On Invariance (1990-1999)

"Axiomatic theories of choice introduce preference as a primitive relation, which is interpreted through specific empirical procedures such as choice or pricing. Models of rational choice assume a principle of procedure invariance, which requires strategically equivalent methods of elicitation to yield the same preference order." (Amos Tversky et al, "The Causes of Preference Reversal", The American Economic Review Vol. 80 (1), 1990)

"Scaling invariance results from the fact that homogeneous power laws lack natural scales; they do not harbor a characteristic unit (such as a unit length, a unit time, or a unit mass). Such laws are therefore also said to be scale-free or, somewhat paradoxically, 'true on all scales'. Of course, this is strictly true only for our mathematical models. A real spring will not expand linearly on all scales; it will eventually break, at some characteristic dilation length. And even Newton's law of gravitation, once properly quantized, will no doubt sprout a characteristic length." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"The unifying concept underlying fractals, chaos, and power laws is self-similarity. Self-similarity, or invariance against changes in scale or size, is an attribute of many laws of nature and innumerable phenomena in the world around us. Self-similarity is, in fact, one of the decisive symmetries that shape our universe and our efforts to comprehend it." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"Homology theory introduces a new connection between invariants of manifolds. Continuing the 'physical' analogy, we say that a homology theory studies the intrinsic structure of a manifold by breaking it into a system of portions arranged simply, or, more precisely, in a standard way. Then, given certain rules for glueing the portions together, the theory obtains the whole manifold. The main problem consists in proving the resultant geometric quantities that are independent of the decomposition and glueing (i.e., proving the topological invariance of the characteristics)." (Michael I Monastyrsky, "Topology of Gauge Fields and Condensed Matter", 1993)

"It would be preferable to have some property or quantity that could indisputably differentiate between two surfaces. This, however, requires more sophisticated mathematics. [...] we introduce the simplest topological invariant: the Euler characteristic. This is the first step in the algebraization of topology: finding something computable to describe the shape of a space." (L Christine Kinsey. "Topology of Surfaces", 1993)

"The Euler characteristic can be used to shorten the process, but for some cases a lengthy procedure is still necessary. Neither of these options provide a clear accounting for the ways in which the surfaces vary, e.g., which enclose cavities, which are non-orientable, etc., and neither can be completely generalized to higher-dimensional manifolds. Ideally, one would like some sort of algebraic invariant or computable quantity that would codify a lot of information: how many connected pieces a space has, how the gluing directions work, whether the surface is orientable or not, etc. The euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have x = 0." (L Christine Kinsey, "Topology of Surfaces", 1993)

"The Euler characteristic is strikingly easy to compute, and it really seems odd that it should be a topological invariant. After all, it is computed from a cell decomposition, and it is very easy to come up with a lot of different complexes on a single surface, all with varying numbers of faces, edges, and vertices. But somehow by taking the alternating sum, one arrives at a quantity which depends only on the underlying shape and not on the particular complex." (L Christine Kinsey,"Topology of Surfaces", 1993)

"Symmetry is basically a geometrical concept. Mathematically it can be defined as the invariance of geometrical patterns under certain operations. But when abstracted, the concept applies to all sorts of situations. It is one of the ways by which the human mind recognizes order in nature. In this sense symmetry need not be perfect to be meaningful. Even an approximate symmetry attracts one's attention, and makes one wonder if there is some deep reason behind it." (Eguchi Tohru & ‎K Nishijima ," Broken Symmetry: Selected Papers Of Y Nambu", 1995)

"The cliché became, erroneously, 'everything is relative'; whereas the point is that out of the vast flux one can distill the very opposite: 'some things are invariant'." (Gerald Holton, "Einstein, History, and Other Passions: The Rebellion Against Science at the End of the Twentieth Century", 1996)

