these crude, though sound, appreciations of common sense were till then devoid." (Morgan W Crofton, "Probability", Encyclopaedia Britannica 9th Ed,, 1885)

 "The theory most prevalent among teachers is that mathematics affords the best training for the reasoning powers; […] The modem, and to my mind true, theory is that mathematics is the abstract form of the natural sciences; and that it is valuable as a training of the reasoning powers, not because it is abstract, but because it is a representation of actual things." (Truman H Safford, "Mathematical Teaching and Its Modern Methods", 1886)

"There are three aspects of thinking as more or less complete stages of it. These are, conception, judgment, and reasoning. They are not to be considered three distinct acts; not even three successive stages. No one of them could occur without each of the others." (John Dewey, "Psychology", 1887) 

"There is probably no other science which presents such different appearances to one who cultivates it and one who does not, as mathematics. To [the noncultivator] it is ancient, venerable, and complete; a body of dry, irrefutable, unambiguous reasoning. To the mathematician, on the other hand, his science is yet in the purple bloom of vigorous youth, everywhere stretching out after the 'attainable but unattained', and full of the excitement of nascent thoughts; its logic is beset with ambiguities, and its analytic processes, like Bunyan's road, have a quagmire on one side and a deep ditch on the other, and branch off into innumerable by-paths that end in a wilderness." (Charles H Chapman, Bulletin of the New York Mathematical Society 2, 1892) 

"[…] geometry is the art of reasoning well from badly drawn figures; however, these figures, if they are not to deceive us, must satisfy certain conditions; the proportions may be grossly altered, but the relative positions of the different parts must not be upset." (Henri Poincaré, 1895)

"Deduction is that mode of reasoning which examines the state of things asserted in the premises, forms a diagram of that state of things, perceives in the parts of the diagram relations not explicitly mentioned in the premises, satisfies itself by mental experiments upon the diagram that these relations would always subsist, or at least would do so in a certain proportion of cases, and concludes their necessary, or probable, truth." (Charles S Peirce,"Kinds of Reasoning", cca. 1896)

"A system is a set of objects compromising all that stands to one another in a group of connected relations." (Charles S Peirce, "Cambridge Lectures on Reasoning and the Logic of Things: Detached Ideas on Vitally Important Topics", 1898) 

"The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus." (Alfred N Whitehead,"Universal Algebra", 1898) 

"The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning." (Alfred North Whitehead,"A Treatise on Universal Algebra", 1898)

"Many theorems are obvious upon looking at a moderately-sized figure; but the reasoning must be such as to convince the mind of their truth when, from excessive increase or diminution of the scale, the figures themselves have past the boundary even of imagination." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

"The ordinary logic has a great deal to say about genera and species, or in our nineteenth century dialect, about classes. Now a class is a set of objects compromising all that stand to one another in a special relation of similarity. But where ordinary logic talks of classes the logic of relatives talks of systems. A system is a set of objects compromising all that stands to one another in a group of connected relations. Induction according to ordinary logic rises from the contemplation of a sample of a class to that of a whole class; but according to the logic of relatives it rises from the contemplation of a fragment of a system to the envisagement of the complete system." (Charles S Peirce, "Cambridge Lectures on Reasoning and the Logic of Things: Detached Ideas on Vitally Important Topics", 1898) 

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

🪷On Mind: On Reasoning (1850-1874)

"The reasoning of mathematicians is founded on certain and infallible principles. Every word they use conveys a determinate idea, and by accurate definitions they excite the same ideas in the mind of the reader that were in the mind of the writer. When they have defined the terms they intend to make use of, they premise a few axioms, or self-evident principles, that every one must assent to as soon as proposed. They then take for granted certain postulates, that no one can deny them, such as, that a right line may be drawn from any given point to another, and from these plain, simple principles they have raised most astonishing speculations, and proved the extent of the human mind to be more spacious and capacious than any other science." (John Adams,"Diary", 1850)

"The discovery of truth is prevented more effectively, not by the false appearance things present and which mislead into error, not directly by weakness of the reasoning powers, but by preconceived opinion, by prejudice." (Arthur Schopenhauer, "Parerga and Paralipomena", 1851)

"Reasoning from analogy is often most plausible and most deceptive." (Charles Simmons, "A Laconic Manual and Brief Remarker", 1852)

"To deduce the laws of the symbols of Logic from a consideration of those operations of the mind which are implied in the strict use of language as an instrument of reasoning." (George Boole, "An Investigation of the Laws of Thought", 1854)

 "It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"[...] observation is the mind's support in reasoning [...]" (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"An anticipative idea or an hypothesis is, then, the necessary starting point for all experimental reasoning. Without it, we could not make any investigation at all nor learn anything; we could only pile up sterile observations. If we experiment without a preconceived idea, we should move at random […]" (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Considered by itself, the experimental method is nothing but reasoning by whose help we methodically submit our ideas to experience, - the experience of facts." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"Every process of what has been called Universal Geometry - the great creation of Descartes and his successors, in which a single train of reasoning solves whole classes of problems at once, and others common to large groups of them - is a practical lesson in the management of wide generalizations, and abstraction of the points of agreement from those of difference among objects of great and confusing diversity, to which the purely inductive sciences cannot furnish many superior. Even so elementary an operation as that of abstracting from the particular configuration of the triangles or other figures, and the relative situation of the particular lines or points, in the diagram which aids the apprehension of a common geometrical demonstration, is a very useful, and far from being always an easy, exercise of the faculty of generalization so strangely imagined to have no place or part in the processes of mathematics." (John S Mill, "An Examination of Sir William Hamilton’s Philosophy", 1865)

"If the facts used as a basis of reasoning are ill-established or erroneous, everything will crumble or be falsified; and it is thus that errors in scientific theories most often originate in errors of fact." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)

"It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"Speaking concretely, when we say ‘making experiments or making observations’, we mean that we devote ourselves to investigation and to research, that we make attempts and trials in order to gain facts from which the mind, through reasoning, may draw knowledge or instruction. Speaking in the abstract, when we say, ‘relying on observation and gaining experience’, we mean that observation is the mind’s support in reasoning, and experience the mind’s support in deciding, or still better, the fruit of exact reasoning applied to the interpretation of facts. Observation, then, is what shows facts; experiment is what teaches about facts and gives experience in relation to anything." (Claude Bernard, "An Introduction to the Study of Experimental Medicine", 1865)"

"The degree in which each mind habitually substitutes signs for images will be, CETERIS PARIBUS [with other conditions remaining the same], the degree in which it is liable to error. This is not contradicted by the fact that mathematical, astronomical, and physical reasonings may, when complex, be carried on more successfully by the employment of signs; because in these cases the signs themselves accurately represent the abstractness of the relations. Such sciences deal only with relations, and not with objects; hence greater simplification ensures greater accuracy. But no sooner do we quit this sphere of abstractions to enter that of concrete things, than the use of symbols becomes a source of weakness. Vigorous and effective minds habitually deal with concrete images." (George H Lewes, "The Principles of Success in Literature", 1865)

"It has long been a complaint against mathematicians that they are hard to convince: but it is a far greater disqualification both for philosophy, and for the affairs of life, to be too easily convinced; to have too low a standard of proof. The only sound intellects are those which, in the first instance, set their standards of proof high. Practice in concrete affairs soon teaches them to make the necessary abatement: but they retain the consciousness, without which there is no sound practical reasoning, that in accepting inferior evidence because there is no better to be had, they do not by that acceptance raise it to completeness." (John S Mill, "An Examination of Sir William Hamilton's Philosophy", 1865)

"Ideas are substitutions which require a secondary process when what is symbolized by them is translated into the images and experiences it replaces; and this secondary process is frequently not performed at all, generally only performed to a very small extent. Let anyone closely examine what has passed in his mind when he has constructed a chain of reasoning, and he will be surprised at the fewness and faintness of the images which have accompanied the ideas." (George H Lewes "Problems of Life and Mind", 1873)"

