On Motion (2000-2025)

"The motion of the mind is conveyed along a cloud of meaning. There is this paradox that we get to meaning only when we strip the meaning from symbols." (David Berlinski, "The Advent of the Algorithm: The Idea that Rules the World", 2000)

"Entropy [...] is the amount of disorder or randomness present in any system. All non-living systems tend toward disorder; left alone they will eventually lose all motion and degenerate into an inert mass. When this permanent stage is reached and no events occur, maximum entropy is attained. A living system can, for a finite time, avert this unalterable process by importing energy from its environment. It is then said to create negentropy, something which is characteristic of all kinds of life." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"Limiting factors in population dynamics play the role in ecology that friction does in physics. They stop exponential growth, not unlike the way in which friction stops uniform motion. Whether or not ecology is more like physics in a viscous liquid, when the growth-rate-based traditional view is sufficient, is an open question. We argue that this limit is an oversimplification, that populations do exhibit inertial properties that are noticeable. Note that the inclusion of inertia is a generalization - it does not exclude the regular rate-based, first-order theories. They may still be widely applicable under a strong immediate density dependence, acting like friction in physics." (Lev Ginzburg & Mark Colyvan, "Ecological Orbits: How Planets Move and Populations Grow", 2004)

"Physically, the stability of the dynamics is characterized by the sensitivity to initial conditions. This sensitivity can be determined for statistically stationary states, e.g. for the motion on an attractor. If this motion demonstrates sensitive dependence on initial conditions, then it is chaotic. In the popular literature this is often called the 'Butterfly Effect', after the famous 'gedankenexperiment' of Edward Lorenz: if a perturbation of the atmosphere due to a butterfly in Brazil induces a thunderstorm in Texas, then the dynamics of the atmosphere should be considered as an unpredictable and chaotic one. By contrast, stable dependence on initial conditions means that the dynamics is regular." (Ulrike Feudel et al, "Strange Nonchaotic Attractors", 2006)

"Heat is the energy of random chaotic motion, and entropy is the amount of hidden microscopic information." (Leonard Susskind, "The Black Hole War", 2008)

"Perhaps the simplest way to explain symmetry is to follow the operational approach used by mathematicians: a symmetry is a motion. That is, suppose you have an object and pick it up, move it around, and set it down. If it is impossible to distinguish between the object in its original and final positions, we say that it has a symmetry." (Michael Field & Martin Golubitsky, "Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature" 2nd Ed, 2009)

"Chaos provides order. Chaotic agitation and motion are needed to create overall, repetitive order. This ‘order through fluctuations’ keeps dynamic markets stable and evolutionary processes robust. In essence, chaos is a phase transition that gives spontaneous energy the means to achieve repetitive and structural order." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"For a moving object, time contracts. Not only is there no single time for different places - there is not even a single time for any particular place. A duration can be associated only with the movement of something, with a given trajectory." (Carlo Rovelli, "The Order of Time", 2018)

"If one section of the molecules is still, it becomes stirred up by the frenzy of neighboring ones that set them in motion, too: the agitation spreads, the molecules bump into and shove each other. In this way, cold things are heated in contact with hot ones: their molecules become jostled by hot ones and pushed into ferment. That is, they heat up." (Carlo Rovelli, "The Order of Time", 2018)

"The entropy of the world does not depend only on the configuration of the world; it also depends on the way in which we are blurring the world, and this depends on what the variables of the world are that we interact with. That is to say, on the variables with which our part of the world interacts. [...] The difference between past and future does not lie in the elementary laws of motion; it does not reside in the deep grammar of nature. It is the natural disordering that leads to gradually less particular, less special situations."" (Carlo Rovelli, "The Order of Time", 2018)

On Motion (1900-1949)

"Motion consists merely in the occupation of different places at different times, subject to continuity. [...] There is no transition from place to place, no consecutive moment or consecutive position, no such thing as velocity except in the sense of a real number, which is the limit of a certain set of quotients." (Bertrand Russell's, "Principles of Mathematics", 1903)

