🦋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)

Mark Needham - Collected Quotes

 "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)

"At the most abstract level, graph analytics is applied to forecast behavior and pre‐ scribe action for dynamic groups. Doing this requires understanding the relationships and structure within the group. Graph algorithms accomplish this by examining the overall nature of networks through their connections. With this approach, you can understand the topology of connected systems and model their processes." (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)

"Centrality algorithms are used to understand the roles of particular nodes in a graph and their impact on that network. They’re useful because they identify the most important nodes and help us understand group dynamics such as credibility, accessibility, the speed at which things spread, and bridges between groups." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Community formation is common in all types of networks, and identifying them is essential for evaluating group behavior and emergent phenomena. The general prin‐ ciple in finding communities is that its members will have more relationships within the group than with nodes outside their group. Identifying these related sets reveals clusters of nodes, isolated groups, and network structure. This information helps infer similar behavior or preferences of peer groups, estimate resiliency, find nested relationships, and prepare data for other analyses. Community detection algorithms are also commonly used to produce network visualization for general inspection." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Feature extraction is a way to distill large volumes of data and attributes down to a set of representative descriptive attributes. The process derives numerical values (fea‐ tures) for distinctive characteristics or patterns in input data so that we can differenti‐ ate categories in other data. It’s used when data is difficult for a model to analyze directly - perhaps because of size, format, or the need for incidental comparisons. Feature selection is the process of determining the subset of extracted features that are most important or influential to a target goal. It’s used to surface predictive importance as well as for efficiency." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Graph algorithms provide one of the most potent approaches to analyzing connected data because their mathematical calculations are specifically built to operate on relationships. They describe steps to be taken to process a graph to discover its general qualities or specific quantities. Based on the mathematics of graph theory, graph algo‐ rithms use the relationships between nodes to infer the organization and dynamics of complex systems. Network scientists use these algorithms to uncover hidden infomation, test hypotheses, and make predictions about behavior." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Graph embedding is the representation of the nodes and relationships in a graph asfeature vectors. [...] Graph embedding uses graph data slightly differently than in connected feature extraction. It enables us to represent entire graphs, or subsets of graph data, in a numerical format ready for machine learning tasks. This is especially useful for unsu‐pervised learning, where the data is not categorized because it pulls in more contextual information through relationships. Graph embedding is also useful for data exploration, computing similarity between entities, and reducing dimensionality to aid in statistical analysis." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"In classic graph theory, an acyclic graph that is undirected is called a tree. In computer science, trees can also be directed. A more inclusive definition would be a graph where any two nodes are connected by only one path. Trees are significant for understanding graph structures and many algorithms. They play a key role in designing networks, data structures, and search optimizations to improve categorization or organizational hierarchies." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Keep in mind that centrality measures represent the importance of a node in comparison to other nodes. Centrality is a ranking of the potential impact of nodes, not a measure of actual impact. For example, you might identify the two people with the highest cen‐ trality in a network, but perhaps policies or cultural norms are in play that actually shift influence to others. Quantifying actual impact is an active research area to develop additional influence metrics." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Machine learning is not artificial intelligence (AI), but a method for achieving AI. ML uses algorithms to train software through specific examples and progressive improvements based on expected outcome - without explicit programming of how to accomplish these better results. Training involves providing a lot of data to a model and enabling it to learn how to process and incorporate that information." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Make it a habit to run Connected Components to test whether a graph is connected as a preparatory step for general graph analysis. Performing this quick test can avoid accidentally running algorithms on only one disconnected component of a graph and getting incorrect results." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Sometimes the most important cog in the system is not the one with the most overt power or the highest status. Sometimes it’s the middlemen that connect groups or the brokers who the most control over resources or the flow of information. Betweenness Centrality is a way of detecting the amount of influence a node has over the flow of information or resources in a graph. It is typically used to find nodes that serve as a bridge from one part of a graph to another." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Putting together the right mix of features can increase accuracy because it fundamentally influences how our models learn. Because even modest improvements can make a significant difference, our focus in this chapter is on connected features. Connected features are features extracted from the structure of the data. These features can be derived from graph-local queries based on parts of the graph surrounding a node, or graph-global queries that use graph algorithms to identify predictive elements within data based on relationships for connected feature extraction." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Use Degree Centrality if you’re attempting to analyze influence by looking at the number of incoming and outgoing relationships, or find the “popularity” of individual nodes. It works well when you’re concerned with immediate connectedness or near-term probabilities. However, Degree Centrality is also applied to global analysis when you want to evaluate the minimum degree, maximum degree, mean degree, and standard deviation across the entire graph." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"When using community detection algorithms, be conscious of the density of the relationships. If the graph is very dense, you may end up with all nodes congregating in one or just a few clusters. You can counteract this by filtering by degree, relationship weights, or similarity metrics. On the other hand, if the graph is too sparse with few connected nodes, you may end up with each node in its own cluster. In this case, try to incorporate additional relationship types that carry more relevant information." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Without peripheral and related information, solutions that attempt to predict behav‐ ior or make recommendations for varying circumstances require more exhaustive training and prescriptive rules. This is partly why AI is good at specific, well-defined tasks, but struggles with ambiguity. Graph-enhanced ML can help fill in that missing contextual information that is so important for better decisions." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

