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

Antinomia Imediata – experiments in a reaction from the left

march 2018 by nhaliday

https://antinomiaimediata.wordpress.com/lrx/

So, what is the Left Reaction? First of all, it’s reaction: opposition to the modern rationalist establishment, the Cathedral. It opposes the universalist Jacobin program of global government, favoring a fractured geopolitics organized through long-evolved complex systems. It’s profoundly anti-socialist and anti-communist, favoring market economy and individualism. It abhors tribalism and seeks a realistic plan for dismantling it (primarily informed by HBD and HBE). It looks at modernity as a degenerative ratchet, whose only way out is intensification (hence clinging to crypto-marxist market-driven acceleration).

How come can any of this still be in the *Left*? It defends equality of power, i.e. freedom. This radical understanding of liberty is deeply rooted in leftist tradition and has been consistently abhored by the Right. LRx is not democrat, is not socialist, is not progressist and is not even liberal (in its current, American use). But it defends equality of power. It’s utopia is individual sovereignty. It’s method is paleo-agorism. The anti-hierarchy of hunter-gatherer nomads is its understanding of the only realistic objective of equality.

...

In more cosmic terms, it seeks only to fulfill the Revolution’s side in the left-right intelligence pump: mutation or creation of paths. Proudhon’s antinomy is essentially about this: the collective force of the socius, evinced in moral standards and social organization vs the creative force of the individuals, that constantly revolutionize and disrupt the social body. The interplay of these forces create reality (it’s a metaphysics indeed): the Absolute (socius) builds so that the (individualistic) Revolution can destroy so that the Absolute may adapt, and then repeat. The good old formula of ‘solve et coagula’.

Ultimately, if the Neoreaction promises eternal hell, the LRx sneers “but Satan is with us”.

https://antinomiaimediata.wordpress.com/2016/12/16/a-statement-of-principles/

Liberty is to be understood as the ability and right of all sentient beings to dispose of their persons and the fruits of their labor, and nothing else, as they see fit. This stems from their self-awareness and their ability to control and choose the content of their actions.

...

Equality is to be understood as the state of no imbalance of power, that is, of no subjection to another sentient being. This stems from their universal ability for empathy, and from their equal ability for reason.

...

It is important to notice that, contrary to usual statements of these two principles, my standpoint is that Liberty and Equality here are not merely compatible, meaning they could coexist in some possible universe, but rather they are two sides of the same coin, complementary and interdependent. There can be NO Liberty where there is no Equality, for the imbalance of power, the state of subjection, will render sentient beings unable to dispose of their persons and the fruits of their labor[1], and it will limit their ability to choose over their rightful jurisdiction. Likewise, there can be NO Equality without Liberty, for restraining sentient beings’ ability to choose and dispose of their persons and fruits of labor will render some more powerful than the rest, and establish a state of subjection.

https://antinomiaimediata.wordpress.com/2017/04/18/flatness/

equality is the founding principle (and ultimately indistinguishable from) freedom. of course, it’s only in one specific sense of “equality” that this sentence is true.

to try and eliminate the bullshit, let’s turn to networks again:

any nodes’ degrees of freedom is the number of nodes they are connected to in a network. freedom is maximum when the network is symmetrically connected, i. e., when all nodes are connected to each other and thus there is no topographical hierarchy (middlemen) – in other words, flatness.

in this understanding, the maximization of freedom is the maximization of entropy production, that is, of intelligence. As Land puts it:

https://antinomiaimediata.wordpress.com/category/philosophy/mutualism/

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blog
stream
politics
polisci
ideology
philosophy
land
accelerationism
left-wing
right-wing
paradox
egalitarianism-hierarchy
civil-liberty
power
hmm
revolution
analytical-holistic
mutation
selection
individualism-collectivism
tribalism
us-them
modernity
multi
tradeoffs
network-structure
complex-systems
cybernetics
randy-ayndy
insight
contrarianism
metameta
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altruism
list
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cartoons
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distribution
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invariance
government
markets
paying-rent
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frontier
exit-voice
nl-and-so-can-you
war
track-record
usa
history
mostly-modern
world-war
military
justice
protestant-cathol
So, what is the Left Reaction? First of all, it’s reaction: opposition to the modern rationalist establishment, the Cathedral. It opposes the universalist Jacobin program of global government, favoring a fractured geopolitics organized through long-evolved complex systems. It’s profoundly anti-socialist and anti-communist, favoring market economy and individualism. It abhors tribalism and seeks a realistic plan for dismantling it (primarily informed by HBD and HBE). It looks at modernity as a degenerative ratchet, whose only way out is intensification (hence clinging to crypto-marxist market-driven acceleration).

How come can any of this still be in the *Left*? It defends equality of power, i.e. freedom. This radical understanding of liberty is deeply rooted in leftist tradition and has been consistently abhored by the Right. LRx is not democrat, is not socialist, is not progressist and is not even liberal (in its current, American use). But it defends equality of power. It’s utopia is individual sovereignty. It’s method is paleo-agorism. The anti-hierarchy of hunter-gatherer nomads is its understanding of the only realistic objective of equality.

...