"Two divisions are distinguished in all natural sciences - 'statics' which deals with forms in equilibrium, and 'dynamics' which deals with the same forms, as well as their motion, in the process of change. […] Statics always evolves earlier than dynamics, the former being then reconstructed under the influence of the latter. The relationship between mathematics and tektology is seen to be similar: one represents the standpoint of organizational statics and the other - that of organizational dynamics. The latter standpoint is the more general, for equilibrium is only a particular case of motion, and in essence, is just an ideal case resulting from changes which are completely equal but quite opposite in direction." (Alexander Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Our model of Nature […] should be like an engine with movable parts. We need not fix the position of any one lever; that is to be adjusted from time to time as the latest observations indicate. The aim of the theorist is to know the train of wheels which the lever sets in motion - that binding of the parts which is the soul of the engine." (Sir Arthur S Eddington, The Internal Constitution of Stars, Nature Vol. 106 (2603), 1920)

"But the star-glistered salver of infinity, The circle, blind crucible of endless space, Is sluiced by motion,-subjugated never." (Hart Crane. "The Bridge", 1930)

"Let us now discuss the extent of the mathematical quality in Nature. According to the mechanistic scheme of physics or to its relativistic modification, one needs for the complete description of the universe not merely a complete system of equations of motion, but also a complete set of initial conditions, and it is only to the former of these that mathematical theories apply. The latter are considered to be not amenable to theoretical treatment and to be determinable only from observation." (Paul A M Dirac, "The Relation Between Mathematics And Physics", Proceedings of the Royal Society of Edinburgh", 1938-1939)

"An isolated system or a system in a uniform environment (which for the present consideration we do best to include as a part of the system we contemplate) increases its entropy and more or less rapidly approaches the inert state of maximum entropy. We now recognize this fundamental law of physics to be just the natural tendency of things to approach the chaotic state (the same tendency that the books of a library or the piles of papers and manuscripts on a writing desk display) unless we obviate it. (The analogue of irregular heat motion, in this case, is our handling those objects now and again without troubling to put them back in their proper places.) (Erwin Schrödinger, "What is Life?", 1944)

"[…] the universe is not a rigid and inimitable edifice where independent matter is housed in independent space and time; it is an amorphous continuum, without any fixed architecture, plastic and variable, constantly subject to change and distortion. Wherever there is matter and motion, the continuum is disturbed. Just as a fi sh swimming in the sea agitates the water around it, so a star, a comet, or a galaxy distorts the geometry of the spacetime through which it moves." (Lincoln Barnett, "The Universe and Dr. Einstein", 1948)

On Motion (1950-1999)

"Space and time are commonly regarded as the forms of existence of the real world, matter as its substance. A definite portion of matter occupies a definite part of space at a definite moment of time. It is in the composite idea of motion that these three fundamental conceptions enter into intimate relationship." (Hermann Weyl, "Space, Time, Matter", 1952)

"Mathematics, springing from the soil of basic human experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas." (Saunders MacLane, "Of Course and Courses"The American Mathematical Monthly, Vol 61, No 3, 1954)

"The ultimate origin of the difficulty lies in the fact (or philosophical principle) that we are compelled to use the words of common language when we wish to describe a phenomenon, not by logical or mathematical analysis, but by a picture appealing to the imagination. Common language has grown by everyday experience and can never surpass these limits. Classical physics has restricted itself to the use of concepts of this kind; by analysing visible motions it has developed two ways of representing them by elementary processes; moving particles and waves. There is no other way of giving a pictorial description of motions - we have to apply it even in the region of atomic processes, where classical physics breaks down." (Max Born, "Atomic Physics", 1957)

"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"While static stability is concerned with the tendency of a displaced body to return to equilibrium, dynamic stability is concerned with the resulting motion with time. If an object is disturbed from equilibrium, the time history of the resulting motion indicates the dynamic stability of the system. In general, the system will demonstrate positive dynamic stability if the amplitude of the motion decreases with time." (Hugh H Hurt, "Aerodynamics for Naval Aviators", 1960)

"Questions are the engines of intellect, the cerebral machines which convert energy to motion, and curiosity to controlled inquiry." (David H Fischer, "Historians’ Fallacies", 1970)