🕸️On Graph Theory: On Ranndom Walks

"To every event defined for the original random walk there corresponds an event of equal probability in the dual random walk, and in this way almost every probability relation has its dual." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950) 

"A random walk is a stochastic process, which is a general term for a chain of events which change in a random way with time. Sequential binomial sampling is another example of a stochastic. process, since each 'step' in the 'walk' represents the result of sampling one more item from the population to determine whether it is to be accepted or rejected. The decision to be reached by the process is to determine whether or not to accept a certain hypothesis about the population. Such a hypothesis might relate to the proportion of defective members, the proportion of members having a property within specified limits, or some similar statement." (David B MacNeil, "Modern Mathematics for the Practical Man", 1963)

"Random walk processes are significant, not only in many types of physical phenomena, but also in basic statistical methods, including decision making and the related sequential sampling." (David B MacNeil, "Modern Mathematics for the Practical Man", 1963)

"However, random walk theory also tells us that the chance that the balance never returns to zero - that is, that H stays in the lead for ever - is 0. This is the sense in which the 'law of averages' is true. If you wait long enough, then almost surely the numbers of heads and tails will even out. But this fact carries no implications about improving your chances of winning, if you're betting on whether H or T turns up. The probabilities are unchanged, and you don't know how long the 'long run' is going to be. Usually it is very long indeed." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)

"In everyday language, a fair coin is called random, but not a coin that shows head more often than tail. A coin that keeps a memory of its own record of heads and tails is viewed as even less random. This mental picture is present in the term random walk, especially as used in finance." (Benoit B Mandelbrot, "Fractals and Scaling in Finance: Discontinuity, concentration, risk", 1997) 

"A random walk is one in which future steps or directions cannot be predicted on the basis of past history. When the term is applied to the stock market, it means that short-run changes in stock prices are unpredictable. Investment advisory services, earnings forecasts, and chart patterns are useless. [...] What are often called 'persistent patterns' in the stock market occur no more frequently than the runs of luck in the fortunes of any gambler playing a game of chance. This is what economists mean when they say that stock prices behave very much like a random walk." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

"Perhaps the most common complaint about the weakness of the random-walk theory is based on a distrust of mathematics and a misconception of what the theory means. 'The market isn't random', the complaint goes, 'and no mathematician is going to convince me it is'. [...] But, even if markets were dominated during certain periods by irrational crowd behavior, the stock market might still well be approximated by a random walk. The original illustrative analogy of a random walk concerned a drunken man staggering around an empty field. He is not rational, but he's not predictable either." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)

"The concept of a random walk is simple but rich for its many applications, not only in finance but also in physics and the description of natural phenomena. It is arguably one of the most founding concepts in modern physics as well as in finance, as it underlies the theories of elementary particles, which are the building blocks of our universe, as well as those describing the complex organization of matter around us." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)

"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)

"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)

📓On Literature: On Projections (From Fiction to Science-Fiction)

"Engage people with what they expect; it is what they are able to discern and confirms their projections. It settles them into predictable patterns of response, occupying their minds while you wait for the extraordinary moment - that which they cannot anticipate." (Sun Tzu, "The Art of War", 5th century BC)

"The person one loves never really exists, but is a projection focused through the lens of the mind onto whatever screen it fits with least distortion." (Arthur C Clarke, "The Road to the Sea", 1961)

"Much of what you see in perceiving me - 'He pointed to himself for emphasis.' - is a projection from your own mind. To another percept-system I would appear quite different. To the police, for instance. There’re as many worldviews as there are sentient creatures." (Philip K Dick, "Galactic Pot-Healer", 1969)

"To be a god can ultimately become boring and degrading. There'd be reason enough for the invention of free will! A god might wish to escape into sleep and be alive only in the unconscious projections of his dream-creatures." (Frank Herbert,"Children of Dune", 1976)

"Science fiction is the branch of literature that deals with the effects of change on people in the real world as it can be projected into the past, the future, or to distant places. It often concerns itself with scientific or technological change, and it usually involves matters whose importance is greater than the individual or the community; often civilization or the race itself is in danger." (James Gunn, "Road to Science Fiction" Vol. 2, 1979)