In more cosmic terms, it seeks only to fulfill the Revolution’s side in the left-right intelligence pump: mutation or creation of paths. Proudhon’s antinomy is essentially about this: the collective force of the socius, evinced in moral standards and social organization vs the creative force of the individuals, that constantly revolutionize and disrupt the social body. The interplay of these forces create reality (it’s a metaphysics indeed): the Absolute (socius) builds so that the (individualistic) Revolution can destroy so that the Absolute may adapt, and then repeat. The good old formula of ‘solve et coagula’.

Ultimately, if the Neoreaction promises eternal hell, the LRx sneers “but Satan is with us”.

https://antinomiaimediata.wordpress.com/2016/12/16/a-statement-of-principles/

Liberty is to be understood as the ability and right of all sentient beings to dispose of their persons and the fruits of their labor, and nothing else, as they see fit. This stems from their self-awareness and their ability to control and choose the content of their actions.

...

Equality is to be understood as the state of no imbalance of power, that is, of no subjection to another sentient being. This stems from their universal ability for empathy, and from their equal ability for reason.

...

It is important to notice that, contrary to usual statements of these two principles, my standpoint is that Liberty and Equality here are not merely compatible, meaning they could coexist in some possible universe, but rather they are two sides of the same coin, complementary and interdependent. There can be NO Liberty where there is no Equality, for the imbalance of power, the state of subjection, will render sentient beings unable to dispose of their persons and the fruits of their labor[1], and it will limit their ability to choose over their rightful jurisdiction. Likewise, there can be NO Equality without Liberty, for restraining sentient beings’ ability to choose and dispose of their persons and fruits of labor will render some more powerful than the rest, and establish a state of subjection.

https://antinomiaimediata.wordpress.com/2017/04/18/flatness/

equality is the founding principle (and ultimately indistinguishable from) freedom. of course, it’s only in one specific sense of “equality” that this sentence is true.

to try and eliminate the bullshit, let’s turn to networks again:

any nodes’ degrees of freedom is the number of nodes they are connected to in a network. freedom is maximum when the network is symmetrically connected, i. e., when all nodes are connected to each other and thus there is no topographical hierarchy (middlemen) – in other words, flatness.

in this understanding, the maximization of freedom is the maximization of entropy production, that is, of intelligence. As Land puts it:

https://antinomiaimediata.wordpress.com/category/philosophy/mutualism/

march 2018 by nhaliday

Drude model - Wikipedia

september 2017 by nhaliday

The Drude model of electrical conduction was proposed in 1900[1][2] by Paul Drude to explain the transport properties of electrons in materials (especially metals). The model, which is an application of kinetic theory, assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like _a pinball machine_, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.

The two most significant results of the Drude model are an electronic equation of motion,

d<p(t)>/dt = q(E + 1/m <p(t)> x B) - <p(t)>/τ

and a linear relationship between current density J and electric field E,

J = (nq^2τ/m) E

latter is Ohm's law

nibble
physics
electromag
models
local-global
stat-mech
identity
atoms
wiki
reference
ground-up
cartoons
The two most significant results of the Drude model are an electronic equation of motion,

d<p(t)>/dt = q(E + 1/m <p(t)> x B) - <p(t)>/τ

and a linear relationship between current density J and electric field E,

J = (nq^2τ/m) E

latter is Ohm's law

september 2017 by nhaliday

Homo reciprocans

shalizi scitariat essay anthropology cultural-dynamics deep-materialism economics behavioral-econ cooperate-defect cartoons old-anglo big-peeps quotes aphorism n-factor self-interest social-norms justice inequality GT-101 s:* 🌞 🎩 broad-econ decision-making microfoundations axelrod interests

june 2017 by nhaliday

shalizi scitariat essay anthropology cultural-dynamics deep-materialism economics behavioral-econ cooperate-defect cartoons old-anglo big-peeps quotes aphorism n-factor self-interest social-norms justice inequality GT-101 s:* 🌞 🎩 broad-econ decision-making microfoundations axelrod interests

june 2017 by nhaliday

In the first place | West Hunter

may 2017 by nhaliday

We hear a lot about innovative educational approaches, and since these silly people have been at this for a long time now, we hear just as often about the innovative approaches that some idiot started up a few years ago and are now crashing in flames. We’re in steady-state.

I’m wondering if it isn’t time to try something archaic. In particular, mnemonic techniques, such as the method of loci. As far as I know, nobody has actually tried integrating the more sophisticated mnemonic techniques into a curriculum. Sure, we all know useful acronyms, like the one for resistor color codes, but I’ve not heard of anyone teaching kids how to build a memory palace.

https://westhunt.wordpress.com/2013/12/28/in-the-first-place/#comment-20106

I have never used formal mnemonic techniques, but life has recently tested me on how well I remember material from my college days. Turns out that I can still do the sorts of math and physics problems that I could then, in subjects like classical mechanics, real analysis, combinatorics, complex variables, quantum mechanics, statistical mechanics, etc. I usually have to crack the book though. Some of that material I have used from time to time, or even fairly often (especially linear algebra), most not. I’m sure I’m slower than I was then, at least on the stuff I haven’t used.

https://westhunt.wordpress.com/2013/12/28/in-the-first-place/#comment-20109

Long-term memory capacity must be finite, but I know of no evidence that anyone has ever run out of it. As for the idea that you don’t really need a lot of facts in your head to come up with new ideas: pretty much the opposite of the truth, in a lot of fields.

https://en.wikipedia.org/wiki/Method_of_loci

Mental Imagery > Ancient Imagery Mnemonics: https://plato.stanford.edu/entries/mental-imagery/ancient-imagery-mnemonics.html

In the Middle Ages and the Renaissance, very elaborate versions of the method evolved, using specially learned imaginary spaces (Memory Theaters or Palaces), and complex systems of predetermined symbolic images, often imbued with occult or spiritual significances. However, modern experimental research has shown that even a simple and easily learned form of the method of loci can be highly effective (Ross & Lawrence, 1968; Maguire et al., 2003), as are several other imagery based mnemonic techniques (see section 4.2 of the main entry).

