Alexander von Humboldt - Collected Quotes

"Whatever relates to extent and quantity may be represented by geometrical figures. Statistical projections which speak to the senses without fatiguing the mind, possess the advantage of fixing the attention on a great number of important facts." (Alexander von Humboldt, 1811)

"Regardless of communication between man and man, speech is a necessary condition for the thinking of the individual in solitary seclusion. In appearance, however, language develops only socially, and man understands himself only once he has tested the intelligibility of his words by trial upon others." (Wilhelm von Humboldt, "On Language", 1836)

"At the limits of circumscribed knowledge, as from some lofty island shore, the eye delights to penetrate to distant regions." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe", 1845)

"Conclusions based upon analogies may fill up a portion of the vast chasm which separates the certain results of a mathematical natural philosophy from conjectures verging on the extreme, and therefore obscure and barren confines of all scientific development of mind." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe", 1845)

"Impressions change with the varying movements of the mind, and we are led by a happy illusion to believe that we receive from the external world that with which we have ourselves invested it." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe", 1845)

"Nature, in the manifold signification of the word - whether considered as the universality of all that is and ever will be - as the inner moving force of all phenomena, or as their mysterious prototype - reveals itself to the simple mind and feelings of man as something earthly, and closely allied to himself."(Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe", 1845)

"The mere accumulation of unconnected observations of details, devoid of generalization of ideas, may doubtlessly have tended to create and foster the deeply rooted prejudice, that the study of the exact sciences must necessarily chill the feelings, and diminish the nobler enjoyments attendant upon a contemplation of nature." (Alexander von Humboldt, "Views of Nature: Or Contemplation of the Sublime Phenomena of Creation", 1850)

"The philosophical study of nature rises above the requirements of mere delineation, and does not consist in the sterile accumulation of isolated facts. The active and inquiring spirit of man may therefore be occasionally permitted to escape from the present into the domain of the past, to conjecture that which cannot yet be clearly determined, and thus to revel amid the ancient and ever-recurring myths of geology." (Alexander von Humboldt, "Views of Nature: Or Contemplation of the Sublime Phenomena of Creation", 1850)

"In order to depict nature in its exalted sublimity, we must not dwell exclusively on its external manifestations, but we must trace its image, reflected in the mind of man, at one time filling the dreamy land of physical myths with forms of grace and beauty, and at another developing the noble germ of artistic creations." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe" Vol. 2, 1869)

"Nature considered rationally, that is to say, submitted to the process of thought, is a unity in diversity of phenomena; a harmony, blending together all created things, however dissimilar in form and attributes; one great whole animated by the breath of life." (Alexander von Humboldt)

"The most dangerous worldview is the worldview of those who have not viewed the world." (Alexander von Humboldt [attributed to])

"With the simplest statements of scientific facts there must ever mingle a certain eloquence. Nature herself is sublimely eloquent. The stars as they sparkle in the firmament fill us with delight and ecstasy, and yet they all move in orbits marked out with mathematical precision." (Alexander von Humboldt)

Peter B Stevens - Collected Quotes

"All patterns, whether drawn by artists, calculated by mathematicians, or produced by natural forces are shaped by the same spatial environment. All are subject to the tyranny of space. Synthetic patterns of lines and dots are engaging in their own right but, more importantly, they speak eloquently of the order that all things inevitably share." (Peter B Stevens, "Patterns in Nature", 1974)

"Although we expect to find eddies in turbulent flow, we do not know when any specific eddy will come into being or die away . We cannot yet predict how eddies interact. Similarly, we know as a general rule that any particle within a turbulent flow gets knocked about in an aimless fashion by the swirls, so that it describes an erratic meandering path, but at any given moment we cannot predict the precise location or velocity of the particle." (Peter B Stevens, "Patterns in Nature", 1974)

"But even if we cannot predict all the details, we can predict something about the average case. We can consider the unpredictable local velocities and pressures as chance or random occurrences and then, with the aid of probability theory, take the mean of those occurrences and obtain mathematical descriptions of average motions in average flows." (Peter B Stevens, "Patterns in Nature", 1974)

"In order to describe meanders, then, we can invoke a model involving scour and centrifugal force, a model that describes the uniform expenditure of energy, or a model based on probability theory. All three models describe the same phenomenon. As far as meanders are concerned, all three models happen to be interrelated - but not out of any fundamental necessity. That is to say, the cross circulation induced by centrifugal force need not necessarily result in a uniform distribution of effort or produce a path that is especially probable. In a world of limited patterns, however, the meander answers several entirely different sets of specifications, so that scour, uniform effort, and probability produce the same design." (Peter B Stevens, "Patterns in Nature", 1974)

"The analysis of turbulence in terms of probability reveals several interesting things about eddies. For instance, the average eddy moves a distance about equal to its own diameter before it generates small eddies that move, more often than not, in the opposite direction. Those smaller eddies generate still smaller eddies and the process continues until all the energy dissipates as heat through molecular motion." (Peter B Stevens, "Patterns in Nature", 1974)

"Of all constraints of nature, the most far-reaching are imposed by space. For space itself has a structure that influences the shape of every existing thing." (Peter B Stevens, "Patterns in Nature", 1974)

"The effect of magnitude or absolute size as a determinant of form shows again how space shapes the things around us. In studying polyhedrons we are unconcerned with magnitude. We assume that a cube is a cube no matter what its size. We find, however, that the geometric relations that arise from a difference in size affect structural behavior, and that a large cube is relatively weaker than a small cube. We also find, as a corollary, that in order to maintain the same structural characteristics a difference in size must be accompanied by a difference in shape." (Peter B Stevens, "Patterns in Nature", 1974)

"The spiral is beautifully uniform; it curves around on itself in a perfectly regular manner. It can fill all of two-dimensional space, being capable of infinite expansion, and it is also quite short. But [...], as measured by the mean of distances to its center, the spiral is extremely indirect." (Peter B Stevens, "Patterns in Nature", 1974)

"Turbulence forms the primordial pattern, the chaos that was 'in the beginning'." (Peter B Stevens, "Patterns in Nature", 1974)

"We might note in passing that the lines and circles adopted by natural forms are never perfect. Neither the stream nor the meteor runs perfectly straight, nor is the pond or orbital trajectory a perfect circle. Straight lines and circles are only the pure forms. They occur under only the simplest conditions. In nature, however, conditions are never entirely simple, and any 'elementary' or 'isolated' part is embedded in a larger system that operates in turn within other still larger systems. To some extent, then, the part is acted upon by the whole - by the totality of all the systems - and it never exactly fits an easily definable pattern. The warning is clear: nature never conforms precisely to our simple models; she introduces modifications as dictated by her lawful response to a multiplicity of demands." (Peter B Stevens, "Patterns in Nature", 1974)

On Laws IV: The Laws of Nature

"Nature always uses the simplest means to accomplish its effects." (Pierre L Maupertuis, "Accord between different laws of Nature that seemed incompatible", Mémoires de l'académie royale des sciences, 1744)

"The supreme Being is everywhere; but He is not equally visible everywhere. Let us seek Him in the simplest things, in the most fundamental laws of Nature, in the universal rules by which movement is conserved, distributed or destroyed; and let us not seek Him in phenomena that are merely complex consequences of these laws." (Pierre L Maupertuis, "Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique", 1746)

"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)

"Systems in physical science […] are no more than appropriate instruments to aid the weakness of our organs: they are, properly speaking, approximate methods which put us on the path to the solution of the problem; these are the hypotheses which, successively modified, corrected, and changed in proportion as they are found false, should lead us infallibly one day, by a process of exclusion, to the knowledge of the true laws of nature." (Antoine L Lavoisier, "Mémoires de l’Académie Royale des Sciences", 1777)

"[…] we are far from having exhausted all the applications of analysis to geometry, and instead of believing that we have approached the end where these sciences must stop because they  have reached the limit of the forces of the human spirit, we ought to avow rather we are only at the first steps of an immense career. These new [practical] applications, independently of the utility which they may have in themselves, are necessary to the progress of analysis in general; they give birth to questions which one would not think to propose; they demand that one create new methods. Technical processes are the children of need; one can say the same for the methods of the most abstract sciences. But we owe the latter to the needs of a more noble kind, the need to discover the new truths or to know better the laws of nature." (Nicolas de Condorcet, 1781)

"The laws of nature are the rules according to which the effects are produced; but there must be a cause which operates according to these rules." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"The end of natural philosophy is to increase either the knowledge or power of man, and enable him to understand the ways and procedure of nature. By discovering the laws of nature, he acquires knowledge, and obtains power; for when these laws are discovered, he can use them as rules of practice, to equal, subdue, or even excel nature by art." (George Adams, "Lectures on Natural and Experimental Philosophy" Vol. 2, 1794)

