🏛️On Art: On Poetry & Mathematics V

"The true mathematician is always a great deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have created an ideal world which they attempt to develop into the most perfect of all worlds, and which is being explored in every direction. None has the faintest conception of this world except him who knows it; only presumptuous ignorance can assert that the mathematician moves in a narrow circle. The truth which he seeks is, to be sure, broadly considered, neither more nor less than consistency; but does not his mastership show, indeed, in this very limitation? To solve questions of this kind he passes unenviously over others." (Alfred Pringsheim, Jaresberichte der Deutschen Mathematiker Vereinigung Vol 13, 1904)

"Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite. " (Ezra Pound, "The Spirit of Romance", 1910)

"[...] mathematics and poetry move together between two extremes of mysticism, the mysticism of the commonplace where ideas illuminate and create facts, and the mysticism of the extraordinary where God, the Infinite, the Real, poses the riddles of desire and disappointment, sin and salvation, effort and failure, question and paradoxical answer [...]" (Scott Buchanan, "Poetry and Mathematics", 1929)

"[…] the major mathematical research acquires an organization and orientation similar to the poetical function which, adjusting by means of metaphor disjunctive elements, displays a structure identical to the sensitive universe. Similarly, by means of its axiomatic or theoretical foundation, mathematics assimilates various doctrines and serves the instructive purpose, the one set up by the unifying moral universe of concepts." (Dan Barbilian, "The Autobiography of the Scientist", 1940)

"Mathematics is one component of any plan for liberal education. Mother of all the sciences, it is a builder of the imagination, a weaver of patterns of sheer thought, an intuitive dreamer, a poet. The study of mathematics cannot be replaced by any other activity that will train and develop man's purely logical faculties to the same level of rationality. Through countless dimensions, riding high the winds of intellectual adventure and filled with the zest of discovery, the mathematician tracks the heavens for harmony and eternal verity. There is not wholly unexpected surprise, but surprise nevertheless, that mathematics has direct application to the physical world about us. For mathematics, in a wilderness of tragedy and change, is a creature of the mind, born to the cry of humanity in search of an invariant reality, immutable in substance, unalterable with time. Mathematics is an infinity of flexibles forcing pure thought into a cosmos. It is an arc of austerity cutting realms of reason with geodesic grandeur. Mathematics is crystallized clarity, precision personified, beauty distilled and rigorously sublimated. The life of the spirit is a life of thought; the ideal of thought is truth; everlasting truth is the goal of mathematics." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)

"The structures with which mathematics deals are more like lace, the leaves of trees, and the play of light and shadow on a human face, than they are like buildings and machines, the least of their representatives. The best proofs in mathematics are short and crisp like epigrams, and the longest have swings and rhythms that are like music. The structures of mathematics and the propositions about them are ways for the imagination to travel and the wings, or legs, or vehicles to take you where you want to go." (Scott Buchanan, "Poetry and Mathematics", 1975)

"The theory of number is the epipoem of mathematics." (Scott Buchanan, "Poetry and Mathematics", 1975)

"To survive, mathematical ideas must be beautiful, they must be seductive, and they must be illuminating, they must help us to understand, they must inspire us. […] Part of that beauty, an essential part, is the clarity and sharpness that the mathematical way of thinking about things promotes and achieves. Yes, there are also mystic and poetic ways of relating to the world, and to create a new math theory, or to discover new mathematics, you have to feel comfortable with vague, unformed, embryonic ideas, even as you try to sharpen them."  (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"The relationship of math to the real world has been a conundrum for philosophers for centuries, but it is also an inspiration for poets. The patterns of mathematics inhabit a liminal space - they were initially derived from the natural world and yet seem to exist in a separate, self-contained system standing apart from that world. This makes them a source of potential metaphor: mapping back and forth between the world of personal experience and the world of mathematical patterns opens the door to novel connections." (Alice Major, "Mapping from e to Metaphor", 2018)

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On Geometrical Figures: On Lines