"How deep truths can be defined as invariants – things that do not change no matter what; how invariants are defined by symmetries, which in turn define which properties of nature are conserved, no matter what. These are the selfsame symmetries that appeal to the senses in art and music and natural forms like snowflakes and galaxies. The fundamental truths are based on symmetry, and there’s a deep kind of beauty in that." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"Homology theory studies properties of manifolds by decomposing them into simpler parts. The structure of these parts can be investigated easily by introducing algebraic characteristics associated with these decompositions. The main difficulty lies in proving that the corresponding characteristics of the decomposition, in fact, do not depend on the particular choice of the decomposition but are rather a topological invariant of the manifold itself." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"Cybernetics is the science of effective organization, of control and communication in animals and machines. It is the art of steersmanship, of regulation and stability. The concern here is with function, not construction, in providing regular and reproducible behaviour in the presence of disturbances. Here the emphasis is on families of solutions, ways of arranging matters that can apply to all forms of systems, whatever the material or design employed. [...] This science concerns the effects of inputs on outputs, but in the sense that the output state is desired to be constant or predictable – we wish the system to maintain an equilibrium state. It is applicable mostly to complex systems and to coupled systems, and uses the concepts of feedback and transformations (mappings from input to output) to effect the desired invariance or stability in the result." (Chris Lucas, "Cybernetics and Stochastic Systems", 1999)

"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

On Invariance (1980-1989)

"An essential condition for a theory of choice that claims normative status is the principle of invariance: different representations of the same choice problem should yield the same preference. That is, the preference between options should be independent of their description. Two characterizations that the decision maker, on reflection, would view as alternative descriptions of the same problem should lead to the same choice-even without the benefit of such reflection." (Amos Tversky & Daniel Kahneman, "Rational Choice and the Framing of Decisions", The Journal of Business Vol. 59 (4), 1986)

"Unlike an architect, Nature does not go around expounding on the wondrous symmetries of Her design. Instead, theoretical physicists must deduce them. Some symmetries, such as parity and rotational invariances, are intuitively obvious. We expect Nature to possess these symmetries, and we are shocked if She does not. Other symmetries, such as Lorentz invariance and general covariance, are more subtle and not grounded in our everyday perceptions. But, in any case, in order to find out if Nature employs a certain symmetry, we must compare the implications of the symmetry with observation." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The impossibility of defining absolute motion can be seen as the manifestation of a symmetry known as relativistic invariance. In the same way that parity invariance tells us that we cannot distinguish the mirror-image world from our world, relativistic invariance tells us that it is impossible to decide whether we are at rest or moving steadily." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"The power and glory of symmetry allow us to bypass completely the construction of strong interaction theories of dubious utility. We are able to contain and isolate our ignorance. [...] Symmetry tells us that states in the same multiplet must have the same energy, but it cannot tell us what that energy is." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"To detect a symmetry in the fundamental design, one would have to check the covariance of each of the many equations of motion in the differential formulation. With the action formulation, on the other hand, one has the considerably easier task of checking the invariance of the action." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Unlike an architect, Nature does not go around expounding on the wondrous symmetries of Her design. Instead, theoretical physicists must deduce them. Some symmetries, such as parity and rotational invariances, are intuitively obvious. We expect Nature to possess these symmetries, and we are shocked if She does not. Other symmetries, such as Lorentz invariance and general covariance, are more subtle and not grounded in our everyday perceptions. But, in any case, in order to find out if Nature employs a certain symmetry, we must compare the implications of the symmetry with observation." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"Fractal geometry is concerned with the description, classification, analysis, and observation of subsets of metric spaces (X, d). The metric spaces are usually, but not always, of an inherently 'simple' geometrical character; the subsets are typically geometrically 'complicated'. There are a number of general properties of subsets of metric spaces, which occur over and over again, which are very basic, and which form part of the vocabulary for describing fractal sets and other subsets of metric spaces. Some of these properties, such as openness and closedness, which we are going to introduce, are of a topological character. That is to say, they are invariant under homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)

"In deterministic fractal geometry the focus is on those subsets of a space which are generated by, or possess invariance properties under, simple geometrical transformations of the space into itself. A simple geometrical transformation is one which is easily conveyed or explained to someone else. Usually they can be completely specified by a small set of parameters." (Michael Barnsley, "Fractals Everwhere", 1988)

On Invariance (1925-1949)

"The sequence of different positions of the same particle at different times forms a one-dimensional continuum in the four-dimensional space-time, which is called the world-line of the particle. All that physical experiments or observations can teach us refers to intersections of world-lines of different material particles, light-pulsations, etc., and how the course of the world-line is between these points of intersection is entirely irrelevant and outside the domain of physics. The system of intersecting world-lines can thus be twisted about at will, so long as no points of intersection are destroyed or created, and their order is not changed. It follows that the equations expressing the physical laws must be invariant for arbitrary transformations." (Willem de Sitter, "Kosmos", 1932) 