"This is the awe-inspiring universe of magic: There are no atoms, only waves and motions all around. Here, you discard all belief in barriers to understanding. You put aside understanding itself. This universe cannot be seen, cannot be heard, cannot be detected in any way by fixed perceptions. It is the ultimate void where no preordained screens occur upon which forms may be projected. You have only one awareness here - the screen of the magi: Imagination! Here, you learn what it is to be human. You are a creator of order, of beautiful shapes and systems, an organizer of chaos." (Frank Herbert, “Heretics of Dune”, 1984)

"We know so little about each other. We lie mostly submerged, like ice floes, with our visible social selves projecting only cool and white." (Ian McEwan, "Amsterdam", 1998)

"Things outside you are projections of what's inside you, and what's inside you is a projection of what's outside. So when you step into the labyrinth outside you, at the same time you're stepping into the labyrinth inside." (Haruki Murakami, "Kafka on the Shore", 2002)

🥀On Grace (Trivia)

"[Intuitive] Understanding is consequent upon deliberation, and firmly embraces the better part. For [intuitive] understanding concerns itself with divine truths, and the relish, love, and observance of the latter constitutes true wisdom. Rather than being the [mere] product of nature, these successive steps are the result of grace. The latter, according to its own free determination, derives the various rivulets of the sciences and wisdom from the fountainhead of sense perception. Grace reveals hidden divine truths by means of those things which have been made, and by that unity which belongs to love, communicates what it has made manifest, thus uniting man to God." (, "Metalogicon", 1159)

*Orators and Philosophers treat Nature after a very different manner […] with all her graces and ornaments, and if there be anything which is not capable of that, they dissemble it, or pass it over slightly. But Philosophers view Nature with a more impartial eye, and without favor or prejudice give a just and free account [of] how they find all the parts of the Universe, some more, some less perfect." (Thomas Burnet,"The Sacred Theory of the Earth" 2nd ed.,1691)

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

"[...] science gives Man from time to time a greatly increased mastery over Nature; science, with its analytical triumphs, ever tends to diminish, in the shallowminded, the saving grace of wonder; and science is ever dispelling the darkness that oppresses the mind." (J Arthur Thomson, "The System of Animate Nature" Vol 1, 1920) 

"What the world needs is a fusion of the sciences and the humanities. The humanities express the symbolic, poetic, and prophetic qualities of the human spirit. Without them we would not be conscious of our history; we would lose our aspirations and the grace of expression that move men’s hearts. The sciences express the creative urge in man to construct a universe which is comprehensible in terms of the human intellect. Without them, mankind would find itself bewildered in a world of natural forces beyond comprehension, victims of ignorance, superstition and fear." (Isidor I Rabi, [address] 1954)

"There was a moment when I knew how nature worked. It had elegance and beauty. The goddam thing was gleaming." (Paul K Feyerabend, 1967)

"Ever bigger machines, entailing ever bigger concentrations of economic power and exerting ever greater violence against the environment, do not represent progress: they are a denial of wisdom. Wisdom demands a new orientation of science and technology towards the organic, the gentle, the non-violent, the elegant and beautiful." (Ernst F Schumacher,"Small Is Beautiful", 1973)

"While Occam’s razor is a useful tool in the physical sciences, it can be a very dangerous implement in biology. It is thus very rash to use simplicity and elegance as a guide in biological research." (Francis H C Crick, "What Mad Pursuit: A Personal View of Scientific Discovery", 1988) 

"We move from part to whole and back again, and in that dance of comprehension, in that amazing circle of understanding, we come alive to meaning, to value, and to vision: the very circle of understanding guides our way, weaving together the pieces, healing the fractures, mending the torn and tortured fragments, lighting the way ahead - this extraordinary movement from part to whole and back again, with healing the hallmark of each and every step, and grace the tender reward." (Ken Wilber, "The Eye of Spirit: An Integral Vision for a World Gone Slightly Mad", 1997)

🥀On Grace in Mathematics

"The mathematician is perfect only in so far as he is a perfect man, in so far as he senses in himself the beauty of truth; only then will his work be thorough, transparent, prudent, pure, clear, graceful, indeed elegant." (Johann Wolfgang von Goethe, 1829)

"Proportion, or symmetry, is the basis of beauty; propriety, of grace." (Henry Fuseli [in "The Life and Writings of Henry Fuseli", 1831])

"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)

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess – it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1940)