The advantages of organizing knowledge in terms of country and place: http://marginalrevolution.com/marginalrevolution/2018/02/advantages-organizing-knowledge-terms-country-place.html

https://www.quora.com/What-are-the-best-books-on-Memory-Palace

fascinating aside:

US vs Nazi army, Vietnam, the draft: https://westhunt.wordpress.com/2013/12/28/in-the-first-place/#comment-20136

You think I know more about this than a retired major general and former head of the War College? I do, of course, but that fact itself should worry you.

He’s not all wrong, but a lot of what he says is wrong. For example, the Germany Army was a conscript army, so conscription itself can’t explain why the Krauts were about 25% more effective than the average American unit. Nor is it true that the draft in WWII was corrupt.

The US had a different mix of armed forces – more air forces and a much larger Navy than Germany. Those services have higher technical requirements and sucked up a lot of the smarter guys. That was just a product of the strategic situation.

The Germans had better officers, partly because of better training and doctrine, partly the fruit of a different attitude towards the army. The US, much of the time, thought of the Army as a career for losers, but Germans did not.

The Germans had an enormous amount of relevant combat experience, much more than anyone in the US. Spend a year or two on the Eastern Front and you learn.

And the Germans had better infantry weapons.

The US tooth-to-tail ratio was , I think, worse than that of the Germans: some of that was a natural consequence of being an expeditionary force, but some was just a mistake. You want supply sergeants to be literate, but it is probably true that we put too many of the smarter guys into non-combat positions. That changed some when we ran into manpower shortages in late 1944 and combed out the support positions.

This guy is back-projecting Vietnam problems into WWII – he’s mostly wrong.

more (more of a focus on US Marines than Army): https://www.quora.com/Were-US-Marines-tougher-than-elite-German-troops-in-WW2/answer/Joseph-Scott-13

west-hunter
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visual-understanding
cartoons
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comparison
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ability-competence
leadership
elite
higher-ed
math
physics
linear-algebra
cost-benefit
prioritizing
defense
martial
war-nerd
worrydream
I’m wondering if it isn’t time to try something archaic. In particular, mnemonic techniques, such as the method of loci. As far as I know, nobody has actually tried integrating the more sophisticated mnemonic techniques into a curriculum. Sure, we all know useful acronyms, like the one for resistor color codes, but I’ve not heard of anyone teaching kids how to build a memory palace.

https://westhunt.wordpress.com/2013/12/28/in-the-first-place/#comment-20106

I have never used formal mnemonic techniques, but life has recently tested me on how well I remember material from my college days. Turns out that I can still do the sorts of math and physics problems that I could then, in subjects like classical mechanics, real analysis, combinatorics, complex variables, quantum mechanics, statistical mechanics, etc. I usually have to crack the book though. Some of that material I have used from time to time, or even fairly often (especially linear algebra), most not. I’m sure I’m slower than I was then, at least on the stuff I haven’t used.

https://westhunt.wordpress.com/2013/12/28/in-the-first-place/#comment-20109

Long-term memory capacity must be finite, but I know of no evidence that anyone has ever run out of it. As for the idea that you don’t really need a lot of facts in your head to come up with new ideas: pretty much the opposite of the truth, in a lot of fields.

https://en.wikipedia.org/wiki/Method_of_loci

Mental Imagery > Ancient Imagery Mnemonics: https://plato.stanford.edu/entries/mental-imagery/ancient-imagery-mnemonics.html

In the Middle Ages and the Renaissance, very elaborate versions of the method evolved, using specially learned imaginary spaces (Memory Theaters or Palaces), and complex systems of predetermined symbolic images, often imbued with occult or spiritual significances. However, modern experimental research has shown that even a simple and easily learned form of the method of loci can be highly effective (Ross & Lawrence, 1968; Maguire et al., 2003), as are several other imagery based mnemonic techniques (see section 4.2 of the main entry).

The advantages of organizing knowledge in terms of country and place: http://marginalrevolution.com/marginalrevolution/2018/02/advantages-organizing-knowledge-terms-country-place.html

https://www.quora.com/What-are-the-best-books-on-Memory-Palace

fascinating aside:

US vs Nazi army, Vietnam, the draft: https://westhunt.wordpress.com/2013/12/28/in-the-first-place/#comment-20136

You think I know more about this than a retired major general and former head of the War College? I do, of course, but that fact itself should worry you.

He’s not all wrong, but a lot of what he says is wrong. For example, the Germany Army was a conscript army, so conscription itself can’t explain why the Krauts were about 25% more effective than the average American unit. Nor is it true that the draft in WWII was corrupt.

The US had a different mix of armed forces – more air forces and a much larger Navy than Germany. Those services have higher technical requirements and sucked up a lot of the smarter guys. That was just a product of the strategic situation.