"The theory [of categories] also emphasizes that, whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these new objects as an integral part of their definition. The pursuit of this program entails a simultaneous consideration of objects and their mappings (in our terminology, this means the consideration not of individual objects but of categories). This emphasis on the specification of the type of mappings employed gives more insight onto the degree of invariance of the various concepts involved." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"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." (Cletus O Oakley, The American Mathematical Monthly, 1949)

"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)

"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)

On Invariance (1900-1925)

"It [mathematics] is in the inner world of pure thought, where all entia dwell, where is every type of order and manner of correlation and variety of relationship, it is in this infinite ensemble of eternal verities whence, if there be one cosmos or many of them, each derives its character and mode of being, - it is there that the spirit of mathesis has its home and its life. [/] Is it a restricted home, a narrow life, static and cold and grey with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of solar light, not clad in the colours that liven and glorify the things of sense, but it is an illuminated world, and over it all and everywhere throughout are hues and tints transcending sense, painted there by radiant pencils of psychic light, the light in which it lies. It is a silent world, and, nevertheless, in respect to the highest principle of art - the interpenetration of content and form, the perfect fusion of mode and meaning - it even surpasses music. In a sense, it is a static world, but so, too, are the worlds of the sculptor and the architect. The figures, however, which reason constructs and the mathematic vision beholds, transcend the temple and the statue, alike in simplicity and in intricacy, in delicacy and in grace, in symmetry and in poise. Not only are this home and this life thus rich in aesthetic interests, really controlled and sustained by motives of a sublimed and supersensuous art, but the religious aspiration, too, finds there, especially in the beautiful doctrine of invariants, the most perfect symbols of what it seeks - the changeless in the midst of change, abiding things hi a world of flux, configurations that remain the same despite the swirl and stress of countless hosts of curious transformations." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal, 1904-1906)

"The equations of Newton's mechanics exhibit a two-fold invariance. Their form remains unaltered, firstly, if we subject the underlying system of spatial coordinates to any arbitrary change of position ; secondly, if we change its state of motion, namely, by imparting to it any uniform translatory motion ; furthermore, the zero point of time is given no part to play. We are accustomed to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of mechanics, and for that reason the two invariances are probably rarely mentioned in the same breath. Each of them by itself signifies, for the differential equations of mechanics, a certain group of transformations. The existence of the first group is looked upon as a fundamental characteristic of space. The second group is preferably treated with disdain, so that we with un-troubled minds may overcome the difficulty of never being able to decide, from physical phenomena, whether space, which is supposed to be stationary, may not be after all in a state of uniform translation. Thus the two groups, side by side, lead their lives entirely apart. Their utterly heterogeneous character may have discouraged any attempt to compound them. But it is precisely when they are compounded that the complete group, as a whole, gives us to think." (Hermann Minkowski, "Space and Time", [Address to the 80th Assembly of German Natural Scientists and Physicians] 1908)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"Two Riemann surfaces which can be mapped conformally onto each other are (conformally) equivalent and are to be regarded as different representations of one and the same ideal Riemann surface. The intrinsic properties of a Riemann surface will include only those properties which are invariant under conformal maps; that is, those properties which, if possessed by one Riemann surface are possessed by every equivalent surface. Obviously all topological properties are intrinsic properties of a Riemann surface; similarly with those properties belonging to the surface by virtue of its smoothness." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"There appears a fundamental principle which can serve to characterize all possible geometries. […] Given any group of transformations in space which includes the principal group as a sub-group, then the invariant theory of this group gives a definite kind of geometry, and every possible geometry can be obtained in this way." (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1919)

"A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line. In the four-dimensional continuum of the special theory of relativity (with real time co-ordinate) this a real straight line. The natural, that is, the simplest, generalization of the straight line which is meaningful in the system of concepts of the general (Riemannian) theory of invariants is that of the straightest, or geodesic, line."(Albert Einstein, "The Meaning of Relativity", 1922)

On Invariance (2010-)

"In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes." (Gregory Faye, "An introduction to bifurcation theory", 2011)

