🦋On Chaos Theory: On Paths

"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic" (that is fixed) rules" (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order" (a pattern) within disorder" (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)

"One reason nature pleases us is its endless use of a few simple principles: the cube-square law; fractals; spirals; the way that waves, wheels, trig functions, and harmonic oscillators are alike; the importance of ratios between small primes; bilateral symmetry; Fibonacci series, golden sections, quantization, strange attractors, path-dependency, all the things that show up in places where you don’t expect them [...] these rules work with and against each other ceaselessly at all levels, so that out of their intrinsic simplicity comes the rich complexity of the world around us. That tension - between the simple rules that describe the world and the complex world we see - is itself both simple in execution and immensely complex in effect. Thus exactly the levels, mixtures, and relations of complexity that seem to be hardwired into the pleasure centers of the human brain - or are they, perhaps, intrinsic to intelligence and perception, pleasant to anything that can see, think, create? - are the ones found in the world around us." (John Barnes, "Mother of Storms", 1994)

"Complexity is the characteristic property of complicated systems we don’t understand immediately. It is the amount of difficulties we face while trying to understand it. In this sense, complexity resides largely in the eye of the beholder - someone who is familiar with s.th. often sees less complexity than someone who is less familiar with it. [...] A complex system is created by evolutionary processes. There are multiple pathways by which a system can evolve. Many complex systems are similar, but each instance of a system is unique." (Jochen Fromm, The Emergence of Complexity, 2004)

"Chaos can leave statistical footprints that look like noise. This can arise from simple systems that are deterministic and not random. [...] The surprising mathematical fact is that most systems are chaotic. Change the starting value ever so slightly and soon the system wanders off on a new chaotic path no matter how close the starting point of the new path was to the starting point of the old path. Mathematicians call this sensitivity to initial conditions but many scientists just call it the butterfly effect. And what holds in math seems to hold in the real world - more and more systems appear to be chaotic." (Bart Kosko, "Noise", 2006)

"'Chaos' refers to systems that are very sensitive to small changes in their inputs. A minuscule change in a chaotic communication system can flip a 0 to a 1 or vice versa. This is the so-called butterfly effect: Small changes in the input of a chaotic system can produce large changes in the output. Suppose a butterfly flaps its wings in a slightly different way. can change its flight path. The change in flight path can in time change how a swarm of butterflies migrates." (Bart Kosko, "Noise", 2006)

⚛On Physics: On Paths

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"Experimental physicists [...] walk a narrow path with pitfalls on either side. If we spend all our time developing equipment, we risk the appellation of 'plumber', and if we merely use the tools developed by others, we risk the censure of our peers for being parasitic." (Luis W Alvarez, "Recent Developments in Particle Physics", [Nobel] 1968)

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

"The Feynman method has the virtue that it provides us with a vivid picture of nature’s quantum trickery at work. The idea is that the path of a particle through space is not generally well defined in quantum mechanics. […] So when an electron arrives at a point in space - say a target screen - many different histories must be integrated together to create this one event. Feynman’s so-called path-integral, or sum-over-histories approach to quantum mechanics, set this remarkable concept out as a mathematical procedure. It remained more or less a curiosity for many years, but as physicists pushed quantum mechanics to its limits - applying it to gravitation and even cosmology - so the Feynman approach turned out to offer the best calculational tool for describing a quantum universe. History may well judge that, among his many outstanding contributions to physics, the path-integral formulation of quantum mechanics is the most significant." (Paul Davies , [Introduction to Richard P Feynman's "Six Easy Pieces"] 1994)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)