The Germans had better officers, partly because of better training and doctrine, partly the fruit of a different attitude towards the army. The US, much of the time, thought of the Army as a career for losers, but Germans did not.

The Germans had an enormous amount of relevant combat experience, much more than anyone in the US. Spend a year or two on the Eastern Front and you learn.

And the Germans had better infantry weapons.

The US tooth-to-tail ratio was , I think, worse than that of the Germans: some of that was a natural consequence of being an expeditionary force, but some was just a mistake. You want supply sergeants to be literate, but it is probably true that we put too many of the smarter guys into non-combat positions. That changed some when we ran into manpower shortages in late 1944 and combed out the support positions.

This guy is back-projecting Vietnam problems into WWII – he’s mostly wrong.

more (more of a focus on US Marines than Army): https://www.quora.com/Were-US-Marines-tougher-than-elite-German-troops-in-WW2/answer/Joseph-Scott-13

may 2017 by nhaliday

A cube, a starfish, a thin shell, and the central limit theorem – Libres pensées d'un mathématicien ordinaire

mathtariat org:bleg nibble math acm probability concentration-of-measure high-dimension cartoons limits dimensionality measure yoga hi-order-bits synthesis exposition spatial geometry math.MG curvature convexity-curvature

february 2017 by nhaliday

mathtariat org:bleg nibble math acm probability concentration-of-measure high-dimension cartoons limits dimensionality measure yoga hi-order-bits synthesis exposition spatial geometry math.MG curvature convexity-curvature

february 2017 by nhaliday

Do grad school students remember everything they were taught in college all the time? - Quora

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february 2017 by nhaliday

q-n-a qra grad-school learning synthesis hi-order-bits neurons physics lens analogy cartoons links 🎓 scholar gowers mathtariat feynman giants quotes games nibble thinking zooming retention meta:research big-picture skeleton s:** p:whenever wire-guided narrative intuition lesswrong commentary ground-up limits examples problem-solving info-dynamics knowledge studying ideas the-trenches chart

february 2017 by nhaliday

254A, Supplement 4: Probabilistic models and heuristics for the primes (optional) | What's new

february 2017 by nhaliday

among others, the Cramér model for the primes (basically kinda looks like primality is independently distributed w/ Pr[n is prime] = 1/log n)

gowers
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lecture-notes
exposition
math
math.NT
probability
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models
cartoons
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org:bleg
pseudorandomness
borel-cantelli
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multiplicative
truth
guessing
february 2017 by nhaliday

ho.history overview - History of the high-dimensional volume paradox - MathOverflow

q-n-a overflow math math.MG geometry spatial dimensionality limits measure concentration-of-measure history stories giants cartoons soft-question nibble paradox novelty high-dimension examples gotchas recruiting

january 2017 by nhaliday

q-n-a overflow math math.MG geometry spatial dimensionality limits measure concentration-of-measure history stories giants cartoons soft-question nibble paradox novelty high-dimension examples gotchas recruiting

january 2017 by nhaliday

pr.probability - "Entropy" proof of Brunn-Minkowski Inequality? - MathOverflow

q-n-a overflow math information-theory wormholes proofs geometry math.MG estimate gowers mathtariat dimensionality limits intuition insight stat-mech concentration-of-measure 👳 cartoons math.FA additive-combo measure entropy-like nibble tensors coarse-fine brunn-minkowski boltzmann high-dimension curvature convexity-curvature

january 2017 by nhaliday

q-n-a overflow math information-theory wormholes proofs geometry math.MG estimate gowers mathtariat dimensionality limits intuition insight stat-mech concentration-of-measure 👳 cartoons math.FA additive-combo measure entropy-like nibble tensors coarse-fine brunn-minkowski boltzmann high-dimension curvature convexity-curvature

january 2017 by nhaliday

Richard Feynman: Physics is fun to imagine | TED Talk | TED.com

feynman physics curiosity :) video interview classic insight org:edge lens giants nibble virtu communication cartoons exposition metameta thinking hi-order-bits science meta:science synthesis visual-understanding worrydream vitality dynamic org:anglo thermo mechanics electromag phys-energy better-explained teaching the-world-is-just-atoms presentation wisdom waves space gravity wordlessness oscillation quantum concrete minimum-viable s:*** new-religion energy-resources big-picture 🔬 info-dynamics elegance

january 2017 by nhaliday

feynman physics curiosity :) video interview classic insight org:edge lens giants nibble virtu communication cartoons exposition metameta thinking hi-order-bits science meta:science synthesis visual-understanding worrydream vitality dynamic org:anglo thermo mechanics electromag phys-energy better-explained teaching the-world-is-just-atoms presentation wisdom waves space gravity wordlessness oscillation quantum concrete minimum-viable s:*** new-religion energy-resources big-picture 🔬 info-dynamics elegance

january 2017 by nhaliday

reference request - Why are two "random" vectors in $mathbb R^n$ approximately orthogonal for large $n$? - MathOverflow

q-n-a overflow math probability tidbits intuition cartoons math.MG spatial geometry linear-algebra mathtariat dimensionality magnitude concentration-of-measure probabilistic-method random separation inner-product nibble relaxation paradox novelty high-dimension direction guessing