"One of the most important skills you will need to acquire in order to use manifold theory effectively is an ability to switch back and forth easily between invariant descriptions and their coordinate counterparts." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The fact that manifolds do not come with any predetermined choice of coordinates is both a blessing and a curse. The flexibility to choose coordinates more or less arbitrarily can be a big advantage in approaching problems in manifold theory, because the coordinates can often be chosen to simplify some aspect of the problem at hand. But we pay for this flexibility by being obliged to ensure that any objects we wish to define globally on a manifold are not dependent on a particular choice of coordinates. There are generally two ways of doing this: either by writing down a coordinate-dependent definition and then proving that the definition gives the same results in any coordinate chart, or by writing down a definition that is manifestly coordinate-independent (often called an invariant definition)." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The invariance of physical laws with respect to position or orientation (i.e., the symmetry of space) gives rise to conservation laws for linear and angular momentum. Sometimes the implications of symmetry invariance are far more complicated or sophisticated than might at first be supposed; the invariance of the forces predicted by electromagnetic theory when measurements are made in observation frames moving uniformly at different speeds (inertial frames) was an important clue leading Einstein to the discovery of special relativity. With the advent of quantum mechanics, considerations of angular momentum and spin introduced new symmetry concepts into physics. These ideas have since catalyzed the modern development of particle theory." (George B Arfken et al, "Mathematical Methods for Physicists: A comprehensive guide", 2013)

"In Euclidean vector spaces, there is the concept of orthonormal bases. But the non-orthonormal bases are as legitimate as those which are. On the other hand, in the geometry of Euclidean spaces, orthonormal bases have a legitimacy that arbitrary bases do not. Why? Because in geometry the role of groups becomes paramount. Euclidean geometry is the study of figures and properties that are invariant under the Euclidean group, the basic figures being the orthonormal bases at all points. Affine geometry is the study of figures and properties invariant under the affine group, the basic figures being all vector bases at all points." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Intersections of lines, for example, remain intersections, and the hole in a torus (doughnut) cannot be transformed away. Thus a doughnut may be transformed topologically into a coffee cup (the hole turning into a handle) but never into a pancake. Topology, then, is really a mathematics of relationships, of unchangeable, or 'invariant', patterns." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"One sometimes finds in the literature the statement that a tensor transforms in such and such way under a coordinate transformation. Tensors do not transform. They are invariants. Their components do, under changes of section of the frame bundle." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"We say that a set of quantities transforms vectorially under a group of transformations of the bases of a vector space if, under an element of the group, they transform like the components of vectors, whether contravariant or covariant. We say that some quantity is a scalar if it is an invariant under the transformations of a group. A quantity (respectively, a set of quantities) may be scalar (respectively vectorial) under the transformation of a group, while not being so under the transformations of another group." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements - nothing more." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015)

"At every level of the fractal hierarchy, variation enters the system in different ways, so there is no scale invariance. All of the information is stored in the relationships of and in each level in the computational geometry." (Rick Delmonico, "Fractals all the way down", 2018)

"Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. 'Structure' has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Symmetries directly point to invariants, which pinpoint intrinsic properties of the data and of the background empirical domain of interest. As our data models change, so too do our perspectives on analysing data." (Fionn Murtagh, "Data Science Foundations: Geometry and Topology of Complex Hierarchic Systems and Big Data Analytics", 2018)

On Invariance (2000-2009)

