**finiteness**14

Attributes of God in Christianity - Wikipedia

article list trivia wiki reference religion christianity theos ideology properties paradox heterodox time morality good-evil love-hate emotion philosophy universalism-particularism number whole-partial-many truth power intervention space finiteness envy embodied values descriptive things knowledge justice virtu leviathan exegesis-hermeneutics lexical

june 2018 by nhaliday

article list trivia wiki reference religion christianity theos ideology properties paradox heterodox time morality good-evil love-hate emotion philosophy universalism-particularism number whole-partial-many truth power intervention space finiteness envy embodied values descriptive things knowledge justice virtu leviathan exegesis-hermeneutics lexical

june 2018 by nhaliday

An Untrollable Mathematician Illustrated

ratty lesswrong comics infographic ai-control ai thinking skeleton miri-cfar big-picture synthesis hi-order-bits interdisciplinary lens logic iteration-recursion probability decision-theory decision-making values flux-stasis formal-values bayesian axioms cs computation math truth uncertainty finiteness nibble cartoons visual-understanding machine-learning troll internet volo-avolo hypothesis-testing telos-atelos inference apollonian-dionysian

april 2018 by nhaliday

ratty lesswrong comics infographic ai-control ai thinking skeleton miri-cfar big-picture synthesis hi-order-bits interdisciplinary lens logic iteration-recursion probability decision-theory decision-making values flux-stasis formal-values bayesian axioms cs computation math truth uncertainty finiteness nibble cartoons visual-understanding machine-learning troll internet volo-avolo hypothesis-testing telos-atelos inference apollonian-dionysian

april 2018 by nhaliday

Best Topology Olympiad ***EVER*** - Affine Mess - Quora

october 2017 by nhaliday

Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!

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Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!

october 2017 by nhaliday

Archimedes Palimpsest - Wikipedia

may 2017 by nhaliday

Using this method, Archimedes was able to solve several problems now treated by integral calculus, which was given its modern form in the seventeenth century by Isaac Newton and Gottfried Leibniz. Among those problems were that of calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines. (For explicit details, see Archimedes' use of infinitesimals.)

When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In "On the Sphere and Cylinder," he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the area of the sphere considered as a surface of revolution.

But there are two essential differences between Archimedes' method and 19th-century methods:

1. Archimedes did not know about differentiation, so he could not calculate any integrals other than those that came from center-of-mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time; he never figured out how to change variables or integrate by parts.

2. When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the Greeks lacked algebraic methods that could establish that error terms in an approximation are small.

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When rigorously proving theorems, Archimedes often used what are now called Riemann sums. In "On the Sphere and Cylinder," he gives upper and lower bounds for the surface area of a sphere by cutting the sphere into sections of equal width. He then bounds the area of each section by the area of an inscribed and circumscribed cone, which he proves have a larger and smaller area correspondingly. He adds the areas of the cones, which is a type of Riemann sum for the area of the sphere considered as a surface of revolution.

But there are two essential differences between Archimedes' method and 19th-century methods:

1. Archimedes did not know about differentiation, so he could not calculate any integrals other than those that came from center-of-mass considerations, by symmetry. While he had a notion of linearity, to find the volume of a sphere he had to balance two figures at the same time; he never figured out how to change variables or integrate by parts.

2. When calculating approximating sums, he imposed the further constraint that the sums provide rigorous upper and lower bounds. This was required because the Greeks lacked algebraic methods that could establish that error terms in an approximation are small.

may 2017 by nhaliday

Sacred Principles As Exhaustible Resources | Slate Star Codex

ratty yvain ssc commentary current-events politics culture-war harvard civil-liberty toxoplasmosis info-dynamics sanctity-degradation absolute-relative exit-voice murray higher-ed polarization tribalism volo-avolo finiteness flexibility unintended-consequences social-norms

april 2017 by nhaliday

ratty yvain ssc commentary current-events politics culture-war harvard civil-liberty toxoplasmosis info-dynamics sanctity-degradation absolute-relative exit-voice murray higher-ed polarization tribalism volo-avolo finiteness flexibility unintended-consequences social-norms

april 2017 by nhaliday

What is the relationship between information theory and Coding theory? - Quora

february 2017 by nhaliday

basically:

- finite vs. asymptotic

- combinatorial vs. probabilistic (lotsa overlap their)

- worst-case (Hamming) vs. distributional (Shannon)

Information and coding theory most often appear together in the subject of error correction over noisy channels. Historically, they were born at almost exactly the same time - both Richard Hamming and Claude Shannon were working at Bell Labs when this happened. Information theory tends to heavily use tools from probability theory (together with an "asymptotic" way of thinking about the world), while traditional "algebraic" coding theory tends to employ mathematics that are much more finite sequence length/combinatorial in nature, including linear algebra over Galois Fields. The emergence in the late 90s and first decade of 2000 of codes over graphs blurred this distinction though, as code classes such as low density parity check codes employ both asymptotic analysis and random code selection techniques which have counterparts in information theory.

They do not subsume each other. Information theory touches on many other aspects that coding theory does not, and vice-versa. Information theory also touches on compression (lossy & lossless), statistics (e.g. large deviations), modeling (e.g. Minimum Description Length). Coding theory pays a lot of attention to sphere packing and coverings for finite length sequences - information theory addresses these problems (channel & lossy source coding) only in an asymptotic/approximate sense.

