nhaliday + elegance   153

The Future of Mathematics? [video] | Hacker News
Kevin Buzzard (the Lean guy)

- general reflection on proof asssistants/theorem provers
- Kevin Hale's formal abstracts project, etc
- thinks of available theorem provers, Lean is "[the only one currently available that may be capable of formalizing all of mathematics eventually]" (goes into more detail right at the end, eg, quotient types)
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8 weeks ago by nhaliday
The Law of Leaky Abstractions – Joel on Software
[TCP/IP example]

All non-trivial abstractions, to some degree, are leaky.


- Something as simple as iterating over a large two-dimensional array can have radically different performance if you do it horizontally rather than vertically, depending on the “grain of the wood” — one direction may result in vastly more page faults than the other direction, and page faults are slow. Even assembly programmers are supposed to be allowed to pretend that they have a big flat address space, but virtual memory means it’s really just an abstraction, which leaks when there’s a page fault and certain memory fetches take way more nanoseconds than other memory fetches.

- The SQL language is meant to abstract away the procedural steps that are needed to query a database, instead allowing you to define merely what you want and let the database figure out the procedural steps to query it. But in some cases, certain SQL queries are thousands of times slower than other logically equivalent queries. A famous example of this is that some SQL servers are dramatically faster if you specify “where a=b and b=c and a=c” than if you only specify “where a=b and b=c” even though the result set is the same. You’re not supposed to have to care about the procedure, only the specification. But sometimes the abstraction leaks and causes horrible performance and you have to break out the query plan analyzer and study what it did wrong, and figure out how to make your query run faster.


- C++ string classes are supposed to let you pretend that strings are first-class data. They try to abstract away the fact that strings are hard and let you act as if they were as easy as integers. Almost all C++ string classes overload the + operator so you can write s + “bar” to concatenate. But you know what? No matter how hard they try, there is no C++ string class on Earth that will let you type “foo” + “bar”, because string literals in C++ are always char*’s, never strings. The abstraction has sprung a leak that the language doesn’t let you plug. (Amusingly, the history of the evolution of C++ over time can be described as a history of trying to plug the leaks in the string abstraction. Why they couldn’t just add a native string class to the language itself eludes me at the moment.)

- And you can’t drive as fast when it’s raining, even though your car has windshield wipers and headlights and a roof and a heater, all of which protect you from caring about the fact that it’s raining (they abstract away the weather), but lo, you have to worry about hydroplaning (or aquaplaning in England) and sometimes the rain is so strong you can’t see very far ahead so you go slower in the rain, because the weather can never be completely abstracted away, because of the law of leaky abstractions.

One reason the law of leaky abstractions is problematic is that it means that abstractions do not really simplify our lives as much as they were meant to. When I’m training someone to be a C++ programmer, it would be nice if I never had to teach them about char*’s and pointer arithmetic. It would be nice if I could go straight to STL strings. But one day they’ll write the code “foo” + “bar”, and truly bizarre things will happen, and then I’ll have to stop and teach them all about char*’s anyway.


The law of leaky abstractions means that whenever somebody comes up with a wizzy new code-generation tool that is supposed to make us all ever-so-efficient, you hear a lot of people saying “learn how to do it manually first, then use the wizzy tool to save time.” Code generation tools which pretend to abstract out something, like all abstractions, leak, and the only way to deal with the leaks competently is to learn about how the abstractions work and what they are abstracting. So the abstractions save us time working, but they don’t save us time learning.

People think a lot about abstractions and how to design them well. Here’s one feature I’ve recently been noticing about well-designed abstractions: they should have simple, flexible and well-integrated escape hatches.
techtariat  org:com  working-stiff  essay  programming  cs  software  abstraction  worrydream  thinking  intricacy  degrees-of-freedom  networking  examples  traces  no-go  volo-avolo  tradeoffs  c(pp)  pls  strings  dbs  transportation  driving  analogy  aphorism  learning  paradox  systems  elegance  nitty-gritty  concrete  cracker-prog  metal-to-virtual  protocol-metadata  design  system-design  multi  ratty  core-rats  integration-extension  composition-decomposition  flexibility  parsimony  interface-compatibility 
july 2019 by nhaliday
C++ Core Guidelines
This document is a set of guidelines for using C++ well. The aim of this document is to help people to use modern C++ effectively. By “modern C++” we mean effective use of the ISO C++ standard (currently C++17, but almost all of our recommendations also apply to C++14 and C++11). In other words, what would you like your code to look like in 5 years’ time, given that you can start now? In 10 years’ time?