🕸️On Graph Theory: On Paths

"Euler's proof that in Königsberg there is no path crossing all seven bridges only once was based on a simple observation. Nodes with an odd number of links must be either the starting or the end point of the journey. A continuous path that goes through all the bridges can have only one starting and one end point. Thus, such a path cannot exist on a graph that has more than two nodes with an odd number of links. As the Königsberg graph had four such nodes, one could not find the desired path." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Average path length reflects the global structure; it depends on the way the entire network is connected, and cannot be inferred from any local measurement. Clustering reflects the local structure; it depends only on the interconnectedness of a typical neighborhood, the inbreeding among nodes tied to a common center. Roughly speaking, path length measures how big the network is. Clustering measures how incestuous it is." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"The concept of path-connectedness, in which it is required that it be possible to reach any point in the space from any other point along a continuous path is necessary for the notion of fundamental group. This approach is especially useful in studying connectivity properties from an algebraic point of view, e.g., via homotopy theory." (D Chatterjee, "Topology: General & Algebraic", 2003)

"If a network is solely composed of neighborhood connections, information must traverse a large number of connections to get from place to place. In a small-world network, however, information can be transmitted between any two nodes using, typically, only a small number of connections. In fact, just a small percentage of random, long-distance connections is required to induce such connectivity. This type of network behavior allows the generation of 'six degrees of separation' type results, whereby any agent can connect to any other agent in the system via a path consisting of only a few intermediate nodes." (John H Miller & Scott E Page, "Complex Adaptive Systems", 2007)

"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)

"A random walk, in general, is sometimes described as being similar to how a drunk person traverses a city. They know what direction or end point they want to reach but may take a very circuitous route to get there. The algorithm starts at one node and somewhat randomly follows one of the relationships forward or backward to a neighbor node. It then does the same from that node and so on, until it reaches the set path length. ('We say somewhat randomly because the number of relationships a node has, and its neighbors have, influences the probability a node will be walked through.)'" (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Betweenness Centrality makes the assumption that all communication between nodes happens along the shortest path and with the same frequency, which isn’t always the case in real life. Therefore, it doesn’t give us a perfect view of the most influential nodes in a graph, but rather a good representation." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"A widely adopted technique for solving the data sparsity issue and the cold-start problem is based on graph representation, navigation, and processing. Graph navigation methods (like the pathfinding example [...]) and graph algorithms (such as PageRank) are applied to fill some gaps and create a denser representation of [a] dataset." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)

💠On Problem Solving: On Paths

"Having gotten, with God’s help, to the very desired place, i.e. the mother of all cases called by the people 'the rule of the thing' or the 'Greater Art', i.e. speculative practice; otherwise called Algebra and Almucabala in the Arab language or Chaldean according to some, which in our [language] amounts to saying 'restaurationis et oppositionis', Algebra id est Restau­ ratio. Almucabala id est Oppositio vel contemptio et Solutio, because by this path one solves infinite questions. And one picks out those which cannot yet be solved." (Luca Pacioli, "Summa de arithmetica geometria proportioni et proportionalita", 1494)

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

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"The philosophical emphasis here is: to solve a geometrical problem of a global nature, one first reduces it to a homotopy theory problem; this is in turn reduced to an algebraic problem and is solved as such. This path has historically been the most fruitful one in algebraic topology." (Brayton Gray, "Homotopy Theory", Pure and Applied Mathematics Vol. 64, 1975)

"In the real world, none of these assumptions are uniformly valid. Often people want to know why mathematics and computers cannot be used to handle the meaningful problems of society, as opposed, let us say, to the moon boondoggle and high energy-high cost physics. The answer lies in the fact that we don't know how to describe the complex systems of society involving people, we don't understand cause and effect, which is to say the consequences of decisions, and we don't even know how to make our objectives reasonably precise. None of the requirements of classical science are met. Gradually, a new methodology for dealing with these 'fuzzy' problems is being developed, but the path is not easy." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"It is actually impossible in theory to determine exactly what the hidden mechanism is without opening the box, since there are always many different mechanisms with identical behavior. Quite apart from this, analysis is more difficult than invention in the sense in which, generally, induction takes more time to perform than deduction: in induction one has to search for the way, whereas in deduction one follows a straightforward path." (Valentino Braitenberg, "Vehicles: Experiments in Synthetic Psychology", 1984)

