Mathematics
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Mathematics

  • Bertrand Russell, Mysticism and Logic: And Other Essays
    • Mathematical functions describe the behaviour of complex systems. Some say that mathematics is a higher or purer form of science, or that it is something fundamental about the universe.
    • Yet I think that they cannot exist independently, the functions are born from the existence of these systems like time is born from matter. Because of this, I think that I personally lean towards mathematics as an invention, rather than a discovery.
    • Mathematics, rightly viewed, possesses not only truth but supreme beauty.
  • Michio Kaku, Reddit
    • In France, eventually you have to learn the French language, including how to conjugate verbs, etc. In physics, eventually you have learn the math. Math is the language of nature. Unfortunately, math is the dividing line separating amateurs from professionals. Sorry to say.

Recurring patterns

All of these mathematical functions describe similar patterns that emerge at many different magnitudes of size or complexity, from the universe and physics to the interior of the cell to the body to the ecosystem, to human society and economy.

The individual units in different kinds complex systems often have similar relationships with each other. So the systems resemble each other.

  • #[[oscillation]] (Like sin or cos)
  • Parabola
  • Exponential increase (self reinforcing)
  • Exponential decrease (downward spiral)
  • Logarithmic increase (starts fast then slows)
  • Logarithmic decrease (starts fast then slows)
  • #branching
  • Thresholds

Distributions

  • Bell curve

The universe isn't 'fractal'. What happens is patterns go through the magnitudes of size of the universe in so far as that what is stable is most prevalent. It is only fractal in so far as different 'levels' can mirror each other (which they can do if their macro conditions are similar). As an aside: [Molecules that actively conspire to make themselves more prevalent is a curious oddity.]

Oscillation

  • A large number of systems, including organic systems, exist on a spectrum that constantly oscillates between two or more extremes. Feedback loops maintain the system and correct it when imbalances occur.
  • It seems that ancient Taoist philosophy noticed the phenomenon, and called one end Yin and the other end Yang. It also corresponds to the mathematical sin wave.
  • Many patterns within systems of the Earth, of nature, and of human society seem to oscillate naturally. The system moves to one extreme, where pressure builds, and then is released to be balanced by another, and return. In a system such as this, no point is ever sustained - there is only momentum, and pressure to start moving in the other direction, before it swaps.
  • This concept could be used in engineering, or in large scale social engineering to create self regulating and sustainable systems.
  • Wikipedia, Yin and yang
    • In Daoist philosophy, dark and light, yin and yang, arrive in the Dàodéjīng at chapter 42. It becomes sensible from an initial quiescence or emptiness wuji, sometimes symbolized by an empty circle), and continues moving until quiescence is reached again. For instance, dropping a stone in a calm pool of water will simultaneously raise waves and lower troughs between them, and this alternation of high and low points in the water will radiate outward until the movement dissipates and the pool is calm once more. Yin and yang thus are always opposite and equal qualities. Further, whenever one quality reaches its peak, it will naturally begin to transform into the opposite quality: for example, grain that reaches its full height in summer (fully yang) will produce seeds and die back in winter (fully yin) in an endless cycle.
    • Yin and yang transform each other: like an undertow in the ocean, every advance is complemented by a retreat, and every rise transforms into a fall. Thus, a seed will sprout from the earth and grow upwards towards the sky—an intrinsically yang movement. Then, when it reaches its full potential height, it will fall. Also, the growth of the top seeks light, while roots grow in darkness.
    • The relationship between yin and yang is often described in terms of sunlight playing over a mountain and a valley. Yin (literally the 'shady place' or 'north slope') is the dark area occluded by the mountain's bulk, while yang (literally the 'sunny place' or 'south slope') is the brightly lit portion. As the sun moves across the sky, yin and yang gradually trade places with each other, revealing what was obscured and obscuring what was revealed.