"The eye projects and focuses the inner image (idea) onto the physical world in the same manner that a motion picture camera transfers an image onto a screen. The mouth creates words. The ears create sound. The difficulty in understanding this principle is due to the fact that we’ve taken it for granted that the image and sound already exist for the senses to interpret. Actually the senses are the channels of creation by which idea is projected into material expression." (Jane Roberts,"The Seth Material", 1970)

"Systems, acting dynamically, produce (and incidentally, reproduce) their own boundaries, as structures which are complementary (necessarily so) to their motion and dynamics. They are liable, for all that, to instabilities chaos, as commonly interpreted of chaotic form, where nowadays, is remote from the random. Chaos is a peculiar situation in which the trajectories of a system, taken in the traditional sense, fail to converge as they approach their limit cycles or 'attractors' or 'equilibria'. Instead, they diverge, due to an increase, of indefinite magnitude, in amplification or gain." (Gordon Pask, "Different Kinds of Cybernetics", 1992)

"The classic example of chaos at work is in the weather. If you could measure the positions and motions of all the atoms in the air at once, you could predict the weather perfectly. But computer simulations show that tiny differences in starting conditions build up over about a week to give wildly different forecasts. So weather predicting will never be any good for forecasts more than a few days ahead, no matter how big (in terms of memory) and fast computers get to be in the future. The only computer that can simulate the weather is the weather; and the only computer that can simulate the Universe is the Universe." (John Gribbin, "The Little Book of Science", 1999)

On Motion (1800-1899)

"The determination of the average man is not merely a matter of speculative curiosity; it may be of the most important service to the science of man and the social system. It ought necessarily to precede every other inquiry into social physics, since it is, as it were, the basis. The average man, indeed, is in a nation what the centre of gravity is in a body; it is by having that central point in view that we arrive at the apprehension of all the phenomena of equilibrium and motion." (Adolphe Quetelet, "A Treatise on Man and the Development of his Faculties", 1842)

"Science gains from it [the pendulum] more than one can expect. With its huge dimensions, the apparatus presents qualities that one would try in vain to communicate by constructing it on a small [scale], no matter how carefully. Already the regularity of its motion promises the most conclusive results. One collects numbers that, compared with the predictions of theory, permit one to appreciate how far the true pendulum approximates or differs from the abstract system called 'the simple pendulum'." (Jean-Bernard-Léon Foucault, "Demonstration Experimentale du Movement de Rotation de la Terre", 1851)

"The Mathematics, like language, (of which indeed they may be considered a species,) comprehending under that designation the whole science of number, space, form, time, and motion, as far as it can be expressed in abstract formulas, are evidently not only one of the most useful, but one of the grandest of studies." (Edward Everett, [address] 1857)

"When I have before my eyes a pair of stereoscopic drawings which are hard to combine, it is difficult to bring the lines and points that correspond, to cover each other, and with every little motion of the eyes they glide apart. But if I chance to gain a lively mental image (Anschauungsbild) of the represented solid form (a thing that often occurs by lucky chance), I then move my two eyes with perfect certainty over the figure without the picture separating again." (Hermann von Helmholtz, "Tonempfindungen" ["Sensations of Tone"], 1863)

"Consider an arbitrary figure in general position, indeterminate in the sense that it can be chosen from all such figures without upsetting the laws, conditions, and connections among the different parts of the system; suppose that given these data we have found one or more relations or properties, metric or descriptive, of that figure using the usual obvious inference (i.e., in a way regarded in certain cases as the only rigorous argument). Is it not obvious that if, preserving these very data, one begins to change the initial figure by insensible steps, or applies to some parts of the figure an arbitrary continuous motion, then is it not obvious that the properties and relations established for the initial system remain applicable to subsequent states of this system provided that one is mindful of particular changes, when, say, certain magnitudes vanish, change direction or sign, and so on - changes which one can always anticipate a priori on the basis of reliable rules." (Jean V Poncelet,"Treatise on Projective Properties of Figures", 1865)

"A strict materialist believes that everything depends on the motion of matter. He knows the form of the laws of motion though he does not know all their consequences when applied to systems of unknown complexity." (James C Maxwell, [Letter to Mark Pattison] 1868)

"Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number." (George Holmes Howison, "The Departments of Mathematics, and their Mutual Relations", Journal of Speculative Philosophy Vol. 5, No. 2, 1871)