"Although a long series of rules might be added and a long discourse given about them, we conclude our detailed consideration with the cubic, others being merely mentioned, even if generally, in passing. For as positio refers to a line, quadratum to the surface, and cubum to a solid body, it would be very foolish for us to go beyond this point. Nature does no permit it." (Girolamo Cardano, "The Great Rules of Algebra" ["Ars Magna" aka"Artis Magnae, sive de Regulis Algebraicis Liber Unus"], 1545)

"Thus, all unknown quantities can be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth." (René Descartes, "La Géométrie", 1637)

"The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn." (Isaac Newton, 1687)

"Can a surface (perhaps a square including its boundary) be put into one-to-one correspondence with a line (perhaps a straight line segment including its endpoints) so that to each point of the surface there corresponds a point of the line and conversely to each point of the line there corresponds a point of the surface?" (Georg Cantor, [leter to Richard Dedekind] 1874)

"If one succeeds in setting up a one-to-one and complete correspondence between the points of a continuous manifold A of a dimensions on the one hand and the points of a continuous manifold B of b dimensions on the other, then this correspondence itself must necessarily be discontinuous throughout if a and b are unequal." (Georg Cantor, [leter to Richard Dedekind] 1877)

"Now it seems to me that all philosophical or mathematical deductions which make use of this mistaken assumption [on the number of coordinates] are inadmissible. Rather the distinction which exists between figures of different dimension numbers must be sought in entirely different aspects than in the number of independent coordinates, which is normally held to be characteristic." (Georg Cantor, [leter to Richard Dedekind] 1877)

"[...] surfaces, solids, even continuous figures of p dimensions can be put into one-to-one correspondence with confinuous lines, thus figures of only one dimension; therefore, that surfaces, solids, even figures of p dimensions have the same power as curves." (Georg Cantor, [leter to Richard Dedekind] 1877)

"It strikes me that all investigations taken up in this field begin for their part from an unproved assumption which does not appear to me to be obvious, but rather seems to need a proof. I mean the assumption that a ^-fold extended continuous manifold requires q independent real coordinates for the determination of its element and that this number of coordinates can be neither increased nor decreased for one and the same manifold."  (Georg Cantor, [leter to Richard Dedekind] 1877)

"Mathematics [...] would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude." (Friedrich Nietzsche, "Human, All Too Human", 1878)

"Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"And here is what makes this analysis situs interesting to us; it is that geometric intuition really intervenes there. When, in a theorem of metric geometry, one appeals to this intuition, it is because it is impossible to study the metric properties of a figure as abstractions of its qualitative properties, that is, of those which are the proper business of analysis situs. It has often been said that geometry is the art of reasoning correctly from badly drawn figures. This is not a capricious statement; it is a truth that merits reflection. But what is a badly drawn figure? It is what might be executed by the unskilled draftsman spoken of earlier; he alters the properties more or less grossly; his straight lines have disquieting zigzags; his circles show awkward bumps. But this does not matter; this will by no means bother the geometer; this will not prevent him from reasoning." (Henri Poincaré, "Dernières pensées", 1913)

"But it is a third geometry from which quantity is completely excluded and which is purely qualitative; this is analysis situs. In this discipline, two figures are equivalent whenever one can pass from one to the other by a continuous deformation; whatever else the law of this deformation may be, it must be continuous. Thus, a circle is equivalent to an ellipse or even to an arbitrary closed curve, but it is not equivalent to a straight-line segment since this segment is not closed. A sphere is equivalent to any convex surface; it is not equivalent to a torus since there is a hole in a torus and in a sphere there is not. Imagine an arbitrary design and a copy of this same design executed by an unskilled draftsman; the properties are altered, the straight lines drawn by an inexperienced hand have suffered unfortunate deviations and contain awkward bends. From the point of view of metric geometry, and even of projective geometry, the two figures are not equivalent; on the contrary, from the point of view of analysis situs, they are.” (Henri Poincaré, “Dernières pensées”, 1913)

"Geometric lines are fictions in the sense that, while we draw them, we think of them as having no width, simply length. Lines may be curved or straight." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume - as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures - the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