Strange loops

  • A strange loop arises when, by moving only upwards or downwards through a hierarchical system, one finds oneself back where one started.
  • Strange loops may involve self-reference and paradox. The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter in Gödel, Escher, Bach, and is further elaborated in Hofstadter's book ITags:: #universe, #epistemologya Strange Loop, published in 2007.
  • A tangled hierarchy is a hierarchical consciousness system in which a strange loop appears.
  • Definitions**
  • A strange loop is a hierarchy of levels, each of which is linked to at least one other by some type of relationship. A strange loop hierarchy is "tangled" (Hofstadter refers to this as a "heterarchy"), in that there is no well defined highest or lowest level; moving through the levels, one eventually returns to the starting point, i.e., the original level. Examples of strange loops that Hofstadter offers include: many of the works of M. C. Escher, the information flow network between DNA and enzymes through protein synthesis and DNA replication, and self-referential Gödelian statements in formal systems.
  • In ITags:: #universe, #epistemologya Strange Loop, Hofstadter defines strange loops as follows:
  • And yet when I say "strange loop", I have something else in mind — a less concrete, more elusive notion. What I mean by "strange loop" is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive "upward" shifts turn out to give rise to a closed cycle. That is, despite one's sense of departing ever further from one's origin, one winds up, to one's shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop. (pp. 101-102)
  • In cognitive science**
  • Strange loops take form in human consciousness as the complexity of active symbols in the brain inevitably leads to the same kind of self-reference which Gödel proved was inherent in any complex logical or arithmetical system in his incompleteness theorem.(https://en.wikipedia.org/wiki/Strange_loop#cite_note-1) Gödel showed that mathematics and logic contain strange loops: propositions that not only refer to [mathematical and logical truths, but also to the symbol systems expressing those truths. This leads to the sort of paradoxes seen in statements such as "This statement is false," wherein the sentence's basis of truth is found in referring to itself and its assertion, causing a logical paradox.(https://en.wikipedia.org/wiki/Strange_loop#cite_note-oreilly-2)
  • Hofstadter argues that the psychological self arises out of a similar kind of paradox. We are not born with an "I" – the ego emerges only gradually as experience shapes our dense web of active symbols into a tapestry rich and complex enough to begin twisting back upon itself. According to this view the psychological "I" is a narrative fiction, something created only from intake of symbolic data and its own ability to create stories about itself from that data. The consequence is that a perspective (a mind) is a culmination of a unique pattern of symbolic activity in our nervous systems, which suggests that the pattern of symbolic activity that makes identity, that constitutes subjectivity, can be replicated within the brains of others, and perhaps even in artificial brains.(https://en.wikipedia.org/wiki/Strange_loop#cite_note-oreilly-2)
  • Strangeness**
  • The "strangeness" of a strange loop comes from our way of perception, because we categorize our input in a small number of 'symbols' (by which Hofstadter means groups of neurons standing for one thing in the outside world). So the difference between the video-feedback loop and our strange loops, our "I"s, is that while the former converts light to the same pattern on a screen, the latter categorizes a pattern and outputs its essence, so that as we get closer and closer to our essence, we get further down our strange loop.(https://en.wikipedia.org/wiki/Strange_loop#cite_note-Hofstadter-3)
  • Downward causality**
  • Hofstadter thinks our minds appear to us to determine the world by way of "downward causality", which refers to a situation where a cause-and-effect relationship in a system gets flipped upside-down. Hofstadter says this happens in the proof of Gödel's incompleteness theorem:
  • Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false. (pp. 169-170)
  • Hofstadter claims a similar "flipping around of causality" appears to happen in minds possessing self-consciousness. The mind perceives itself as the cause of certain feelings, ("I am he source of my desires), while according to popular scientific models, feelings and desires are strictly caused by the interactions of neurons.
  • Examples**
  • Hofstadter points to Bach's Canon per Tonos, M. C. Escher's drawings Waterfall, Drawing Hands, Ascending and Descending, and the liar paradox as examples that illustrate the idea of strange loops, which is expressed fully in the proof of Gödel's incompleteness theorem.
  • The "chicken or the egg" paradox is perhaps the best-known strange loop problem.
  • The "ouroboros", which depicts a dragon eating its own tail, is perhaps one of the most ancient and universal symbolic representations of the reflexive loop concept.
  • A Shepard tone is another illustrative example of a strange loop. Named after Roger Shepard, it is a sound consisting of a superposition of tones separated by octaves. When played with the base pitch of the tone moving upwards or downwards, it is referred to as the Shepard scale. This creates the auditory illusion of a tone that continually ascends or descends in pitch, yet which ultimately seems to get no higher or lower. See Barberpole illusion.
  • Another musical example can be found in the soundtrack themes of many early Sonic the Hedgehog games; designed to support the game's motif of speed and constant acceleration, the songs appear to grow continuously in intensity, despite the fact that they are actually on a continuous loop.[citation needed]
  • A quine in software programming is a program that produces a new version of itself without any input from the outside. A similar concept is metamorphic code.
  • Efron's dice are four dice that are intransitive under gambler's preference. In other words, the dice are ordered A > B > C > D > A, where x > y means "a gambler prefers x to y".
  • The liar paradox and Russell's paradox also involve strange loops, as does René Magritte's painting The Treachery of Images.
  • The mathematical phenomenon of polysemy has been observed to be a strange loop. At the denotational level, the term refers to situations where a single entity can be seen to mean more than one mathematical object. See Tanenbaum (1999).
  • The Stonecutter is an old Japanese fairy tale with a story that explains social and natural hierarchies as a strange loop.
  • Midnight in Paris is a film by Woody Allen which can be considered another example.
  • Wikipedia, Strange loop