“Nor do I know any study which can compete with mathematics in general in furnishing matter for severe and continued thought. Metaphysical problems may be even more difficult; but then they are far less definite, and, as they rarely lead to any precise conclusion, we miss the power of checking our own operations, and of discovering whether we are thinking and reasoning or merely fancying and dreaming." (Isaac Todhunter,"Conflict of Studies and Other Essays", 1873)

"With Algebra we enter a new sphere, that of symbolical quantities; here letters are symbols of any values we please; all we deal with in them is the relations of equality which the letters symbolise. Although the values are changeable, jet, once assigned, they must remain fixed throughout the operation. Illogical reasoning, in philosophic as in ordinary minds, is not due to any irregularity in the normal operation, but to a departure &om the values assigned." (George H Lewes "Problems of Life and Mind", 1873)

"By deductive reasoning and calculation, we must endeavor to anticipate such new phenomena, especially those of a singular and exceptional nature, as would necessarily happen if the hypothesis be true." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874) 

"Deduction is certain and infallible, in the sense that each step in deductive reasoning will lead us to some result, as certain as the law itself. But it does not follow that deduction will lead the reasoner to every result of a law or combination of laws." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"It must be the ground of all reasoning and inference that what is true of one thing will be true of its equivalent, and that under carefully ascertained conditions Nature repeats herself." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. [...] Numerical precision is the soul of science [...]" (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

🪷On Mind: On Reasoning (1825-1949)

"It is true that of far the greater part of things, we must content ourselves with such knowledge as description may exhibit, or analogy supply; but it is true likewise, that these ideas are always incomplete, and that at least, till we have compared them with realities, we do not know them to be just. As we see more, we become possessed of more certainties, and consequently gain more principles of reasoning, and found a wider base of analogy." (Samuel Johnson, 1825)

"Geometry is a true natural science: - only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Everybody studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone." (Auguste Comte,"Course of Positive Philosophy", 1830)  "

"The errors which arise from the absence of facts are far more numerous and more durable than those which result from unsound reasoning respecting true data." (Charles Babbage, "On the Economy of Machinery and Manufactures", 1832)

"Every syllogism, then, consists of three propositions; the two first of which are called the premises and the third the conclusion. Now, the advantage of the all these [valid] forms to direct our reasoning is this, that if the premises are both true, the conclusion infallibly is so.This is likewise the only method of discovering unknown truths. Every truth must always be the conclusion of a syllogism, whose premises are indubitably true." (Leonhard Euler, "Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess" Vol. 1, 1833)

"Theories usually result from the precipitate reasoning of an impatient mind which would like to be rid of phenomena and replace them with images, concepts, indeed often with mere words." (Johann Wolfgang von Goethe, "Maxims and Reflections", 1833) 

"Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art are constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic." (William Chambers & Robert Chambers, "Chambers's Information for the People" Vol. 2, 1835)

"Scientific Ideas can often be adequately exhibited for all the purposes of reasoning, by means of Definitions and Axioms; all attempts to reason by means of Definitions from common Notions, lead to empty forms or entire confusion." (William Whewell, "History of the Inductive Sciences from the Earliest to the Present Time", 1837)

"The connected course of reasoning by which any Geometrical truth is established is called a demonstration." (Robert Potts, "Euclid's Elements of Geometry", 1845)

🪷On Mind: On Reasoning (1800-1824)

"[…] although the symbol √-1 be beyond the power of arithmetical computation, the operations in which it is introduced are intelligible, and deserve, if any operations do, the name of reasoning." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

 "Mathematical science has been at times embarrassed with contradictions and paradoxes; yet they are not to be imputed to imaginary symbols, rather than to any other symbols invented for the purpose of rendering demonstration compendious, and expeditious. It may; however, be justly remarked, that mathematicians, neglecting to exercise mental superintendance, are too prone to trust to mechanical dexterity; and that some, instead of establishing the truth of conclusions on antecedent reasons, have endeavoured to prop it by imperfect analogies or mere algebraic forms. On the other hand, there are mathematicians, whose zeal for just reasoning has been alarmed at a verbal absurdity and, from a name improperly applied, or a definition incautiously given, l have been hurried to the precipitate conclusion, that operations with symbols of which the mind can form no idea, must necessarily be doubtful and unintelligible." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

 "The introduction of impossible quantities, is assigned as a great and primary cause of the evils under which mathematical science labours. During the operation of these quantities, it is said, all just reasoning is suspended, and the mind is bewildered by exhibitions that resemble the juggling tricks of mechanical dexterity." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

 "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)

 "The symbol √-1 might arise from translating questions of which the statement involved a contradiction of ideas into algebraic language, and reasoning on them, as if they really admitted a solution." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"Mathematical science has been at times embarrassed with contradictions and paradoxes; yet they are not to be imputed to imaginary symbols, rather than to any other symbols invented for the purpose of rendering demonstration compendious, and expeditious. It may; however, be justly remarked, that mathematicians, neglecting to exercise mental superintendance, are too prone to trust to mechanical dexterity; and that some, instead of establishing the truth of conclusions on antecedent reasons, have endeavoured to prop it by imperfect analogies or mere algebraic forms. On the other hand, there are mathematicians, whose zeal for just reasoning has been alarmed at a verbal absurdity and, from a name improperly applied, or a definition incautiously given, l have been hurried to the precipitate conclusion, that operations with symbols of which the mind can form no idea, must necessarily be doubtful and unintelligible." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"The introduction of impossible quantities, is assigned as a great and primary cause of the evils under which mathematical science labours. During the operation of these quantities, it is said, all just reasoning is suspended, and the mind is bewildered by exhibitions that resemble the juggling tricks of mechanical dexterity." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

[…] although the symbol √-1 be beyond the power of arithmetical computation, the operations in which it is introduced are intelligible, and deserve, if any operations do, the name of reasoning." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

The introduction of impossible quantities, is assigned as a great and primary cause of the evils under which mathematical science labours. During the operation of these quantities, it is said, all just reasoning is suspended, and the mind is bewildered by exhibitions that resemble the juggling tricks of mechanical dexterity." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

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)

 The Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things, that has not been enumerated. [...] Besides its uses in mathematical investigations, it not only enables us to form our ideas of the elegant compositions of design, but to contemplate the prodigious variety which constitutes the beauties of nature, and which arises from the combinations of objects, by their number, forms, color, and positions. It has a relation to every species of useful knowledge upon which the mind of man can be employed." (Peter Nicholson, "Essays on the Combinatorial Analysis", 1818) 

"Genius and science have burst the limits of space, and few observations, explained by just reasoning, have unveiled the mechanism of the universe. Would it not also be glorious for man to burst the limits of time, and, by a few observations, to ascertain the history of this world, and the series of events which preceded the birth of the human race?" (Georges Cuvier, "Essays on the Theory of the Earth", 1822)"

“Mathematical proofs, like diamonds, are hard and clear, and will be touched with nothing but strict reasoning." (John Locke, 1824)

🪷On Mind: On Reasoning (1750-1799)

"Especially when we investigate the general laws of Nature, induction has very great power; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law; although, whether they can be made to agree in this way better than in any Other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law." (Roger J Boscovich, "De Lege Continuitatis" ["On the law of continuity"], 1754)

"We think only through the medium of words. Languages are true analytical methods. Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method. The art of reasoning is nothing more than a language well arranged." (Abbé de Condillac, "System of Logic", cca. 1781)

 "The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"In every chain of reasoning, the evidence of the last conclusion can be no greater than that of the weakest link of the chain, whatever may be the strength of the rest." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"In mathematics it [sophistry] had no place from the beginning: Mathematicians having had the wisdom to define accurately the terms they use, and to lay down, as axioms, the first principles on which their reasoning is grounded. Accordingly we find no parties among mathematicians, and hardly any disputes." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"In metaphysical reasoning, the process is always short. The conclusion is but a step or two, seldom more, from the first principles or axioms on which it is grounded, and the different conclusions depend not one upon another.

