nhaliday + concrete   31

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.
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july 2019 by nhaliday
Lateralization of brain function - Wikipedia
Language
Language functions such as grammar, vocabulary and literal meaning are typically lateralized to the left hemisphere, especially in right handed individuals.[3] While language production is left-lateralized in up to 90% of right-handers, it is more bilateral, or even right-lateralized, in approximately 50% of left-handers.[4]

Broca's area and Wernicke's area, two areas associated with the production of speech, are located in the left cerebral hemisphere for about 95% of right-handers, but about 70% of left-handers.[5]:69

Auditory and visual processing
The processing of visual and auditory stimuli, spatial manipulation, facial perception, and artistic ability are represented bilaterally.[4] Numerical estimation, comparison and online calculation depend on bilateral parietal regions[6][7] while exact calculation and fact retrieval are associated with left parietal regions, perhaps due to their ties to linguistic processing.[6][7]

...

Depression is linked with a hyperactive right hemisphere, with evidence of selective involvement in "processing negative emotions, pessimistic thoughts and unconstructive thinking styles", as well as vigilance, arousal and self-reflection, and a relatively hypoactive left hemisphere, "specifically involved in processing pleasurable experiences" and "relatively more involved in decision-making processes".

Chaos and Order; the right and left hemispheres: https://orthosphere.wordpress.com/2018/05/23/chaos-and-order-the-right-and-left-hemispheres/
In The Master and His Emissary, Iain McGilchrist writes that a creature like a bird needs two types of consciousness simultaneously. It needs to be able to focus on something specific, such as pecking at food, while it also needs to keep an eye out for predators which requires a more general awareness of environment.

These are quite different activities. The Left Hemisphere (LH) is adapted for a narrow focus. The Right Hemisphere (RH) for the broad. The brains of human beings have the same division of function.

The LH governs the right side of the body, the RH, the left side. With birds, the left eye (RH) looks for predators, the right eye (LH) focuses on food and specifics. Since danger can take many forms and is unpredictable, the RH has to be very open-minded.

The LH is for narrow focus, the explicit, the familiar, the literal, tools, mechanism/machines and the man-made. The broad focus of the RH is necessarily more vague and intuitive and handles the anomalous, novel, metaphorical, the living and organic. The LH is high resolution but narrow, the RH low resolution but broad.

The LH exhibits unrealistic optimism and self-belief. The RH has a tendency towards depression and is much more realistic about a person’s own abilities. LH has trouble following narratives because it has a poor sense of “wholes.” In art it favors flatness, abstract and conceptual art, black and white rather than color, simple geometric shapes and multiple perspectives all shoved together, e.g., cubism. Particularly RH paintings emphasize vistas with great depth of field and thus space and time,[1] emotion, figurative painting and scenes related to the life world. In music, LH likes simple, repetitive rhythms. The RH favors melody, harmony and complex rhythms.

...

Schizophrenia is a disease of extreme LH emphasis. Since empathy is RH and the ability to notice emotional nuance facially, vocally and bodily expressed, schizophrenics tend to be paranoid and are often convinced that the real people they know have been replaced by robotic imposters. This is at least partly because they lose the ability to intuit what other people are thinking and feeling – hence they seem robotic and suspicious.

Oswald Spengler’s The Decline of the West as well as McGilchrist characterize the West as awash in phenomena associated with an extreme LH emphasis. Spengler argues that Western civilization was originally much more RH (to use McGilchrist’s categories) and that all its most significant artistic (in the broadest sense) achievements were triumphs of RH accentuation.

The RH is where novel experiences and the anomalous are processed and where mathematical, and other, problems are solved. The RH is involved with the natural, the unfamiliar, the unique, emotions, the embodied, music, humor, understanding intonation and emotional nuance of speech, the metaphorical, nuance, and social relations. It has very little speech, but the RH is necessary for processing all the nonlinguistic aspects of speaking, including body language. Understanding what someone means by vocal inflection and facial expressions is an intuitive RH process rather than explicit.

...

RH is very much the center of lived experience; of the life world with all its depth and richness. The RH is “the master” from the title of McGilchrist’s book. The LH ought to be no more than the emissary; the valued servant of the RH. However, in the last few centuries, the LH, which has tyrannical tendencies, has tried to become the master. The LH is where the ego is predominantly located. In split brain patients where the LH and the RH are surgically divided (this is done sometimes in the case of epileptic patients) one hand will sometimes fight with the other. In one man’s case, one hand would reach out to hug his wife while the other pushed her away. One hand reached for one shirt, the other another shirt. Or a patient will be driving a car and one hand will try to turn the steering wheel in the opposite direction. In these cases, the “naughty” hand is usually the left hand (RH), while the patient tends to identify herself with the right hand governed by the LH. The two hemispheres have quite different personalities.

The connection between LH and ego can also be seen in the fact that the LH is competitive, contentious, and agonistic. It wants to win. It is the part of you that hates to lose arguments.

Using the metaphor of Chaos and Order, the RH deals with Chaos – the unknown, the unfamiliar, the implicit, the emotional, the dark, danger, mystery. The LH is connected with Order – the known, the familiar, the rule-driven, the explicit, and light of day. Learning something means to take something unfamiliar and making it familiar. Since the RH deals with the novel, it is the problem-solving part. Once understood, the results are dealt with by the LH. When learning a new piece on the piano, the RH is involved. Once mastered, the result becomes a LH affair. The muscle memory developed by repetition is processed by the LH. If errors are made, the activity returns to the RH to figure out what went wrong; the activity is repeated until the correct muscle memory is developed in which case it becomes part of the familiar LH.

Science is an attempt to find Order. It would not be necessary if people lived in an entirely orderly, explicit, known world. The lived context of science implies Chaos. Theories are reductive and simplifying and help to pick out salient features of a phenomenon. They are always partial truths, though some are more partial than others. The alternative to a certain level of reductionism or partialness would be to simply reproduce the world which of course would be both impossible and unproductive. The test for whether a theory is sufficiently non-partial is whether it is fit for purpose and whether it contributes to human flourishing.

...

Analytic philosophers pride themselves on trying to do away with vagueness. To do so, they tend to jettison context which cannot be brought into fine focus. However, in order to understand things and discern their meaning, it is necessary to have the big picture, the overview, as well as the details. There is no point in having details if the subject does not know what they are details of. Such philosophers also tend to leave themselves out of the picture even when what they are thinking about has reflexive implications. John Locke, for instance, tried to banish the RH from reality. All phenomena having to do with subjective experience he deemed unreal and once remarked about metaphors, a RH phenomenon, that they are “perfect cheats.” Analytic philosophers tend to check the logic of the words on the page and not to think about what those words might say about them. The trick is for them to recognize that they and their theories, which exist in minds, are part of reality too.

The RH test for whether someone actually believes something can be found by examining his actions. If he finds that he must regard his own actions as free, and, in order to get along with other people, must also attribute free will to them and treat them as free agents, then he effectively believes in free will – no matter his LH theoretical commitments.

...

We do not know the origin of life. We do not know how or even if consciousness can emerge from matter. We do not know the nature of 96% of the matter of the universe. Clearly all these things exist. They can provide the subject matter of theories but they continue to exist as theorizing ceases or theories change. Not knowing how something is possible is irrelevant to its actual existence. An inability to explain something is ultimately neither here nor there.

If thought begins and ends with the LH, then thinking has no content – content being provided by experience (RH), and skepticism and nihilism ensue. The LH spins its wheels self-referentially, never referring back to experience. Theory assumes such primacy that it will simply outlaw experiences and data inconsistent with it; a profoundly wrong-headed approach.

...

Gödel’s Theorem proves that not everything true can be proven to be true. This means there is an ineradicable role for faith, hope and intuition in every moderately complex human intellectual endeavor. There is no one set of consistent axioms from which all other truths can be derived.

Alan Turing’s proof of the halting problem proves that there is no effective procedure for finding effective procedures. Without a mechanical decision procedure, (LH), when it comes to … [more]
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september 2018 by nhaliday
Who We Are | West Hunter
I’m going to review David Reich’s new book, Who We Are and How We Got Here. Extensively: in a sense I’ve already been doing this for a long time. Probably there will be a podcast. The GoFundMe link is here. You can also send money via Paypal (Use the donate button), or bitcoins to 1Jv4cu1wETM5Xs9unjKbDbCrRF2mrjWXr5. In-kind donations, such as orichalcum or mithril, are always appreciated.

This is the book about the application of ancient DNA to prehistory and history.

height difference between northern and southern europeans: https://westhunt.wordpress.com/2018/03/29/who-we-are-1/
mixing, genocide of males, etc.: https://westhunt.wordpress.com/2018/03/29/who-we-are-2-purity-of-essence/
rapid change in polygenic traits (appearance by Kevin Mitchell and funny jab at Brad Delong ("regmonkey")): https://westhunt.wordpress.com/2018/03/30/rapid-change-in-polygenic-traits/
schiz, bipolar, and IQ: https://westhunt.wordpress.com/2018/03/30/rapid-change-in-polygenic-traits/#comment-105605
Dan Graur being dumb: https://westhunt.wordpress.com/2018/04/02/the-usual-suspects/
prediction of neanderthal mixture and why: https://westhunt.wordpress.com/2018/04/03/who-we-are-3-neanderthals/
New Guineans tried to use Denisovan admixture to avoid UN sanctions (by "not being human"): https://westhunt.wordpress.com/2018/04/04/who-we-are-4-denisovans/
also some commentary on decline of Out-of-Africa, including:
"Homo Naledi, a small-brained homonin identified from recently discovered fossils in South Africa, appears to have hung around way later that you’d expect (up to 200,000 years ago, maybe later) than would be the case if modern humans had occupied that area back then. To be blunt, we would have eaten them."

Live Not By Lies: https://westhunt.wordpress.com/2018/04/08/live-not-by-lies/
Next he slams people that suspect that upcoming genetic genetic analysis will, in most cases, confirm traditional stereotypes about race – the way the world actually looks.

The people Reich dumps on are saying perfectly reasonable things. He criticizes Henry Harpending for saying that he’d never seen an African with a hobby. Of course, Henry had actually spent time in Africa, and that’s what he’d seen. The implication is that people in Malthusian farming societies – which Africa was not – were selected to want to work, even where there was no immediate necessity to do so. Thus hobbies, something like a gerbil running in an exercise wheel.

He criticized Nicholas Wade, for saying that different races have different dispositions. Wade’s book wasn’t very good, but of course personality varies by race: Darwin certainly thought so. You can see differences at birth. Cover a baby’s nose with a cloth: Chinese and Navajo babies quietly breathe through their mouth, European and African babies fuss and fight.

Then he attacks Watson, for asking when Reich was going to look at Jewish genetics – the kind that has led to greater-than-average intelligence. Watson was undoubtedly trying to get a rise out of Reich, but it’s a perfectly reasonable question. Ashkenazi Jews are smarter than the average bear and everybody knows it. Selection is the only possible explanation, and the conditions in the Middle ages – white-collar job specialization and a high degree of endogamy, were just what the doctor ordered.

Watson’s a prick, but he’s a great prick, and what he said was correct. Henry was a prince among men, and Nick Wade is a decent guy as well. Reich is totally out of line here: he’s being a dick.

Now Reich may be trying to burnish his anti-racist credentials, which surely need some renewal after having pointing out that race as colloquially used is pretty reasonable, there’s no reason pops can’t be different, people that said otherwise ( like Lewontin, Gould, Montagu, etc. ) were lying, Aryans conquered Europe and India, while we’re tied to the train tracks with scary genetic results coming straight at us. I don’t care: he’s being a weasel, slandering the dead and abusing the obnoxious old genius who laid the foundations of his field. Reich will also get old someday: perhaps he too will someday lose track of all the nonsense he’s supposed to say, or just stop caring. Maybe he already has… I’m pretty sure that Reich does not like lying – which is why he wrote this section of the book (not at all logically necessary for his exposition of the ancient DNA work) but the required complex juggling of lies and truth required to get past the demented gatekeepers of our society may not be his forte. It has been said that if it was discovered that someone in the business was secretly an android, David Reich would be the prime suspect. No Talleyrand he.

https://westhunt.wordpress.com/2018/04/12/who-we-are-6-the-americas/
The population that accounts for the vast majority of Native American ancestry, which we will call Amerinds, came into existence somewhere in northern Asia. It was formed from a mix of Ancient North Eurasians and a population related to the Han Chinese – about 40% ANE and 60% proto-Chinese. Is looks as if most of the paternal ancestry was from the ANE, while almost all of the maternal ancestry was from the proto-Han. [Aryan-Transpacific ?!?] This formation story – ANE boys, East-end girls – is similar to the formation story for the Indo-Europeans.

https://westhunt.wordpress.com/2018/04/18/who-we-are-7-africa/
In some ways, on some questions, learning more from genetics has left us less certain. At this point we really don’t know where anatomically humans originated. Greater genetic variety in sub-Saharan African has been traditionally considered a sign that AMH originated there, but it possible that we originated elsewhere, perhaps in North Africa or the Middle East, and gained extra genetic variation when we moved into sub-Saharan Africa and mixed with various archaic groups that already existed. One consideration is that finding recent archaic admixture in a population may well be a sign that modern humans didn’t arise in that region ( like language substrates) – which makes South Africa and West Africa look less likely. The long-continued existence of homo naledi in South Africa suggests that modern humans may not have been there for all that long – if we had co-existed with homo naledi, they probably wouldn’t lasted long. The oldest known skull that is (probably) AMh was recently found in Morocco, while modern humans remains, already known from about 100,000 years ago in Israel, have recently been found in northern Saudi Arabia.