Droste effect

Self-teaching mathematics

  • Mathematics*
  • Mathematics is very important to becoming an educated person.. When I was in high school I didn't enjoy mathematics at all and I know I wasn't the only one. Mathematics is a lot like a good book. You learn it for the purpose of knowing how to understand the world, not because you'll use it your entire life. Mathematics on it's own is useless, it's application however is useful in almost, if not, every field.
  • Start Here*
  • If you are like me and didn't pay much attention in mathematics during high school you should start at Khan Academy | non-ref. Start where you need to, you know where you're at better than I do. Go up to the Algebra II section and then stop. You want a solid foundation before going up any further, your foundations are very important in mathematics. This should take anywhere from a day to 3 months depending where you are at.
  • Next use Stewart's Book. Do one-two chapters a day and do all odd number problems. Do this every day, on the seventh day review what you have learned. Skip focus on problem solving until you finish the whole thing. If you need help go back to Khan Academy for any assistance you may need.
  • This next one I'm sorry I do not have any free links to, you'll have to pay for it on your own or get it from an alternative source. Read either Velleman's or Rotman's introduction to proofs. Tbh most intro to proof books will do, but these are the best (in my opinion) This should take a week or two.
  • After you are finished with that get Spivak's Book and finish the entire thing.
  • Again I do not have links for these next few books, if I were you I'd simply buy them or obtain from less than legal sources. Get a book on Multivariable and Vector Calculus. This is my recommended, but most books will do. This will take around 3-4 months. Next take Strang's Book on Linear Algebra to make sure you get a good run on Differential Equations somewhere else. Use Apostol for a more rigorous calculus review and to keep up on your proofs. Volume 1 Volume 2. If you want a challenge move onto abstract algebra with Foote's book. If you want to continue Linear Algebra then read and complete Lay's Book
  • Next you want to start Baby Rudin. This may take a few months but if you should be very prepared and it will be a breeze. This may take 2-4 months.If you didn't do abstract algebra with Foote's book then now is the time.
  • Next you want to do Papa Rudin. Sorry once again for not having the free link, but again you always have less than legal sources. This should take half a year or more.
  • Congratz, you now have all the knowledge of a math undergrad. The total price if you bought everything should be around 250$ and it should take around 2 years.
  • What Next?*
  • Once you've finished rudin you want to do the following in this order.
  • Ode (half a year)
  • Pde (half a year)
  • Rings/Fields
  • Number Theory
  • Discrete Math
  • Logic (Half a year for those 4 combined)
  • Topology + Geometry (algebraic, differential; half a year for all)
  • Numerical Analysis
  • Number Theory
  • Statistics (Half a year)
  • If you need any help use physicsforum or math overflow (/r/math might help)
  • I'll go more in depth on what to do next later. My next guide will be on philosophy, probably in the next day or two.
  • Thanks for reading!
  • Reddit community, Mathematics

Chaos theory

Butterfly effect

  • October 7, 1916. The battle of the Somme, France.
  • An unknown artillery bombardier
  • The shell explodes in a dugot
  • Foir soldiers were inside, X were injured, one in the thigh. One of them was Adolf Hitler.
  • This is the probable artillery equipment t tang would have been used. if this dial had been set just one degree to the left, our world would be radically different.
  • Adolf Hitler served in World War I. Usually, he was in administration well behind the front lines. But for a period of time, he was at The battle of the Somme, which was the largest and deadliest battle of World War One. He was in a trench when and artillery shell explode next to him and he wasn't killed, he was only injured when a piece of Shrapnell stuck him in his left thigh. Imagine if the artillery bombardier and just one quarter of a degree to his left. Maybe there would've been no world War two, and then no Cold War, but instead a completely different sequence of major international politics throughout the 20th-century. Or maybe some other charismatic leader might have carried Germany into a Second World War, but perhaps 6 million Jews may have been left alive, or even to serve in the German army. One quarter of a degree change in angle at the right time at the right place may have had all of these changes. Most of the time, your actions aren't going to kill an upcoming major political figure. But sometimes they might. This is the butterfly effect.
  • In complex systems, like the weather, the stock market, or our civilisation, tiny effects can have dramatic consequences.