"It is otherwise in mathematical reasoning. Here the field has no limits. One proposition leads on to another, that to a third, and so on without end. If it should be asked, why demonstrative reasoning has so wide a field in mathematics, while, in other abstract subjects, it is confined within very narrow limits, I conceive this is chiefly owing to the nature of quantity, […] mathematical quantities being made up of parts without number, can touch in innumerable points, and be compared in innumerable different ways." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"The mathematician pays not the least regard either to testimony or conjecture, but deduces everything by demonstrative reasoning, from his definitions and axioms. Indeed, whatever is built upon conjecture, is improperly called science; for conjecture may beget opinion, but cannot produce knowledge." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure." (Joseph-Louis de Lagrange, "Mechanique Analytique", 1788)

 "The only method of preventing such errors from taking place, and of correcting them when formed, is to restrain and simplify our reasoning as much as possible. This depends entirely upon ourselves, and the neglect of it is the only source of our mistakes." " (Antoine-Laurent de Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

 "We think only through the medium of words. Languages are true analytical methods. Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method. The art of reasoning is nothing more than a language well arranged." (Antoine-Laurent Lavoisier,"Elements of Chemistry", 1790)

"We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoin-Laurent de Lavoisiere, "Elements of Chemistry", 1790)

[...] mathematicians obtain the solution of a problem by the mere arrangement of data, and by reducing their reasoning to such simple steps, to conclusions so very obvious, as never to lose sight of the evidence which guides them." (Antoine-Laurent Lavoisier, "Elements of Chemistry In a New Systematic Order", 1790)

We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoine Lavoisier, "Elements of Chemistry", 1790)

We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoine-Laurent Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

In mathematics, such processes [of reasoning] are much longer than in any other science; and hence the study of it is peculiarly calculated to strengthen the power of steady and concatenated thinking, - a power which, in all the pursuits of life, whether speculative or active, is one of the most valuable endowments we can possess. This command of attention, however, it may be proper to add, is to be acquired, not by the practice of modern methods, but by the study of Greek geometry, more particularly, by accustoming ourselves to pursue long trains of demonstration, without availing ourselves of the aid of any sensible diagrams; the thoughts being directed solely by those ideal delineations which the powers of conception and of memory enable us to form." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"Whoever limits his exertions to the gratification of others, whether by personal exhibition, as in the case of the actor and of the mimic, or by those kinds of literary composition which are calculated for no end but to please or to entertain, renders himself, in some measure, dependent on their caprices and humours. The diversity among men, in their judgments concerning the objects of taste, is incomparably greater than in their speculative conclusions; and accordingly, a mathematician will publish to the world a geometrical demonstration, or a philosopher, a process of abstract reasoning, with a confidence very different from what a poet would feel, in communicating one of his productions even to a friend." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

🪷On Mind: On Reason (1725-1749)

"For chemistry is no science form’d à priori; ’tis no production of the human mind, framed by reasoning and deduction: it took its rise from a number of experiments casually made, without any expectation of what follow’d; and was only reduced into an art or system, by collecting and comparing the effects of such unpremeditated experiments, and observing the uniform tendency thereof. So far, then, as a number of experimenters agree to establish any undoubted truth; so far they may be consider'd as constituting the theory of chemistry." (Hermann Boerhaave, "A New Method of Chemistry", 1727)

"[…] our reasonings about the wonderful and intricate operations of Nature are so full of uncertainty, that, as the wise-man truly observes, hardly do we guess aright at the things that are upon earth, and with labour do we find the things that are before us." (Stephen Hales, "Vegetable Staticks", 1727)

"It hath been an old remark, that Geometry is an excellent Logic. And it must be owned that when the definitions are clear; when the postulata cannot be refused, nor the axioms denied; when from the distinct contemplation and comparison of figures, their properties are derived, by a perpetual well-connected chain of consequences, the objects being still kept in view, and the attention ever fixed upon them; there is acquired a habit of reasoning, close and exact and methodical; which habit strengthens and sharpens the mind, and being transferred to other subjects is of general use in the inquiry after truth." (George Berkeley, "The Analyst; Or, A Discourse Addressed to an Infidel Mathematician", 1734)

"All the perceptions of the human mind resolve themselves into two distinct kinds, which I shall call impressions and ideas. The difference betwixt these consists in the degrees of force and liveliness, with which they strike upon the mind, and make their way into our thought or consciousness. Those perceptions, which enter with most force and violence, we may name impressions; and under this name I comprehend all our sensations, passions and emotions, as they make their first appearance in the soul. By ideas I mean the faint images of these in thinking and reasoning. I believe it will not be very necessary to employ many words in explaining this distinction." (David Hume, "A Treatise of Human Nature", 1738)

"But to form the idea of an object, and to form an idea simply is the same thing; the reference of the idea to an object being an extraneous denomination, of which in itself it bears no mark or character. Now as it is impossible to form an idea of an object, that is possessed of quantity and quality, and yet is possessed of no precise degree of either; it follows, that there is an equal impossibility of forming an idea, that is not limited and confined in both these particulars. Abstract ideas are therefore in themselves individual, however they may become general in their representation. The image in the mind is only that of a particular object, though the application of it in our reasoning be the same, as if it were universal." (David Hume, "Treatise of Human Nature", 1738)

 "Do not expect to arrive at certainty in every subject which you pursue. There are a hundred things wherein we mortals […] must be content with probability, where our best light and reasoning will reach no farther." (Isaac Watts," The Improvement of the Mind", 1741) 

 "It is not therefore the business of philosophy, in our present situation in the universe, to attempt to take in at once, in one view, the whole scheme of nature; but to extend, with great care and circumspection, our knowledge, by just steps, from sensible things, as far as our observations or reasonings from them will carry us, in our enquiries concerning either the greater motions and operations of nature, or her more subtile and hidden works." (Colin Maclaurin, "An Account of Sir Isaac Newton's Philosophical Discoveries, in Four Books", 1748)

It seems evident, that men are carried, by a natural instinct or prepossession, to repose faith in their senses; and that, without any reasoning, or even almost before the use of reason, we always suppose an external universe, which depends not on our perception, but would exist, though we and every sensible creature were absent or annihilated. […] It seems also evident, that, when men follow this blind and powerful instinct of nature, they always suppose the very images, presented by the senses, to be the external objects." (David Hume, "An Enquiry Concerning Human Understanding", 1748)

There is no method of reasoning more common, and yet none more blameable, than, in philosophical disputes, to endeavour the refutation of any hypothesis, by a pretence of its dangerous consequences to religion and morality. When any opinion leads to absurdities, it is certainly false; but it is not certain that an opinion is false, because it is of dangerous consequence. Such topics, therefore, ought entirely to be forborne; as serving nothing to the discovery of truth, but only to make the person of an antagonist odious." (David Hume, "An Enquiry Concerning Human Understanding", 1748) 

🪷On Mind: On Reason (1700-1724)

"The mathematics are the friends to religion, inasmuch as they charm the passions, restrain the impetuosity of the imagination, and purge the mind from error and prejudice. Vice is error, confusion and false reasoning; and all truth is more or less opposite to it. Besides, mathematical truth may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness of them being such as to make solitude not only easy but desirable." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)

"[…] even if someone refuses to admit infinite and infinitesimal lines in a rigorous metaphysical sense and as real things, he can still use them with confidence as ideal concepts" (notions ideales) which shorten his reasoning, similar to what we call imaginary roots in the ordinary algebra, for example, v-2." (Gottfried W Leibniz, [letter to Varignon], 1702)