“Within C++ is a smaller, simpler, safer language struggling to get out.” – Bjarne Stroustrup


The guidelines are focused on relatively higher-level issues, such as interfaces, resource management, memory management, and concurrency. Such rules affect application architecture and library design. Following the rules will lead to code that is statically type safe, has no resource leaks, and catches many more programming logic errors than is common in code today. And it will run fast - you can afford to do things right.

We are less concerned with low-level issues, such as naming conventions and indentation style. However, no topic that can help a programmer is out of bounds.

Our initial set of rules emphasize safety (of various forms) and simplicity. They may very well be too strict. We expect to have to introduce more exceptions to better accommodate real-world needs. We also need more rules.


The rules are designed to be supported by an analysis tool. Violations of rules will be flagged with references (or links) to the relevant rule. We do not expect you to memorize all the rules before trying to write code.

This will be a long wall of text, and kinda random! My main points are:
1. C++ compile times are important,
2. Non-optimized build performance is important,
3. Cognitive load is important. I don’t expand much on this here, but if a programming language or a library makes me feel stupid, then I’m less likely to use it or like it. C++ does that a lot :)
programming  engineering  pls  best-practices  systems  c(pp)  guide  metabuch  objektbuch  reference  cheatsheet  elegance  frontier  libraries  intricacy  advanced  advice  recommendations  big-picture  novelty  lens  philosophy  state  error  types  concurrency  memory-management  performance  abstraction  plt  compilers  expert-experience  multi  checking  devtools  flux-stasis  safety  system-design  techtariat  time  measure  dotnet  comparison  examples  build-packaging  thinking  worse-is-better/the-right-thing  cost-benefit  tradeoffs  essay  commentary  oop  correctness  computer-memory  error-handling  resources-effects  latency-throughput 
june 2019 by nhaliday
What's the expected level of paper for top conferences in Computer Science - Academia Stack Exchange
Top. The top level.

My experience on program committees for STOC, FOCS, ITCS, SODA, SOCG, etc., is that there are FAR more submissions of publishable quality than can be accepted into the conference. By "publishable quality" I mean a well-written presentation of a novel, interesting, and non-trivial result within the scope of the conference.


There are several questions that come up over and over in the FOCS/STOC review cycle:

- How surprising / novel / elegant / interesting is the result?
- How surprising / novel / elegant / interesting / general are the techniques?
- How technically difficult is the result? Ironically, FOCS and STOC committees have a reputation for ignoring the distinction between trivial (easy to derive from scratch) and nondeterministically trivial (easy to understand after the fact).
- What is the expected impact of this result? Is this paper going to change the way people do theoretical computer science over the next five years?
- Is the result of general interest to the theoretical computer science community? Or is it only of interest to a narrow subcommunity? In particular, if the topic is outside the STOC/FOCS mainstream—say, for example, computational topology—does the paper do a good job of explaining and motivating the results to a typical STOC/FOCS audience?
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june 2019 by nhaliday
Theories of humor - Wikipedia
There are many theories of humor which attempt to explain what humor is, what social functions it serves, and what would be considered humorous. Among the prevailing types of theories that attempt to account for the existence of humor, there are psychological theories, the vast majority of which consider humor to be very healthy behavior; there are spiritual theories, which consider humor to be an inexplicable mystery, very much like a mystical experience.[1] Although various classical theories of humor and laughter may be found, in contemporary academic literature, three theories of humor appear repeatedly: relief theory, superiority theory, and incongruity theory.[2] Among current humor researchers, there is no consensus about which of these three theories of humor is most viable.[2] Proponents of each one originally claimed their theory to be capable of explaining all cases of humor.[2][3] However, they now acknowledge that although each theory generally covers its own area of focus, many instances of humor can be explained by more than one theory.[2][3][4][5] Incongruity and superiority theories, for instance, seem to describe complementary mechanisms which together create humor.[6]