"We wrote down all the states and legal moves (here it turned out to be helpful to have a systematic notation, but that's not essential). Then we formed a graph whose nodes correspond to states and whose edges correspond to legal moves. The solution of the puzzle is then a path through the graph that joins the start to the finish. Such a path is usually obvious to the eye, provided the puzzle is sufficiently simple for the entire graph to be drawn. Puzzles of this type are really mazes, for a maze is just a graph drawn in a slightly different fashion. Metaphorically, they are logical mazes - you have to find the right sequence of moves to solve them. The graph turns the logical maze into a genuine maze, turning the metaphor into reality. The fact that solving the real maze also solves the logical maze is one of the magical features of the maze that is mathematics." (Ian Stewart, "The Magical Maze: Seeing the World Through Mathematical Eyes", 1997)

"When a book is being written, it is a maze of possibilities, most of which are never realised. Reading the resulting book, once all decisions have been taken, is like tracing one particular path through that maze. The writer's job is to choose that path, define it clearly, and make it as smooth as possible for those who follow. Mathematics is much the same. Mathematical ideas form a network. The interconnections between ideas are logical deductions. If we assume this, then that must follow - a logical path from this to that. When mathematicians try to understand a problem, they have to thread a maze of logic. The body of knowledge that we call mathematics is a catalogue of interesting excursions through the logical maze."(Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"An algorithm refers to a successive and finite procedure by which it is possible to solve a certain problem. Algorithms are the operational base for most computer programs. They consist of a series of instructions that, thanks to programmers’ prior knowledge about the essential characteristics of a problem that must be solved, allow a step-by-step path to the solution." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Monte Carlo is able to discover practical solutions to otherwise intractable problems because the most efficient search of an unmapped territory takes the form of a random walk. Today’s search engines, long descended from their ENIAC-era ancestors, still bear the imprint of their Monte Carlo origins: random search paths being accounted for, statistically, to accumulate increasingly accurate results. The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

🪸On Systems Thinking: On Paths

"The state of a system at a given moment depends on two things - its initial state, and the law according to which that state varies. If we know both this law and this initial state, we have a simple mathematical problem to solve, and we fall back upon our first degree of ignorance. Then it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets? We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow recti-linear paths and obey the laws of impact and elastic bodies; yet as we know nothing of their initial velocities, we know nothing of their present velocities. The calculus of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible, that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance, and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known, and we want to find the law; or that, instead of deducing effects from causes, we wish to deduce the causes." (Henri Poincaré, "Science and Hypothesis", 1902)

"It is not enough to know the critical stress, that is, the quantitative breaking point of a complex design; one should also know as much as possible of the qualitative geometry of its failure modes, because what will happen beyond the critical stress level can be very different from one case to the next, depending on just which path the buckling takes. And here catastrophe theory, joined with bifurcation theory, can be very helpful by indicating how new failure modes appear." (Alexander Woodcock & Monte Davis, "Catastrophe Theory", 1978)

"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)

"In a free-market economy, then, uncertainty is a necessary element. Only when the economy is in a state of uncertainty can the participants efficiently search for solutions to problems and find creative answers. In addition, only a system that depends on uncertainty can survive unexpected shocks. A complex process can take multiple paths to an optimal solution. It does not require 'ideal' conditions; in fact, shocks often force it to find a better solution, a higher hill in the fitness landscape. The 'creative destruction' identified by the Austrian school suggests that a free-market economy is not only resilient to shocks, but is also creative and capable of generating innovation. It can only do so while in a high state of uncertainty." (Edgar E Peters, "Patterns in the dark: understanding risk and financial crisis with complexity theory", 1999)

"It is, however, fair to say that very few applications of swarm intelligence have been developed. One of the main reasons for this relative lack of success resides in the fact that swarm-intelligent systems are hard to 'program', because the paths to problem solving are not predefined but emergent in these systems and result from interactions among individuals and between individuals and their environment as much as from the behaviors of the individuals themselves. Therefore, using a swarm-intelligent system to solve a problem requires a thorough knowledge not only of what individual behaviors must be implemented but also of what interactions are needed to produce such or such global behavior." (Eric Bonabeau et al, "Swarm Intelligence: From Natural to Artificial Systems", 1999)

"Complexity theory shows that great changes can emerge from small actions. Change involves a belief in the possible, even the 'impossible'. Moreover, social innovators don’t follow a linear pathway of change; there are ups and downs, roller-coaster rides along cascades of dynamic interactions, unexpected and unanticipated divergences, tipping points and critical mass momentum shifts. Indeed, things often get worse before they get better as systems change creates resistance to and pushback against the new. Traditional evaluation approaches are not well suited for such turbulence. Traditional evaluation aims to control and predict, to bring order to chaos. Developmental evaluation accepts such turbulence as the way the world of social innovation unfolds in the face of complexity. Developmental evaluation adapts to the realities of complex nonlinear dynamics rather than trying to impose order and certainty on a disorderly and uncertain world." (Michael Q Patton, "Developmental Evaluation", 2010)

"In the 'computation' that is the economy, large and small probabilistic events at particular non-repeatable moments determine the attractors fallen into, the temporal structures that form and die away, the technologies that are brought to life, the economic structures and institutions that result from these, the technologies and structures that in turn build upon these; indeed the future shape of the economy - the future path taken. The economy at all levels and at all times is path dependent. History again becomes important. And time reappears." (W Brian Arthur, "Complexity and the Economy", 2015)

"Feedback systems are closed loop systems, and the inputs are changed on the basis of output. A feedback system has a closed loop structure that brings back the results of the past action to control the future action. In a closed system, the problem is perceived, action is taken and the result influences the further action. Thus, the distinguishing feature of a closed loop system is a feedback path of information, decision and action connecting the output to input." (Bilash K Bala et al, "System Dynamics: Modelling and Simulation", 2017)

On Paths: On Shortest Path

"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900)

"Imagine the forehead of a bull, with the protuberances from which the horns and ears start, and with the collars hollowed out between these protuberances; but elongate these horns and ears without limit so that they extend to infinity; then you will have one of the surfaces we wish to study. On such a surface geodesics may show many different aspects. There are, first of all, geodesics which close on themselves. There are some also which are never infinitely distant from their starting point even though they never exactly pass through it again; some turn continually around the right horn, others around the left horn, or right ear, or left ear; others, more complicated, alternate, in accordance with certain rules, the turns they describe around one horn with the turns they describe around the other horn, or around one of the ears. Finally, on the forehead of our bull with his unlimited horns and ears there will be geodesics going to infinity, some mounting the right horn, others mounting the left horn, and still others following the right or left ear. [...] If, therefore, a material point is thrown on the surface studied starting from a geometrically given position with a geometrically given velocity, mathematical deduction can determine the trajectory of this point and tell whether this path goes to infinity or not. But, for the physicist, this deduction is forever useless. When, indeed, the data are no longer known geometrically, but are determined by physical procedures as precise as we may suppose, the question put remains and will always remain unanswered." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)

"A variety of natural phenomena exhibit what is called the minimum principle. The principle is displayed where the amount of energy expended in performing a given action is the least required for its execution, where the path of a particle or wave in moving from one point to another is the shortest possible, where a motion is completed in the shortest possible time, and so on." (James R Newman, "The World of Mathematics" Vol. II, 1956)

"We frequently find that nature acts in such a way as to minimize certain magnitudes. The soap film will take the shape of a surface of smallest area. Light always follows the shortest path, that is, the straight line, and, even when reflected or broken, follows a path which takes a minimum of time. In mechanical systems we find that the movements actually take place in a form which requires less effort in a certain sense than any other possible movement would use. There was a period, about 150 years ago, when physicists believed that the whole of physics might be deduced from certain minimizing principles, subject to calculus of variations, and these principles were interpreted as tendencies - so to say, economical tendencies of nature. Nature seems to follow the tendency of economizing certain magnitudes, of obtaining maximum effects with given means, or to spend minimal means for given effects." (Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)