january 2017 by nhaliday

q-n-a overflow math probability tidbits intuition cartoons math.MG spatial geometry linear-algebra mathtariat dimensionality magnitude concentration-of-measure probabilistic-method random separation inner-product nibble relaxation paradox novelty high-dimension direction guessing

january 2017 by nhaliday

A cheap version of the Kabatjanskii-Levenstein bound for almost orthogonal vectors | What's new

gowers mathtariat exposition tidbits math geometry spatial math.CO magnitude probabilistic-method cartoons linear-algebra math.MG dimensionality random separation inner-product nibble org:bleg relaxation high-dimension direction

january 2017 by nhaliday

gowers mathtariat exposition tidbits math geometry spatial math.CO magnitude probabilistic-method cartoons linear-algebra math.MG dimensionality random separation inner-product nibble org:bleg relaxation high-dimension direction

january 2017 by nhaliday

fa.functional analysis - Almost orthogonal vectors - MathOverflow

january 2017 by nhaliday

- you can pick exp(Θ(nε^2)) ε-almost orthogonal unit vectors in R^n w/ probabilistic method

- can also use Johnson-Lindenstrauss

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high-dimension
direction
shift
- can also use Johnson-Lindenstrauss

january 2017 by nhaliday

Oh, they were looking for their Missing Piece – spottedtoad

january 2017 by nhaliday

Assuming that the value of an offspring’s trait are determined by averaging the value of both parents and then adding some random error due to mutation or developmental noise, the ideal mate for each individual in the population isn’t the one that is closest to the ideal value, but one that is “complementary”- ie, equally distant from the ideal value, but from the opposite side.

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tails
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complement-substitute
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signum
ecology
EGT
january 2017 by nhaliday

"Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character ... - Richard P. Feynman - Google Books

january 2017 by nhaliday

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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s:**
quotes
gbooks
elegance
january 2017 by nhaliday

pr.probability - What is convolution intuitively? - MathOverflow

january 2017 by nhaliday

I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences when taking off his or her glasses (and, indeed, if one works through the geometric optics, convolution is not a bad first approximation for this effect). I found this to be very helpful, not just for understanding convolution per se, but as a lesson that one should try to use physical intuition to model mathematical concepts whenever one can.

More generally, if one thinks of functions as fuzzy versions of points, then convolution is the fuzzy version of addition (or sometimes multiplication, depending on the context). The probabilistic interpretation is one example of this (where the fuzz is a a probability distribution), but one can also have signed, complex-valued, or vector-valued fuzz, of course.

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nibble
yoga
neurons
retrofit
optics
concrete
s:*
multiplicative
fourier
More generally, if one thinks of functions as fuzzy versions of points, then convolution is the fuzzy version of addition (or sometimes multiplication, depending on the context). The probabilistic interpretation is one example of this (where the fuzz is a a probability distribution), but one can also have signed, complex-valued, or vector-valued fuzz, of course.

january 2017 by nhaliday

soft question - Thinking and Explaining - MathOverflow

january 2017 by nhaliday

- 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)

q-n-a
soft-question
big-list
intuition
communication
teaching
math
thinking
writing
thurston
lens
overflow
synthesis
hi-order-bits
👳
insight
meta:math
clarity
nibble
giants
cartoons
gowers
mathtariat
better-explained
stories
the-trenches
problem-solving
homogeneity
symmetry
fedja
examples
philosophy
big-picture
vague
isotropy
reflection
spatial
ground-up
visual-understanding
polynomials
dimensionality
math.GR
worrydream
scholar
🎓
neurons
metabuch
yoga
retrofit
mental-math
metameta
wisdom
wordlessness
oscillation
operational
adversarial
quantifiers-sums
exposition
explanation
tricki
concrete
s:***
manifolds
invariance
dynamical
info-dynamics
cool
direction
elegance
heavyweights
analysis
guessing
grokkability-clarity
technical-writing
- 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)

january 2017 by nhaliday

gt.geometric topology - Intuitive crutches for higher dimensional thinking - MathOverflow

december 2016 by nhaliday

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".

q-n-a
intuition
math
visual-understanding
list
discussion
thurston
tidbits
aaronson
tcs
geometry
problem-solving
yoga
👳
big-list
metabuch
tcstariat
gowers
mathtariat
acm
overflow
soft-question
levers
dimensionality
hi-order-bits
insight
synthesis
thinking
models
cartoons
coding-theory
information-theory
probability
concentration-of-measure
magnitude
linear-algebra
boolean-analysis
analogy
arrows
lifts-projections
measure
markov
sampling
shannon
conceptual-vocab
nibble
degrees-of-freedom
worrydream
neurons
retrofit
oscillation
paradox
novelty
tricki
concrete
high-dimension
s:***
manifolds
direction
curvature
convexity-curvature
elegance
guessing
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".

december 2016 by nhaliday

On “local” and “global” errors in mathematical papers, and how to detect them

november 2016 by nhaliday

local vs. global errors in technical papers

old:

https://plus.google.com/+TerenceTao27/posts/78aoEHoPhpS

gowers
social
metabuch
thinking
problem-solving
math
advice
reflection
scholar
🎓
expert
mathtariat
lens
local-global
meta:math
cartoons
learning
the-trenches
meta:research
s:**
info-dynamics
studying
expert-experience
meta:reading
multi
heavyweights
old:

https://plus.google.com/+TerenceTao27/posts/78aoEHoPhpS

november 2016 by nhaliday

On “compilation errors” in mathematical reading, and how to resolve them

november 2016 by nhaliday

compilation errors in academic papers

old:

[Google Buzz closed down for good recently, so I will be reprinting a small n...

https://plus.google.com/u/0/+TerenceTao27/posts/TGjjJPUdJjk

gowers
social
advice
reflection
math
thinking
problem-solving
metabuch
expert
scholar
🎓
mathtariat
lens
meta:math
cartoons
learning
lifts-projections
the-trenches
meta:research
s:**
info-dynamics
studying
expert-experience
meta:reading
analogy
compilers
multi
heavyweights
zooming
old:

[Google Buzz closed down for good recently, so I will be reprinting a small n...

https://plus.google.com/u/0/+TerenceTao27/posts/TGjjJPUdJjk

november 2016 by nhaliday

The Castle and the Forest Sauvage – spottedtoad

november 2016 by nhaliday

So meritocracy might rule when it comes to who ends up nearer or further from the castle (or then it might again not), but it will never determine who is allowed to come over the drawbridge, in through the gate, and allowed to see the Castle’s inside.

thinking
ratty
government
speculation
parable
essay
unaffiliated
gedanken
metabuch
analogy
cartoons
wonkish
inequality
mobility
winner-take-all
managerial-state
s-factor
chart
november 2016 by nhaliday

The probabilistic heuristic justification of the ABC conjecture | What's new

open-problems gowers thinking tricks intuition probability math tidbits yoga mathtariat models heuristic math.NT cartoons nibble org:bleg borel-cantelli big-surf additive multiplicative questions guessing

october 2016 by nhaliday

open-problems gowers thinking tricks intuition probability math tidbits yoga mathtariat models heuristic math.NT cartoons nibble org:bleg borel-cantelli big-surf additive multiplicative questions guessing

october 2016 by nhaliday

nt.number theory - When has the Borel-Cantelli heuristic been wrong? - MathOverflow

intuition counterexample list q-n-a tidbits math thinking tricks synthesis yoga probability models big-list heuristic overflow levers math.NT rigor cartoons nibble borel-cantelli guessing truth error

october 2016 by nhaliday

intuition counterexample list q-n-a tidbits math thinking tricks synthesis yoga probability models big-list heuristic overflow levers math.NT rigor cartoons nibble borel-cantelli guessing truth error

october 2016 by nhaliday

What should one who is creating a 'summary note' (scribe note, lecture note, etc.) for an advanced mathematics course care about? - Quora

q-n-a notetaking advice learning productivity reflection qra soft-question scholar oly mathtariat nibble studying s:null hi-order-bits synthesis cartoons problem-solving summary metabuch checklists yoga zooming visual-understanding retention metameta math top-n list the-trenches meta:research knowledge skeleton chart big-picture visualization

september 2016 by nhaliday

q-n-a notetaking advice learning productivity reflection qra soft-question scholar oly mathtariat nibble studying s:null hi-order-bits synthesis cartoons problem-solving summary metabuch checklists yoga zooming visual-understanding retention metameta math top-n list the-trenches meta:research knowledge skeleton chart big-picture visualization

september 2016 by nhaliday

Why Information Grows – Paul Romer

september 2016 by nhaliday

thinking like a physicist:

The key element in thinking like a physicist is being willing to push simultaneously to extreme levels of abstraction and specificity. This sounds paradoxical until you see it in action. Then it seems obvious. Abstraction means that you strip away inessential detail. Specificity means that you take very seriously the things that remain.

Abstraction vs. Radical Specificity: https://paulromer.net/abstraction-vs-radical-specificity/

books
summary
review
economics
growth-econ
interdisciplinary
hmm
physics
thinking
feynman
tradeoffs
paul-romer
econotariat
🎩
🎓
scholar
aphorism
lens
signal-noise
cartoons
skeleton
s:**
giants
electromag
mutation
genetics
genomics
bits
nibble
stories
models
metameta
metabuch
problem-solving
composition-decomposition
structure
abstraction
zooming
examples
knowledge
human-capital
behavioral-econ
network-structure
info-econ
communication
learning
information-theory
applications
volo-avolo
map-territory
externalities
duplication
spreading
property-rights
lattice
multi
government
polisci
policy
counterfactual
insight
paradox
parallax
reduction
empirical
detail-architecture
methodology
crux
visual-understanding
theory-practice
matching
analytical-holistic
branches
complement-substitute
local-global
internet
technology
cost-benefit
investing
micro
signaling
limits
public-goodish
interpretation
elegance
meta:reading
intellectual-property
writing
The key element in thinking like a physicist is being willing to push simultaneously to extreme levels of abstraction and specificity. This sounds paradoxical until you see it in action. Then it seems obvious. Abstraction means that you strip away inessential detail. Specificity means that you take very seriously the things that remain.