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- finite vs. asymptotic

- combinatorial vs. probabilistic (lotsa overlap their)

- worst-case (Hamming) vs. distributional (Shannon)

Information and coding theory most often appear together in the subject of error correction over noisy channels. Historically, they were born at almost exactly the same time - both Richard Hamming and Claude Shannon were working at Bell Labs when this happened. Information theory tends to heavily use tools from probability theory (together with an "asymptotic" way of thinking about the world), while traditional "algebraic" coding theory tends to employ mathematics that are much more finite sequence length/combinatorial in nature, including linear algebra over Galois Fields. The emergence in the late 90s and first decade of 2000 of codes over graphs blurred this distinction though, as code classes such as low density parity check codes employ both asymptotic analysis and random code selection techniques which have counterparts in information theory.

They do not subsume each other. Information theory touches on many other aspects that coding theory does not, and vice-versa. Information theory also touches on compression (lossy & lossless), statistics (e.g. large deviations), modeling (e.g. Minimum Description Length). Coding theory pays a lot of attention to sphere packing and coverings for finite length sequences - information theory addresses these problems (channel & lossy source coding) only in an asymptotic/approximate sense.

february 2017 by nhaliday

general topology - What should be the intuition when working with compactness? - Mathematics Stack Exchange

january 2017 by nhaliday

http://math.stackexchange.com/questions/485822/why-is-compactness-so-important

The situation with compactness is sort of like the above. It turns out that finiteness, which you think of as one concept (in the same way that you think of "Foo" as one concept above), is really two concepts: discreteness and compactness. You've never seen these concepts separated before, though. When people say that compactness is like finiteness, they mean that compactness captures part of what it means to be finite in the same way that shortness captures part of what it means to be Foo.

--

As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about how something behaves on an open set", and for compact spaces this means that there are only finitely many possible behaviors.

--

Compactness does for continuous functions what finiteness does for functions in general.

If a set A is finite then every function f:A→R has a max and a min, and every function f:A→R^n is bounded. If A is compact, the every continuous function from A to R has a max and a min and every continuous function from A to R^n is bounded.

If A is finite then every sequence of members of A has a subsequence that is eventually constant, and "eventually constant" is the only kind of convergence you can talk about without talking about a topology on the set. If A is compact, then every sequence of members of A has a convergent subsequence.

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The situation with compactness is sort of like the above. It turns out that finiteness, which you think of as one concept (in the same way that you think of "Foo" as one concept above), is really two concepts: discreteness and compactness. You've never seen these concepts separated before, though. When people say that compactness is like finiteness, they mean that compactness captures part of what it means to be finite in the same way that shortness captures part of what it means to be Foo.

--

As many have said, compactness is sort of a topological generalization of finiteness. And this is true in a deep sense, because topology deals with open sets, and this means that we often "care about how something behaves on an open set", and for compact spaces this means that there are only finitely many possible behaviors.

--

Compactness does for continuous functions what finiteness does for functions in general.

If a set A is finite then every function f:A→R has a max and a min, and every function f:A→R^n is bounded. If A is compact, the every continuous function from A to R has a max and a min and every continuous function from A to R^n is bounded.

If A is finite then every sequence of members of A has a subsequence that is eventually constant, and "eventually constant" is the only kind of convergence you can talk about without talking about a topology on the set. If A is compact, then every sequence of members of A has a convergent subsequence.

january 2017 by nhaliday

Memento mori - Wikipedia

europe history wiki art death medieval nihil religion christianity philosophy ideology classic occident iron-age mediterranean reference concept foreign-lang article the-classics eden-heaven morality symbols zeitgeist flux-stasis humility finiteness lexical afterlife

january 2017 by nhaliday

europe history wiki art death medieval nihil religion christianity philosophy ideology classic occident iron-age mediterranean reference concept foreign-lang article the-classics eden-heaven morality symbols zeitgeist flux-stasis humility finiteness lexical afterlife

january 2017 by nhaliday

Soft analysis, hard analysis, and the finite convergence principle | What's new

january 2017 by nhaliday

It is fairly well known that the results obtained by hard and soft analysis respectively can be connected to each other by various “correspondence principles” or “compactness principles”. It is however my belief that the relationship between the two types of analysis is in fact much closer[3] than just this; in many cases, qualitative analysis can be viewed as a convenient abstraction of quantitative analysis, in which the precise dependencies between various finite quantities has been efficiently concealed from view by use of infinitary notation. Conversely, quantitative analysis can often be viewed as a more precise and detailed refinement of qualitative analysis. Furthermore, a method from hard analysis often has some analogue in soft analysis and vice versa, though the language and notation of the analogue may look completely different from that of the original. I therefore feel that it is often profitable for a practitioner of one type of analysis to learn about the other, as they both offer their own strengths, weaknesses, and intuition, and knowledge of one gives more insight[4] into the workings of the other. I wish to illustrate this point here using a simple but not terribly well known result, which I shall call the “finite convergence principle” (thanks to Ben Green for suggesting this name; Jennifer Chayes has also suggested the “metastability principle”). It is the finitary analogue of an utterly trivial infinitary result – namely, that every bounded monotone sequence converges – but sometimes, a careful analysis of a trivial result can be surprisingly revealing, as I hope to demonstrate here.

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january 2017 by nhaliday

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