“It is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men, and which constitutes the greatest part of the beauty of the universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other. But, because the number of causes that concur in producing a given event, or effect, is oftentimes so immensely great, and the causes themselves are so different one from another, that it is extremely difficult to reckon up all the different ways in which they may be arranged, or combined together, it often happens that men, even of the best understandings and the greatest circumspection, ale guilty of that fault in reasoning which the writers on logic call the insufficient, or imperfect enumeration of parts, or cases: insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the knowledge of nature, or the merits and motives of human actions. It must therefore be acknowledged, that that art which affords a cure to this weakness, or defect, of our understandings, and teaches us so to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted any one arrangement of them that can lead to the object of our inquiry, deserves to be considered as most eminently useful and worthy of our highest esteem and attention. And this is the business of the art, or doctrine of combinations." (Jakob Bernoulli,"Ars Conjectandi" [“The Art of Conjecturing"], 1713) 

"[...] this subject] has a relation to almost every species of useful knowledge that the mind of man can be employed upon." (Jacob Bernoulli,"Ars Conjectandi"[“The Art of Conjecturing"], 1713)

"There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible." (Gottfried W Leibniz, "Monadology", 1714)

“Reasoning will never make a Man correct an ill Opinion, which by Reasoning he never acquired." ( Jonathan Swift,"A Letter to a Young Gentleman, Lately Enter’d Into Holy Orders by a Person of Quality", 1721)

🪷On Mind: On Reasoning (1600-1699)

"The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another." (René Descartes, "Discourse on the Method", 1637)

"So there arise two kinds of variation: complexion [combinations] and situs [permutations]. And viewed in themselves, both complexion and situs belong to metaphysics, or to the science of whole and parts. If we look at their variability, however, that is, at the quantity of variation, we must turn to numbers and to arithmetic. I am inclined to think that the science of complexions pertains more to pure arithmetic, and that of situs to an arithmetic of figure." (Gottfried Leibniz,"Dissertatio de arte combinatoria" [“Dissertation on the Art of Combinations"], Leipzig, 1666)

"It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters, insofar as they are subject to reasoning, all that we can do in arithmetic and geometry." (Gottfried W Leibniz, 1677)

" Mathematics in gross, it is plain, are a grievance in natural philosophy, and with reason: for mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning. Mathematical proofs are out of the reach of topical arguments; and are not to be attacked by the equivocal use of words or declaration, that make so great a part of other discourses, - nay, even of controversies." (John Locke, "An Essay Concerning Human Understanding", 1690)

“Even the wisest and most prudent people often suffer from what Logicians call insufficient enumeration of cases." (Jacob Bernoulli , 1692)

"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)

🪷On Mind: On Reasoning (600-1499)

"To master this instrument the religious thinker must make a preliminary study of logic, just as the lawyer must study legal reasoning. This is no more heretical in the one case than in the other." (Averroes, "The Decisive Treatise", 1178)

"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)

"There are two modes of acquiring knowledge, namely, by reasoning and experience. 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", 1267)

"Geometry enlightens the intellect and sets one’s mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence." (Ibn Khaldun, "The Muqaddimah: An Introduction to History", 1377)

"Geometry enlightens the intellect and sets one's mind right. All of its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this convenient way, the person who knows geometry acquires intelligence." (Ibn Khaldun, cca. 14th century)

"Among all the studies of natural causes and reasoning, Light chiefly delights the beholder; and among the great features of mathematics the certainty of its demonstrations is what preeminently tends to elevate the mind of the investigator: Perspective, therefore, might be preferred to all the discourses and systems of the human learning." (Leonardo da Vinci, 1497)

🪷On Mind: On Reasoning (-599)

"Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." (Eudoxus, cca. 4th c. BC)

"But in synthesis, reversing the process, we start from the point which we reached last of all in the analysis, from the thing already known or admittedly true. We derive from it what preceded it in the analysis, and go on making derivations until, retracing our steps, we finally succeed in arriving at what is required. This procedure we call synthesis, or constructive solution, or progressive reasoning." (Pappus of Alexandria, cca. 4th c. BC)

"In analysis, we start from what is required, we take it for granted, and we draw consequences from it, and consequences from the consequences, till we reach a point that we can use as starting point in synthesis. For in analysis we assume what is required to be done as already done" (what is sought as already found, what we have to prove as true). We inquire from what antecedent the desired result could be derived; then we inquire again what could be the antecedent of that antecedent, and so on, until passing from antecedent to antecedent, we come eventually upon something already known or admittedly true. This procedure we call analysis, or solution backwards, or regressive reasoning." (Pappus of Alexandria, cca. 4th c. BC)

"It is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician demonstrative proofs." (Aristotle, cca. 4th c. BC)

"Thus, of all the honorable arts, which are carried out either naturally or proceed in imitation of nature, geometry takes the skill of reasoning as its field. It is hard at the beginning and difficult of access, delightful in its order, full of beauty, unsurpassable in its effect. For with its clear processes of reasoning it illuminates the field of rational thinking, so that it may be understood that geometry belongs to the arts or that the arts are from geometry." (Agennius Urbicus,"Controversies about Fields", cca. 4th c. BC)

[...] so external sensation is the image of this perception of the soul, which is in its essence truer and is a contemplation of forms alone without being affected. From these forms, from which the soul alone receives its lordship over the living being, come reasonings, and opinions and noetic acts; and this is precisely where ‘we’ are." (Plotinus, "Enneads", cca. 270 AD)

📁On Data: On Clusters

"The logarithmic transformation serves several purposes: (1) The resulting regression coefficients sometimes have a more useful theoretical interpretation compared to a regression based on unlogged variables. (2) Badly skewed distributions - in which many of the observations are clustered together combined with a few outlying values on the scale of measurement - are transformed by taking the logarithm of the measurements so that the clustered values are spread out and the large values pulled in more toward the middle of the distribution. (3) Some of the assumptions underlying the regression model and the associated significance tests are better met when the logarithm of the measured variables is taken." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"Exploratory data analysis (EDA) is a collection of techniques that reveal (or search for) structure in a data set before calculating any probabilistic model. Its purpose is to obtain information about the data distribution (univariate or multivariate), about the presence of outliers and clusters, to disclose relationships and correlations between objects and/or variables." (Ildiko E  Frank & Roberto Todeschini, "The Data Analysis Handbook", 1994)

"While classification is important, it can certainly be overdone. Making too fine a distinction between things can be as serious a problem as not being able to decide at all. Because we have limited storage capacity in our brain (we still haven't figured out how to add an extender card), it is important for us to be able to cluster similar items or things together. Not only is clustering useful from an efficiency standpoint, but the ability to group like things together (called chunking by artificial intelligence practitioners) is a very important reasoning tool. It is through clustering that we can think in terms of higher abstractions, solving broader problems by getting above all of the nitty-gritty details." (Joseph P Bigus,"Data Mining with Neural Networks: Solving business problems from application development to decision support", 1996)

"Data clusters are everywhere, even in random data. Someone who looks for an explanation will inevitably find one, but a theory that fits a data cluster is not persuasive evidence. The found explanation needs to make sense and it needs to be tested with uncontaminated data." (Gary Smith, "Standard Deviations", 2014)

"The general design choice of hierarchical aggregation is to construct the derived data of a hierarchical clustering of items in the original dataset and allow the user to interactively control the level of detail to show based on this hierarchy."  (Tamara Munzner, "Visualization Analysis and Design", 2014)

"Cluster analysis refers to the grouping of observations so that the objects within each cluster share similar properties, and properties of all clusters are independent of each other. Cluster algorithms usually optimize by maximizing the distance among clusters and minimizing the distance between objects in a cluster. Cluster analysis does not complete in a single iteration but goes through several iterations until the model converges. Model convergence means that the cluster memberships of all objects converge and don’t change with every new iteration." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"A form of machine learning in which the goal is to identify regularities in the data. These regularities may include clusters of similar instances within the data or regularities between attributes. In contrast to supervised learning, in unsupervised learning no target attribute is defined in the data set." (John D Kelleher & Brendan Tierney, "Data science", 2018)