"[…] we must not measure the simplicity of the laws of nature by our facility of conception; but when those which appear to us the most simple, accord perfectly with observations of the phenomena, we are justified in supposing them rigorously exact." (Pierre-Simon Laplace, "The System of the World", 1809)

"In all speculations on the origin, or agents that have produced the changes on this globe, it is probable that we ought to keep within the boundaries of the probable effects resulting from the regular operations of the great laws of nature which our experience and observation have brought within the sphere of our knowledge. When we overleap those limits, and suppose a total change in nature's laws, we embark on the sea of uncertainty, where one conjecture is perhaps as probable as another; for none of them can have any support, or derive any authority from the practical facts wherewith our experience has brought us acquainted." (William Maclure, "Observations on the Geology of the United States of America", 1817)

"There are no rules or models; that is, there are no rules except general laws of nature which hover over art and special laws which apply to specific subjects." (Victor M Hugo, "Cromwell", 1827)

"Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

"The Laws of Nature are merely truths or generalized facts, in regard to matter, derived by induction from experience, observation, arid experiment. The laws of mathematical science are generalized truths derived from the consideration of Number and Space." (Charles Davies, "The Logic and Utility of Mathematics", 1850)

"If we knew all the laws of Nature, we should need only one fact, or the description of one actual phenomenon, to infer all the particular results at that point. Now we know only a few laws, and our result is vitiated, not, of course, by any confusion or irregularity in Nature, but by our ignorance of essential elements in the calculation. Our notions of law and harmony are commonly confined to those instances which we detect; but the harmony which results from a far greater number of seemingly conflicting, but really concurring, laws, which we have not detected, is still more wonderful. The particular laws are as our points of view, as to the traveler, a mountain outline varies with every step, and it has an infinite number of profiles, though absolutely but one form. Even when cleft or bored through it is not comprehended in its entireness." (Henry D Thoreau, "Walden; or, Life in the Woods", 1854)

"This science, Geometry, is one of indispensable use and constant reference, for every student of the laws of nature; for the relations of space and number are the alphabet in which those laws are written. But besides the interest and importance of this kind which geometry possesses, it has a great and peculiar value for all who wish to understand the foundations of human knowledge, and the methods by which it is acquired. For the student of geometry acquires, with a degree of insight and clearness which the unmathematical reader can but feebly imagine, a conviction that there are necessary truths, many of them of a very complex and striking character; and that a few of the most simple and self-evident truths which it is possible for the mind of man to apprehend, may, by systematic deduction, lead to the most remote and unexpected results." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"A law of nature, however, is not a mere logical conception that we have adopted as a kind of memoria technical to enable us to more readily remember facts. We of the present day have already sufficient insight to know that the laws of nature are not things which we can evolve by any speculative method. On the contrary, we have to discover them in the facts; we have to test them by repeated observation or experiment, in constantly new cases, under ever-varying circumstances; and in proportion only as they hold good under a constantly increasing change of conditions, in a constantly increasing number of cases with greater delicacy in the means of observation, does our confidence in their trustworthiness rise." (Hermann von Helmholtz, "Popular Lectures on Scientific Subjects", 1873)

"Our books of science, as they improve in accuracy, are in danger of losing the freshness and vigor and readiness to appreciate the real laws of Nature, which is a marked merit in the ofttimes false theories of the ancients." (Henry D Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)

"The history of thought should warn us against concluding that because the scientific theory of the world is the best that has yet been formulated, it is necessarily complete and final. We must remember that at bottom the generalizations of science or, in common parlance, the laws of nature are merely hypotheses devised to explain that ever-shifting phantasmagoria of thought which we dignify with the high-sounding names of the world and the universe." (Sir James G Frazer, "The Golden Bough: A Study in Magic and Religion", 1890)

"The strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

"The history of thought should warn us against concluding that because the scientific theory of the world is the best that has yet been formulated, it is necessarily complete and final. We must remember that at bottom the generalizations of science or, in common parlance, the laws of nature are merely hypotheses devised to explain that ever-shifting phantasmagoria of thought which we dignify with the high-sounding names of the world and the universe." (Sir James G Frazer, "The Golden Bough: A Study in Magic and Religion", 1890)

"Education is the instruction of the intellect in the laws of Nature, under which name I include not merely things and their forces, but men and their ways; and the fashioning of the affections and of the will into an earnest and loving desire to move in harmony with those laws." (Thomas H Huxley, "Science and Education", 1891)

"The laws of nature are drawn from experience, but to express them one needs a special language: for, ordinary language is too poor and too vague to express relations so subtle, so rich, so precise. Here then is the first reason why a physicist cannot dispense with mathematics: it provides him with the one language he can speak […]. Who has taught us the true analogies, the profound analogies which the eyes do not see, but which reason can divine? It is the mathematical mind, which scorns content and clings to pure form." (Henri Poincaré, "The Value of Science", 1905)

"Pythagoras says that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe. But the law of number possesses an immanent order, which is at first sight mystifying, but on a more intimate acquaintance we easily understand it to be intrinsically necessary; and this law of number explains the wondrous consistency of the laws of nature. (Paul Carus, "Reflections on Magic Squares", Monist Vol. 16, 1906)

"An exceedingly small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation 'approximately'. If that enabled us to predict the succeeding situation with 'the same approximation', that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. (Jules H Poincaré, "Science and Method", 1908)

"The regularities in the phenomena which physical science endeavors to uncover are called the laws of nature. The name is actually very appropriate. Just as legal laws regulate actions and behavior under certain conditions but do not try to regulate all action and behavior, the laws of physics also determine the behavior of its objects of interest only under certain well-defined conditions but leave much freedom otherwise." (Eugene P Wigner, "Events, Laws of Nature, and Invariance principles", [Nobel lecture] 1914)

"But it is just this characteristic of simplicity in the laws of nature hitherto discovered which it would be fallacious to generalize, for it is obvious that simplicity has been a part cause of their discovery, and can, therefore, give no ground for the supposition that other undiscovered laws are equally simple." (Bertrand Russell, "'On the Scientific Method in Philosophy", 1918)

"The laws of nature cannot be intelligently applied until they are understood, and in order to understand them, many experiments bearing upon the ultimate nature of things must be made, in order that all may be combined in a far-reaching generalization impossible without the detailed knowledge upon which it rests." (Theodore W Richards, "The Problem of Radioactive Lead", 1918)

"'Causation' has been popularly used to express the condition of association, when applied to natural phenomena. There is no philosophical basis for giving it a wider meaning than partial or absolute association. In no case has it been proved that there is an inherent necessity in the laws of nature. Causation is correlation. [...] perfect correlation, when based upon sufficient experience, is causation in the scientific sense." (Henry E. Niles, "Correlation, Causation and Wright's Theory of 'Path Coefficients'", Genetics, 1922)

"Architecture is the first manifestation of man creating his own universe, creating it in the image of nature, submitting to the laws of nature, the laws which govern our own nature, our universe. The laws of gravity, of statics and of dynamics, impose themselves by a reductio ad absurdum: everything must hold together or it will collapse." (Charles-Edouard Jeanneret [Le Corbusier], "Towards a New Architecture", 1923)

"For establishing the laws of nature one resorts (not deliberately but involuntarily) to the simplest formulas that seem to describe the phenomena with reasonable accuracy. […] Even those laws of nature that are the most general and important for the world view have always been proved experimentally only in a confined ambit and with limited accuracy. […] The exact formulation of the laws of nature by simple formulas is based on the desire to master external phenomena with the simplest tools possible." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"It goes without saying that the laws of nature are in themselves independent of the properties of the instruments with which they are measured. Therefore in every observation of natural phenomena we must remember the principle that the reliability of the measuring apparatus must always play an important role." (Max Planck,"Where is Science Going?", 1932)

"Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature." (Albert Einstein, [Obituary for Emmy Noether], 1935)

"The researcher worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty. […] It often happens that the requirements of simplicity and beauty are the same, but where they clash the latter must take precedence." (Paul A M Dirac, "The Relation Between Mathematics and Physics", Proceedings of the Royal Society , Volume LIX, 1939)

"The fundamental difference between engineering with and without statistics boils down to the difference between the use of a scientific method based upon the concept of laws of nature that do not allow for chance or uncertainty and a scientific method based upon the concepts of laws of probability as an attribute of nature." (Walter A Shewhart, 1940)

"The laws of nature may be operative up to a certain limit, beyond which they turn against themselves to give birth to the absurd." (Albert Camus, "The Myth of Sisyphus", 1942)