"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"A physical system is said to possess a symmetry if one can make a change in the system such that, after the change, the system is exactly the same as it was before. We call the change we are making to the system a symmetry operation or a symmetry transformation. If a system stays the same when we do a transformation to it, we say that the system is invariant under the transformation." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"So, a scientist's definition of symmetry would be something like this: symmetry is an invariance of an object or system to a transformation. The invariance is the sameness or constancy of the system in form, appearance, composition, arrangement, and so on, and a transformation is the abstract action we apply to the system that takes it from one state into another, equivalent, one. There are often numerous transformations we can apply on a given system that take it into an equivalent state." (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"Together with plane Euclidean geometry, spherical and hyperbolic geometry are 2-dimensional geometries with the following properties: (1) distance, lines and angles are defined and invariant under motions; (2) the motions act transitively on points and directions at a point; (3) locally, incidence properties are as in plane Euclidean geometry." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Like moonlight itself, Monstrous Moonshine is an indirect phenomenon. Just as in the theory of moonlight one must introduce the sun, so in the theory of Moonshine one must go well beyond the Monster. Much as a book discussing moonlight may include paragraphs on sunsets or comet tails, so do we discuss fusion rings, Galois actions and knot invariants." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Numerical invariants and invariant properties enable us to distinguish certain topological spaces. We can go further and associate with a topological space a set having an algebraic structure. The fundamental group is the most basic of such possibilities. It not only provides a useful invariant for topological spaces, but the algebraic operation of multiplication defined for this group reflects the global structure of the space." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"Each of the most basic physical laws that we know corresponds to some invariance, which in turn is equivalent to a collection of changes which form a symmetry group. […] whilst leaving some underlying theme unchanged. […] for example, the conservation of energy is equivalent to the invariance of the laws of motion with respect to translations backwards or forwards in time […] the conservation of linear momentum is equivalent to the invariance of the laws of motion with respect to the position of your laboratory in space, and the conservation of angular momentum to an invariance with respect to directional orientation… discovery of conservation laws indicated that Nature possessed built-in sustaining principles which prevented the world from just ceasing to be." (John D Barrow, "New Theories of Everything", 2007)

"The concept of symmetry (invariance) with its rigorous mathematical formulation and generalization has guided us to know the most fundamental of physical laws. Symmetry as a concept has helped mankind not only to define ‘beauty’ but also to express the ‘truth’. Physical laws tries to quantify the truth that appears to be ‘transient’ at the level of phenomena but symmetry promotes that truth to the level of ‘eternity’." (Vladimir G Ivancevic & Tijana T Ivancevic,"Quantum Leap", 2008)

"Mathematical symmetries and invariants underlie nearly all physical laws in nature, suggesting that the search for many natural laws is inseparably a search for conserved quantities and invariant equations [...]" (Michael Schmidt & Hod Lipson, "Distilling Free-Form Natural Laws from Experimental Data", Science, Vol 324 (5923), 2009)

"The concept of symmetry is used widely in physics. If the laws that determine relations between physical magnitudes and a change of these magnitudes in the course of time do not vary at the definite operations (transformations), they say, that these laws have symmetry (or they are invariant) with respect to the given transformations. For example, the law of gravitation is valid for any points of space, that is, this law is in variant with respect to the system of coordinates." (Alexey Stakhov et al, "The Mathematics of Harmony", 2009)

On Invariance (19700-1979)

 "As a metaphor - and I stress that it is intended as a metaphor - the concept of an invariant that arises out of mutually or cyclically balancing changes may help us to approach the concept of self. In cybernetics this metaphor is implemented in the ‘closed loop’, the circular arrangement of feedback mechanisms that maintain a given value within certain limits. They work toward an invariant, but the invariant is achieved not by a steady resistance, the way a rock stands unmoved in the wind, but by compensation over time. Whenever we happen to look in a feedback loop, we find the present act pitted against the immediate past, but already on the way to being compensated itself by the immediate future. The invariant the system achieves can, therefore, never be found or frozen in a single element because, by its very nature, it consists in one or more relationships - and relationships are not in things but between them."  (Ernst von Glasersfeld German, "Cybernetics, Experience and the Concept of Self", 1970)

"The theory of rings and ideals was put on a more systematic and axiomatic basis by Emmy Noether, one of the few great women mathematicians [...] Many results on rings and ideals were already known [...] but by properly formulating the abstract notions she was able to subsume these results under the abstract theory. Thus she reexpressed Hilbert's basic theorem [...] as follows: A ring of polynomials in any number of variables over a ring of coeffcients that has an identity element and a finite basis, itself has a finite basis. In this reforumulation she made the theory of invariants a part of abstract algebra." (Morris Kline, "Mathematical Thought From Ancient to Modern Times", 1972)

"Non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather, there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future." (Kurt Gödel, "Remark on Non-standard Analysis", 1974)