"A line normally represents space, but if we reinterpret that line as representing the passage of time, then presumably we are looking at 'shapes that repeat indefinitely along a temporal line at regular intervals'. Well, nearly. 'Shape' is the wrong word for something that exists in time. What word should we use? Let's reason it out by analogy - one of the sharpest weapons in the mathematical armoury. A shape is a (more or less) complicated collection of points in space; the temporal analogue is a complicated collection of points in time. A point in time is usually called an 'event'. So instead of 'shape' we have a 'series of events'. That is, the temporal pattern corresponding to the simplest frieze is a series of events that repeats indefinitely at regular intervals of time." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"Double periodicity is more interesting than single periodicity, because it is more varied. There is really only one periodic line, since all circles are the same up to a scale factor. However, there are infinitely many doubly periodic planes, even if we ignore scale. This is because the angle between the two periodic axes can vary, and so can the ratio of period lengths. The general picture of a doubly periodic plane is given by a lattice in the plane of complex numbers: a set of points of the form mA + nB, where A and B are nonzero complex numbers in different directions from O, and m and n run through all the integers. A and B are said to generate the lattice because it consists of all their sums and differences. […] The shape of the lattice of points mA + nB can therefore be represented by the complex number A/B. It is not hard to see that any nonzero complex number represents a lattice shape, so in some sense there is whole plane of lattice shapes. Even more interesting: the plane of lattice shapes is a periodic plane, because different numbers represent the same lattice." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"The engineer and the mathematician have a completely different understanding of the number pi. In the eyes of an engineer, pi is simply a value of measurement between three and four, albeit fiddlier than either of these whole numbers. [...] Mathematicians know the number pi differently, more intimately. What is pi to them? It is the length of a circle’s round line (its circumference) divided by the straight length (its diameter) that splits the circle into perfect halves. It is an essential response to the question, ‘What is a circle?’ But this response – when expressed in digits – is infinite: the number has no last digit, and therefore no last-but-one digit, no antepenultimate digit, no third-from-last digit, and so on." (Daniel Tammet, "Thinking in Numbers" , 2012)

On Geometrical Figures: On Squares

"Can a surface (perhaps a square including its boundary) be put into one-to-one correspondence with a line (perhaps a straight line segment including its endpoints) so that to each point of the surface there corresponds a point of the line and conversely to each point of the line there corresponds a point of the surface?" (Georg Cantor, [leter to Richard Dedekind] 1874)

"Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!" (Paul Carus, "The Nature of Logical and Mathematical Thought"; Monist Vol 20, 1910)

"The circle is the synthesis of the greatest oppositions. It combines the concentric and the eccentric in a single form and in equilibrium. Of the three primary forms [triangle, square, circle], it points most clearly to the fourth dimension." (Wassily Kandinsky, [letter] 1926)

"Pure mathematics are concerned only with abstract propositions, and have nothing to do with the realities of nature. There is no such thing in actual existence as a mathematical point, line or surface. There is no such thing as a circle or square. But that is of no consequence. We can define them in words, and reason about them. We can draw a diagram, and suppose that line to be straight which is not really straight, and that figure to be a circle which is not strictly a circle. It is conceived therefore by the generality of observers, that mathematics is the science of certainty." (William Godwin, "Thoughts on Man", 1969)

"The Squaring of the Circle is of great importance to the geometer-cosmologist because for him the circle represents pure, unmanifest spirit-space, while the square represents the manifest and comprehensible represents world. When a near-equality is drawn between the circle and square, the infinite is able to express its dimensions or qualities through the finite." (Robert Lawlor, "Sacred Geometry", 1982)

"Geometry is the study of form and shape. Our first encounter with it usually involves such figures as triangles, squares, and circles, or solids such as the cube, the cylinder, and the sphere. These objects all have finite dimensions of length, area, and volume - as do most of the objects around us. At first thought, then, the notion of infinity seems quite removed from ordinary geometry. That this is not so can already be seen from the simplest of all geometric figures - the straight line. A line stretches to infinity in both directions, and we may think of it as a means to go 'far out' in a one-dimensional world." (Eli Maor, "To Infinity and Beyond: A Cultural History of the Infinite", 1987)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems - the determination of the diagonal of a square and that of the circumference of a circle - revealed the existence of new mathematical beings for which no place could be found within the rational domain." (Tobias Dantzig)