"We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens in that phenomenon which we term the motion of matter'." (William K Clifford, Richard C Rowe & Karl Pearson,"The Common Sense of the Exact Sciences", 1885)

"Since a given system can never of its own accord go over into another equally probable state but into a more probable one, it is likewise impossible to construct a system of bodies that after traversing various states returns periodically to its original state, that is a perpetual motion machine." (Ludwig E Boltzmann, "The Second Law of Thermodynamics", [Address to a Formal meeting of the Imperial Academy of Science], 1886)

"That branch of physics which is at once the oldest and the simplest and which is therefore treated as introductory to other departments of this science, is concerned with the motions and equilibrium of masses. It bears the name of mechanics." (Ernst Mach, "The Science of Mechanics: A Critical and Historical Account of Its Development", 1893)

"The first step, whenever a practical problem is set before a mathematician, is to form the mathematical hypothesis. It is neither needful nor practical that we should take account of the details of the structure as it will exist. We have to reason about a skeleton diagram in which bearings are reduced to points, pieces to lines, etc. and [in] which it is supposed that certain relations between motions are absolutely constrained, irrespective of forces. Some writers call such a hypothesis a fiction, and say that the mathematician does not solve the real problem, but only a fictitious one. That is one way of looking at the matter, to which I have no objection to make: only, I notice, that in precisely the same sense in which the mathematical hypothesis is 'false', so also is this statement 'false', that it is false. Namely, both representations are false in the sense that they omit subsidiary elements of the fact, provided that element of the case can be said to be subsidiary which those writers overlook, namely, that the skeleton diagram is true in the only sense in which from the nature of things any mental representation, or understanding, of the brute existent can be true. For every possible conception, by the very nature of thought, involves generalization; now generalization omits, means to omit, and professes to omit, the differences between the facts generalized." (Charles S Peirce, "Report on Live Loads", cca. 1895)

"Certainly, if a system moves under the action of given forces and its initial conditions have given values in the mathematical sense, its future motion and behavior are exactly known. But, in astronomical problems, the situation is quite different: the constants defining the motion are only physically known, that is with some errors; their sizes get reduced along the progresses of our observing devices, but these errors can never completely vanish." (Jacques Hadamard, "Les surfaces à courbures opposées et leurs lignes géodésiques", Journal de mathématiques pures et appliqués 5e (4), 1898)

On Motion (1700-1799)

"Moreover, it must be confessed that perception and that which depends upon it are inexplicable on mechanical grounds, that is to say, by means of figures and motions. And supposing there were a machine, so constructed as to think, feel, and have perception, it might be conceived as increased in size, while keeping the same proportions, so that one might go into it as into a mill. That being so, we should, on examining its interior, find only parts which work one upon another, and never anything by which to explain a perception. Thus it is in a simple substance, and not in a compound or in a machine, that perception must be sought for." (Gottfried W Leibniz,  "Monadology", 1714)

"Is it 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 subtle and hidden works." (Colin Maclaurin, "An Account of Sir Isaac Newton’s Philosophical Discoveries", 1748)

"We only find, that the one does actually, in fact, follow the other. The impulse of one billiard-ball is attended with motion in the second. This is the whole that appears to the outward senses. The mind feels no sentiment or inward impression from this succession of objects: consequently, there is not, in any single, particular instance of cause and effect, any thing which can suggest the idea of power or necessary connexion." (David Hume, "An Enquiry Concerning Human Understanding", 1748)

"The man of system, on the contrary, is apt to be very wise in his own conceit; and is often so enamoured with the supposed beauty of his own ideal plan of government, that he cannot suffer the smallest deviation from any part of it. He goes on to establish it completely and in all its parts, without any regard either to the great interests, or to the strong prejudices which may oppose it. He seems to imagine that he can arrange the different members of a great society with as much ease as the hand arranges the different pieces upon a chess-board. He does not consider that the pieces upon the chess-board have no other principle of motion besides that which the hand impresses upon them; but that, in the great chess-board of human society, every single piece has a principle of motion of its own, altogether different from that which the legislature might choose to impress upon it. If those two principles coincide and act in the same direction, the game of human society will go on easily and harmoniously, and is very likely to be happy and successful. If they are opposite or different, the game will go on miserably, and the society must be at all times in the highest degree of disorder." (Adam Smith, "The Theory of Moral Sentiments", 1759)