"Because of its foundation in topology, catastrophe theory is qualitative, not quantitative. Just as geometry treated the properties of a triangle without regard to its size, so topology deals with properties that have no magnitude, for example, the property of a given point being inside or outside a closed curve or surface. This property is what topologists call 'invariant' -it does not change even when the curve is distorted. A topologist may work with seven-dimensional space, but he does not and cannot measure (in the ordinary sense) along any of those dimensions. The ability to classify and manipulate all types of form is achieved only by giving up concepts such as size, distance, and rate. So while catastrophe theory is well suited to describe and even to predict the shape of processes, its descriptions and predictions are not quantitative like those of theories built upon calculus. Instead, they are rather like maps without a scale: they tell us that there are mountains to the left, a river to the right, and a cliff somewhere ahead, but not how far away each is, or how large." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"Every branch of geometry can be defined as the study of properties that are unaltered when a specified figure is given specified symmetry transformations. Euclidian plane geometry, for instance, concerns the study of properties that are 'invariant' when a figure is moved about on the plane, rotated, mirror reflected, or uniformly expanded and contracted. Affine geometry studies properties that are invariant when a figure is "stretched" in a certain way. Projective geometry studies properties invariant under projection. Topology deals with properties that remain unchanged even when a figure is radically distorted in a manner similar to the deformation of a figure made of rubber." (Martin Gardner, "Aha! Insight", 1978)

"So far as is feasible, diagrams should be planned so that (a) departures from "standard" conditions should be revealed as departures from linearity, or departures from totally random scatter, or as departures of contours from circular form; (b) different points should have approximately independent errors; (c) points should have approximately equal errors, preferably known and indicated, or, if equal errors cannot be achieved, major differences in the precision of individual points should be indicated, at least roughly; (d) individual points should have clearcut interpretation; (e) variables plotted should have clearcut physical interpretation; (f) any non-linear transformations applied should not accentuate uninteresting ranges; (g) any reasonable invariance should be exploited." (David R Cox,"Some Remarks on the Role in Statistics of Graphical Methods", Applied Statistics 27 (1), 1978)

"[…] in trying to prove a concrete geometrical result such as the classification theorem for surfaces, the purely topological structure of the surface (that it be locally euclidean) does not give us much leverage from which to start. On the other hand, although we can define algebraic invariants, such as the fundamental group, for topological spaces in general, they are not a great deal of use to us unless we can calculate them for a reasonably large collection of spaces. Both of these problems may be dealt with effectively by working with spaces that can be broken up into pieces which we can recognize, and which fit together nicely, the so called triangulable spaces." (Mark A Armstrong, "Basic Topology", 1979)

"Showing that two spaces are homeomorphic is a geometrical problem, involving the construction of a specific homeomorphism between them. The techniques used vary with the problem. […] Attempting to prove that two spaces are not homeomorphic to one another is a problem of an entirely different nature. We cannot possibly examine each function between the two spaces individually and check that it is not a homeomorphism. Instead we look for 'topological invariants' of spaces: an invariant may be a geometrical property of the space, a number like the Euler number defined for the space, or an algebraic system such as a group or a ring constructed from the space. The important thing is that the invariant be preserved by a homeomorphism- hence its name. If we suspect that two spaces are not homeomorphic, we may be able to confirm our suspicion by computing some suitable invariant and showing that we obtain different answers." (Mark A Armstrong, "Basic Topology", 1979)

On Invariance (1950-1969)

"Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects. From this point of view, the present universally used system of concepts in which particles and waves appear simultaneously, can be completely justified. The latest research on nuclei and elementary particles has led us, however, to limits beyond which this system of concepts itself does not appear to suffice. The lesson to be learned from what I have told of the origin of quantum mechanics is that probable refinements of mathematical methods will not suffice to produce a satisfactory theory, but that somewhere in our doctrine is hidden a concept, unjustified by experience, which we must eliminate to open up the road." (Max Born, "The Statistical Interpretations of Quantum Mechanics", [Nobel lecture] 1954)

"[…] as every law of nature implies the existence of an invariant, it follows that every law of nature is a constraint. […] Science looks for laws; it is therefore much concerned with looking for constraints. […] the world around us is extremely rich in constraints. We are so familiar with them that we take most of them for granted, and are often not even aware that they exist. […] A world without constraints would be totally chaotic." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"Many of the activities of living organisms permit this double aspect. On the one hand the observer can notice the great deal of actual movement and change that occurs, and on the other hand he can observe that throughout these activities, so far as they are coordinated or homeostatic, there are invariants and constancies that show the degree of regulation that is being achieved." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"[...] the existence of any invariant over a set of phenomena implies a constraint, for its existence implies that the full range of variety does not occur. The general theory of invariants is thus a part of the theory of constraints. Further, as every law of nature implies the existence of an invariant, it follows that every law of nature is a constraint." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"Through all the meanings runs the basic idea of an 'invariant': that although the system is passing through a series of changes, there is some aspect that is unchanging; so some statement can be made that, in spite of the incessant changing, is true unchangingly." (W Ross Ashby, "An Introduction to Cybernetics", 1956)

"It is common knowledge today that in general a symmetry principle (or equivalently an invariance principle) generates a conservation law. For example, the invariance of physical laws under space displacement has as a consequence the conservation of momentum, the invariance under space rotation has as a consequence the conservation of angular momentum." (Chen-Ning Yang, "The Law of Parity Conservation and Other Symmetry Laws of Physics", [Nobel lecture] 1957)

"A satisfactory prediction of the sequential properties of learning data from a single experiment is by no means a final test of a model. Numerous other criteria - and some more demanding - can be specified. For example, a model with specific numerical parameter values should be invariant to changes in independent variables that explicitly enter in the model." (Robert R Bush & Frederick Mosteller,"A Comparison of Eight Models?", Studies in Mathematical Learning Theory, 1959)

"When either the translation or the rotation transformations are applied to an equation, they change the coordinates of all points in the plane (except the origin in the case of a rotation). These transformations actually move the axes with relation to the curve, but they do not alter the shape of the curve represented by the original equation and by the transformed equation. Beside this constant or unvarying feature, i.e., the preservation of the geometric form of the curve, there are also certain algebraic expressions whose values remain unchanged in both equations. Mathematical forms which are preserved or do not vary when other changes take place are called invariants." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"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)

"[..] principle of equipresence: A quantity present as an independent variable in one constitutive equation is so present in all, to the extent that its appearance is not forbidden by the general laws of Physics or rules of invariance. […] The principle of equipresence states, in effect, that no division of phenomena is to be laid down by constitutive equations." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"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)

On Invariance (-1899)

"The calculation of probabilities is of the utmost value, […] but in statistical inquiries there is need not so much of mathematical subtlety as of a precise statement of all the circumstances. The possible contingencies are too numerous to be covered by a finite number of experiments, and exact calculation is, therefore, out of the question. Although nature has her habits, due to the recurrence of causes, they are general, not invariable. Yet empirical calculation, although it is inexact, may be adequate in affairs of practice." (Gottfried W Leibniz [letter to Bernoulli], 1703)

"The analytical equations, unknown to the ancient geometers, which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and from obscurities, that is to say more worthy to express the invariable relations of natural things." (Joseph Fourier, "The Analytical Theory of Heat", 1822)

"There cannot be a language more universal and more simple, more free from errors and obscurities [...] more worthy to express the invariable relations of all natural things [than mathematics]. [It interprets] all phenomena by the same language, as if to attest the unity and simplicity of the plan of the universe, and to make still more evident that unchangeable order which presides over all natural causes." (Joseph Fourier, "The Analytical Theory of Heat", 1822)

"Every scientific task aims at the determination of equivalences and at the discovery of their invariants and for it the following line of poetry holds: 'the wise one seeks a stable pole amid the flight of phenomena'." (Leopold Kronecker,"Zur Theorie der elliptischen Functionen", 1889-1890)

"The following investigations aim to continue the study of the properties of a group in its abstract formulation. In particular, this will pose the question of the extent to which these properties have an invariant character present in all the different realizations of the group, and the question of what leads to the exact determination of their essential group-theoretic content." (Walther von Dyck, "Group-theoretic studies", 1882)

"With the happy expression 'Invariants' chosen by Mr. Sylvester, and quite appropriate to the meaning of the matter, one originally denotes only rational functions of the coefficients of forms that remain unchanged under certain linear transformations of the variables of the forms. But the same expression has since then also been extended to some other entities [Bildungen] that remain unchanged under transformation. This multiple applicability of the concept of invariants rests upon the fact that it belongs to a much more general and abstract realm of ideas. In fact, when the concept of invariants is separated from the direct formal relation to a process of transformation and it is tied rather to the general concept of equivalence, then the concept of invariants reaches the most general realm of thought. For, every abstraction, - say an abstraction from certain differences that are presented by a number of objects, - states an equivalence and the concept originating from the abstraction, for instance the concept of a species, represents the 'invariant of the equivalence'." (Leopold Kronecker,"Zur Theorie der elliptischen Functionen", 1889-1890)

"[...] for each arbitrary equivalence there are always invariants in the above mentioned sense. This is indeed a trivial and completely uninformative truth, for it is the broad concept of function that puts us in a position to give an answer. Let us consider again the already mentioned equivalence for a system of three whole numbers, then we can conceive of the elements of an arbitrarily selected system as characteristic invariants. The elements are invariants because I can reach this system from any other system and they are the characteristic invariants because they represent a system that belongs to one definite class but to none of the other classes. In this typical case of quadratic forms [which are identified as triples of numbers, PM], one chooses as the characteristic invariants of the class the elements of that system that have the smallest value and calls, since Lagrange’s time, the form represented by them the reduced form (forme réduite). From it one can of course generate the entire class of forms. I distinguish the invariants as arithmetical, algebraical, and analytical, according to the method through which they are derived from the elements of a system." (Leopold Kronecker, 1891)

"In our century the conceptions substitution and substitution group, transformation and transformation group, operation and operation group, invariant, differential invariant and differential parameter, appear more and more clearly as the most important conceptions of mathematics." (Sophus Lie, Leipziger Berichte No. 47, 1896)

"In mathematics there is no understanding. In mathematics there are only necessities, laws of existence, invariant relationships. Thus any mathematico-mechanistic outlook must, in the last analysis, waive all understanding. For, we only understand when we know the motives; where there are no motives, all understanding ceases." (Friedrich W Nietzsche) [attributed]

On Concepts IX

"The symbols organized by knowledge, or concepts, themselves belong to social nature as its ideological elements. Therefore, by operating upon them, knowledge is able to expand its organizing function much more broadly than labour in its technological operation of real things; and as we have already seen that many things, which are not organized in practice, can be organized by knowledge, i.e. in symbols: where the ingression of things is absent, the ingression of their concepts is still possible." (Alexander A Bogdanov, "Tektology: The Universal Organizational Science" Vol. I, 1913)

"Every object that we perceive appears in innumerable aspects. The concept of the object is the invariant of all these aspects." (Max Born physicist, "The Statistical Interpretations of Quantum Mechanics", [Nobel lecture] 1954)

"It is one of the consolations of philosophy that the benefit of showing how to dispense with a concept does not hinge on dispensing with it." (Willard v O Quine, "Word and Object", 1960)

"The idea that one can 'introduce' a kind of objects simply by laying down an identity criterion for them really inverts the proper order of explanation. As Locke clearly understood, one must first have a clear conception of what kind of objects one is dealing with in order to extract a criterion of identity for them from that conception. […] So, rather than 'abstract' a kind of object from a criterion of identity, one must in general 'extract' a criterion of identity from a metaphysically defensible conception of a given kind of objects." (Edward J Lowe," The metaphysics of abstract objects", Journal of Philosophy 92(10), 1995)

"The realm of the particularity of each experienced item differs from the formal realm of concepts. [...] The power of paradigmatic thought is to bring order to experience by seeing individual things as belonging to a category." (Donald E Polkinghorne, “Narrative configuration in qualitative analysis", International Journal of Qualitative Studies in Education Vol. 8 (1), 1995)

"In the new systems thinking, the metaphor of knowledge as a building is being replaced by that of the network. As we perceive reality as a network of relationships, our descriptions, too, form an interconnected network of concepts and models in which there are no foundations. For most scientists such a view of knowledge as a network with no firm foundations is extremely unsettling, and today it is by no means generally accepted. But as the network approach expands throughout the scientific community, the idea of knowledge as a network will undoubtedly find increasing acceptance." (Fritjof Capra, "The Web of Life: a new scientific understanding of living systems", 1996)

"Abstraction is an essential knowledge process, the process (or, to some, the alleged process) by which we form concepts. It consists in recognizing one or several common features or attributes (properties, predicates) in individ­uals, and on that basis stating a concept subsuming those common features or attributes. Concept is an idea, associated with a word expressing a prop­erty or a collection of properties inferred or derived from different samples. Subsumption is the logical technique to get generality from particulars." (Hourya B Sinaceur," Facets and Levels of Mathematical Abstraction", Standards of Rigor in Mathematical Practice 18-1, 2014)