☰On Number Theory (1900-1924)

"If one intends to base arithmetic on the theory of natural numbers as finite cardinals, one has to deal mainly with the definition of finite set; for the cardinal is, according to its nature, a property of a set, and any proposition about finite cardinals can always be expressed as a proposition about finite sets. In the following I will try to deduce the most important property of natural numbers, namely the principle of complete induction, from a definition of finite set which is as simple as possible, at the same time showing that the different definitions [of finite set] given so far are equivalent to the one given here." (Ernst Zermelo,  "Ueber die Grundlagen der Arithmetik", Atti del IV Congresso Internazionale dei Matematici, 1908)

[…] theory of numbers lies remote from those who are indifferent; they show little interest in its development, indeed they positively avoid it. [..] the pure theory of numbers is an extremely abstract thing, and one does not often find the gift of ability to understand with pleasure anything so abstract. […] I believe that the theory of numbers would be made more accessible, and would awaken more general interest, if it mere presented in connection with graphical elements and appropriate figures.”  (Felix Klein, “Elementary Mathematics from an Advanced Standpoint”, 1908)

"Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them." (Henri Poincaré, "Annual Report of the Board of Regents of the Smithsonian Institution", 1909)

"The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found it in it a mysterious attraction impossible to resist." (Godfrey H Hardy, 1915)

"The theory of numbers is unrivalled for the number and variety of its results and for the beauty and wealth of its demonstrations. The Higher Arithmetic seems to include most of the romance of mathematics." (Louis Mordell, 1917)

"The function of a mathematician, then, is simply to observe the facts about his own intricate system of reality, that astonishingly beautiful complex of logical relations which forms the subject-matter of his science, as if he were an explorer looking at a distant range of mountains, and to record the results of his observations in a series of maps, each of which is a branch of pure mathematics. […] Among them there perhaps none quite so fascinating, with quite the astonishing contrasts of sharp outline and shade, as that which constitutes the theory of numbers." (Godfrey H. Hardy, "The Theory of Numbers", Nature 1922)

⚖️On Logic (Unsourced)

"[It used to be that] geometry must, like logic, rely on formal reasoning in order to rebut the quibblers. But the tables have turned. All reasoning concerned with what common sense knows in advance, serves only to conceal the truth and to weary the reader and is today disregarded." (Alexis C Clairaut)

"Arithmetic, then, means dealing logically with certain facts that we know, about numbers, with a view to arriving at knowledge which as yet we do not possess." (Anonymous)

"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)

"Intuition is the supra-logic that cuts out all routine processes of thought and leaps straight from the problem to the answer." (Robert Graves)

"Logic merely sanctions the conquests of the intuition." (Jacques S Hadamard)

"Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry." (Friedrich von Schlegel)

"No discovery has been made in mathematics, or anywhere else for that matter, by an effort of deductive logic; it results from the work of creative imagination which builds what seems to be truth, guided sometimes by analogies, sometimes by an esthetic ideal, but which does not hold at all on solid logical bases. Once a discovery is made, logic intervenes to act as a control; it is logic that ultimately decides whether the discovery is really true or is illusory; its role therefore, though considerable, is only secondary." (Henri Lebesgue)

"Some problems are just too complicated for rational logical solutions. They admit of insights, not answers." (Jerome B Wiesner, The New Yorker, 1963)

"The art of observation and that of experimentation are very distinct. In the first case, the fact may either proceed from logical reasons or be mere good fortune; it is sufficient to have some penetration and a sense of truth in order to profit by it. But the art of experimentation leads from the first to the last link of the chain, without hesitation and without a blank, making successive use of Reason, which suggests an alternative, and of Experience, which decides on it, until, starting from a faint glimmer, the full blaze of light is reached." (Jean Baptiste-Andre Dumas)

"The supreme task is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can lead to them." (Albert Einstein)

"We especially need imagination in science. It is not all mathematics, nor all logic, but it is somewhat beauty and poetry." (Maria Mitchell)

"What I’m really interested in is whether God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all." (Albert Einstein)

"What truly is logic? Who decides reason? […] It's only in the mysterious equations of love that any logical reasons can be found." (John Forbes Nash Jr.)