"Unsupervised learning or clustering is a way of discovering hidden structures in unlabeled data. Clustering algorithms aim to discover latent patterns in unlabeled data using features to organize instances into meaningfully dissimilar groups." (Benjamin Bengfort et al, "Applied Text Analysis with Python: Enabling Language-Aware Data Products with Machine Learning", 2018)

📓On Literature: On Hallucinations (From Fiction to Science-Fiction)

"Kings are not born: they are made by artificial hallucination." (George B Shaw, "Man and Superman", 1903)

"It must be, I thought, one of the race's most persistent and comforting hallucinations to trust that 'it can't happen here' - that one's own time and place is beyond cataclysm." (John Wyndham, "The Day of the Triffids", 1951)

"An hallucination is not, strictly speaking, manufactured in the brain; it is received by the brain, like any 'real' sense datum, and the patient act in response to this to-him-very-real perception of reality in as logical a way as we do to our sense data. In any way to suppose he only 'thinks he sees it' is to misunderstand totally the experience of psychosis."" (Philip K. Dick, "Drugs, Hallucinations, and the Quest for Reality", 1964)

"Cyberspace. A consensual hallucination experienced daily by billions of legitimate operators, in every nation, by children being taught mathematical concepts. [...] A graphic representation of data abstracted from banks of every computer in the human system. Unthinkable complexity. Lines of light ranged in the nonspace of the mind, clusters and constellations of data." (William Gibson, "Neuromancer", 1984)

"He’d used decks in school, toys that shuttled you through the infinite reaches of that space that wasn’t space, mankind’s unthinkably complex consensual hallucination, the matrix, cyberspace, where the great corporate hotcores burned like neon novas, data so dense you suffered sensory overload if you tried to apprehend more than the merest outline." (William Gibson, "Count Zero", 1986)

📓On Literature: On Cyberspace (From Fiction to Science-Fiction)

"Cyberspace. A consensual hallucination experienced daily by billions of legitimate operators, in every nation, by children being taught mathematical concepts. [...] A graphic representation of data abstracted from banks of every computer in the human system. Unthinkable complexity. Lines of light ranged in the nonspace of the mind, clusters and constellations of data." (William Gibson, "Neuromancer", 1984)

"He’d used decks in school, toys that shuttled you through the infinite reaches of that space that wasn’t space, mankind’s unthinkably complex consensual hallucination, the matrix, cyberspace, where the great corporate hotcores burned like neon novas, data so dense you suffered sensory overload if you tried to apprehend more than the merest outline." (William Gibson, "Count Zero", 1986) 

"In cyberspace, she noted, there are no shadows." (William Gibson, "Mona Lisa Overdrive", 1988)

"All I knew about the word 'cyberspace'when I coined it, was that it seemed like an effective buzzword. It seemed evocative and essentially meaningless. It was suggestive of something, but had no real semantic meaning, even for me, as I saw it emerge on the page." (William Gibson, No Maps for These Territories", 2000)

"Although personally, I think cyberspace means the end of our species." (Michael Crichton, "The Lost World: A Novel", 2001)

"We need to bridge our sense of loneliness and disconnection with a sense of community and continuity even if we must manufacture it from our time on the Web and our use of calling cards to connect long distance. We must log on somewhere, and if it is only in cyberspace, that is still far better than nowhere at all." (Julia Cameron, "God is No Laughing Matter", 2001)

"That's how it is online - there's no time in cyberspace. It's almost like everything physical evaporates, and it's just your mind and the different sites floating in a void." (Tim Tharp, "The Spectacular Now", 2008)

🌌On Cosmology: On Planets (1850-1899)

"The circle is one of the noblest representation of the Deity, in his noble works of human nature. It bounds, determines, governs, and dictates bounds, dictates space, bounds latitude and longitude, refers to space, the sun, moon, and all the planets, in direction, brings to the mind thoughts of eternity, and concentrates the mind to imagine for itself the distance and space it comprehends. It rectifies all boundaries; it is the key to information of the knowledge of God." (John Davis, "The Measure of the Circle", 1854)

"The genuine spirit of Mathesis is devout. No intellectual pursuit more truly leads to profound impressions of the existence and attributes of a Creator, and to a deep sense of our filial relations to him, than the study of these abstract sciences. Who can understand so well how feeble are our conceptions of Almighty Power, as he who has calculated the attraction of the sun and the planets, and weighed in his balance the irresistible force of the lightning? Who can so well understand how confused is our estimate of the Eternal Wisdom, as he who has traced out the secret laws which guide the hosts of heaven, and combine the atoms on earth? Who can so well understand that man is made in the image of his Creator, as he who has sought to frame new laws and conditions to govern imaginary worlds, and found his own thoughts similar to those on which his Creator has acted?" (Thomas Hill, "The Imagination in Mathematics", The North American Review Vol. 85 (176), 1857)

"SYSTEM (σύστημα, σύν ἵστημιavu, to place together) - is a full and connected view of all the truths of some department of knowledge. An organized body of truth, or truths arranged under one and the same idea, which idea is as the life or soul which assimilates all those truths. No truth is altogether isolated. Every truth has relation to some other. And we should try to unite the facts of our knowledge so as to see them in their several bearings. This we do when we frame them into a system. To do so legitimately we must begin by analysis and end with synthesis. But system applies not only to our knowledge, but to the objects of our knowledge. Thus we speak of the planetary system, the muscular system, the nervous system. We believe that the order to which we would reduce our ideas has a foundation in the nature of things. And it is this belief that encourages us to reduce our knowledge of things into systematic order. The doing so is attended with many advantages. At the same time a spirit of systematizing may be carried too far. It is only in so far as it is in accordance with the order of nature that it can be useful or sound." (William Fleming, "Vocabulary of philosophy, mental, moral, and metaphysical; with quotations and references; for the use of students", 1857) 

"The pursuit of mathematical science makes its votary appear singularly indifferent to the ordinary interests and cares of men. Seeking eternal truths, and finding his pleasures in the realities of form and number, he has little interest in the disputes and contentions of the passing hour. His views on social and political questions partake of the grandeur of his favorite contemplations, and, while careful to throw his mite of influence on the side of right and truth, he is content to abide the workings of those general laws by which he doubts not that the fluctuations of human history are as unerringly guided as are the perturbations of the planetary hosts." (Thomas Hill, "The Imagination in Mathematics", The North American Review Vol. 85 (176), 1857)

"Although the astronomer has achieved many successes in studying comets, yet these objects still remain outside the surveyed fields of astronomy - now, as in the old days when men spoke of sun and moon, planet and stars, as including all the members of the heavenly host." (Richard A Proctor, "The Twentieth Century, Whence Came the Comets?", Vol. 29 (111), 1886)

🌌On Cosmology: On Planets (1800-1849)

"Nothong is more simple than the plan of the Solar System: we have only to conceive the Sun in one of the focuses of the eliptic orbits, Which the planets and comets describe around him. We will bestow our chief attention on comets, as on them will depend all the merit of novelty to be found in the following theory." (Johann H Lambert, "The System of the World", 1800)

"Surrounded as we are by an infinite variety of phenomena, which continually succeed each other in the heavens and on the earth, philosophers have succeeded in recognizing the small number of general laws to which matter is subject in its motions. To them, all nature is obedient; and everything is as necessarily derived from them, as the return of the seasons; so that the curve which is described by the lightest atom that seems to be driven at random by the winds, is regulated by laws as certain as those which confine the planets to their orbits." (Pierre-Simon Laplace, "System of the World" Vol. 1, 1809)