"The responsibility for the creation of new scientific knowledge - and for most of its application - rests on that small body of men and women who understand the fundamental laws of nature and are skilled in the techniques of scientific research. We shall have rapid or slow advance on any scientific frontier depending on the number of highly qualified and trained scientists exploring it."(Vannevar Bush, "Science: The Endless Frontier", 1945)

"In classical physics, most of the fundamental laws of nature were concerned either with the stability of certain configurations of bodies, e.g. the solar system, or else with the conservation of certain properties of matter, e.g. mass, energy, angular momentum or spin. The outstanding exception was the famous Second Law of Thermodynamics, discovered by Clausius in 1850. This law, as usually stated, refers to an abstract concept called entropy, which for any enclosed or thermally isolated system tends to increase continually with lapse of time. In practice, the most familiar example of this law occurs when two bodies are in contact: in general, heat tends to flow from the hotter body to the cooler. Thus, while the First Law of Thermodynamics, viz. the conservation of energy, is concerned only with time as mere duration, the Second Law involves the idea of trend." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"We have assumed that the laws of nature must be capable of expression in a form which is invariant for all possible transformations of the space-time co-ordinates." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"The world is not made up of empirical facts with the addition of the laws of nature: what we call the laws of nature are conceptual devices by which we organize our empirical knowledge and predict the future." (Richard B Braithwaite, "Scientific Explanation", 1953)

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that ‘laws of nature’ exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." (Eugene P Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," 1960)

"We know many laws of nature and we hope and expect to discover more. Nobody can foresee the next such law that will be discovered. Nevertheless, there is a structure in laws of nature which we call the laws of invariance. This structure is so far-reaching in some cases that laws of nature were guessed on the basis of the postulate that they fit into the invariance structure." (Eugene P Wigner, "The Role of Invariance Principles in Natural Philosophy", 1963)

"It is now natural for us to try to derive the laws of nature and to test their validity by means of the laws of invariance, rather than to derive the laws of invariance from what we believe to be the laws of nature." (Eugene P Wigner, "Symmetries and Reflections", 1967)

"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

"[...] it seems self-evident that mathematics is not likely to be much help in discovering laws of nature. If a mathematician wants to make a contribution on this (and I admit it is the highest) level, he will have to master so much experimental material and train himself to think in a way so different from the one he has been accustomed to that he will, in effect, cease to be a mathematician." (Mark Kac, "On Applying Mathematics: Reflections and Examples", Quarterly of Applied Mathematics, 1972)

"Laws of nature are human inventions, like ghosts. Laws of logic, or mathematics are also human inventions, like ghosts. The whole blessed thing is a human invention, including the idea that it isn't a human invention." (Robert M Pirsig, "Zen and the Art of Motorcycle Maintenance", 1974)

"It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances." (Hannah Arendt, "The Life of the Mind", 1977)

"The ideas that are basic to [my work] often bear witness to my amazement and wonder at the laws of nature which operate in the world around us. He who wonders discovers that this is in itself a wonder. By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics." (Maurits C Escher, "The Graphic Work", 1978)

"Our form of life depends, in delicate and subtle ways, on several apparent ‘coincidences’ in the fundamental laws of nature which make the Universe tick. Without those coincidences, we would not be here to puzzle over the problem of their existence […] What does this mean? One possibility is that the Universe we know is a highly improbable accident, ‘just one of those things’." (John R Gribbin, "Genesis: The Origins of Man and the Universe", 1981)

"Scientific theories must tell us both what is true in nature, and how we are to explain it. I shall argue that these are entirely different functions and should be kept distinct. […] Scientific theories are thought to explain by dint of the descriptions they give of reality. […] The covering-law model supposes that all we need to know are the laws of nature - and a little logic, perhaps a little probability theory - and then we know which factors can explain which others." (Nancy Cartwright, "How the Laws of Physics Lie", 1983)

"Human beings are very conservative in some ways and virtually never change numerical conventions once they grow used to them. They even come to mistake them for laws of nature." (Isaac Asimov, "Foundation and Earth", 1986)

"Where chaos begins, classical science stops. For as long as the world has had physicists inquiring into the laws of nature, it has suffered a special ignorance about disorder in the atmosphere, in the fluctuations of the wildlife populations, in the oscillations of the heart and the brain. The irregular side of nature, the discontinuous and erratic side these have been puzzles to science, or worse, monstrosities." (James Gleick, "Chaos", 1987)

"The principle of maximum diversity operates both at the physical and at the mental level. It says that the laws of nature and the initial conditions are such as to make the universe as interesting as possible.  As a result, life is possible but not too easy. Always when things are dull, something new turns up to challenge us and to stop us from settling into a rut. Examples of things which make life difficult are all around us: comet impacts, ice ages, weapons, plagues, nuclear fission, computers, sex, sin and death.  Not all challenges can be overcome, and so we have tragedy. Maximum diversity often leads to maximum stress. In the end we survive, but only by the skin of our teeth." (Freeman J Dyson, "Infinite in All Directions", 1988)

"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)

"Somehow the breathless world that we witness seems far removed from the timeless laws of Nature which govern the elementary particles and forces of Nature. The reason is clear. We do not observe the laws of Nature: we observe their outcomes. Since these laws find their most efficient representation as mathematical equations, we might say that we see only the solutions of those equations not the equations themselves. This is the secret which reconciles the complexity observed in Nature with the advertised simplicity of her laws." (John D Barrow, "New Theories of Everything", 1991)

"How surprising it is that the laws of nature and the initial conditions of the universe should allow for the existence of beings who could observe it. Life as we know it would be impossible if any one of several physical quantities had slightly different values." (Steven Weinberg, Life in the Quantum Universe", Scientific American, 1995)

"It suggests to me that consciousness and our ability to do mathematics are no mere accident, no trivial detail, no insignificant by-product of evolution that is piggy-backing on some other mundane property. It points to what I like to call the cosmic connection, the existence of a really deep relationship between minds that can do mathematics and the underlying laws of nature that produce them. We have a closed system of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can encode [...] the very laws of physics that gave rise to it." (Paul Davies, "Are We Alone?: Philosophical Implications of the Discovery of Extraterrestrial Life", 1995)

"Riemann concluded that electricity, magnetism, and gravity are caused by the crumpling of our three-dimensional universe in the unseen fourth dimension. Thus a 'force' has no independent life of its own; it is only the apparent effect caused by the distortion of geometry. By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would become one of the dominant themes in modern theoretical physics, that the laws of nature appear simple when expressed in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed." (Michio Kaku, "Hyperspace", 1995)

"The problems associated with the initial singularity of the universe bring us to what is called the theory of everything. It is an all-encompassing theory that would completely explain me origin of the universe and everything in it. It would bring together general relativity and quantum mechanics, and explain everything there is to know about the elementary particles of the universe, and the four basic forces of nature (gravitational, electromagnetic, weak, and strong nuclear forces). Furthermore, it would explain the basic laws of nature and the fundamental constants of nature such as the speed of light and Planck's constant." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"Knowledge is encoded in models. Models are synthetic sets of rules, pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns. As argued by philosophers and shown by scientists, we do not have access to 'reality', only to some of its manifestations, whose regularities are used to determine rules, which when widely applicable become 'laws of nature'. These laws are constantly tested in the scientific march, and they evolve, develop and transmute as the frontier of knowledge recedes further away."  (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

"Chaos theory reconciles our intuitive sense of free will with the deterministic laws of nature. However, it has an even deeper philosophical ramification. Not only do we have freedom to control our actions, but also the sensitivity to initial conditions implies that even our smallest act can drastically alter the course of history, for better or for worse. Like the butterfly flapping its wings, the results of our behavior are amplified with each day that passes, eventually producing a completely different world than would have existed in our absence!" (Julien C Sprott, "Strange Attractors: Creating Patterns in Chaos", 2000)

"We have come, in our time, to systematize our understanding of the rules of nature. We say that these rules are the laws of physics. The language of the laws of nature is mathematics. We acknowledge that our understanding of the laws is still incomplete, yet we know how to proceed to enlarge our understanding by means of the 'scientific method' - a logical process of observation and reason that distills the empirically true statements we can make about nature." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"The reality is that without mathematics, modern-day cosmologists could not have progressed even one step in attempting to understand the laws of nature. Mathematics provides the solid scaffolding that holds together any theory of the universe. […] Mathematics appears to be almost too effective in describing and explaining not only the cosmos at large, but even some of the most chaotic of human enterprises." (Mario Livio, "Is God a Mathematician?", 2011)