Relief theory
Relief theory maintains that laughter is a homeostatic mechanism by which psychological tension is reduced.[2][3][7] Humor may thus for example serve to facilitate relief of the tension caused by one's fears.[8] Laughter and mirth, according to relief theory, result from this release of nervous energy.[2] Humor, according to relief theory, is used mainly to overcome sociocultural inhibitions and reveal suppressed desires. It is believed that this is the reason we laugh whilst being tickled, due to a buildup of tension as the tickler "strikes".[2][9] According to Herbert Spencer, laughter is an "economical phenomenon" whose function is to release "psychic energy" that had been wrongly mobilized by incorrect or false expectations. The latter point of view was supported also by Sigmund Freud.

Superiority theory
The superiority theory of humor traces back to Plato and Aristotle, and Thomas Hobbes' Leviathan. The general idea is that a person laughs about misfortunes of others (so called schadenfreude), because these misfortunes assert the person's superiority on the background of shortcomings of others.[10] Socrates was reported by Plato as saying that the ridiculous was characterized by a display of self-ignorance.[11] For Aristotle, we laugh at inferior or ugly individuals, because we feel a joy at feeling superior to them.[12]

Incongruous juxtaposition theory
The incongruity theory states that humor is perceived at the moment of realization of incongruity between a concept involved in a certain situation and the real objects thought to be in some relation to the concept.[10]

Since the main point of the theory is not the incongruity per se, but its realization and resolution (i.e., putting the objects in question into the real relation), it is often called the incongruity-resolution theory.[10]


Detection of mistaken reasoning
In 2011, three researchers, Hurley, Dennett and Adams, published a book that reviews previous theories of humor and many specific jokes. They propose the theory that humor evolved because it strengthens the ability of the brain to find mistakes in active belief structures, that is, to detect mistaken reasoning.[46] This is somewhat consistent with the sexual selection theory, because, as stated above, humor would be a reliable indicator of an important survival trait: the ability to detect mistaken reasoning. However, the three researchers argue that humor is fundamentally important because it is the very mechanism that allows the human brain to excel at practical problem solving. Thus, according to them, humor did have survival value even for early humans, because it enhanced the neural circuitry needed to survive.

Misattribution theory
Misattribution is one theory of humor that describes an audience's inability to identify exactly why they find a joke to be funny. The formal theory is attributed to Zillmann & Bryant (1980) in their article, "Misattribution Theory of Tendentious Humor", published in Journal of Experimental Social Psychology. They derived the critical concepts of the theory from Sigmund Freud's Wit and Its Relation to the Unconscious (note: from a Freudian perspective, wit is separate from humor), originally published in 1905.

Benign violation theory
The benign violation theory (BVT) is developed by researchers A. Peter McGraw and Caleb Warren.[47] The BVT integrates seemingly disparate theories of humor to predict that humor occurs when three conditions are satisfied: 1) something threatens one's sense of how the world "ought to be", 2) the threatening situation seems benign, and 3) a person sees both interpretations at the same time.

From an evolutionary perspective, humorous violations likely originated as apparent physical threats, like those present in play fighting and tickling. As humans evolved, the situations that elicit humor likely expanded from physical threats to other violations, including violations of personal dignity (e.g., slapstick, teasing), linguistic norms (e.g., puns, malapropisms), social norms (e.g., strange behaviors, risqué jokes), and even moral norms (e.g., disrespectful behaviors). The BVT suggests that anything that threatens one's sense of how the world "ought to be" will be humorous, so long as the threatening situation also seems benign.


Sense of humor, sense of seriousness
One must have a sense of humor and a sense of seriousness to distinguish what is supposed to be taken literally or not. An even more keen sense is needed when humor is used to make a serious point.[48][49] Psychologists have studied how humor is intended to be taken as having seriousness, as when court jesters used humor to convey serious information. Conversely, when humor is not intended to be taken seriously, bad taste in humor may cross a line after which it is taken seriously, though not intended.[50]

Philosophy of humor bleg: http://marginalrevolution.com/marginalrevolution/2017/03/philosophy-humor-bleg.html

Inside Jokes: https://mitpress.mit.edu/books/inside-jokes
humor as reward for discovering inconsistency in inferential chain



People of all ages and cultures respond to humour. Most people are able to experience humour—be amused, smile or laugh at something funny—and thus are considered to have a sense of humour. The hypothetical person lacking a sense of humour would likely find the behaviour inducing it to be inexplicable, strange, or even irrational.