"While most of us were just trying to learn to arrange logical statements into coherent arguments, Ted was quietly solving open problems and creating new mathematics. It was as if he could write poetry while the rest of us were trying to learn grammar." (Joel Shapiro) 

On Mysticism V: Mathematics & Mysticism II

"In order to know the curvature of a curve, the determination of the radius of the osculating circle furnishes us the best measure, where for each point of the curve we find a circle whose curvature is precisely the same. However, when one looks for the curvature of a surface, the question is very equivocal and not at all susceptible to an absolute response, as in the case above. There are only spherical surfaces where one would be able to measure the curvature, assuming the curvature of the sphere is the curvature of its great circles, and whose radius could be considered the appropriate measure. But for other surfaces one doesn’t know even how to compare a surface with a sphere, as when one can always compare the curvature of a curve with that of a circle. The reason is evident, since at each point of a surface there are an infinite number of different curvatures. One has to only consider a cylinder, where along the directions parallel to the axis, there is no curvature, whereas in the directions perpendicular to the axis, which are circles, the curvatures are all the same, and all other oblique sections to the axis give a particular curvature. It’s the same for all other surfaces, where it can happen that in one direction the curvature is convex, and in another it is concave, as in those resembling a saddle." (Leonhard Euler, "Recherches sur la courbure des surfaces", 1767)

"A circle no doubt has a certain appealing simplicity at the first glance, but one look at a healthy ellipse should have convinced even the most mystical of astronomers that that the perfect simplicity of the circle is akin to the vacant smile of complete idiocy. Compared to what an ellipse can tell us, a circle has nothing to say." (Eric T Bell, "The Handmaiden of the Sciences", 1937)

"Here, then, in mathematics we have a universal language, valid, useful, intelligible everywhere in place and in time - in banks and insurance companies, on the parchments of the architects who raised the Temple of Solomon, and on the blueprints of the engineers who, with their calculus of chaos, master the winds. Here is a discipline of a hundred branches, fabulously rich, literally without limit in its sphere of application, laden with honors for an unbroken record of magnificent accomplishment. Here is a creation of the mind, both mystic and pragmatic in appeal. Austere and imperious as logic, it is still sufficiently sensitive and flexible to meet each new need." (Edward Kasner & James R Newman, "Mathematics and the Imagination", 1940)

"[...] I find both a special pleasure and constraint in describing the progress of mathematics, because it has been part of so much speculation: a ladder for mystical as well as rational thought in the intellectual ascent of man." (Jacob Bronowski, "The Ascent of Man", 1973)

"[...] mathematics and poetry move together between two extremes of mysticism, the mysticism of the commonplace where ideas illuminate and create facts, and the mysticism of the extraordinary where God, the Infinite, the Real, poses the riddles of desire and disappointment, sin and salvation, effort and failure, question and paradoxical answer [...]" (Scott Buchanan, "Poetry and Mathematics", 1975)

"[The Riemann] zeros did not appear to be scattered at random. Riemann’s calculations indicated that they were lining up as if along some mystical ley line running through the landscape."  (Marcus du Sautoy, "The Music of the Primes", 2003)

"Until [the RH is proved], we shall listen enthralled by this unpredictable mathematical music, unable to master its twists and turns. The primes have been a constant companion in our exploration of the mathematical world yet they remain the most enigmatic of all numbers. Despite the best efforts of the greatest mathematical minds to explain the modulation and transformation of this mystical music, the primes remain an unanswered riddle. We still await the person whose name will live for ever as the mathematician who made the primes sing." (Marcus du Sautoy, "The Music of the Primes", 2003)