While work by Nick Patterson suggests that modern humans were formed by a fusion between two long-isolated populations, a bit less than half a million years ago.

So: genomics had made recent history Africa pretty clear. Bantu agriculuralists expanded and replaced hunter-gatherers, farmers and herders from the Middle East settled North Africa, Egypt and northeaat Africa, while Nilotic herdsmen expanded south from the Sudan. There are traces of earlier patterns and peoples, but today, only traces. As for questions back further in time, such as the origins of modern humans – we thought we knew, and now we know we don’t. But that’s progress.

https://westhunt.wordpress.com/2018/04/18/reichs-journey/
David Reich’s professional path must have shaped his perspective on the social sciences. Look at the record. He starts his professional career examining the role of genetics in the elevated prostate cancer risk seen in African-American men. Various social-science fruitcakes oppose him even looking at the question of ancestry ( African vs European). But they were wrong: certain African-origin alleles explain the increased risk. Anthropologists (and human geneticists) were sure (based on nothing) that modern humans hadn’t interbred with Neanderthals – but of course that happened. Anthropologists and archaeologists knew that Gustaf Kossina couldn’t have been right when he said that widespread material culture corresponded to widespread ethnic groups, and that migration was the primary explanation for changes in the archaeological record – but he was right. They knew that the Indo-European languages just couldn’t have been imposed by fire and sword – but Reich’s work proved them wrong. Lots of people – the usual suspects plus Hindu nationalists – were sure that the AIT ( Aryan Invasion Theory) was wrong, but it looks pretty good today.

Some sociologists believed that caste in India was somehow imposed or significantly intensified by the British – but it turns out that most jatis have been almost perfectly endogamous for two thousand years or more…

It may be that Reich doesn’t take these guys too seriously anymore. Why should he?

varnas, jatis, aryan invastion theory: https://westhunt.wordpress.com/2018/04/22/who-we-are-8-india/

europe and EEF+WHG+ANE: https://westhunt.wordpress.com/2018/05/01/who-we-are-9-europe/

https://www.nationalreview.com/2018/03/book-review-david-reich-human-genes-reveal-history/
The massive mixture events that occurred in the recent past to give rise to Europeans and South Asians, to name just two groups, were likely “male mediated.” That’s another way of saying that men on the move took local women as brides or concubines. In the New World there are many examples of this, whether it be among African Americans, where most European ancestry seems to come through men, or in Latin America, where conquistadores famously took local women as paramours. Both of these examples are disquieting, and hint at the deep structural roots of patriarchal inequality and social subjugation that form the backdrop for the emergence of many modern peoples.
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march 2018 by nhaliday
Resonance in a Pendulum - YouTube
The vibration of any given washer is able to transmit its energy only to another washer with exactly the same frequency. Since the length of a pendulum determines its frequency of vibration, each pendulum can only set another pendulum vibrating if it has the same length.
nibble  video  social  physics  mechanics  waves  oscillation  synchrony  flux-stasis  increase-decrease  concrete  ground-up  dirty-hands  phys-energy  frequency  spreading 
september 2017 by nhaliday
Suspicious Banana on Twitter: ""platonic forms" seem more sinister when you realize that integers were reaching down into his head and giving him city planning advice https://t.co/4qaTdwOlry"
https://en.wikipedia.org/wiki/5040_(number)
Plato mentions in his Laws that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a state) into lesser parts. He remarks that this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11 (however, it is not the smallest number to have this property; 2520 is). He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is divisible by 11. Plato also took notice of the fact that 5040 can be divided by 12 twice over. Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that it is written, "Plato, writing under Pythagorean influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on justice and moderation."[1]

https://en.wikipedia.org/wiki/Plato%27s_number
"Now for divine begettings there is a period comprehended by a perfect number, and for mortal by the first in which augmentations dominating and dominated when they have attained to three distances and four limits of the assimilating and the dissimilating, the waxing and the waning, render all things conversable and commensurable [546c] with one another, whereof a basal four-thirds wedded to the pempad yields two harmonies at the third augmentation, the one the product of equal factors taken one hundred times, the other of equal length one way but oblong,-one dimension of a hundred numbers determined by the rational diameters of the pempad lacking one in each case, or of the irrational lacking two; the other dimension of a hundred cubes of the triad. And this entire geometrical number is determinative of this thing, of better and inferior births."[3]

Shortly after Plato's time his meaning apparently did not cause puzzlement as Aristotle's casual remark attests.[6] Half a millennium later, however, it was an enigma for the Neoplatonists, who had a somewhat mystic penchant and wrote frequently about it, proposing geometrical and numerical interpretations. Next, for nearly a thousand years, Plato's texts disappeared and it is only in the Renaissance that the enigma briefly resurfaced. During the 19th century, when classical scholars restored original texts, the problem reappeared. Schleiermacher interrupted his edition of Plato for a decade while attempting to make sense of the paragraph. Victor Cousin inserted a note that it has to be skipped in his French translation of Plato's works. In the early 20th century, scholarly findings suggested a Babylonian origin for the topic.[7]

https://en.wikipedia.org/wiki/Pythagoreanism
https://www.jstor.org/stable/638781

Socrates: Surely we agree nothing more virtuous than sacrificing each newborn infant while reciting the factors of 39,916,800?

Turgidas: Uh

different but interesting: https://aeon.co/essays/can-we-hope-to-understand-how-the-greeks-saw-their-world
Another explanation for the apparent oddness of Greek perception came from the eminent politician and Hellenist William Gladstone, who devoted a chapter of his Studies on Homer and the Homeric Age (1858) to ‘perceptions and use of colour’. He too noticed the vagueness of the green and blue designations in Homer, as well as the absence of words covering the centre of the ‘blue’ area. Where Gladstone differed was in taking as normative the Newtonian list of colours (red, orange, yellow, green, blue, indigo, violet). He interpreted the Greeks’ supposed linguistic poverty as deriving from an imperfect discrimination of prismatic colours. The visual organ of the ancients was still in its infancy, hence their strong sensitivity to light rather than hue, and the related inability to clearly distinguish one hue from another. This argument fit well with the post-Darwinian climate of the late 19th century, and came to be widely believed. Indeed, it prompted Nietzsche’s own judgment, and led to a series of investigations that sought to prove that the Greek chromatic categories do not fit in with modern taxonomies.

Today, no one thinks that there has been a stage in the history of humanity when some colours were ‘not yet’ being perceived. But thanks to our modern ‘anthropological gaze’ it is accepted that every culture has its own way of naming and categorising colours. This is not due to varying anatomical structures of the human eye, but to the fact that different ocular areas are stimulated, which triggers different emotional responses, all according to different cultural contexts.
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june 2017 by nhaliday
Why I see academic economics moving left | askblog
http://www.arnoldkling.com/blog/on-the-state-of-economics/
http://www.nationalaffairs.com/publications/detail/how-effective-is-economic-theory
I have a long essay on the scientific status of economics in National Affairs. A few excerpts from the conclusion:

In the end, can we really have effective theory in economics? If by effective theory we mean theory that is verifiable and reliable for prediction and control, the answer is likely no. Instead, economics deals in speculative interpretations and must continue to do so.

Young economists who employ pluralistic methods to study problems are admired rather than marginalized, as they were in 1980. But economists who question the wisdom of interventionist economic policies seem headed toward the fringes of the profession.

This is my essay in which I say that academic economics is on the road to sociology.

example...?:
Property Is Only Another Name for Monopoly: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2818494
Hanson's take more positive: http://www.overcomingbias.com/2017/10/for-stability-rents.html

women:
http://www.arnoldkling.com/blog/college-women-and-the-future-of-economics/
http://www.arnoldkling.com/blog/road-to-sociology-watch-2/
http://www.arnoldkling.com/blog/road-to-sociology-watch-3/
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may 2017 by nhaliday
pr.probability - What is convolution intuitively? - MathOverflow
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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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
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
soft question - How do you not forget old math? - MathOverflow
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-d‌​one

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.
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june 2016 by nhaliday
For potential Ph.D. students
Ravi Vakil's advice for PhD students

General advice:
Think actively about the creative process. A subtle leap is required from undergraduate thinking to active research (even if you have done undergraduate research). Think explicitly about the process, and talk about it (with me, and with others). For example, in an undergraduate class any Ph.D. student at Stanford will have tried to learn absolutely all the material flawlessly. But in order to know everything needed to tackle an important problem on the frontier of human knowledge, one would have to spend years reading many books and articles. So you'll have to learn differently. But how?

Don't be narrow and concentrate only on your particular problem. Learn things from all over the field, and beyond. The facts, methods, and insights from elsewhere will be much more useful than you might realize, possibly in your thesis, and most definitely afterwards. Being broad is a good way of learning to develop interesting questions.

When you learn the theory, you should try to calculate some toy cases, and think of some explicit basic examples.

Talk to other graduate students. A lot. Organize reading groups. Also talk to post-docs, faculty, visitors, and people you run into on the street. I learn the most from talking with other people. Maybe that's true for you too.

Specific topics:
- seminars
- giving talks
- writing
- links to other advice
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may 2016 by nhaliday

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