"Symmetries are transformations that keep certain parameters (properties, equations, and so on) invariant, that is, the parameters they refer to are conserved under these transformations. It is to be expected, therefore, that the identification of conserved quantities is inseparable from the identification of fundamental symmetries in the laws of nature. Symmetries single out 'privileged' operations, conservation laws single out 'privileged' quantities or properties that correspond to these operations. Yet the specific connections between a particular symmetry and the invariance it entails are far from obvious. For instance, the isotropy of space (the indistinguishability of its directions) is intuitive enough, but the conservation of angular momentum based on that symmetry, and indeed, the concept of angular momentum, are far less intuitive." (Yemima Ben-Menahem, "Causation in Science", 2018)

"It is impossible to transcend the laws of nature. You can only determine that your understanding of nature has changed." (Nick Powers)

"Modern man lacks a unified conception of the world. He lives in a dual world: in his environment, which is naturally given to him, and, at the same time, in the world which since the beginning of the modern era has been created for him by sciences founded upon the principle that the laws of nature are, in essence, mathematical. The non-unity which has thus come to penetrate our entire life is the true source of the spiritual crisis we are going through today." (Jan Patočka) 

"Nothing is too wonderful to be true, if it be consistent with the laws of nature, and in such things as these, experiment is the best test of such consistency." (Michael Faraday)

"The highest triumph of the human mind, the true knowledge of the most general laws of nature, ought not to remain the private possession of a privileged class of learned men, but ought to become the common property of all mankind."  (Ernst Häckel)

"The laws of nature are but the mathematical thoughts of God." (Euclid)

"The laws of Nature are written in the language of mathematics […]" (Galileo Galilei)

"The secret of nature is symmetry. When searching for new and more fundamental laws of nature, we should search for new symmetries." (David Gross)

On Beauty: Nature

"Nature builds up by her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (Sir John F W Herschel, "The Cabinet of Natural Philosophy", 1831)

"As long as men inquire, they will find opportunities to know more upon these topics than those who have gone before them, so inexhaustibly rich is nature in the innermost diversity of her treasures of beauty, order, and intelligence." (J Louis R Agassiz, "Essay on Classification", 1859)

"In order to depict nature in its exalted sublimity, we must not dwell exclusively on its external manifestations, but we must trace its image, reflected in the mind of man, at one time filling the dreamy land of physical myths with forms of grace and beauty, and at another developing the noble germ of artistic creations." (Alexander von Humboldt, "Cosmos: A Sketch of a Physical Description of the Universe" Vol. 2, 1869)

"The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Of course I do not here speak of that beauty that strikes the senses, the beauty of qualities and appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts, and which a pure intelligence can grasp." (Henri Poincaré, "The Value of Science", 1905)

"In the grandeur of its sweep in space and time, and the beauty and simplicity of the relations which it discloses between the greatest and the smallest things of which we know, it reveals as perhaps nothing else does, the majesty of the order about us which we call nature, and, as I believe, of that Power behind the order, of which it is but a passing shadow." (Henry N Russell, "Annual Report of the Board of Regents of the Smithsonian Institution", 1923)

"It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better." (Paul Dirac, "The Evolution of the Physicist's Picture of Nature", 1963)

"The study of symmetry was born out of art and mathematics; art as the comprehension of the beauty of nature and mathematics as the comprehension of the world's harmony. " (N F Ovchinnikov, "Principles of Preservation", 1966)

"To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. […] If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in." (Richard P Feynman, "The Character of Physical Law", 1967)

"Beauty is our weapon against nature; by it we make objects, giving them limit, symmetry, proportion. Beauty halts and freezes the melting flux of nature." (Camille Paglia, "Sexual Personae", 1990)

"The immediate evidence from the natural world may seem to be chaotic and without any inner regularity, but mathematics reveals that under the surface the world of nature has an unexpected simplicity - an extraordinary beauty and order." (William Byers, "How Mathematicians Think", 2007)

"The beauty of nature insists on taking its time. Everything is prepared. Nothing is rushed. The rhythm of emergence is a gradual, slow beat; always inching its way forward, change remains faithful to itself until the new unfolds in the full confidence of true arrival. Because nothing is abrupt, the beginning of spring nearly always catches us unawares. It is there before we see it; and then we can look nowhere without seeing it. (John O'Donohue, "To Bless the Space Between Us: A Book of Blessings", 2008)

"If nature leads to mathematical forms of great simplicity and beauty - to forms that no one has previously encountered - we cannot help thinking that they are true and that they revealed genuine features of Nature." (Werner K Heisenberg)

"Nature builds up her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (John Herschel)

"Whence is it that nature does nothing in vain; and whence arises all that order and beauty which we see in the world?" (Sir Isaac Newton)

Georges-Louis Leclerc - Collected Quotes

"In general, the more one augments the number of divisions of the productions of nature, the more one approaches the truth, since in nature only individuals exist, while genera, orders, and classes only exist in our imagination." (Georges-Louis Leclerc, "Natural History, General and Particular", 1749)

"Let us gather facts in order to get ourselves thinking." (Georges-Louis Leclerc, "Natural History, General and Particular" Vol. 2, 1749)

"Let us investigate more closely this property common to animal and plant, this power of producing its likeness, this chain of successive existences of individuals, which constitutes the real existence of the species." (Georges-Louis Leclerc, "Natural History, General and Particular" Vol 2., 1749)

"Natural History is the most extensive, and perhaps the most instructive and entertaining of all the sciences. It is the chief source from which human knowledge is derived. To recommend the study of it from motives of utility were to affront the understanding of mankind. Its importance, accordingly, in the arts of life, and in storing the mind with just ideas of external objects, as well as of their relations to the human race, was early perceived by all nations in their progress from rudeness to refinement." (Georges-Louis Leclerc, "Natural History, General and Particular" Vol. 1, 1749)

"Only those works which are well-written will pass to posterity: the amount of knowledge, the uniqueness of the facts, even the novelty of the discoveries are no guarantees of immortality...These things are exterior to a man but style is the man himself." (Georges-Louis Leclerc, "Natural History, General and Particular" Vol. 7, 1749)

"Nature, displayed in its full extent, presents us with an immense tableau, in which all the order of beings are each represented by a chain which sustains a continuous series of objects, so close and so similar that their difference would be difficult to define. This chain is not a simple thread which is only extended in length, it is a large web or rather a network, which, from interval to interval, casts branches to the side in order to unite with the networks of another order." (Comte Georges-Louis Leclerc de Buffon, "Les Oiseaux Qui Ne Peuvent Voler", Histoire Naturelle des Oiseaux Vol. I, 1770)

"As we can judge only in proportion as we compare, and as all our knowledge turns upon the relations by which one object differs from another, if there existed no brute animals, the nature of the human beings would be still more incomprehensible." (Georges-Louis Leclerc, "Natural History, General and Particular" Vol. 5, 1781)

"Nature is that system of laws established by the Creator for regulating the existence of bodies, and the succession of beings. Nature is not a body; for this body would comprehend everything. Either is it a being; for this being would necessarily be God. But nature may be considered as an immense living power, which animates the universe, and which, in subordination to the first and supreme Being, began to act by his command, and its action is still continued by his concurrence or consent." (Georges-Louis Leclerc, "Natural History, General and Particular" Vol. 6, 1781)

"Nature turns upon two steady pivots, unlimited fecundity which she has given to all species; and those innumerable causes of destruction which reduce the product of this fecundity [...]" (Georges-Louis Leclerc, "Natural History, General and Particular" Vol. 5, 1781)

"The only good science is the knowledge of facts, and mathematical truths are only truths of definition, and completely arbitrary, quite unlike physical truths." (Georges-Louis Leclerc)

Scientific Experience III: Nature

"The phenomena of nature are most often enveloped by so many strange circumstances, and so great a number of disturbing causes mix their influence, that it is very difficult to recognize them. We may arrive at them only by multiplying the observations or the experiences, so that the strange effects finally destroy reciprocally each other." (Pierre-Simon Laplace, "A Philosophical Essays on Probabilities", 1814)

"The Laws of Nature are merely truths or generalized facts, in regard to matter, derived by induction from experience, observation, arid experiment. The laws of mathematical science are generalized truths derived from the consideration of Number and Space." (Charles Davies, "The Logic and Utility of Mathematics", 1850)

"The laws of nature are drawn from experience, but to express them one needs a special language: for, ordinary language is too poor and too vague to express relations so subtle, so rich, so precise. Here then is the first reason why a physicist cannot dispense with mathematics: it provides him with the one language he can speak […]. Who has taught us the true analogies, the profound analogies which the eyes do not see, but which reason can divine? It is the mathematical mind, which scorns content and clings to pure form." (Henri Poincaré, "The Value of Science", 1905)

"The scientific worker has elected primarily to know, not do. He does not directly seek, like the practical man, to realize the ideal of exploiting nature and controlling life – though he makes this more possible; he seeks rather to idealize – to conceptualize – the real, or at least those aspects of reality that are available in his experience. He thinks more of lucidity and formulae than of loaves and fishes. He is more concerned with knowing Nature than with enjoying her. His main intention is to describe the sequences in Nature in the simplest possible formulae, to make a working thought-model of the known world. He would make the world translucent, not that emotion may catch the glimmer of the indefinable light that shines through, but for other reasons – because of his inborn inquisitiveness, because of his dislike of obscurities, because of his craving for a system – an intellectual system in which phenomena are at least provisionally unified." (Sir John A Thomson," Introduction to Science", 1911)

"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)

"Everything we think we know about the world is a model. Every word and every language is a model. All maps and statistics, books and databases, equations and computer programs are models. So are the ways I picture the world in my head - my mental models. None of these is or ever will be the real world. […] Our models usually have a strong congruence with the world. That is why we are such a successful species in the biosphere. Especially complex and sophisticated are the mental models we develop from direct, intimate experience of nature, people, and organizations immediately around us." (Donella Meadows, "Limits to Growth", 1972)

On Series XII: Nature

"Further we believe that GOD Himself is present everywhere throughout the whole of the undoubtedly divisible space that all bodies occupy; & yet He is onefold in the highest degree & admits not of any composite nature whatever. Moreover, the same idea seems to depend on an analogy between space & time. For, just as rest is a conjunction with a continuous series of all the instants In the interval of time during which the rest endures; so also this virtual extension is a conjunction of one instant of time with a continuous series of all the points of space throughout which this one-fold entity extends virtually. Hence, just as rest is believed to exist in Nature, so also are we bound to admit virtual extension; & if this is admitted, then it will be possible for the primary elements of matter to be simple, & yet not absolutely non-extended." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Nature, displayed in its full extent, presents us with an immense tableau, in which all the order of beings are each represented by a chain which sustains a continuous series of objects, so close and so similar that their difference would be difficult to define. This chain is not a simple thread which is only extended in length, it is a large web or rather a network, which, from interval to interval, casts branches to the side in order to unite with the networks of another order." (Comte Georges-Louis Leclerc de Buffon, "Les Oiseaux Qui Ne Peuvent Voler", Histoire Naturelle des Oiseaux Vol. I, 1770)

"[…] without the theory of evolution all the big general series of phenomena of organic nature remain completely incomprehensible and inexplicable riddles, while by means of this theory they can be explained simply and consistently. This holds especially true for two complexes of biological phenomena which we now in conclusion wish to single out in a few words. These form the subject of two special branches of physiology which so far have been largely neglected, namely, the ecology and chorology of organisms." (Ernst Häckel, "Generelle Morphologie der Organismen", 1866)

"Nature's action is complex: and nothing is gained in the long run by pretending that it is simple, and trying to describe it in a series of elementary propositions." (Alfred Marshall, "Principles of Economics", 1890)

"We define law, using the word in the philosophic sense, as the constant relation discoverable in a series of phenomena." (Michel Bréal, "Essai de semantique", 1897)

"Nature is an infinitely complex series of facts; it is not an object lesson, and it is not a ready-made sermon on conduct or morality." (Hal Borland, "The Enduring Pattern, A Place to Live: Time", 1959)

On Puzzles (Unsourced)

"It is an outcome of faith that nature - as she is perceptible to our five senses - takes the character of such a well formulated puzzle." (Albert Einstein)

"Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man." (Maria Goeppert-Mayer)

"Science is a game - but a game with reality, a game with sharpened knives [..] If a man cuts a picture carefully into 1000 pieces, you solve the puzzle when you reassemble the pieces into a picture; in the success or failure, both your intelligences compete. In the presentation of a scientific problem, the other player is the good Lord. He has not only set the problem but also has devised the rules of the game - but they are not completely known, half of them are left for you to discover or to deduce. The experiment is the tempered blade which you wield with success against the spirits of darkness - or which defeats you shamefully. The uncertainty is how many of the rules God himself has permanently ordained, and how many apparently are caused by your own mental inertia, while the solution generally becomes possible only through freedom from its limitations." (Erwin Schrödinger)

"The art of simplicity is a puzzle of complexity." (Douglas Horton)

"Throughout science there is a constant alternation between periods when a particular subject is in a state of order, with all known data falling neatly into their places, and a state of puzzlement and confusion, when new observations throw all neatly arranged ideas into disarray." (Sir Hermann Bondi)

"While the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what anyone man will be up to, but you can say with precision what an average number will be up to. Individuals vary, but percentages remain constant. So says the statistician." (Sir Arthur C Doyle)

Roger J Boscovich - Collected Quotes

"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)

"Any point has a real mode of existence, through which it is where it is; & another, due to which it exists at the time when it does exist. These real modes of existence are to me real time & space ; the possibility of these modes, hazily apprehended by us, is, to my mind, empty space & again empty time, so to speak ; in other words, space & imaginary time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Further we believe that GOD Himself is present everywhere throughout the whole of the undoubtedly divisible space that all bodies occupy; & yet He is onefold in the highest degree & admits not of any composite nature whatever. Moreover, the same idea seems to depend on an analogy between space & time. For, just as rest is a conjunction with a continuous series of all the instants In the interval of time during which the rest endures; so also this virtual extension is a conjunction of one instant of time with a continuous series of all the points of space throughout which this one-fold entity extends virtually. Hence, just as rest is believed to exist in Nature, so also are we bound to admit virtual extension; & if this is admitted, then it will be possible for the primary elements of matter to be simple, & yet not absolutely non-extended." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Hence I acknowledge continuity in motion only, which is something successive and not co-existent ; & also in it alone, or because of it alone, in corporeal entities at any rate, lies my reason for admitting the Law of Continuity. From this it will be all the more clear that, as I remarked above, Nature accurately observes the Law of Continuity, or at least tries to do so. Nature observes it in motions & in distance, & tries to in many other cases, with which continuity, as we have defined it above, is in no wise in agreement; also in certain other cases, in which continuity cannot be completely obtained. This continuity does not present itself to us at first sight, unless we consider the subjects somewhat more deeply & study them closely." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"Hence the whole of geometry is imaginary; but the hypothetical propositions that are deduced from it are true, if the conditions assumed by it exist, & also the conditional things deduced from them, in every case; & the relations between the imaginary distances of points, derived by the help of geometry from certain conditions, will always be real, & such as they are found to be by geometry, when those conditions exist for real distances of points." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"If matter is continuous, it may & must be subject to infinite divisibility; but actual division carried on indefinitely brings in its train difficulties that are truly inextricable; however, this infinite division is required by those who do not admit that there are any particles, no matter how small, in bodies that are perfectly free from, & incapable of, compression." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"In the same way, this should also happen with regard to time, namely, that between a preceding continuous time & the next following there should be a single instant, which is the indivisible boundary of either. There cannot be two instants, as we intimated above, contiguous to one another; but between one instant & another there must always intervene some interval of continuous time divisible indefinitely. In the same way, in any quantity which lasts for a continuous interval of time, there must be obtained a series of magnitudes of such a kind that to each instant of time there is its corresponding magnitude; & this magnitude connects the one that precedes with the one that follows it, & differs from the former by some definite magnitude. Nay even in that class of quantities, in which we cannot have two magnitudes at the same time, this very point can be deduced far more clearly, namely, that there cannot be any sudden change from one to another. For at that instant, when the sudden change should take place, & the series be broken by some momentary definite addition, two -magnitudes would necessarily be obtained, namely, the last of the first series & the first of the next. Now this very point is still more clearly seen in those states of things, in which on the one hand there must be at any instant some state so that at no time can the thing be without some state of the kind, whilst on the other hand it can never have two states of the kind simultaneously." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"The Law of Continuity, as we here deal with it, consists in the idea that [...] any quantity, in passing from one magnitude to another, must pass through all intermediate magnitudes of the same class. The same notion is also commonly expressed by saying that the passage is made by intermediate stages or steps; [...] the idea should be interpreted as follows: single states correspond to single instants of time, but increments or decrements only to small areas of continuous time." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"The theory of non-extension is also convenient for eliminating from Nature all idea of a coexistent continuum — to explain which philosophers have up till now laboured so very hard & generally in vain. Assuming non-extension, no division of a real entity can be carried on indefinitely ; we shall not be brought to a standstill when we seek to find out whether the number of parts that are actually distinct & separable is finite or infinite ; nor with it will there come in any of those other truly innumerable difficulties that, with the idea of continuous composition, have given so much trouble- to philosophers. For if the primary elements of matter are perfectly non-extended & indivisible points separated from one another by some definite interval, then the number of points in any given mass must bc finite ; because all the distances are finite." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"(1) There is absolutely no argument that can be brought forward to prove that matter has continuous extension, that it is not rather made up of perfectly indivisible points separated from one another by a definite interval ; nor is there any reason apart from prejudice in favour of continuous extension in preference to composition from points that are perfectly indivisible, non-extended, forming no extended continuum of any sort. (2) There are arguments, & fairly strong ones too, which will prove that this composition from indivisible points is preferable to continuous extension." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