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)

Architecture in Systems

"Thus, the central theme that runs through my remarks is that complexity frequently takes the form of hierarchy, and that hierarchic systems have some common properties that are independent of their specific content. Hierarchy, I shall argue, is one of the central structural schemes that the architect of complexity uses." (Herbert A Simon, "The Architecture of Complexity", Proceedings of the American Philosophical Society Vol. 106 (6), 1962)

"From a functional point of view, mental models can be described as symbolic structures which permit people: to generate descriptions of the purpose of a system, to generate descriptions of the architecture of a system, to provide explanations of the state of a system, to provide explanations of the functioning of a system, to make predictions of future states of a system." (Gert Rickheit & Lorenz Sichelschmidt, "Mental Models: Some Answers, Some Questions, Some Suggestions", 1999)

"Most systems displaying a high degree of tolerance against failures are a common feature: Their functionality is guaranteed by a highly interconnected complex network. A cell's robustness is hidden in its intricate regulatory and metabolic network; society's resilience is rooted in the interwoven social web; the economy's stability is maintained by a delicate network of financial and regulator organizations; an ecosystem's survivability is encoded in a carefully crafted web of species interactions. It seems that nature strives to achieve robustness through interconnectivity. Such universal choice of a network architecture is perhaps more than mere coincidences." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"The simplest basic architecture of an artificial neural network is composed of three layers of neurons - input, output, and intermediary (historically called perceptron). When the input layer is stimulated, each node responds in a particular way by sending information to the intermediary level nodes, which in turn distribute it to the output layer nodes and thereby generate a response. The key to artificial neural networks is in the ways that the nodes are connected and how each node reacts to the stimuli coming from the nodes it is connected to. Just as with the architecture of the brain, the nodes allow information to pass only if a specific stimulus threshold is passed. This threshold is governed by a mathematical equation that can take different forms. The response depends on the sum of the stimuli coming from the input node connections and is 'all or nothing'." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"A key discovery of network science is that the architecture of networks emerging in various domains of science, nature, and technology are similar to each other, a consequence of being governed by the same organizing principles. Consequently we can use a common set of mathematical tools to explore these systems."  (Albert-László Barabási, "Network Science", 2016)

Architecture and Mathematics

"The arts which are useful, and absolutely necessary to the architect are painting and mathematics." (Leon Battista Alberti, "Treatise on Architecture", 1452)

"For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated to use with sufficient dexterity, without the aid and intervention of Mathematic: of which sort are Perspective, Music, Astronomy, cosmography, Architecture, Machinery, and some others." (Sir Francis Bacon, "De Augmentis"Bk. 3 ["The Advancement of Learning"], 1605)

"The true mathematician is always a great deal of an artist, an architect, yes, of a poet. Beyond the real world, though perceptibly connected with it, mathematicians have created an ideal world which they attempt to develop into the most perfect of all worlds, and which is being explored in every direction. None has the faintest conception of this world except him who knows it; only presumptuous ignorance can assert that the mathematician moves in a narrow circle. The truth which he seeks is, to be sure, broadly considered, neither more nor less than consistency; but does not his mastership show, indeed, in this very limitation? To solve questions of this kind he passes unenviously over others." (Alfred Pringsheim, Jaresberichte der Deutschen Mathematiker Vereinigung Vol 13, 1904)

"Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood, no more than painting is the art of mixing colors on a palette, no more than the science of geology is the art of breaking rocks, or the science of anatomy the art of butchering." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"Architecture is geometry made visible in the same sense that music is number made audible." (Claude F Bragdon, "The Beautiful Necessity: Seven Essays on Theosophy and Architecture", 1910)

"Architecture is the masterly, correct and magnificent play of masses brought together in light. Our eyes are made to see forms in light; light and shade reveal these forms; cubes, cones, spheres, cylinders or pyramids are the great primary forms which light reveals to advantage; the image of these is distinct and tangible within us without ambiguity. It is for this reason that these are beautiful forms, the most beautiful forms. Everybody is agreed to that, the child, the savage and the metaphysician." (Charles-Edouard Jeanneret [Le Corbusier], "Towards a New Architecture", 1923)