Ancient Greece
Western humour theory begins with Plato, who attributed to Socrates (as a semi-historical dialogue character) in the Philebus (p. 49b) the view that the essence of the ridiculous is an ignorance in the weak, who are thus unable to retaliate when ridiculed. Later, in Greek philosophy, Aristotle, in the Poetics (1449a, pp. 34–35), suggested that an ugliness that does not disgust is fundamental to humour.


Confucianist Neo-Confucian orthodoxy, with its emphasis on ritual and propriety, has traditionally looked down upon humour as subversive or unseemly. The Confucian "Analects" itself, however, depicts the Master as fond of humorous self-deprecation, once comparing his wanderings to the existence of a homeless dog.[10] Early Daoist philosophical texts such as "Zhuangzi" pointedly make fun of Confucian seriousness and make Confucius himself a slow-witted figure of fun.[11] Joke books containing a mix of wordplay, puns, situational humor, and play with taboo subjects like sex and scatology, remained popular over the centuries. Local performing arts, storytelling, vernacular fiction, and poetry offer a wide variety of humorous styles and sensibilities.


Physical attractiveness
90% of men and 81% of women, all college students, report having a sense of humour is a crucial characteristic looked for in a romantic partner.[21] Humour and honesty were ranked as the two most important attributes in a significant other.[22] It has since been recorded that humour becomes more evident and significantly more important as the level of commitment in a romantic relationship increases.[23] Recent research suggests expressions of humour in relation to physical attractiveness are two major factors in the desire for future interaction.[19] Women regard physical attractiveness less highly compared to men when it came to dating, a serious relationship, and sexual intercourse.[19] However, women rate humorous men more desirable than nonhumorous individuals for a serious relationship or marriage, but only when these men were physically attractive.[19]

Furthermore, humorous people are perceived by others to be more cheerful but less intellectual than nonhumorous people. Self-deprecating humour has been found to increase the desirability of physically attractive others for committed relationships.[19] The results of a study conducted by McMaster University suggest humour can positively affect one’s desirability for a specific relationship partner, but this effect is only most likely to occur when men use humour and are evaluated by women.[24] No evidence was found to suggest men prefer women with a sense of humour as partners, nor women preferring other women with a sense of humour as potential partners.[24] When women were given the forced-choice design in the study, they chose funny men as potential … [more]
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april 2018 by nhaliday
Transcendentals - Wikipedia
The transcendentals (Latin: transcendentalia) are the properties of being that correspond to three aspects of the human field of interest and are their ideals; science (truth), the arts (beauty) and religion (goodness).[citation needed] Philosophical disciplines that study them are logic, aesthetics and ethics.

See also: Proto-Indo-European religion, Asha, and Satya

Parmenides first inquired of the properties co-extensive with being.[1] Socrates, spoken through Plato, then followed (see Form of the Good).

Aristotle's substance theory (being a substance belongs to being qua being) has been interpreted as a theory of transcendentals.[2] Aristotle discusses only unity ("One") explicitly because it is the only transcendental intrinsically related to being, whereas truth and goodness relate to rational creatures.[3]

In the Middle Ages, Catholic philosophers elaborated the thought that there exist transcendentals (transcendentalia) and that they transcended each of the ten Aristotelian categories.[4] A doctrine of the transcendentality of the good was formulated by Albert the Great.[5] His pupil, Saint Thomas Aquinas, posited five transcendentals: res, unum, aliquid, bonum, verum; or "thing", "one", "something", "good", and "true".[6] Saint Thomas derives the five explicitly as transcendentals,[7] though in some cases he follows the typical list of the transcendentals consisting of the One, the Good, and the True. The transcendentals are ontologically one and thus they are convertible: e.g., where there is truth, there is beauty and goodness also.

In Christian theology the transcendentals are treated in relation to theology proper, the doctrine of God. The transcendentals, according to Christian doctrine, can be described as the ultimate desires of man. Man ultimately strives for perfection, which takes form through the desire for perfect attainment of the transcendentals. The Catholic Church teaches that God is Himself truth, goodness, and beauty, as indicated in the Catechism of the Catholic Church.[8] Each transcends the limitations of place and time, and is rooted in being. The transcendentals are not contingent upon cultural diversity, religious doctrine, or personal ideologies, but are the objective properties of all that exists.
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march 2018 by nhaliday
All models are wrong - Wikipedia
Box repeated the aphorism in a paper that was published in the proceedings of a 1978 statistics workshop.[2] The paper contains a section entitled "All models are wrong but some are useful". The section is copied below.

Now it would be very remarkable if any system existing in the real world could be exactly represented by any simple model. However, cunningly chosen parsimonious models often do provide remarkably useful approximations. For example, the law PV = RT relating pressure P, volume V and temperature T of an "ideal" gas via a constant R is not exactly true for any real gas, but it frequently provides a useful approximation and furthermore its structure is informative since it springs from a physical view of the behavior of gas molecules.

For such a model there is no need to ask the question "Is the model true?". If "truth" is to be the "whole truth" the answer must be "No". The only question of interest is "Is the model illuminating and useful?".
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august 2017 by nhaliday
Introduction to Scaling Laws

Galileo’s Discovery of Scaling Laws: https://www.mtholyoke.edu/~mpeterso/classes/galileo/scaling8.pdf
Days 1 and 2 of Two New Sciences

An example of such an insight is “the surface of a small solid is comparatively greater than that of a large one” because the surface goes like the square of a linear dimension, but the volume goes like the cube.5 Thus as one scales down macroscopic objects, forces on their surfaces like viscous drag become relatively more important, and bulk forces like weight become relatively less important. Galileo uses this idea on the First Day in the context of resistance in free fall, as an explanation for why similar objects of different size do not fall exactly together, but the smaller one lags behind.
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august 2017 by nhaliday
Shtetl-Optimized » Blog Archive » Logicians on safari
So what are they then? Maybe it’s helpful to think of them as “quantitative epistemology”: discoveries about the capacities of finite beings like ourselves to learn mathematical truths. On this view, the theoretical computer scientist is basically a mathematical logician on a safari to the physical world: someone who tries to understand the universe by asking what sorts of mathematical questions can and can’t be answered within it. Not whether the universe is a computer, but what kind of computer it is! Naturally, this approach to understanding the world tends to appeal most to people for whom math (and especially discrete math) is reasonably clear, whereas physics is extremely mysterious.

the sequel: http://www.scottaaronson.com/blog/?p=153
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january 2017 by nhaliday
probability - How to prove Bonferroni inequalities? - Mathematics Stack Exchange
- integrated version of inequalities for alternating sums of (N choose j), where r.v. N = # of events occuring
- inequalities for alternating binomial coefficients follow from general property of unimodal (increasing then decreasing) sequences, which can be gotten w/ two cases for increasing and decreasing resp.
- the final alternating zero sum property follows for binomial coefficients from expanding (1 - 1)^N = 0
- The idea of proving inequality by integrating simpler inequality of r.v.s is nice. Proof from CS 150 was more brute force from what I remember.
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january 2017 by nhaliday
ho.history overview - Proofs that require fundamentally new ways of thinking - MathOverflow
my favorite:
Although this has already been said elsewhere on MathOverflow, I think it's worth repeating that Gromov is someone who has arguably introduced more radical thoughts into mathematics than anyone else. Examples involving groups with polynomial growth and holomorphic curves have already been cited in other answers to this question. I have two other obvious ones but there are many more.

I don't remember where I first learned about convergence of Riemannian manifolds, but I had to laugh because there's no way I would have ever conceived of a notion. To be fair, all of the groundwork for this was laid out in Cheeger's thesis, but it was Gromov who reformulated everything as a convergence theorem and recognized its power.