"There really must be, in the commencement of contact, in that indivisible instant of time which is an indivisible limit between the continuous time that preceded the contact & that subsequent to it (just in the same way as a point in geometry is an indivisible limit between two segments of a continuous line), a change of velocity taking place suddenly, without any passage through intermediate stages; & this violates the Law of Continuity, which absolutely denies the possibility of a passage from one magnitude to another without passing through intermediate stages." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)

Nature's Architecture

"Whoever surveys the curious fabric of the universe can never imagine, that so noble a structure should be fram’d for no other use, than barely for mankind to live and breathe in. It was certainly the design of the great Architect, that his creatures should afford not only necessaries and accommodations to our animal part, but also instructions to our intellectual." (Sir Thomas P Blount, "A Natural History", 1693)

"Nature builds up by her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (Sir John F W Herschel, "The Cabinet of Natural Philosophy", 1831)

"[…] the lifeless symmetry of architecture, however beautiful the design and proportion, no man would be so mad as to put in competition with the animated charms of nature." (Fanny Burney, "Evelina", 1909)

"The pleasure derived from the discovery of some secret of Nature unknown before except to the architect of the universe surpasses all the rewards the world can give." (Richard Gregory, "Discovery: or, The Spirit and Service of Science", 1916)

"Architecture is the first manifestation of man creating his own universe, creating it in the image of nature, submitting to the laws of nature, the laws which govern our own nature, our universe. The laws of gravity, of statics and of dynamics, impose themselves by a reductio ad absurdum: everything must hold together or it will collapse." (Charles-Edouard Jeanneret [Le Corbusier], "Towards a New Architecture", 1923)

"As the complexity of the structure of matter became revealed through research, its basic simplicity, unity, and dependability became equally evident. So we now see ourselves in a world governed by natural laws instead of by capricious deities and devils. This does not necessarily mean that God has been ruled out of the picture, but it does mean that the architect and engineer of the universe is a far different type of being from the gods assumed by the ancients, and that man lives and dies in a world of logical system and orderly performance." (Karl T Compton, cca. 1930–1949)

"[…] 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)

"Nature builds up her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (John Herschel)

Freeman J Dyson - Collected Quotes

"The reason why new concepts in any branch of science are hard to grasp is always the same; contemporary scientists try to picture the new concept in terms of ideas which existed before." (Freeman J Dyson, "Innovation in Physics" , Scientific American, 1958)

"It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex." (Freeman Dyson, "The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics" , Journal of Mathematical Physics Vol. 3, 1962)

"For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American, 1964)

"Mathematical intuition is more often conservative than revolutionary, more often hampering than liberating." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American Vol,. 211 (3), 1964)

"The trouble with group theory is that it leaves so much unexplained that one would like to explain. It isolates in a beautiful way those aspects of nature that can be understood in terms of abstract symmetry alone. It does not offer much hope of explaining the messier facts of life, the numerical values of particle lifetimes and interaction strengths - the great bulk of quantitative experimental data that is now waiting for explanation. The process of abstraction seems to have been too drastic, so that many essential and concrete features of the real world have been left out of consideration. Altogether group theory succeeds just because its aims are modest. It does not try to explain everything, and it does not seem likely that it will grow into a complete or comprehensive theory of the physical world." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American Vol. 211 (3), 1964)

"A good scientist is a person with original ideas. A good engineer is a person who makes a design that works with as few original ideas as possible. There are no prima donnas in engineering." (Freeman J Dyson, "Disturbing the Universe", 1979)

"In the long run, qualitative changes always outweigh quantitative ones. Quantitative predictions of economic and social trends are made obsolete by qualitative changes in the rules of the game. Quantitative predictions of technological progress are made obsolete by unpredictable new inventions. I am interested in the long run, the remote future, where quantitative predictions are meaningless. The only certainty in that remote future is that radically new things will be happening." (Freeman J Dyson, "Disturbing the Universe", 1979)

"I have found a universe growing without limit in richness and complexity, a universe of life surviving forever and making itself known to its neighbors across unimaginable gulfs of space and time. Whether the details of my calculations turn out to be correct or not, there are good scientific reasons for taking seriously the possibility that life and intelligence can succeed in molding this universe of ours to their own purposes." (Freeman J Dyson, "Infinite in All Directions", 1988)

"If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed. I would feel that the Creator had been uncharacteristically lacking in imagination." (Freeman J Dyson, "Infinite in All Directions", 1988)

"The principle of maximum diversity operates both at the physical and at the mental level. It says that the laws of nature and the initial conditions are such as to make the universe as interesting as possible.  As a result, life is possible but not too easy. Always when things are dull, something new turns up to challenge us and to stop us from settling into a rut. Examples of things which make life difficult are all around us: comet impacts, ice ages, weapons, plagues, nuclear fission, computers, sex, sin and death.  Not all challenges can be overcome, and so we have tragedy. Maximum diversity often leads to maximum stress. In the end we survive, but only by the skin of our teeth." (Freeman J Dyson, "Infinite in All Directions", 1988)

"Theoretical physicists are accustomed to living in a world which is removed from tangible objects by two levels of abstraction. From tangible atoms we move by one level of abstraction to invisible fields and particles. A second level of abstraction takes us from fields and particles to the symmetry-groups by which fields and particles are related. The superstring theory takes us beyond symmetry-groups to two further levels of abstraction. The third level of abstraction is the interpretation of symmetry-groups in terms of states in ten-dimensional space-time. The fourth level is the world of the superstrings by whose dynamical behavior the states are defined." (Freeman J Dyson, "Infinite in All Directions", 1988)

"The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design." (Freeman J Dyson, [interview] 1994)

"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103 (7), 1996)

"The total disorder in the universe, as measured by the quantity that physicists call entropy, increases steadily steadily as we go from past to future. On the other hand, the total order in the universe, as measured by the complexity and permanence of organized structures, also increases steadily as we go from past to future." (Freeman J Dyson, [Page-Barbour lecture], 2004)

"The progress of science requires the growth of understanding in both directions, downward from the whole to the parts and upward from the parts to the whole." (Freeman J Dyson, "The Scientist As Rebel", 2006)

"The total disorder in the universe, as measured by the quantity that physicists call entropy, increases steadily as we go from past to future. On the other hand, the total order in the universe, as measured by the complexity and permanence of organized structures, also increases steadily as we go from past to future." (Freeman J Dyson, "A Many-Colored Glass: Reflections on the Place of Life in the Universe", 2007)

"Recreational mathematics is a splendid hobby which young and old can equally enjoy. The popularity of Sudoku shows that an aptitude for recreational mathematics is widespread in the population. From Sudoku it is easy to ascend to mathematical pursuits that offer more scope for imagination and originality." (Freeman Dyson, 2011)

"The whole point of science is that most of it is uncertain. That’s why science is exciting–because we don’t know. Science is all about things we don’t understand. The public, of course, imagines science is just a set of facts. But it’s not. Science is a process of exploring, which is always partial. We explore, and we find out things that we understand. We find out things we thought we understood were wrong. That’s how it makes progress." (Freeman Dyson,  [interview] 2014)

"A model is done when nothing else can be taken out." (Freeman J Dyson)

"Much of the history of science, like the history of religion, is a history of struggles driven by power and money. And yet, this is not the whole story. Genuine saints occasionally play an important role, both in religion and science. For many scientists, the reward for being a scientist is not the power and the money but the chance of catching a glimpse of the transcendent beauty of nature." (Freeman J Dyson)

"It often happens that understanding of the mathematical nature of an equation is impossible without a detailed understanding of its solution." (Freeman J Dyson)

"One factor that has remained constant through all the twists and turns of the history of physics is the decisive importance of the mathematical imagination." (Freeman J Dyson)

On Patterns (1940-1949)

"A mathematician, like a painter or a poet, is a maker of patterns. [...]. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey H Hardy, "A Mathematician's Apology", 1941)

"It will probably be the new mathematical discoveries which are suggested through physics that will always be the most important, for, from the beginning Nature has led the way and established the pattern which mathematics, the Language of Nature, must follow." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"One may generalize upon these processes in terms of group equilibrium. The group may be said to be in equilibrium when the interactions of its members fall into the customary pattern through which group activities are and have been organized. The pattern of interactions may undergo certain modifications without upsetting the group equilibrium, but abrupt and drastic changes destroy the equilibrium." (William F Whyte, "Street Corner Society", 1943)

"Those who are content with a positivist conception of the aims of science will feel that he is in an entirely satisfactory position; he has discovered the pattern of events, and so can predict accurately; what more can he want? A mental picture would be an added luxury, but also a useless luxury. For if the picture did not bear any resemblance at all to the reality it would be valueless, and if it did it would be unintelligible […]" (James H Jeans," Physics and Philosophy" 3rd Ed., 1943)

"Without falling into the trap of attempting a precise definition, we may suggest a theory as to the general nature of symbolism, viz. that it is the ability of processes to parallel or imitate each other, or the fact that they can do so since there are recurrent patterns in reality." (Kenneth Craik, "The Nature of Explanation", 1943)

"Science in general […] does not consist in collecting what we already know and arranging it in this or that kind of pattern. It consists in fastening upon something we do not know, and trying to discover it. (Robin G Collingwood, "The Idea of History", 1946)

On Nature (-1599)

"They say that the greatest and fairest things are the work of nature and of chance, the lesser of art, which, receiving from nature the greater and primeval creations, molds and fashions all those lesser works which are generally termed artificial." (Plato, "Nomoi" ["Laws"], cca. 360 BC)

"Everything that depends on the action of nature is by nature as good as it can be, and similarly everything that depends on art or any rational cause, and especially if it depends on the best of all causes. To entrust to chance what is greatest and most noble would be a very defective arrangement." (Aristotle, "Nicomachean Ethics", cca. 350 BC)

"The proof that the state is a creation of nature and prior to the individual is that the individual, when isolated, is not self-sufficing; and therefore he is like a part in relation to the whole." (Aristotle, "Politics", 4th century 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. 4 century BC)

"No species remains constant: that great renovator of matter Nature, endlessly fashions new forms from old: there’s nothing in the whole universe that perishes, believe me; rather it renews and varies its substance. What we describe as birth is no more than incipient change from a prior state, while dying is merely to quit it. Though the parts may be transported hither and thither, the sum of all matter is constant." (Publius Ovidius Naso [Ovid], "Metamorphoses", 8 AD)

"All that is superfluous displeases God and nature. All that displeases God and nature is evil." (Dante Alighieri, "De Monarchia", cca. 1312-1313)

"Given that annihilation of nature in its entirety is impossible, and that death and dissolution are not appropriate to the whole mass of this entire globe or star, from time to time, according to an established order, it is renewed, altered, changed, and transformed in all its parts." (Giordano Bruno, "The Ash Wednesday Supper", 1584)

"Nature that framed us of four elements, Warring within our breasts for regiment, Doth teach us all to have aspiring minds: Our souls, whose faculties can comprehend The wondrous architecture of the world: And measure every wand’ring planet’s course, Still climbing after knowledge infinite, And always moving as the restless spheres, Wills us to wear ourselves and never rest, Until we reach the ripest fruit of all, That perfect bliss and sole felicity, The sweet fruition of an earthly crown.""  (Christopher Marlowe, "Tamburlaine the Great", 1590)

"The diversity of the phenomena of Nature is so great, and the treasures hidden in the heavens so rich, precisely in order that the human mind shall never be lacking in fresh nourishment." (Johannes Kepler, "Mysterium Cosmographicum", 1596)

On Nature: The Nature of Things

"Mathematicians have, in many cases, proved some things to be possible and others to be impossible, which, without demonstration, would not have been believed […] Mathematics afford many instances of impossibilities in the nature of things, which no man would have believed, if they had not been strictly demonstrated. Perhaps, if we were able to reason demonstratively in other subjects, to as great extent as in mathematics, we might find many things to be impossible, which we conclude, without hesitation, to be possible." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"How awkward is the human mind in divining the nature of things, when forsaken by the analogy of what we see and touch directly?" (Ludwig Boltzmann, "Certain Questions of the Theory of Gasses", Nature Vol. 51 (1322), 1895)

"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)

"A statistical estimate may be good or bad, accurate or the reverse; but in almost all cases it is likely to be more accurate than a casual observer’s impression, and the nature of things can only be disproved by statistical methods." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"Great scientific discoveries have been made by men seeking to verify quite erroneous theories about the nature of things." (Aldous L Huxley, "Life and Letters and the London Mercury" Vol. 1, 1928)

"It is his intuition, his mystical insight into the nature of things, rather than his reasoning which makes a great scientist." (Karl R Popper, "The Open Society and Its Enemies", 1945)

"Science does not need mysticism and mysticism does not need science, but man needs both. Mystical experience is necessary to understand the deepest nature of things, and science is essential for modern life. What we need, therefore, is not a synthesis, but a dynamic interplay between mystical intuition and scientific analysis." (Fritjof Capra, "The Tao of Physics: An Exploration of the Parallels Between Modern Physics and Eastern Mysticism", 1975)

"Owing to his lack of knowledge, the ordinary man cannot attempt to resolve conflicting theories of conflicting advice into a single organized structure. He is likely to assume the information available to him is on the order of what we might think of as a few pieces of an enormous jigsaw puzzle. If a given piece fails to fit, it is not because it is fraudulent; more likely the contradictions and inconsistencies within his information are due to his lack of understanding and to the fact that he possesses only a few pieces of the puzzle. Differing statements about the nature of things […] are to be collected eagerly and be made a part of the individual's collection of puzzle pieces. Ultimately, after many lifetimes, the pieces will fit together and the individual will attain clear and certain knowledge." (Alan R Beals, "Strategies of Resort to Curers in South India" [contributed in Charles M. Leslie (ed.), "Asian Medical Systems: A Comparative Study", 1976])

"In strategic thinking, one first seeks a clear understanding of the particular character of each element of a situation and then makes the fullest possible use of human brainpower to restructure the elements in the most advantageous way. Phenomena and events in the real word do not always fit a linear model. Hence the most reliable means of dissecting a situation into its constituent parts and reassembling then in the desired pattern is not a step-by-step methodology such as systems analysis. Rather, it is that ultimate nonlinear thinking tool, the human brain. True strategic thinking thus contrasts sharply with the conventional mechanical systems approach based on linear thinking. But it also contrasts with the approach that stakes everything on intuition, reaching conclusions without any real breakdown or analysis. [...] No matter how difficult or unprecedented the problem, a breakthrough to the best possible solution can come only from a combination of rational analysis, based on the real nature of things, and imaginative reintegration of all the different items into a new pattern, using nonlinear brainpower. This is always the most effective approach to devising strategies for dealing successfully with challenges and opportunities, in the market arena as on the battlefield." (Kenichi Ohmae, "The Mind Of The Strategist", 1982)

"All great theories are expansive, and all notions so rich in scope and implication are underpinned by visions about the nature of things. You may call these visions ‘philosophy’, or ‘metaphor’, or ‘organizing principle’, but one thing they are surely not - they are not simple inductions from observed facts of the natural world." (Stephen J Gould, "Time’s Arrow, Time’s Cycle", 1987)

"If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things." (Robert Kaplan, "The Nothing that Is: A Natural History of Zero", 2000)

"Metaphor is evidence of the human ability to visualize the universe as a coherent organism. Proof of our capacity, not just to see one thing in another but to change the very nature of things. When a metaphor is accepted as fact, it enters groupthink, taking on an existence in the real world. [...] Metaphor is the default form of thought, providing many angles from which to literally 'see' the world." (Marcel Danesi, "Poetic Logic: The Role of Metaphor in Thought, Language, and Culture", 2004)

On Nature (Unsourced)

"[All phenomena] are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough." (Nicolas de Condorcet)

"All mathematical laws which we find in Nature are always suspect to me, in spite of their beauty. They give me no pleasure. They are merely auxiliaries. At close range it is all not true." (Georg C Lichtenberg)

"All the effects of nature are only mathematical results of a small number of immutable laws."  (Pierre Simon de Laplace)

"All the pictures which science draws of Nature, and which alone seem capable of according with observational facts, are mathematical pictures." (Sir James Jeans)

"Although to penetrate into the intimate mysteries of nature and hence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena." (Leonhard Euler)

"Among all of the mathematical disciplines the theory of differential  equations is the most important […]. It furnishes the explanation of all those elementary manifestations of nature which involve time."  (Sophus Lie)

"Beauty is our weapon against nature; by it we make objects, giving them limit, symmetry, proportion. Beauty halts and freezes the melting flux of nature." (Camille Paglia)

"But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end." (Aristotle, "Generation of Animals")

"Engineering is the art or science of utilizing, directing or instructing others in the utilization of the principles, forces, properties and substances of nature in the production, manufacture, construction, operation and use of things [...] or of means, methods, machines, devices and structures [...]"  (Alfred W Kiddle)

"Ever since the observation of nature has existed, it has held a vague notion of its ultimate goal as the composition of the colorful multiplicity of phenomena in a uniform system, where possible, in a single formula." (Max Planck)

"I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author." (Gottfried W Leibniz)

"I believe we can attach mathematically everything in nature and in the world of change."  (Iambilichus)

"If nature leads to mathematical forms of great simplicity and beauty - to forms that no one has previously encountered - we cannot help thinking that they are true and that they revealed genuine features of Nature." (Werner K Heisenberg)

"If this seems complex, the reason is because Tao [nature] is both simple and complex. It is complex when we try to understand it, and simple when we allow ourselves to experience it." (Stanley Rosenthal)

"In the study of Nature conjecture must be entirely put aside, and vague hypothesis carefully guarded against. The study of Nature begins with facts, ascends to laws, and raises itself, as far as the limits of man’s intellect will permit, to the knowledge of causes, by the threefold means of observation, experiment and logical deduction." (Jean Baptiste-Andre Dumas)

"It is an outcome of faith that nature - as she is perceptible to our five senses - takes the character of such a well formulated puzzle." (Albert Einstein)

"It is impossible to transcend the laws of nature. You can only determine that your understanding of nature has changed." (Nick Powers)

"it is the most widely accepted axiom in the natural science that Nature makes use of the fewest possible means" (Johannes Kepler)

"It stands to the everlasting credit of science that by acting on the human mind it has overcome man's insecurity before himself and before nature." (Albert Einstein)

"Like a great poet, Nature produces the greatest results with the simplest means." (Heinrich Heine)

"Look deep, deep, deep into nature, and then you will understand everything." (Albert Einstein)

"Mathematics began to seem too much like puzzle solving. Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man." (Maria Goeppert-Mayer)

"Mathematics is a spirit of rationality. It is this spirit that challenges, simulates, invigorates and drives human minds to exercise themselves to the fullest. It is this spirit that seeks to influence decisively the physical, normal and social life of man, that seeks to answer the problems posed by our very existence, that strives to understand and control nature and that exerts itself to explore and establish the deepest and utmost implications of knowledge already obtained." (Morris Kline)

"Much of the history of science, like the history of religion, is a history of struggles driven by power and money. And yet, this is not the whole story. Genuine saints occasionally play an important role, both in religion and science. For many scientists, the reward for being a scientist is not the power and the money but the chance of catching a glimpse of the transcendent beauty of nature." (Freeman J Dyson)

"Natural Philosophy consists in discovering the frame and operations of Nature, and reducing them, as far as may be, to general Rules or Laws - establishing these rules by observations and experiments, and thence deducing the causes and effects of things." (Isaac Newton)

"Nature always acts in the mathematically shortest and best possible way." (Robert Grosseteste)

"Nature builds up her refined and invisible architecture, with a delicacy eluding our conception, yet with a symmetry and beauty which we are never weary of admiring." (John Herschel)

"Nature considered rationally, that is to say, submitted to the process of thought, is a unity in diversity of phenomena; a harmony, blending together all created things, however dissimilar in form and attributes; one great whole animated by the breath of life." (Alexander von Humboldt)

"Nature imitates mathematics." (Gian-Carlo Rota)

"Nature is pleased with simplicity, and affects not the pomp of superfluous causes." (Sir Issac Newton)

"Nature responds only to questions posed in mathematical language, because nature is the domain of measure and order." (Alexandre Koyré)

"Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop." (Chen Ning Yang)

"Nature, despite her seeming diversity, is always a unity, a whole; and thus, when she manifests herself in any part of that whole, the rest must serve as a basis for that particular manifestation, and the latter must have a relationship to the rest of the system." (Johann Wolfgang von Goethe)

"One might describe the mathematical quality in Nature by saying that the universe is so constituted that mathematics is a useful tool in its description. However, recent advances in physical science show that this statement of the case is too trivial. The connection between mathematics and the description of the universe goes far deeper than this, and one can get an appreciation of it only from a thorough examination of the various facts that make it up." (Paul A M Dirac)

"One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics." (Henri Poincare)

"Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas." (Albert Einstein)

"Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect." (Hermann Weyl)

"The closer we get to know the creatures around us, the clearer is the understanding we obtain of the chain of nature, and its harmony and system, according to which all things appear to have been created." (Carl Linnaeus)

"[…] the desire to understand nature has had on the development of mathematics the most important and happiest influence." (Henri Poincare)

"The equations that really work in describing nature with the most generality and the greatest simplicity are very elegant and subtle." (Edward Witten )

"The human understanding is of its own nature prone to abstractions and gives us a substance and reality to thing which are fleeting. But to resolve nature into abstractions is less to our purpose than to dissect her into parts." (Francis Bacon)

"The laws of nature are but the mathematical thoughts of God." (Euclid)

"The laws of Nature are written in the language of mathematics […]" (Galileo Galilei)

"The most general law in nature is equity-the principle of balance and symmetry which guides the growth of forms along the lines of the greatest structural efficiency." (Herbert Read)

"The object of education is not only to produce a man who knows, but one who does; who makes his mark in the straggle of life and succeeds well in whatever he undertakes: who can solve the problems of nature and of humanity as they arise, and who, when he knows he is right, can boldly convince the world of the fact." (Henry A Rowland)

"The present state of the system of nature is evidently a consequence of what is in the preceding moment, and if we conceive of an intelligence which at a given instant knew all the forces acting in nature and the position of every object in the universe - if endowed with a brain sufficiently vast to make all necessary calculations - could describe with a single formula the motions of the largest astronomical bodies and those of the smallest atoms. To such an intelligence, nothing would be uncertain; the future, like the past, would be an open book." (Pierre-Simon Laplace)

"The profound study of nature is the most fertile source of mathematical discoveries." (Joseph Fourier)

"The randomness which lies at the very foundations of pure mathematics of necessity permeates every human description of nature" (Joseph Ford)

"The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Of course I do not here speak of that beauty that strikes the senses, the beauty of qualities and appearances; not that I undervalue such beauty, far from it, but it has nothing to do with science; I mean that profounder beauty which comes from the harmonious order of the parts, and which a pure intelligence can grasp." (Henri Poincaré)

"The secret of nature is symmetry. When searching for new and more fundamental laws of nature, we should search for new symmetries." (David Gross)

"The simplicity of nature is not that which may easily be read, but is inexhaustible. The last analysis can no wise be made." (Ralph W Emerson)

"The way physics explains Nature is to speak in terms of the consequences of a few very basic equations." (Ekkehard Peik)

"The way to determine the secret workings of Nature is from analogous cases where one has caught her in act." (Georg C Lichtenberg)

"There is poetry in science and the cultivation of the imagination is an essential prerequisite to the successful investigation of nature." (Joseph Henry)

"We know many laws of nature and we hope and expect to discover more. Nobody can foresee the next such law that will be discovered. Nevertheless, there is a structure in laws of nature which we call the laws of invariance. This structure is so far-reaching in some cases that laws of nature were guessed on the basis of the postulate that they fit into the invariance structure." (Eugene P Wigner)

"What we can observe is not nature itself, but nature exposed to our method of questioning."  (Werner Heisenberg)

"Whatever Nature undertakes, she can only accomplish it in a sequence. She never makes a leap. For example she could not produce a horse if it were not preceded by all the other animals on which she ascends to the horse’s structure as if on the rungs of a ladder. Thus every one thing exists for the sake of all things and all for the sake of one; for the one is of course the all as well. Nature, despite her seeming diversity, is always a unity, a whole; and thus, when she manifests herself in any part of that whole, the rest must serve as a basis for that particular manifestation, and the latter must have a relationship to the rest of the system." (Johann Wolfgang von Goethe)

"When confronted with the order and beauty of the universe and the strange coincidences of nature, it's very tempting to take the leap of faith from science into religion. I am sure many physicists want to. I only wish they would admit it." (Tony Rothman)

"When the consequences of either assumption are the same, we should always assume that things are finite rather than infinite in number, since in things constituted by nature that which is infinite and that which is better ought, if possible, to be present rather than the reverse […]" (Aristotle)

"Whence is it that nature does nothing in vain; and whence arises all that order and beauty which we see in the world?" (Sir Isaac Newton)

"Whether man is disposed to yield to nature or to oppose her, he cannot do without a correct understanding of her language." (Jean Rostand)