"The perturbations which the motions of planets suffer from the influence other planets, are so small and so slow that they only become sensible after a long interval of time; within a shorter time, or even within one or several revolutions, according to circumstances, the motion would differ so little from motion exactly described, according to the laws of Kepler, in a perfect ellipse, that observations cannot show the difference. As long as this is true, it not be worth while to undertake prematurely the computation of the perturbations, but it will be sufficient to adapt to the observations what we may call an osculating conic section: but, afterwards, when the planet has been observed for a longer time, the effect of the perturbations will show itself in such a manner, that it will no longer be possible to satisfy exactly all the observations by a purely elliptic motion; then, accordingly, a complete and permanent agreement cannot be obtained, unless the perturbations are properly connected with the elliptic motion." (Carl F Gauss, "Theoria motus corporum coelestium in sectionibus conicis solem ambientium", 1809) 

"The chemical compounds are comparable to a system of planets in that the atoms are held together by chemical affinity. They may be more or less numerous, simple or complex in composition, and in the constitution of the materials, they play the same role as Mars and Venus do in our planetary system, or the compound members such as our earth with its moon, or Jupiter with its satellites. [...] If in such a system a particle is replaced by one of different character, the equilibrium can persist, and then the new compound will exhibit properties similar to those shown by the original substance." (Jean-Baptiste-André Dumas, "Mémoire sur la loi des substitutions et la théorie des types" [*Memoir on the Law of Substitutions and the Theory of Types"], 1810)

"Physical astronomy is the science which compares and identifies the laws of motion observed on earth with the motions that take place in the heavens; and which traces, by an uninterrupted chain of deduction from the great principle that governs the universe, the revolutions and rotations of the planets, and the oscillations of the fluids at their surfaces; and which estimates the changes the system has hitherto undergone, or may hereafter experience - changes which require millions of years for their accomplishment." (Mary Somerville, "The Connection of the Physical Sciences", 1834)

"We can allow satellites, planets, suns, universe, nay whole systems of universe to be governed by laws, but the smallest insect, we wish to be created at once by special act […] Our faculties are more fitted to recognize the wonderful structure of a beetle than a Universe." (Charles Darwin, "Notebook N", 1838)

"If proof were wanted of the inexhaustible fertility of astronomical science in points of novelty and interest, it would suffice to adduce the addition to the list of members of our system of no less than eight new planets and satellites during the preparation of these sheets for the press." (John F W Herschel,"Outlines of Astronomy", 1849)

🌌On Cosmology: On Planets (1975-1999)

"Apart from an understanding of the solar system as a whole, it is becoming clear that information about any planet or satellite illuminates our knowledge of the others. In particular, if we are to understand the Earth, we must have a comprehensive knowledge of the other planets."(Carl Sagan,"The Solar System", Scientific American Vol. 233 (3), 1975)

"Each of the major sciences has contributed an essential ingredient in our long retreat from an initial belief in our own cosmic importance. Astronomy defined our home as a small planet tucked away in one corner of an average galaxy among millions; biology took away our status as paragons created in the image of God; geology gave us the immensity of time and taught us how little of it our own species has occupied." (Stephen J Gould, "Uniformity and Catastrophe", 1977)

"It is unscientific to say that within the many billions of galactic systems, ours is the only planet that supports life in advanced form. Nature shuns one of a kind as much as it abhors a vacuum. Given infinite time and space, anything that occurs at one place or time in the universe will occur elsewhere or ‘elsewhen’." (Norman Cousins, "Rendezvous with Infinity", Cosmic Search Magazine Vol. 1 (1), 1979)

"To call a model an idealization is to suggest that the model is a simplification of what occurs in reality, usually a simplification which omits some relevant features, such as the extended mass of the planets or, in the example of the circuit model, the resistance in the bypass capacitor. Sometimes the omitted factors make only an insignificant contribution to the effect under study. But that does not seem to be essential to idealizations, especially to the idealizations that in the end are applied by engineers to study real things. In calling something an idealization it seems not so important that the contributions from omitted factors be small, but that they be ones for which we know how to correct. If the idealization is to be of use, when the time comes to apply it to a real system we had better know how to add back the contributions of the factors that have been left out. In that case the use of idealizations does not seem to counter realism: either the omitted factors do not matter much, or in principle we know how to treat them." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"A law explains a set of observations; a theory explains a set of laws. […] a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. […] Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism." (John L Casti, "Searching for Certainty: How Scientists Predict the Future", 1990)

"In practice, the intelligibility of the world amounts to the fact that we find it to be algorithmically compressible. We can replace sequences of facts and observational data by abbreviated statements which contain the same information content. These abbreviations we often call 'laws of Nature.' If the world were not algorithmically compressible, then there would exist no simple laws of nature. Instead of using the law of gravitation to compute the orbits of the planets at whatever time in history we want to know them, we would have to keep precise records of the positions of the planets at all past times; yet this would still not help us one iota in predicting where they would be at any time in the future. This world is potentially and actually intelligible because at some level it is extensively algorithmically compressible. At root, this is why mathematics can work as a description of the physical world. It is the most expedient language that we have found in which to express those algorithmic compressions." (John D Barrow, "New Theories of Everything", 1991)

"There is a multilayering of global networks in the key strategic activities that structure and destructure the planet. When these multilayered networks overlap in some node, when there is a node that belongs to different networks, two major consequences follow. First, economies of synergy between these different networks take place in that node: between financial markets and media businesses; or between academic research and technology development and innovation; between politics and media." (Manuel Castells, "The Rise of the Network Society", 1996)

🌌On Cosmology: On Planets (2000-)

"The mathematics of physics resides in physical phenomena themselves - there are ellipses in the elliptical orbits of the planets, fractals in the fractal shapes of leaves and branches, logarithms in the logarithmic spirals of snails. This means that 'the books of nature is written in mathematics', which implies that the language of mathematics is the language of nature and that only those who lznow mathematics can truly understand nature." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"In string theory one studies strings moving in a fixed classical spacetime. […] what we call a background-dependent approach. […] One of the fundamental discoveries of Einstein is that there is no fixed background. The very geometry of space and time is a dynamical system that evolves in time. The experimental observations that energy leaks from binary pulsars in the form of gravitational waves - at the rate predicted by general relativity to the […] accuracy of eleven decimal place - tell us that there is no more a fixed background of spacetime geometry than there are fixed crystal spheres holding the planets up." (Lee Smolin, "Loop Quantum Gravity", The New Humanists: Science at the Edge, 2003)

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The calculus is a theory of continuous change - processes that move smoothly and that do not stop, jerk, interrupt themselves, or hurtle over gaps in space and time. The supreme example of a continuous process in nature is represented by the motion of the planets in the night sky as without pause they sweep around the sun in elliptical orbits; but human consciousness is also continuous, the division of experience into separate aspects always coordinated by some underlying form of unity, one that we can barely identify and that we can describe only by calling it continuous." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"At all events, our world pictures may have components of all three kinds: perceptual, conceptual, and praxiological (action-theoretical).  This is because there are three gates to the outer world: perception, conception, and action. However, ordinarily only one or two of them need be opened: combinations of all three, as in building a house according to a blueprint, are the exception.  We may contemplate a landscape without forming either a conceptual model of it or a plan to act upon it.  And we may build a theoretical model of an imperceptible thing, such as an invisible extrasolar planet, on which we cannot act." (Mario Bunge, "Chasing Reality: Strife over Realism", 2006)

"Physicists use mathematics to study what they amusingly call the real world. It is real, in a sense, but much of physics addresses rather artificial aspects of reality, such as a lone electron or a solar system with only one planet. Physicists are often scathing about proofs, partly out of fear, but also because experiment gives them a very effective way to check their assumptions and calculations." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"Differential equations have found their way into all areas of physics from the motion of planets around the Sun to standing waves on a rope or a drum, to electrical properties of conductors, and the behavior of electromagnetic fields and beyond. As is always the case, no mathematics can draw more attention than that which deals directly with Nature. The urgency of finding solutions to these differential equations prompted many mathematicians of the latter part of the eighteenth and the beginning of the nineteenth centuries to concentrate heavily on certain specific differential equations. It appeared that every differential equation dictated by Nature gave rise to a new function. The most common scheme for solving these differential equations was to assume a power series solution, substitute the assumed solution in the differential equation, and determine the (unknown) coefficients from the resulting equality of power series." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"One of the great advantages of vectors is their ability to express results independent of any specific coordinate systems. Physical laws are always coordinate-independent. For example, when we write F = ma both F and a could be expressed in terms of Cartesian, spherical, cylindrical, or any other convenient coordinate system. This independence allows us the freedom to choose the coordinate systems most convenient for the problem at hand. For example, it is extremely difficult to solve the planetary motions in Cartesian coordinates, while the use of spherical coordinates facilitates the solution of the problem tremendously." (Sadri Hassani, "Mathematical Methods: For Students of Physics and Related Fields", 2009)