Abstraction vs. Radical Specificity: https://paulromer.net/abstraction-vs-radical-specificity/

september 2016 by nhaliday

machine learning - Why is Euclidean distance not a good metric in high dimensions? - Cross Validated

thinking machine-learning math acm synthesis intuition q-n-a overflow soft-question dimensionality hi-order-bits curiosity cartoons concentration-of-measure norms nibble novelty high-dimension direction metric-space yoga measure best-practices

september 2016 by nhaliday

thinking machine-learning math acm synthesis intuition q-n-a overflow soft-question dimensionality hi-order-bits curiosity cartoons concentration-of-measure norms nibble novelty high-dimension direction metric-space yoga measure best-practices

september 2016 by nhaliday

soft question - How do you not forget old math? - MathOverflow

june 2016 by nhaliday

Terry Tao:

I find that blogging about material that I would otherwise forget eventually is extremely valuable in this regard. (I end up consulting my own blog posts on a regular basis.) EDIT: and now I remember I already wrote on this topic: terrytao.wordpress.com/career-advice/write-down-what-youve-done

fedja:

The only way to cope with this loss of memory I know is to do some reading on systematic basis. Of course, if you read one paper in algebraic geometry (or whatever else) a month (or even two months), you may not remember the exact content of all of them by the end of the year but, since all mathematicians in one field use pretty much the same tricks and draw from pretty much the same general knowledge, you'll keep the core things in your memory no matter what you read (provided it is not patented junk, of course) and this is about as much as you can hope for.

Relating abstract things to "real life stuff" (and vice versa) is automatic when you work as a mathematician. For me, the proof of the Chacon-Ornstein ergodic theorem is just a sandpile moving over a pit with the sand falling down after every shift. I often tell my students that every individual term in the sequence doesn't matter at all for the limit but somehow together they determine it like no individual human is of any real importance while together they keep this civilization running, etc. No special effort is needed here and, moreover, if the analogy is not natural but contrived, it'll not be helpful or memorable. The standard mnemonic techniques are pretty useless in math. IMHO (the famous "foil" rule for the multiplication of sums of two terms is inferior to the natural "pair each term in the first sum with each term in the second sum" and to the picture of a rectangle tiled with smaller rectangles, though, of course, the foil rule sounds way more sexy).

One thing that I don't think the other respondents have emphasized enough is that you should work on prioritizing what you choose to study and remember.

Timothy Chow:

As others have said, forgetting lots of stuff is inevitable. But there are ways you can mitigate the damage of this information loss. I find that a useful technique is to try to organize your knowledge hierarchically. Start by coming up with a big picture, and make sure you understand and remember that picture thoroughly. Then drill down to the next level of detail, and work on remembering that. For example, if I were trying to remember everything in a particular book, I might start by memorizing the table of contents, and then I'd work on remembering the theorem statements, and then finally the proofs. (Don't take this illustration too literally; it's better to come up with your own conceptual hierarchy than to slavishly follow the formal hierarchy of a published text. But I do think that a hierarchical approach is valuable.)

Organizing your knowledge like this helps you prioritize. You can then consciously decide that certain large swaths of knowledge are not worth your time at the moment, and just keep a "stub" in memory to remind you that that body of knowledge exists, should you ever need to dive into it. In areas of higher priority, you can plunge more deeply. By making sure you thoroughly internalize the top levels of the hierarchy, you reduce the risk of losing sight of entire areas of important knowledge. Generally it's less catastrophic to forget the details than to forget about a whole region of the big picture, because you can often revisit the details as long as you know what details you need to dig up. (This is fortunate since the details are the most memory-intensive.)

Having a hierarchy also helps you accrue new knowledge. Often when you encounter something new, you can relate it to something you already know, and file it in the same branch of your mental tree.

thinking
math
growth
advice
expert
q-n-a
🎓
long-term
tradeoffs
scholar
overflow
soft-question
gowers
mathtariat
ground-up
hi-order-bits
intuition
synthesis
visual-understanding
decision-making
scholar-pack
cartoons
lens
big-picture
ergodic
nibble
zooming
trees
fedja
reflection
retention
meta:research
wisdom
skeleton
practice
prioritizing
concrete
s:***
info-dynamics
knowledge
studying
the-trenches
chart
expert-experience
quixotic
elegance
heavyweights
I find that blogging about material that I would otherwise forget eventually is extremely valuable in this regard. (I end up consulting my own blog posts on a regular basis.) EDIT: and now I remember I already wrote on this topic: terrytao.wordpress.com/career-advice/write-down-what-youve-done

fedja:

The only way to cope with this loss of memory I know is to do some reading on systematic basis. Of course, if you read one paper in algebraic geometry (or whatever else) a month (or even two months), you may not remember the exact content of all of them by the end of the year but, since all mathematicians in one field use pretty much the same tricks and draw from pretty much the same general knowledge, you'll keep the core things in your memory no matter what you read (provided it is not patented junk, of course) and this is about as much as you can hope for.

Relating abstract things to "real life stuff" (and vice versa) is automatic when you work as a mathematician. For me, the proof of the Chacon-Ornstein ergodic theorem is just a sandpile moving over a pit with the sand falling down after every shift. I often tell my students that every individual term in the sequence doesn't matter at all for the limit but somehow together they determine it like no individual human is of any real importance while together they keep this civilization running, etc. No special effort is needed here and, moreover, if the analogy is not natural but contrived, it'll not be helpful or memorable. The standard mnemonic techniques are pretty useless in math. IMHO (the famous "foil" rule for the multiplication of sums of two terms is inferior to the natural "pair each term in the first sum with each term in the second sum" and to the picture of a rectangle tiled with smaller rectangles, though, of course, the foil rule sounds way more sexy).