Another time Gromov made me laugh was when I was reading what little I could understand of his book Partial Differential Relations. This book is probably full of radical ideas that I don't understand. The one I did was his approach to solving the linearized isometric embedding equation. His radical, absurd, but elementary idea was that if the system is sufficiently underdetermined, then the linear partial differential operator could be inverted by another linear partial differential operator. Both the statement and proof are for me the funniest in mathematics. Most of us view solving PDE's as something that requires hard work, involving analysis and estimates, and Gromov manages to do it using only elementary linear algebra. This then allows him to establish the existence of isometric embedding of Riemannian manifolds in a wide variety of settings.
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january 2017 by nhaliday
"Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character ... - Richard P. Feynman - Google Books
Actually, there was a certain amount of genuine quality to my guesses. I had a scheme, which I still use today when somebody is explaining something that l’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball)—disjoint (two balls). Then the balls tum colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!"
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january 2017 by nhaliday
soft question - Thinking and Explaining - MathOverflow
- good question from Bill Thurston
- great answers by Terry Tao, fedja, Minhyong Kim, gowers, etc.

Terry Tao:
- symmetry as blurring/vibrating/wobbling, scale invariance
- anthropomorphization, adversarial perspective for estimates/inequalities/quantifiers, spending/economy

fedja walks through his though-process from another answer

Minhyong Kim: anthropology of mathematical philosophizing

Per Vognsen: normality as isotropy
comment: conjugate subgroup gHg^-1 ~ "H but somewhere else in G"

gowers: hidden things in basic mathematics/arithmetic
comment by Ryan Budney: x sin(x) via x -> (x, sin(x)), (x, y) -> xy
I kinda get what he's talking about but needed to use Mathematica to get the initial visualization down.
To remind myself later:
- xy can be easily visualized by juxtaposing the two parabolae x^2 and -x^2 diagonally
- x sin(x) can be visualized along that surface by moving your finger along the line (x, 0) but adding some oscillations in y direction according to sin(x)
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january 2017 by nhaliday
soft question - Why does Fourier analysis of Boolean functions "work"? - Theoretical Computer Science Stack Exchange
Here is my point of view, which I learned from Guy Kindler, though someone more experienced can probably give a better answer: Consider the linear space of functions f: {0,1}^n -> R and consider a linear operator of the form σ_w (for w in {0,1}^n), that maps a function f(x) as above to the function f(x+w). In many of the questions of TCS, there is an underlying need to analyze the effects that such operators have on certain functions.

Now, the point is that the Fourier basis is the basis that diagonalizes all those operators at the same time, which makes the analysis of those operators much simpler. More generally, the Fourier basis diagonalizes the convolution operator, which also underlies many of those questions. Thus, Fourier analysis is likely to be effective whenever one needs to analyze those operators.
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december 2016 by nhaliday
gt.geometric topology - Intuitive crutches for higher dimensional thinking - MathOverflow
Terry Tao:
I can't help you much with high-dimensional topology - it's not my field, and I've not picked up the various tricks topologists use to get a grip on the subject - but when dealing with the geometry of high-dimensional (or infinite-dimensional) vector spaces such as R^n, there are plenty of ways to conceptualise these spaces that do not require visualising more than three dimensions directly.

For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, is a point in a million-dimensional vector space; by varying the image, one can explore the space, and various subsets of this space correspond to various classes of images.

One can similarly interpret sound waves, a box of gases, an ecosystem, a voting population, a stream of digital data, trials of random variables, the results of a statistical survey, a probabilistic strategy in a two-player game, and many other concrete objects as states in a high-dimensional vector space, and various basic concepts such as convexity, distance, linearity, change of variables, orthogonality, or inner product can have very natural meanings in some of these models (though not in all).

It can take a bit of both theory and practice to merge one's intuition for these things with one's spatial intuition for vectors and vector spaces, but it can be done eventually (much as after one has enough exposure to measure theory, one can start merging one's intuition regarding cardinality, mass, length, volume, probability, cost, charge, and any number of other "real-life" measures).

For instance, the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers, using the interpretation of a high-dimensional vector space as the state space for a large number of trials of a random variable.

More generally, many facts about low-dimensional projections or slices of high-dimensional objects can be viewed from a probabilistic, statistical, or signal processing perspective.

Scott Aaronson:
Here are some of the crutches I've relied on. (Admittedly, my crutches are probably much more useful for theoretical computer science, combinatorics, and probability than they are for geometry, topology, or physics. On a related note, I personally have a much easier time thinking about R^n than about, say, R^4 or R^5!)