"Nature is the system of laws established by the Creator for the existence of things and for the succession of creatures. Nature is not a thing, because this thing would be everything. Nature is not a creature, because this creature would be God. But one can consider it as an immense vital power, which encompasses all, which animates all, and which, subordinated to the power of the first Being, has begun to act only by his order, and still acts only by his concourse or consent. […] Time, space and matter are its means, the universe its object, motion and life its goal." (Georges-Louis L de Buffon, "Histoire Naturelle, Générale et Particulière, Avec la Description du Cabinet du Roi", 1764)

"Nothing surely is more strongly imprinted in our minds or more closely interwoven with our souls than the idea or sense of order and proportion. Hence all the force of numbers and those powerful arts founded on their management and use! What a difference there is between harmony and discord, cadence and convulsion! What a difference between composed and orderly motion and that which is ungoverned and accidental, between the regular and uniform pile of some noble architect and a hip of sand and stones, between an organized body and a mist or cloud driven by the wind!" (Anthony A Cooper, "Characteristics of Men, Manners, Opinions, Times", 1711)

"Systems in many respects resemble machines. A machine is a little system, created to perform, as well as to connect together, in reality, those different movements and effects which the artist has occasion for.  A system is an imaginary machine invented to connect together in the fancy those different movements and effects which are already in reality performed. […] The machines that are first invented to perform any particular movement are always the most complex, and succeeding artists generally discover that, with fewer wheels, with fewer principles of motion, than had originally been employed, the fame effects may be more easily produced. The first systems, in the fame manner, are always the most complex, and a particular connecting chain, or principle, is generally thought necessary to unite every two seemingly disjointed appearances: but it often happens, that one great connecting principle is afterwards found to be sufficient to bind together all the discordant phænomena that occur in a whole species of things." (Adam Smith, "The Wealth of Nations", 1776)

On Motion (-1699)

"We invoke the imagination and the intervals that it furnishes, since the form itself is without motion or genesis, indivisible and free of all underlying matter, though the elements latent in the form are produced distinctly and individually on the screen of imagination. What projects the images is the understanding; the source of what is projected is the form in the understanding; and what they are projected in is this 'passive nous' that unfolds in revolution about the partlessness of genuine Nous." (Proclus Lycaeus, "A Commentary on the First Book of Euclid’s Elements", cca 5th century)

"[…] the mind orders nothing by its own motions, but lies merely receptive under the impressions of bodies, reflecting empty images in a mirror in place of reality." (Anicius Manlius Severinus Boethius, "The Consolation of Philosophy", cca. 524)

"[H]ere negation is […] contrariety […] that is to say, in the contrary direction. As the west is contrary of east; and the south the converse of north. Thus, of two countries, east and west, if one be taken as positive, the other is relatively negative. So when motion to the east is assumed to be positive, if a planets motion be westward, then the number of degrees equivalent to the planets motion is negative." (Bhaskara II, "Bijaganita", 12th century)

"Sound is generated by motion, since it belongs to the class of successive things. For this reason, while it exists when it is made, it no longer exists once it has been made. […] All music, especially mensurable music, is founded in perfection, combining in itself number and sound." (Jean de Muris, "Ars novae musicae", 1319)

"Those who devised the eccentrics seen thereby in large measure to have solved the problem of apparent motions with approximate calculations. But meanwhile they introduced a good many ideas which apparently contradict the first principles of uniform motion. Nor could they elicit or deduce from the eccentrics the principal consideration, that is, the structure of the universe and the true symmetry of its parts."  (Nicolaus Copernicus, "De revolutionibus orbium coelestium", 1543)

"I tell you that if natural bodies have it from Nature to be moved by any movement, this can only be circular motion, nor is it possible that Nature has given to any of its integral bodies a propensity to be moved by straight motion. I have many confirmations of this proposition, but for the present one alone suffices, which is this. I suppose the parts of the universe to be in the best arrangement, so that none is out of its place, which is to say that Nature and God have perfectly arranged their structure. This being so, it is impossible for those parts to have it from Nature to be moved in straight, or in other than circular motion, because what moves straight changes place, and if it changes place naturally, then it was at first in a place preternatural to it, which goes against the supposition. Therefore, if the parts of the world are well ordered, straight motion is superfluous and not natural, and they can only have it when some body is forcibly removed from its natural place, to which it would then return by a straight line, for thus it appears that a part of the earth does [move] when separated from its whole. I said 'it appears to us', because I am not against thinking that not even for such an effect does Nature make use of straight line motion." (Galileo Galilei, [Letter to Francesco Ingoli] 1624)

"Although time, space, place, and motion are very familiar to everyone, it must be noted that these quantities are popularly conceived solely with reference to the objects of sense perception." (Isaac Newton, "The Principia: Mathematical Principles of Natural Philosophy", 1687)

"To measure motion, space is as necessary to be considered as time....[They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)

On Continuity XVI (The Beginnings)

"Since [...] nature is a principle of motion and mutation [...] it is necessary that we should not be ignorant of what motion is [...] But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. [..] frequently [...] those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous." (Aristotle, "Physics", cca. 350 BC)

"Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two. […] Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one."(Aristotle, "Physics", cca. 350 BC)

"When what surrounds, then, is not separate from the thing, but is in continuity with it, the thing is said to be in what surrounds it, not in the sense of in place, but as a part in a whole. But when the thing is separate or in contact, it is immediately ‘in’ the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for the extremities of things which touch are coincident." (Aristotle, "Physics", cca. 350 BC)

"Things [...] are some of them continuous [...] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end [...] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude [...] or with multitude [...] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus, cca. 100 AD) 

"But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent." (William of Ockham, cca. 1320)

"It is established that every continuum has further parts, and not so many parts finite in number that there are not further parts, and has all its parts actually and simultaneously, and therefore every continuum has simultaneously and actually infinitely many parts." (Gregory of Rimini [Gregorii Ariminensis], "Lectura super primum et secundum sententiarum", cca. 1350)

Sophie Germain - Collected Quotes

"It appears that in everything the interest of ideas is in inverse proportion to the usefulness they have in practice. This is not surprising when we consider that the human intellect, when working for its own satisfaction, should encounter the greatest intellectual beauties rather than when guided by an external motive [...]" (Sophie Germain, [letter to Gauss] 1809)

"But by far the greatest obstacle to the progress of science and to the undertaking of new tasks and provinces therein is found in this: that men despair and think things impossible." (Sophie Germain, 1813)

"If a hypothesis contains all that is part of the problem, if it can be regarded as a true definition, it suffices to introduce this hypothesis into the calculus, in order to obtain all the analytical consequences that belong to the solution of the same problem."(Sophie Germain, 1821) 

"Let me be permitted to recall that the object of mathematics is not to investigate the causes that one can assign to natural phenomena. This science would lose both its character and credit if, renouncing the support of general well-confirmed facts, it sought within the realm of nebulous conjectures, a realm which has always been a fertile source of error for ways of satisfying the thirst fo rexplanation." (Sophie Germain, "Examen des principes qui peuvent conduire a la connaissance des lois de requilibre et du mouvement des solides elastiques", Annales de Chimie 38, 1828)

"The more one reflects, the more one acknowledges that necessity governs the world. At each new progress of science ,that which seemed contingent is recognized as being necessary. Multiple relations are established between the branches that we had thought to be separate; we observe laws where we had thought there were only accidental events. We approach more and more the unity of being […]" (Sophie Germain, "Considerations sur l’etat des sciences et lettres, aux differentes epoques de leur culture", 1833)

"Algebra is but written geometry and geometry is but figured algebra." (Sophie Germain, "Mémoire sur les Surfaces Élastiques", 1880)

"It matters little who first arrives at an idea, rather what is significant is how far that idea can go." (Sophie Germain)

"Space and time: these man proposes to measure. The one circumscribes his momentary existence, the other accompanies his successive stages in life. These two dimensions are tied together through a necessary relationship, namely, motion. When motion is constant and uniform, space is known by time and time is measured by space. Man has nothing within him that is constant and uniform; continually modified every instant. he is changing, irregular. and hardly durable enough to be a measure of duration." (Sophie Germain)