One thing that I don't think the other respondents have emphasized enough is that you should work on prioritizing what you choose to study and remember.

Timothy Chow:

As others have said, forgetting lots of stuff is inevitable. But there are ways you can mitigate the damage of this information loss. I find that a useful technique is to try to organize your knowledge hierarchically. Start by coming up with a big picture, and make sure you understand and remember that picture thoroughly. Then drill down to the next level of detail, and work on remembering that. For example, if I were trying to remember everything in a particular book, I might start by memorizing the table of contents, and then I'd work on remembering the theorem statements, and then finally the proofs. (Don't take this illustration too literally; it's better to come up with your own conceptual hierarchy than to slavishly follow the formal hierarchy of a published text. But I do think that a hierarchical approach is valuable.)

Organizing your knowledge like this helps you prioritize. You can then consciously decide that certain large swaths of knowledge are not worth your time at the moment, and just keep a "stub" in memory to remind you that that body of knowledge exists, should you ever need to dive into it. In areas of higher priority, you can plunge more deeply. By making sure you thoroughly internalize the top levels of the hierarchy, you reduce the risk of losing sight of entire areas of important knowledge. Generally it's less catastrophic to forget the details than to forget about a whole region of the big picture, because you can often revisit the details as long as you know what details you need to dig up. (This is fortunate since the details are the most memory-intensive.)

Having a hierarchy also helps you accrue new knowledge. Often when you encounter something new, you can relate it to something you already know, and file it in the same branch of your mental tree.

june 2016 by nhaliday

Answer to What is it like to understand advanced mathematics? - Quora

may 2016 by nhaliday

thinking like a mathematician

some of the points:

- small # of tricks (echoes Rota)

- web of concepts and modularization (zooming out) allow quick reasoning

- comfort w/ ambiguity and lack of understanding, study high-dimensional objects via projections

- above is essential for research (and often what distinguishes research mathematicians from people who were good at math, or majored in math)

math
reflection
thinking
intuition
expert
synthesis
wormholes
insight
q-n-a
🎓
metabuch
tricks
scholar
problem-solving
aphorism
instinct
heuristic
lens
qra
soft-question
curiosity
meta:math
ground-up
cartoons
analytical-holistic
lifts-projections
hi-order-bits
scholar-pack
nibble
the-trenches
innovation
novelty
zooming
tricki
virtu
humility
metameta
wisdom
abstraction
skeleton
s:***
knowledge
expert-experience
elegance
judgement
advanced
heavyweights
guessing
some of the points:

- small # of tricks (echoes Rota)

- web of concepts and modularization (zooming out) allow quick reasoning

- comfort w/ ambiguity and lack of understanding, study high-dimensional objects via projections

- above is essential for research (and often what distinguishes research mathematicians from people who were good at math, or majored in math)

may 2016 by nhaliday

Making invisible understanding visible

may 2016 by nhaliday

I like the example of cyclic subgroups

visualization
worrydream
thinking
math
yoga
thurston
intuition
algebra
insight
👳
wormholes
visual-understanding
michael-nielsen
water
exocortex
2016
fourier
cartoons
tcstariat
techtariat
clarity
vague
org:bleg
nibble
better-explained
math.GR
bounded-cognition
metameta
wordlessness
meta:math
s:***
composition-decomposition
dynamical
info-dynamics
let-me-see
elegance
heavyweights
guessing
form-design
grokkability-clarity
skunkworks
may 2016 by nhaliday

Notes Essays—Peter Thiel’s CS183: Startup—Stanford, Spring 2012

business startups strategy course thiel contrarianism barons definite-planning entrepreneurialism lecture-notes skunkworks innovation competition market-power winner-take-all usa anglosphere duplication education higher-ed law ranking success envy stanford princeton harvard elite zero-positive-sum war truth realness capitalism markets darwinian rent-seeking google facebook apple microsoft amazon capital scale network-structure tech business-models twitter social media games frontier time rhythm space musk mobile ai transportation examples recruiting venture metabuch metameta skeleton crooked wisdom gnosis-logos thinking polarization synchrony allodium antidemos democracy things exploratory dimensionality nationalism-globalism trade technology distribution moments personality phalanges stereotypes tails plots visualization creative nietzschean thick-thin psych-architecture wealth class morality ethics status extra-introversion info-dynamics narrative stories fashun myth the-classics literature big-peeps crime

february 2016 by nhaliday

business startups strategy course thiel contrarianism barons definite-planning entrepreneurialism lecture-notes skunkworks innovation competition market-power winner-take-all usa anglosphere duplication education higher-ed law ranking success envy stanford princeton harvard elite zero-positive-sum war truth realness capitalism markets darwinian rent-seeking google facebook apple microsoft amazon capital scale network-structure tech business-models twitter social media games frontier time rhythm space musk mobile ai transportation examples recruiting venture metabuch metameta skeleton crooked wisdom gnosis-logos thinking polarization synchrony allodium antidemos democracy things exploratory dimensionality nationalism-globalism trade technology distribution moments personality phalanges stereotypes tails plots visualization creative nietzschean thick-thin psych-architecture wealth class morality ethics status extra-introversion info-dynamics narrative stories fashun myth the-classics literature big-peeps crime

february 2016 by nhaliday

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