On Extrema I

"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of the great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." (Herbert G Wells, "Mankind in the Making", 1903)

"According to Leibniz our world is the best possible. That is why its laws can be described by extremal principles." (Carl L Siegel)

"In Continuity, it is impossible to distinguish phenomena at their merging-points, so we look for them at their extremes." (Charles Fort, "The Book of the Damned", 1919)

"Change is most sluggish at the extremes precisely because the derivative is zero there." (Steven Strogatz, The Joy of X: A Guided Tour of Mathematics, from One to Infinity, 2012)

"Most practical questions can be reduced to problems of largest and smallest magnitudes […] and it is only by solving these problems that we can satisfy the requirements of practice which always seeks the best, the most convenient." (Pafnuty L Chebyshev)

"[…] nothing takes place in the world whose meaning is not that of some maximum or minimum." (Leonhard Euler)

"The world is not dialectical - it is sworn to extremes, not to equilibrium, sworn to radical antagonism, not to reconciliation or synthesis." (Jean Baudrillard)

"There must be a double method for solving mechanical problems: one is the direct method founded on the laws of equilibrium or of motion; but the other one is by knowing which formula must provide a maximum or a minimum. The former way proceeds by efficient causes: both ways lead to the same solution, and it is such a harmony which convinces us of the truth of the solution, even if each method has to be separately founded on indubitable principles. But is often very difficult to discover the formula which must be a maximum or minimum, and by which the quantity of action is represented." (Leonhard Euler)

"We shall consider the simplest maximum and minimum problem that points to a natural transition from functions of a finite number of variables to magnitudes that depend on an infinite number of variables." (Vito Volterra)

"When a quantity is greatest or least, at that moment its flow neither increases nor decreases." (Isaac Newton)

On Topology IV (More on Topology)

"The young mathematical disciple 'topology' might be of some help in making psychology a real science." (Kurt Lewin, Principles of topological psychology, 1936)

"Topology provides the synergetic means of ascertaining the values of any system of experiences. Topology is the science of fundamental pattern and structural relationships of event constellations." (R Buckminster Fuller, "Operating Manual for Spaceship Earth", 1969)

"Topology is not ‘designed to guide us’ in structure. It is this structure." (Jacques Lacan, "L’Étourdit", 1972)

"No other theory known to science [other than superstring theory] uses such powerful mathematics at such a fundamental level. […] because any unified field theory first must absorb the Riemannian geometry of Einstein’s theory and the Lie groups coming from quantum field theory. […] The new mathematics, which is responsible for the merger of these two theories, is topology, and it is responsible for accomplishing the seemingly impossible task of abolishing the infinities of a quantum theory of gravity." (Michio Kaku, "Hyperspace", 1995)

"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I. Monastyrsky, "Riemann, Topology, and Physics" 2nd Ed., 2008)

"At the basis of the distance concept lies, for example, the concept of convergent point sequence and their defined limits, and one can, by choosing these ideas as those fundamental to point set theory, eliminate the notions of distance." (Felix Hausdorff)

"In every subject one looks for the topological and algebraic structures involved, since these structures form a unifying core for the most varied branches of mathematics." (K Weise and H Noack, "Aspects of Topology")

"Mathematicians do not study objects, but relations between objects. Thus, they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only." (Henri Poincaré)

"Poetry and code - and mathematics - make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations." (Stephanie Strickland)

"Topology is the study of the modal relations of spatial figures and the laws of connectivity, mutual position, and ordering of points, lines, surfaces, and solids and their parts independently of measure and magnitude relations." (Johann B Listing)

On Topology III (Topology with a Twist)

"The connection of topology with physics is no passing interlude but rather represents a length affair." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language, whose beauty eventually - in retrospect - compensates for all the drudgery." (Volker Runde, "A Taste of Topology", 2005)

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

"People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook in topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram." (Martin Gardner, "Hexaflexagons and Other Mathematical Diversions", 1988)

"A child[’s …] first geometrical discoveries are topological…If you ask him to copy a square or a triangle, he draws a closed circle." (Jean Piaget)

"If you wear glasses, and you wake up in the morning and you’re not wearing your glasses, and everything is blurred together, that’s what the indiscrete topology is like." (Anonymous)

"In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." (Hermann Weyl)

"Point set topology is a disease from which the human race will soon recover." (Henri Poincaré)

"The true traditional doughnut has the topology of a sphere. It is a matter of taste whether one regards this as having separate internal and external surfaces. The important point is that the inner space should be filled with good raspberry jam. This is also a matter of taste." (Peter B Fellgett)

"Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything." (Edward Witten)

On Topology II (Definitions)

"I believe that we need another analysis properly geometric or linear, which treats PLACE directly the way that algebra treats MAGNITUDE." (Gottfried W Leibniz, 1670s)

"The branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that has been almost unknown up to now; Leibniz spoke of it first, calling it the ‘geometry of position’ (geometria situs). This branch of geometry deals with relations dependent on position; it does not take magnitudes into considerations, nor does it involve calculation with quantities. But as yet no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them." (Leonhard Euler, 1735) 

"Topology is the study of the modal relations of spatial figures and the laws of connectivity, mutual position, and ordering of points, lines, surfaces, and solids and their parts independently of measure and magnitude relations." (Johann B Listing)

"The use of figures is, above all, then, for the purpose of making known certain relations between the objects that we study, and these relations are those which occupy the branch of geometry that we have called Analysis Situs [that is, topology], and which describes the relative situation of points and lines on surfaces, without consideration of their magnitude." (Henri Poincaré, "Analysis Situs", Journal de l'Ecole Polytechnique 1, 1895)

"Imagine any sort of model and a copy of it done by an awkward artist: the proportions are altered, lines drawn by a trembling hand are subject to excessive deviation and go off in unexpected directions. From the point of view of metric or even projective geometry these figures are not equivalent, but they appear as such from the point of view of geometry of position [that is, topology]." (Henri Poincaré, "Dernières pensées", 1920)

"Topology begins where sets are implemented with some cohesive properties enabling one to define continuity." (Solomon Lefschetz, "Introduction to Topology", 1949)

"In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure." (David Hilbert, "Geometry and Imagination", 1952)

"Topology is precisely that mathematical discipline which allows a passage from the local to the global." (René Thom)

"Topology studies the properties of geometrical objects that remain unchanged under transformations called homeomorphisms and deformations." (Victor V Prasolov, "Intuitive Topology", 1995)

"Topology is the mathematical study of properties of objects which are preserved through deformations, twistings, and stretchings but not through breaks or cuts." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table". 2005)

"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I Monastyrsky, "Riemann, Topology, and Physics" 2nd Ed., 2008)

"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)