"When you are told that something is an average you still don't know very much about it unless you can find out which of the common kinds of average it is-mean, median, or mode. [...] The different averages come out close together when you deal with data, such as those having to do with many human characteristics, that have the grace to fall close to what is called the normal distribution. If you draw a curve to represent it you get something shaped like a bell, and mean, median, and mode fall at the same point." (Darell Huff, "How to Lie with Statistics", 1954)

"The theory of the visual display of quantitative information consists of principles that generate design options and that guide choices among options. The principles should not be applied rigidly or in a peevish spirit; they are not logically or mathematically certain; and it is better to violate any principle than to place graceless or inelegant marks on paper. Most principles of design should be greeted with some skepticism, for word authority can dominate our vision, and we may come to see only though the lenses of word authority rather than with our own eyes." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)

📁 On DIKW: On Mastery (-1799)

"All things have a way of adding up together, so that one will become more proficient in any proposed branch of learning to the extent that he has mastered neighboring and related departments of knowledge." (John of Salisbury, "Metalogicon", 1159)

"In our acquisition of [scientific] knowledge, investigation is the first step, and comes before comprehension, analysis, and retention. Innate ability, although it proceeds from nature, is fostered by study and exercise. What is difficult when we first try it, becomes easier after assiduous practice, and once the rules for doing it are mastered, very easy, unless languor creeps in, through lapse of use or carelessness, and impedes our efficiency. This, in short, is how all the arts have originated: Nature, the first fundamental, begets the habit and practice of study, which proceeds to provide an art, and the latter, in turn, finally furnishes the faculty whereof we speak. Natural ability is accordingly effective. So, too, is exercise. And memory likewise, is effective, when employed by the two aforesaid. With the help of the foregoing, reason waxes strong, and produces the arts, which are proportionate to [man’s] natural talents." (John of Salisbury, "Metalogicon", 1159)

"The method of demonstration is therefore generally feeble and ineffective with regard to facts of nature (I refer to corporeal and changeable things). But it quickly recovers its strength when applied to the field of mathematics. For whatever it concludes in regard to such things as numbers, proportions and figures is indubitably true, and cannot be otherwise. One who wishes to become a master of the science of demonstration should first obtain a good grasp of probabilities. Whereas the principles of demonstrative logic are necessary; those of dialectic are probable." (John of Salisbury, "Metalogicon", 1159)

"The truth of voice perishes with the sound; truth latent in the mind is hidden wisdom and invisible treasure; but the truth which illuminates books desires to manifest itself to every disciplinable sense. Let us consider how great a commodity of doctrine exists in books, - how easily, how secretly, how safely, they expose the nakedness of human ignorance without putting it to shame. These are the masters that instruct us without rods and ferules, without hard words and anger, without clothes or money. If you approach them, they are not asleep; if, investigating, you interrogate them, they conceal nothing; if you mistake them, they never grumble; if you are ignorant, they cannot laugh at you." (Richard de Burry, "Philobiblon", 1344)

"Parsimony is enough to make the master of the golden mines as poor as he that has nothing; for a man may be brought to a morsel of bread by parsimony as well as profusion." (Henry Home [Lord Kames] ," Introduction to the Art of Thinking", 1761)

📁 On DIKW: On Mastery (2000-)

 "In an information economy, entrepreneurs master the science of information in order to overcome the laws of the purely physical sciences. They can succeed because of the surprising power of the laws of information, which are conducive to human creativity. The central concept of information theory is a measure of freedom of choice. The principle of matter, on the other hand, is not liberty but limitation - it has weight and occupies space." (George Gilder, "Knowledge and Power: The Information Theory of Capitalism and How it is Revolutionizing our World", 2013)

"Mathematics is a fascinating discipline that calls for creativity, imagination, and the mastery of rigorous standards of proof." (John Meier & Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)

"And that’s what good networkers do. No matter the field, discipline, or industry, if we want to succeed, we must master the networks. Because as the First Law of Success reminds us, the harder it is to measure performance, the less performance matters." (Albert-László Barabási, "The Formula: The Universal Laws of Success", 2018)

"Models are formal structures represented in mathematics and diagrams that help us to understand the world. Mastery of models improves your ability to reason, explain, design, communicate, act, predict, and explore." (Scott E Page,"The Model Thinker", 2018)

"[…] when it comes to interpreting the world around us, we need to realize that our feelings can trump our expertise. […] The more extreme the emotional reaction, the harder it is to think straight. […] It is not easy to master our emotions while assessing information that matters to us, not least because our emotions can lead us astray in different directions. […] We often find ways to dismiss evidence that we don’t like. And the opposite is true, too: when evidence seems to support our preconceptions, we are less likely to look too closely for flaws." (Tim Harford, "The Data Detective: Ten easy rules to make sense of statistics", 2020)