"One expects a mathematical theorem or a mathematical theory not only to describe and to classify in a simple and elegant way numerous and a priori disparate special cases. One also expects ‘elegance’ in its ‘architectural’ structural makeup." (John von Neumann, "The Mathematician" [in "Works of the Mind" Vol. I (1), 1947]) 

"Mathematicians who build new spaces and physicists who find them in the universe can profit from the study of pictorial and architectural spaces conceived and built by men of art." (György Kepes, "The New Landscape In Art and Science", 1956)

"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)

"In mathematics, beauty is a very important ingredient. Beauty exists in mathematics as in architecture and other things. It is a difficult thing to define but it is something you recognise when you see it. It certainly has to have elegance, simplicity, structure and form. All sorts of things make up real beauty. There are many different kinds of beauty and the same is true of mathematical theorems. Beauty is an important criterion in mathematics because basically there is a lot of choice in what you can do in mathematics and science. It determines what you regard as important and what is not." (Michael Atiyah, 2009)

"Architecture is akin to music in that both should be based on the symmetry of mathematics." (Frank L Wright)

"Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence." (Georg W F Hegel)

On Geometrical Figures: On Angles

"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have  real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible. […] How can it be, that a real quantity, a root of the proposed equation, is expressed by the intervention of an imaginary? For this is the remarkable thing, that, as calculation shows, such an imaginary quantity is only observed to enter those cubic equations that have no imaginary root, all their roots being real or possible, as has been shown by trisection of an angle, by Albert Girard and others. […] This difficulty has been too much for all writers on algebra up to the present, and they have all said they that in this case Cardano’s rules fail." (Gottfried W Leibniz, cca. 1675)

"I found the elements of a new characteristic, completely different from Algebra and which will have great advantages for the exact and natural mental representation, although without figures, of everything that depends on the imagination. Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity." (Gottfried W Leibniz, [letter to Varignon], 1702)

"[…] the speculative propositions of mathematics do not relate to facts; […] all that we are convinced of by any demonstration in the science, is of a necessary connection subsisting between certain suppositions and certain conclusions. When we find these suppositions actually take place in a particular instance, the demonstration forces us to apply the conclusion. Thus, if I could form a triangle, the three sides of which were accurately mathematical lines, I might affirm of this individual figure, that its three angles are equal to two right angles; but as the imperfection of my senses puts it out of my power to be, in any case, certain of the exact correspondence of the diagram which I delineate, with the definitions given in the elements of geometry, I never can apply with confidence to a particular figure, a mathematical theorem." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, as, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language." (Thomas Hill, "The Imagination in Mathematics", The North American Review Vol. 85 (176), 1857)

"As an operation, multiplication by i x i has the same effect as multiplication by -1; multiplication by i has the same effect as a rotation by a right angle, and these interpretations […] are consistent. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’." (Eric T Bell, "Gauss, the Prince of Mathematicians", 1956)

"Just as few concrete physical systems are strictly deterministic in their behavior, so very few are strictly linear. The great importance of linearity lies in a combination of two circumstances. First, many tangible phenomena behave approximately linearly over restricted periods of time or restricted ranges of the variables, so that useful linear mathematical models can simulate their behavior. A pendulum swinging through a small angle is a nearly linear system. Second, linear equations can be handled by a wide variety of techniques that do not work with nonlinear equations." (Edward N Lorenz, "The Essence of Chaos", 1993)

"The pinball machine is one of those rare dynamical systems whose chaotic nature we can deduce by pure qualitative reasoning, with fair confidence that we have not wandered astray. Nevertheless, the angles in the paths of the balls that are introduced whenever a ball strikes a pin and rebounds […] render the system some what inconvenient for detailed quantitative study." (Edward N Lorenz, "The Essence of Chaos", 1993)