1. If you're trying to visualize some 4D phenomenon P, first think of a related 3D phenomenon P', and then imagine yourself as a 2D being who's trying to visualize P'. The advantage is that, unlike with the 4D vs. 3D case, you yourself can easily switch between the 3D and 2D perspectives, and can therefore get a sense of exactly what information is being lost when you drop a dimension. (You could call this the "Flatland trick," after the most famous literary work to rely on it.)
2. As someone else mentioned, discretize! Instead of thinking about R^n, think about the Boolean hypercube {0,1}^n, which is finite and usually easier to get intuition about. (When working on problems, I often find myself drawing {0,1}^4 on a sheet of paper by drawing two copies of {0,1}^3 and then connecting the corresponding vertices.)
3. Instead of thinking about a subset S⊆R^n, think about its characteristic function f:R^n→{0,1}. I don't know why that trivial perspective switch makes such a big difference, but it does ... maybe because it shifts your attention to the process of computing f, and makes you forget about the hopeless task of visualizing S!
4. One of the central facts about R^n is that, while it has "room" for only n orthogonal vectors, it has room for exp⁡(n) almost-orthogonal vectors. Internalize that one fact, and so many other properties of R^n (for example, that the n-sphere resembles a "ball with spikes sticking out," as someone mentioned before) will suddenly seem non-mysterious. In turn, one way to internalize the fact that R^n has so many almost-orthogonal vectors is to internalize Shannon's theorem that there exist good error-correcting codes.
5. To get a feel for some high-dimensional object, ask questions about the behavior of a process that takes place on that object. For example: if I drop a ball here, which local minimum will it settle into? How long does this random walk on {0,1}^n take to mix?

Gil Kalai:
This is a slightly different point, but Vitali Milman, who works in high-dimensional convexity, likes to draw high-dimensional convex bodies in a non-convex way. This is to convey the point that if you take the convex hull of a few points on the unit sphere of R^n, then for large n very little of the measure of the convex body is anywhere near the corners, so in a certain sense the body is a bit like a small sphere with long thin "spikes".
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december 2016 by nhaliday
Not Final! | West Hunter
In mathematics we often prove that some proposition is true by showing that  the alternative is false.  The principle can sometimes work in other disciplines, but it’s tricky.  You have to have a very good understanding  to know that some things are impossible (or close enough to impossible).   You can do it fairly often in physics, less often in biology.
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november 2016 by nhaliday
Thick and thin | West Hunter
There is a spectrum of problem-solving, ranging from, at one extreme, simplicity and clear chains of logical reasoning (sometimes long chains) and, at the other, building a picture by sifting through a vast mass of evidence of varying quality. I will give some examples. Just the other day, when I was conferring, conversing and otherwise hobnobbing with my fellow physicists, I mentioned high-altitude lighting, sprites and elves and blue jets. I said that you could think of a thundercloud as a vertical dipole, with an electric field that decreased as the cube of altitude, while the breakdown voltage varied with air pressure, which declines exponentially with altitude. At which point the prof I was talking to said ” and so the curves must cross!”. That’s how physicists think, and it can be very effective. The amount of information required to solve the problem is not very large. I call this a ‘thin’ problem’.


In another example at the messy end of the spectrum, Joe Rochefort, running Hypo in the spring of 1942, needed to figure out Japanese plans. He had an an ever-growing mass of Japanese radio intercepts, some of which were partially decrypted – say, one word of five, with luck. He had data from radio direction-finding; his people were beginning to be able to recognize particular Japanese radio operators by their ‘fist’. He’d studied in Japan, knew the Japanese well. He had plenty of Navy experience – knew what was possible. I would call this a classic ‘thick’ problem, one in which an analyst needs to deal with an enormous amount of data of varying quality. Being smart is necessary but not sufficient: you also need to know lots of stuff.


Nimitz believed Rochefort – who was correct. Because of that, we managed to prevail at Midway, losing one carrier and one destroyer while the the Japanese lost four carriers and a heavy cruiser*. As so often happens, OP-20-G won the bureaucratic war: Rochefort embarrassed them by proving them wrong, and they kicked him out of Hawaii, assigning him to a floating drydock.

The usual explanation of Joe Rochefort’s fall argues that John Redman’s ( head of OP-20-G, the Navy’s main signals intelligence and cryptanalysis group) geographical proximity to Navy headquarters was a key factor in winning the bureaucratic struggle, along with his brother’s influence (Rear Admiral Joseph Redman). That and being a shameless liar.

Personally, I wonder if part of the problem is the great difficulty of explaining the analysis of a thick problem to someone without a similar depth of knowledge. At best, they believe you because you’ve been right in the past. Or, sometimes, once you have developed the answer, there is a ‘thin’ way of confirming your answer – as when Rochefort took Jasper Holmes’s suggestion and had Midway broadcast an uncoded complaint about the failure of their distillation system – soon followed by a Japanese report that ‘AF’ was short of water.

Most problems in the social sciences are ‘thick’, and unfortunately, almost all of the researchers are as well. There are a lot more Redmans than Rocheforts.
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november 2016 by nhaliday
Son of low-hanging fruit | West Hunter
You see, you can think of the thunderstorm, after a ground discharge, as a vertical dipole. Its electrical field drops as the cube of altitude. The threshold voltage for atmospheric breakdown is proportional to pressure, while pressure drops exponentially with altitude: and as everyone knows, a negative exponential drops faster than any power.

The curves must cross. Electrical breakdown occurs. Weird lightning, way above the clouds.

As I said, people reported sprites at least a hundred years ago, and they have probably been observed occasionally since the dawn of time. However, they’re far easier to see if you’re above the clouds – pilots often do.

Pilots also learned not to talk about it, because nobody listened. Military and commercial pilots have to pass periodic medical exams known as ‘flight physicals’, and there was a suspicion that reporting glowing red cephalopods in the sky might interfere with that. Generally, you had to see the things that were officially real (whether they were really real or not), and only those things.

Sprites became real when someone recorded one by accident on a fast camera in 1989. Since then it’s turned into a real subject, full of strangeness: turns out that thunderstorms sometimes generate gamma-rays and even antimatter.
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november 2016 by nhaliday
Paul Krugman Is an "Evolution Groupie" - Evonomics
Let me give you an example. William Hamilton’s wonderfully named paper “Geometry for the Selfish Herd” imagines a group of frogs sitting at the edge of a circular pond, from which a snake may emerge – and he supposes that the snake will grab and eat the nearest frog. Where will the frogs sit? To compress his argument, Hamilton points out that if there are two groups of frogs around the pool, each group has an equal chance of being targeted, and so does each frog within each group – which means that the chance of being eaten is less if you are a frog in the larger group. Thus if you are a frog trying to maximize your choice of survival, you will want to be part of the larger group; and the equilibrium must involve clumping of all the frogs as close together as possible.

Notice what is missing from this analysis. Hamilton does not talk about the evolutionary dynamics by which frogs might acquire a sit-with-the-other-frogs instinct; he does not take us through the intermediate steps along the evolutionary path in which frogs had not yet completely “realized” that they should stay with the herd. Why not? Because to do so would involve him in enormous complications that are basically irrelevant to his point, whereas – ahem – leapfrogging straight over these difficulties to look at the equilibrium in which all frogs maximize their chances given what the other frogs do is a very parsimonious, sharp-edged way of gaining insight.
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november 2016 by nhaliday
Beauty is Fit | Carcinisation
Cage’s music is an example of the tendency for high-status human domains to ignore fit with human nervous systems in favor of fit with increasingly rarified abstract cultural systems. Human nervous systems are limited. Representation of existing forms, and generating pleasure and poignancy in human minds, are often disdained as solved problems. Domains unhinged from the desires and particularities of human nervous systems and bodies become inhuman; human flourishing, certainly, is not a solved problem. However, human nervous systems themselves create and seek out “fit” of the more abstract sort; the domain of abstract systems is part of the natural human environment, and the forms that exist there interact with humans as symbiotes. Theorems and novels and money and cathedrals rely on humans for reproduction, like parasites, but offer many benefits to humans in exchange. Humans require an environment that fits their nervous systems, but part of the definition of “fit” in this case is the need for humans to feel that they are involved in something greater (and perhaps more abstract) than this “animal” kind of fit.
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november 2016 by nhaliday
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