"Standard economists don't seem to understand exponential growth. Ecological economics recognizes that the economy, like any other subsystem on the planet, cannot grow forever. And if you think of an organism as an analogy, organisms grow for a period and then they stop growing. They can still continue to improve and develop, but without physically growing, because if organisms did that you’d end up with nine-billion-ton hamsters." (Robert Costanza, "What is Ecological economics", 2010)

"Probability is often expressed using large but finite numbers: ‘one in a thousand’, ‘one in a million’. But perhaps the probability of life, intelligent life, appearing somewhere in our universe is infinitesimal. If so, a universe would need infinitely many planets to produce even a finite number of civilisations (i.e., one)." (Daniel Tammet, "Thinking in Numbers" , 2012)

"The goal of physics is to explain the workings of the nonliving world. At first, philosophers described the properties of real objects: the wandering of planets across the night sky, the formation of ice, or the sound of a lyre. When attention turned to things that couldn’t be seen or measured so easily, physicists invented mechanical models to take the place of real things." (Hans C von Baeyer, "QBism: The future of quantum physics", 2016)

📓On Literature: On Satellites

"They have likewise discovered two lesser stars, or satellites, which revolve about Mars, whereof the innermost is distant from the centre of the primary planet exactly three of his diameters, and the outermost five; the former revolves in the space of ten hours, and the latter in twenty one and a half." (Jonathan Swift, "Gulliver's Travels", 1726)

"Thanks to the courage and devotion of three men, this project of sending a bullet to the moon, once seen as a futile enterprise, had already produced concrete results, with incalculable consequences. The voyagers, imprisoned in their new satellite, had not reached their destination, but at least they had become part of the lunar world; they were in orbit around the celebrity of the night, and, for the first time, the human eye could penetrate all her mysteries." (Jules Verne, "From the Earth to the Moon", 1865)

"Nothing can be prettier than to see the movement, in perfectly harmonic relations, of planets round their centres, of satellites around planets, of suns, with their planets and satellites, around their centres, and of these in turn around theirs. And to persons who have loved earth as much as I do, and who, while at school there, have studied other worlds and stars, then distant, as carefully as I have, nothing, as I say, can be more charming than to see at once all this play and interplay; to see comets passing from system to system, warming themselves now at one white sun, and then at a party-colored double; to see the people on them changing customs and costumes as they change their light, and to hear their quaint discussions as they justify the new and ridicule the old." (Edward E Hale, "Hands Off", 1881)

"On the earth, satellite of a star speeding through space, living things had arisen under the influence of conditions which were part of the planet's history; and as there had been a beginning of life upon it, so, under the influence of other conditions, there would be an end: man, no more significant than other forms of life, had come not as the climax of creation but as a physical reaction to the environment." (W Somerset Maugham, "Of Human Bondage", 1915)

"I get so scared sometimes. The suddenness of our age! Electronics, rockets, earth satellites, supersonic flight, and now this. It's not so much who invents them. It's the fact they exist. […] Everything's going so fast, I'm just not ready to take it all in. It's, it's all so quick." (James Clavel, "The Fly", [adapted from a 1957 short story by George Langelaan] 1958)

"The Solar System, comprising the nine known worlds of our Sun and their numerous satellites, is a relatively compact structure, a snug little celestial oasis in an endless desert." (Arthur C Clarke, "The Challenge of the Spaceship", 1959) 

"[...] we know now that comet and asteroid impacts have changed history, and soon we will have the technology to avert that disaster. [...] the dinosaurs became extinct because they didn’t have a space program." (Arthur C Clarke, Discover Magazine, [Interview] 1999)

🌌On Cosmology: On Satellites

"The smallest planets are situated nearest the sun and each other; whereas Jupiter and Saturn, that are vastly greater than the rest, and have many satellites about them, are wisely removed to the extreme regions of the system, and placed at an immense distance one from the other." (Richard Bentley, “A Confutation of Atheism from the Origin and Frame of the World",  "Boyle Lecture sermon", 1692)

"Astronomy has already made an important step, in making us acquainted with the motion of the Earth, and the epicycles which the Moon and the satellites describe on the orbits of their respective planets. But if ages were necessary in order to know the motions of the planetary system, what a great length of time must be required for the determination of the motions of the Sun and the stars; notwithstanding this, such motions appear to be already indicated by observations." (Pierre-Simon Laplace, "Exposition du système du monde" ["Exposition of the System of the World"], 1796) 

"We can allow satellites, planets, suns, universe, nay whole systems of universe to be governed by laws, but the smallest insect, we wish to be created at once by special act […] Our faculties are more fitted to recognize the wonderful structure of a beetle than a Universe." (Charles Darwin, "Notebook N", 1838)

"If proof were wanted of the inexhaustible fertility of astronomical science in points of novelty and interest, it would suffice to adduce the addition to the list of members of our system of no less than eight new planets and satellites during the preparation of these sheets for the press." (John F W Herschel,"Outlines of Astronomy", 1849)

"Like a shower of stars the worlds whirl, borne along by the winds of heaven, and are carried down through immensity, suns, earths, satellites, comets, shooting stars, humanities, cradles, graves, atoms of the infi nite, seconds of eternity, perpetually transform beings and things." (Camille Flammarion, "Popular Astronomy: A General Description of the Heavens", 1899)

"To place one’s feet on the soil of asteroids, to lift a stone from the moon with your hand, to construct moving stations in ether space, to organize inhabited rings around Earth, moon and sun, to observe Mars at the distance of several tens of miles, to descend to its satellites or even to its own surface - what could be more insane! However, only at such a time when reactive devices are applied, will a great new era begin in astronomy: the era of more intensive study of the heavens." (Konstantin E Tsiolkovsky,"The Exploration of Cosmic Space by Means of Reaction Devices", 1903

"From the smallest satellite slave of the smallest star to the largest super-galaxy of worlds in space, everything bows to the first law of nature. Chaos and caprice do not exist." (John H Bradley Jr, "Parade of the Living", 1930)

"The satellite is a natural extension of rockets, which are natural extensions of planes and balloons, which are natural extensions of man's climbing trees and mountains in order to get up higher and thus have a better view." (James Van Allen, "Science: Reach Into Space", Time, 1959) 

"The Solar System, comprising the nine known worlds of our Sun and their numerous satellites, is a relatively compact structure, a snug little celestial oasis in an endless desert." (Arthur C Clarke, "The Challenge of the Spaceship", 1959)

"Apart from an understanding of the solar system as a whole, it is becoming clear that information about any planet or satellite illuminates our knowledge of the others. In particular, if we are to understand the Earth, we must have a comprehensive knowledge of the other planets."(Carl Sagan,"The Solar System", Scientific American Vol. 233 (3), 1975)

"New computer calculations, based on observations from the Voyager satellites and ground-based telescopes, show that just about every moon except the earth's-has experienced millions of years [sic] chaotic tumbling. A tumbling moon falls end over end, twists sideways, speeds up, slowsdown, all the time obeying Newton's completely deterministic laws of motion, yet defy [sic] prediction in a way that scientists used to consider impossible." (James Gleick, 1987)

"A law explains a set of observations; a theory explains a set of laws. […] a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. […] Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism." (John L Casti, "Searching for Certainty: How Scientists Predict the Future", 1990)

"[...] we know now that comet and asteroid impacts have changed history, and soon we will have the technology to avert that disaster. [...] the dinosaurs became extinct because they didn’t have a space program." (Arthur C Clarke, Discover Magazine, [Interview] 1999) 

"In every language, from Arabic to Zulu to calligraphy to shorthand to math to music to art to wrought stone, everything from the Unified Field Theory to a curse to a sixpenny nail to an orbiting satellite, anything expressed is a net around some idea." (Richard Bach, "One", 1988)

📡On Technology: On Radio-Television

"One ought to be ashamed to make use of the wonders of science embodied in a radio set, the while appreciating them as little as a cow appreciates the botanic marvels in the plants she munches." (Albert Einstein, [remarks opening the Seventh German Radio Exhibition at Berlin] 1930)

"Typography is not only a technology but is in itself a natural resource or staple, like cotton or timber or radio; and, like any staple, it shapes not only private sense ratios but also patterns of communal interdependence." (Marshall McLuhan, "The Gutenberg Galaxy", 1962)

"Although the medium is the message, the controls go beyond programming. The restraints are always directed to the content, which is always another medium. The content of the press is literary statement, as the content of the book is speech, and the content of the movie is the novel. So the effects of radio are quite independent of its programming."  (Marshall McLuhan, "Understanding Media", 1964)

"If we sit and talk in a dark room, words suddenly aquire new meanings and different textures... and on the radio. Given only the sound of a play, we have to fill in all of the senses, not just the sight of the action. So much do-it-yourself, or completion and closure of action, develops a kind of independent isolation in the young that makes them remote and inaccessible." (Marshall McLuhan, "Understanding Media", 1964)

"Radio comes to us ostensibly with person to person directness that is private and intimate, while in more urgent fact, it is really a subliminal echo chamber of magic power to touch remote and forgotten chords." (Marshall McLuhan, "Understanding Media", 1964)

"The message of radio is one of violent, unified implosion and resonance." (Marshall McLuhan, "Understanding Media", 1964) 

"To the optical astronomer, radio data serves like a good dog on a hunt." (Fred Hoyle, "Galaxies, Nuclei and Quasars". 1965) 

"The important prediction is not the automobile, but the parking problem; not radio, but the soap opera; not the income tax, but the expense account; not the Bomb, but the nuclear stalemate." (Isaac Asimov, "From earth to heaven", 1972)

"It is a striking thought that ten years of radio astronomy have taught humanity more about the creation and organization of the universe than thousands of years of religion and philosophy." (Paul Davies, "Space and Time in the Modern Universe", 1977) 

"The total amount of energy from outside the solar system ever received by all the radio telescopes on the planet Earth is less than the energy of a single snowflake striking the ground." (Carl Sagan, "Cosmos" 1980)

"Television contracts the imagination and radio expands it." (Terry Wogan, Observer, 1984) 

"While amplitude damped spark gap radio waves are suitable for transmitting keyed on-off coded signals, such waves are not suitable for voice (or music) transmission. The reason for this can be understood once it is realized that for information to be transmitted it is necessary for something to change. If nothing changes, then the receiver of the signals can predict with perfect accuracy what the future nature of the signals will be - exactly what they werel It is unnecessary, actually, to transmit anything if nothing changes." (Paul J Nahin, "The Science of Radio", 1995)

"The mind is more powerful than any imaginable particle accelerator, more sensitive than any radio receiver or the largest optical telescope, more complete in its grasp of information than any computer: the human body- its organs, its voice, its powers of locomotion, and its imagination- is a more-than-sufficient means for the exploration of any place, time or energy level in the universe." (Terence McKenna,"True Hallucinations: And, the Archaic Revival", 1998)

"New ways of seeing can disclose new things: the radio telescope revealed quasars and pulsars, and the scanning electron microscope showed the whiskers of the dust mite. But turn the question around: Do new things make for new ways of seeing?" (William L Heat-Moon,"Blue Highways: A Journey into America", 2012) 

📡On Technology: On Internet (2000-2009)

"To get a big company moving fast, especially on a many-headed opportunity like the Internet, you have to have hundreds of people participating and coming up with ideas." (Bill Gates, "Business at the Speed of Thought: Succeeding in the Digital Economy", 2000)

"We must also promote global access to the Internet. We need to bridge the digital divide not just within our country, but among countries. Only by giving people around the world access to this technology can they tap into the potential of the Information Age." (Al Gore, [IPI speech] 2000)

"The fundamental digital divide is no measured by the number of connections to the Internet, but by the consequences of both connection and lack of connection." (Manuel Castells, "The Internet Galaxy: Reflections on the Internet, Business, and Society" 2001) 

"The Internet Culture is the culture of the creators of the Internet." (Manuel Castells, "The Internet Galaxy: Reflections on the Internet, Business, and Society" 2001)

"The Internet gives us everything and forces us to filter it not by the workings of culture, but with our own brains. This risks creating six billion separate encyclopedias, which would prevent any common understanding whatsoever." (Umberto Eco, "This is Not the End of the Book: A conversation curated by Jean-Philippe de Tonnac", 2001)

"The problems with software are just examples of the problems found generally with creativity. Our trend in copyright law has been to enclose as much as we can; the consequence of this enclosure is a stifling of creativity and innovation. If the Internet teaches us anything, it is that great value comes from leaving core resources in a commons, where they're free for people to build upon as they see fit." (Lawrence Lessig, "May the Source Be With You" 2001)

"The new information technology [...] Internet and e-mail [...] have practically eliminated the physical costs of communications." (Peter Drucker, "Managing in the Next Society" 2002)

"In the end, science offers us the only way out of politics. And if we allow science to become politicized, then we are lost. We will enter the Internet version of the dark ages, an era of shifting fears and wild prejudices, transmitted to people who don't know any better." (Michael Crichton, "Environmentalism as a Religion", [speech] 2003)

"One of the powerful things about networking technology like the Internet or the Web or the Semantic Web [...] is that the things we've just done with them far surpass the imagination of the people who invented them." (Tim Berners-Lee, "The Semantic Web: An Introduction", 2003)

"The Internet is a medium only at the bit level. At the human level, it is a conversation that, because of the persistence and linkedness of pages, has elements of a world. It could only be a medium if we absolutely didn't care about it." (David Weinberger, "The Internet is not a medium", [article] 2004) 

"It is not only a metaphor to transform the Internet to a superbrain with self-organizing features of learning and adapting. Information retrieval is already realized by neural networks adapting to the information preferences of a human user with synaptic plasticity. In sociobiology, we can 1 earn from populations of ants and termites how to organize traffic and information processing by swarm intelligence. From a technical point of view, we need intelligent programs distributed in the nets. There are already more or less intelligent virtual organisms {'agents'), learning, self-organizing and adapting to our individual preferences of information, to select our e-mails, to prepare economic transactions or to defend the attacks of hostile computer viruses, like the immune system of our body." (Klaus Mainzer, "Complexity Management in the Age of Globalization", 2006)

"Scale-free networks are particularly vulnerable to intentional attack: if someone wants to wreck the whole network, he simply needs to identify and destroy some of its hubs. And here we see how our world’s increasing connectivity really matters. Scientists have found that as a scale-free network like the Internet or our food-distribution system grows- as it adds more nodes - the new nodes tend to hook up with already highly connected hubs." (Thomas Homer-Dixon, "The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization", 2006)

"The internet's perfect for all manner of things, but productive discussion ain't one of them. It provides scant room for debate and infinite opportunities for fruitless point-scoring: the heady combination of perceived anonymity, gestated responses, random heckling and a notional live audience quickly conspire to create a perfect storm of perpetual bickering." (Charlie Brooker, "Dawn of the Dumb" 2007)

"The real power of tribes has nothing to do with the Internet and everything to do with people." (Seth Godin, "Tribes: We need you to lead us", 2008)