"The concept of homeomorphism appears to be convenient for establishing those important properties of figures which remain unchanged under such deformations. These properties are sometimes referred to as topological, as distinguished from metrical, which are customarily associated with distances between points, angles between lines, edges of a figure, etc." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"There is a certain ambiguity in the concept of angle, for it describes both the qualitative idea of 'separation' between two intersecting lines, and the numerical value of this separation-the measure of the angle. (Note that this is not so with the analogous 'separation' between two points, where the phrases line segment and length make the distinction clear.) Fortunately we need not worry about this ambiguity, for trigonometry is concerned only with the quantitative aspects of line segments and angles." (Eli Maor, "Trigonometric Delights", 1998)

"[…] because imaginary time is at right angles to real time, it behaves like a fourth spatial direction. It can therefore have a much richer range of possibilities than the railroad track of ordinary real time, which can only have a beginning or an end or go around in circles. It is in this imaginary sense that time has a shape." (Stephen W Hawking, "The Universe in a Nutshell", 2001)

"To describe how quantum theory shapes time and space, it is helpful to introduce the idea of imaginary time. Imaginary time sounds like something from science fiction, but it is a well-defined mathematical concept: time measured in what are called imaginary numbers. […] Imaginary numbers can then be represented as corresponding to positions on a vertical line: zero is again in the middle, positive imaginary numbers plotted upward, and negative imaginary numbers plotted downward. Thus imaginary numbers can be thought of as a new kind of number at right angles to ordinary real numbers. Because they are a mathematical construct, they don't need a physical realization […]" (Stephen W Hawking, "The Universe in a Nutshell", 2001)

"Practical geometry is an empirical undertaking, living and breathing and sweating in the real world where measurements are always approximate and things are fudged or smeared or jumbled up. Within Euclidean geometry points are concentrated, lines straightened, angles narrowed; idealizations are made, and some parts of experience discarded and other parts embraced." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"There is weirdness in non-Euclidean geometry, but not because of anything that geometers might say about the ordinary fond familiar world in which space is flat, angles sharp, and only curves are curved. Non-Euclidean geometry is an instrument in the enlargement of the mathematician’s self-consciousness, and so comprises an episode in a long, difficult, and extended exercise in which the human mind attempts to catch sight of itself catching sight of itself, and so without end." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"Double periodicity is more interesting than single periodicity, because it is more varied. There is really only one periodic line, since all circles are the same up to a scale factor. However, there are infinitely many doubly periodic planes, even if we ignore scale. This is because the angle between the two periodic axes can vary, and so can the ratio of period lengths. The general picture of a doubly periodic plane is given by a lattice in the plane of complex numbers: a set of points of the form mA + nB, where A and B are nonzero complex numbers in different directions from O, and m and n run through all the integers. A and B are said to generate the lattice because it consists of all their sums and differences. […] The shape of the lattice of points mA + nB can therefore be represented by the complex number A/B. It is not hard to see that any nonzero complex number represents a lattice shape, so in some sense there is whole plane of lattice shapes. Even more interesting: the plane of lattice shapes is a periodic plane, because different numbers represent the same lattice." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Topology is geometry without distance or angle. The geometrical objects of study, not rigid but rather made of rubber or elastic, are especially stretchy." (Stephen Huggett & David Jordan, “A Topological Aperitif”, 2009)

"[…] the symmetry group of the infinite logarithmic spiral is an infinite group, with one element for each real number. Two such transformations compose by adding the corresponding angles, so this group is isomorphic to the real numbers under addition." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"The trig functions’ input consists of the sizes of angles inside right triangles. Their output consists of the ratios of the lengths of the triangles’ sides. Thus, they act as if they contained phone-directory-like groups of paired entries, one of which is an angle, and the other is a ratio of triangle-side lengths associated with the angle. That makes them very useful for figuring out the dimensions of triangles based on limited information." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"In geometry, shapes like circles and polyhedra are rigid objects; the tools of the trade are lengths, angles and areas. But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable." (David E Richeson, "Topology 101: The Hole Truth", 2021) [source]

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire)