Does Learning to Read Improve Intelligence? A Longitudinal Multivariate Analysis in Identical Twins From Age 7 to 16

september 2017 by nhaliday

Stuart Richie, Bates, Plomin

SEM: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4354297/figure/fig03/

The variance explained by each path in the diagrams included here can be calculated by squaring its path weight. To take one example, reading differences at age 12 in the model shown in FigureFigure33 explain 7% of intelligence differences at age 16 (.262). However, since our measures are of differences, they are likely to include substantial amounts of noise: Measurement error may produce spurious differences. To remove this error variance, we can take an estimate of the reliability of the measures (generally high, since our measures are normed, standardized tests), which indicates the variance expected purely by the reliability of the measure, and subtract it from the observed variance between twins in our sample. Correcting for reliability in this way, the effect size estimates are somewhat larger; to take the above example, the reliability-corrected effect size of age 12 reading differences on age 16 intelligence differences is around 13% of the “signal” variance. It should be noted that the age 12 reading differences themselves are influenced by many previous paths from both reading and intelligence, as illustrated in FigureFigure33.

...

The present study provided compelling evidence that improvements in reading ability, themselves caused purely by the nonshared environment, may result in improvements in both verbal and nonverbal cognitive ability, and may thus be a factor increasing cognitive diversity within families (Plomin, 2011). These associations are present at least as early as age 7, and are not—to the extent we were able to test this possibility—driven by differences in reading exposure. Since reading is a potentially remediable ability, these findings have implications for reading instruction: Early remediation of reading problems might not only aid in the growth of literacy, but may also improve more general cognitive abilities that are of critical importance across the life span.

Does Reading Cause Later Intelligence? Accounting for Stability in Models of Change: http://sci-hub.tw/10.1111/cdev.12669

Results from a state–trait model suggest that reported effects of reading ability on later intelligence may be artifacts of previously uncontrolled factors, both environmental in origin and stable during this developmental period, influencing both constructs throughout development.

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psych-architecture
longitudinal
twin-study
developmental
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studying
🌞
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signal-noise
intervention
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graphs
graphical-models
flexibility
britain
neuro-nitgrit
effect-size
variance-components
measurement
multi
sequential
time
composition-decomposition
biodet
behavioral-gen
direct-indirect
systematic-ad-hoc
debate
hmm
pdf
piracy
flux-stasis
SEM: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4354297/figure/fig03/

The variance explained by each path in the diagrams included here can be calculated by squaring its path weight. To take one example, reading differences at age 12 in the model shown in FigureFigure33 explain 7% of intelligence differences at age 16 (.262). However, since our measures are of differences, they are likely to include substantial amounts of noise: Measurement error may produce spurious differences. To remove this error variance, we can take an estimate of the reliability of the measures (generally high, since our measures are normed, standardized tests), which indicates the variance expected purely by the reliability of the measure, and subtract it from the observed variance between twins in our sample. Correcting for reliability in this way, the effect size estimates are somewhat larger; to take the above example, the reliability-corrected effect size of age 12 reading differences on age 16 intelligence differences is around 13% of the “signal” variance. It should be noted that the age 12 reading differences themselves are influenced by many previous paths from both reading and intelligence, as illustrated in FigureFigure33.

...

The present study provided compelling evidence that improvements in reading ability, themselves caused purely by the nonshared environment, may result in improvements in both verbal and nonverbal cognitive ability, and may thus be a factor increasing cognitive diversity within families (Plomin, 2011). These associations are present at least as early as age 7, and are not—to the extent we were able to test this possibility—driven by differences in reading exposure. Since reading is a potentially remediable ability, these findings have implications for reading instruction: Early remediation of reading problems might not only aid in the growth of literacy, but may also improve more general cognitive abilities that are of critical importance across the life span.

Does Reading Cause Later Intelligence? Accounting for Stability in Models of Change: http://sci-hub.tw/10.1111/cdev.12669

Results from a state–trait model suggest that reported effects of reading ability on later intelligence may be artifacts of previously uncontrolled factors, both environmental in origin and stable during this developmental period, influencing both constructs throughout development.

september 2017 by nhaliday

spaceships - Can there be a space age without petroleum (crude oil)? - Worldbuilding Stack Exchange

june 2017 by nhaliday

Yes...probably

What was really important to our development of technology was not oil, but coal. Access to large deposits of high-quality coal largely fueled the industrial revolution, and it was the industrial revolution that really got us on the first rungs of the technological ladder.

Oil is a fantastic fuel for an advanced civilisation, but it's not essential. Indeed, I would argue that our ability to dig oil out of the ground is a crutch, one that we should have discarded long ago. The reason oil is so essential to us today is that all our infrastructure is based on it, but if we'd never had oil we could still have built a similar infrastructure. Solar power was first displayed to the public in 1878. Wind power has been used for centuries. Hydroelectric power is just a modification of the same technology as wind power.

Without oil, a civilisation in the industrial age would certainly be able to progress and advance to the space age. Perhaps not as quickly as we did, but probably more sustainably.

Without coal, though...that's another matter

What would the industrial age be like without oil and coal?: https://worldbuilding.stackexchange.com/questions/45919/what-would-the-industrial-age-be-like-without-oil-and-coal

Out of the ashes: https://aeon.co/essays/could-we-reboot-a-modern-civilisation-without-fossil-fuels

It took a lot of fossil fuels to forge our industrial world. Now they're almost gone. Could we do it again without them?

But charcoal-based industry didn’t die out altogether. In fact, it survived to flourish in Brazil. Because it has substantial iron deposits but few coalmines, Brazil is the largest charcoal producer in the world and the ninth biggest steel producer. We aren’t talking about a cottage industry here, and this makes Brazil a very encouraging example for our thought experiment.

The trees used in Brazil’s charcoal industry are mainly fast-growing eucalyptus, cultivated specifically for the purpose. The traditional method for creating charcoal is to pile chopped staves of air-dried timber into a great dome-shaped mound and then cover it with turf or soil to restrict airflow as the wood smoulders. The Brazilian enterprise has scaled up this traditional craft to an industrial operation. Dried timber is stacked into squat, cylindrical kilns, built of brick or masonry and arranged in long lines so that they can be easily filled and unloaded in sequence. The largest sites can sport hundreds of such kilns. Once filled, their entrances are sealed and a fire is lit from the top.

q-n-a
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news
org:mag
org:popup
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retrofit
dirty-hands
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duplication
iteration-recursion
latin-america
track-record
trivia
cocktail
data
What was really important to our development of technology was not oil, but coal. Access to large deposits of high-quality coal largely fueled the industrial revolution, and it was the industrial revolution that really got us on the first rungs of the technological ladder.

Oil is a fantastic fuel for an advanced civilisation, but it's not essential. Indeed, I would argue that our ability to dig oil out of the ground is a crutch, one that we should have discarded long ago. The reason oil is so essential to us today is that all our infrastructure is based on it, but if we'd never had oil we could still have built a similar infrastructure. Solar power was first displayed to the public in 1878. Wind power has been used for centuries. Hydroelectric power is just a modification of the same technology as wind power.

Without oil, a civilisation in the industrial age would certainly be able to progress and advance to the space age. Perhaps not as quickly as we did, but probably more sustainably.

Without coal, though...that's another matter

What would the industrial age be like without oil and coal?: https://worldbuilding.stackexchange.com/questions/45919/what-would-the-industrial-age-be-like-without-oil-and-coal

Out of the ashes: https://aeon.co/essays/could-we-reboot-a-modern-civilisation-without-fossil-fuels

It took a lot of fossil fuels to forge our industrial world. Now they're almost gone. Could we do it again without them?

But charcoal-based industry didn’t die out altogether. In fact, it survived to flourish in Brazil. Because it has substantial iron deposits but few coalmines, Brazil is the largest charcoal producer in the world and the ninth biggest steel producer. We aren’t talking about a cottage industry here, and this makes Brazil a very encouraging example for our thought experiment.

The trees used in Brazil’s charcoal industry are mainly fast-growing eucalyptus, cultivated specifically for the purpose. The traditional method for creating charcoal is to pile chopped staves of air-dried timber into a great dome-shaped mound and then cover it with turf or soil to restrict airflow as the wood smoulders. The Brazilian enterprise has scaled up this traditional craft to an industrial operation. Dried timber is stacked into squat, cylindrical kilns, built of brick or masonry and arranged in long lines so that they can be easily filled and unloaded in sequence. The largest sites can sport hundreds of such kilns. Once filled, their entrances are sealed and a fire is lit from the top.

june 2017 by nhaliday

Antibiotic feed/food supplementation | West Hunter

may 2017 by nhaliday

Many domesticated animals show increased growth and improved feed efficiency when given low doses of antibiotics. In fact, this is by far the biggest use of antibiotics. Mostly you hear about this in the context of worries about how this may select for resistant bacteria (which may well be true), but one interesting question is why it even works – and what other applications this technique might have.

It strikes me that it might be useful in food emergencies – famines and so forth. The dosage is low (200 g per ton) and can increase feed efficiency over 10% in some cases. Assuming that antibiotic supplementation works in humans (which is likely, considering that it works in a wide spectrum of domestic animals), you might be able to save 5 or 10% more people with a given food supply. Now if we ever bothered to learn exactly how this works, we might be able to find an equivalent approach that didn’t use antibiotics, some other way of knocking out certain pathogens (phage therapy?) or altering the gut flora.

west-hunter
scitariat
discussion
speculation
ideas
agriculture
food
efficiency
disease
parasites-microbiome
medicine
drugs
pharma
retrofit
questions
dirty-hands
It strikes me that it might be useful in food emergencies – famines and so forth. The dosage is low (200 g per ton) and can increase feed efficiency over 10% in some cases. Assuming that antibiotic supplementation works in humans (which is likely, considering that it works in a wide spectrum of domestic animals), you might be able to save 5 or 10% more people with a given food supply. Now if we ever bothered to learn exactly how this works, we might be able to find an equivalent approach that didn’t use antibiotics, some other way of knocking out certain pathogens (phage therapy?) or altering the gut flora.

may 2017 by nhaliday

One more time | West Hunter

may 2017 by nhaliday

One of our local error sources suggested that it would be impossible to rebuild technical civilization, once fallen. Now if every human were dead I’d agree, but in most other scenarios it wouldn’t be particularly difficult, assuming that the survivors were no more silly and fractious than people are today. So assume a mild disaster, something like the effect of myxomatosis on the rabbits of Australia, or perhaps toe-to-toe nuclear combat with the Russkis – ~90% casualties worldwide.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69221

Books are everywhere. In the type of scenario I sketched out, almost no knowledge would be lost – so Neolithic tech is irrelevant. Look, if a single copy of the 1911 Britannica survived, all would be well.

You could of course harvest metals from the old cities. But even if if you didn’t, the idea that there is no more copper or zinc or tin in the ground is just silly. “recoverable ore” is mostly an economic concept.

Moreover, if we’re talking wiring and electrical uses, one can use aluminum, which makes up 8% of the Earth’s crust.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69368

Some of those book tell you how to win.

Look, assume that some communities strive to relearn how to make automatic weapons and some don’t. How does that story end? Do I have to explain everything?

I guess so!

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69334

Well, perhaps having a zillion times more books around would make a difference. That and all the “X for Dummies” books, which I think the Romans didn’t have.

A lot of Classical civ wasn’t very useful: on the whole they didn’t invent much. On the whole, technology advanced quite a bit more rapidly in Medieval times.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69225

How much coal and oil are in the ground that can still be extracted with 19th century tech? Honest question; I don’t know.

--

Lots of coal left. Not so much oil (using simple methods), but one could make it from low-grade coal, with the Fischer-Tropsch process. Sasol does this.

Then again, a recovering society wouldn’t need much at first.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69223

reply to: https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69220

That’s more like it.

#1. Consider Grand Coulee Dam. Gigawatts. Feeling of power!

#2. Of course.

#3. Might be easier to make superconducting logic circuits with MgB2, starting over.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69325

Your typical biker guy is more mechanically minded than the average Joe. Welding, electrical stuff, this and that.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69260

If fossil fuels were unavailable -or just uneconomical at first- we’d be back to charcoal for our Stanley Steamers and railroads. We’d still have both.

The French, and others, used wood-gasifier trucks during WWII.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69407

Teslas are of course a joke.

west-hunter
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civilization
risk
nihil
gedanken
frontier
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energy-resources
knowledge
the-world-is-just-atoms
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analysis
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gallic
track-record
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barons
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driving
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agriculture
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dirty-hands
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war
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comparison
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class
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encyclopedic
definite-planning
embodied
gnosis-logos
kumbaya-kult
https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69221

Books are everywhere. In the type of scenario I sketched out, almost no knowledge would be lost – so Neolithic tech is irrelevant. Look, if a single copy of the 1911 Britannica survived, all would be well.

You could of course harvest metals from the old cities. But even if if you didn’t, the idea that there is no more copper or zinc or tin in the ground is just silly. “recoverable ore” is mostly an economic concept.

Moreover, if we’re talking wiring and electrical uses, one can use aluminum, which makes up 8% of the Earth’s crust.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69368

Some of those book tell you how to win.

Look, assume that some communities strive to relearn how to make automatic weapons and some don’t. How does that story end? Do I have to explain everything?

I guess so!

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69334

Well, perhaps having a zillion times more books around would make a difference. That and all the “X for Dummies” books, which I think the Romans didn’t have.

A lot of Classical civ wasn’t very useful: on the whole they didn’t invent much. On the whole, technology advanced quite a bit more rapidly in Medieval times.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69225

How much coal and oil are in the ground that can still be extracted with 19th century tech? Honest question; I don’t know.

--

Lots of coal left. Not so much oil (using simple methods), but one could make it from low-grade coal, with the Fischer-Tropsch process. Sasol does this.

Then again, a recovering society wouldn’t need much at first.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69223

reply to: https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69220

That’s more like it.

#1. Consider Grand Coulee Dam. Gigawatts. Feeling of power!

#2. Of course.

#3. Might be easier to make superconducting logic circuits with MgB2, starting over.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69325

Your typical biker guy is more mechanically minded than the average Joe. Welding, electrical stuff, this and that.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69260

If fossil fuels were unavailable -or just uneconomical at first- we’d be back to charcoal for our Stanley Steamers and railroads. We’d still have both.

The French, and others, used wood-gasifier trucks during WWII.

https://westhunt.wordpress.com/2015/05/17/one-more-time/#comment-69407

Teslas are of course a joke.

may 2017 by nhaliday

Extended spider cognition | SpringerLink

april 2017 by nhaliday

Spiders do not seem to be cognitively limited, displaying a large diversity of learning processes, from habituation to contextual learning, including a sense of numerosity. To tease apart the central from the extended cognition, we apply the mutual manipulability criterion, testing the existence of reciprocal causal links between the putative elements of the system. We conclude that the web threads and configurations are integral parts of the cognitive systems. The extension of cognition to the web helps to explain some puzzling features of spider behaviour and seems to promote evolvability within the group, enhancing innovation through cognitive connectivity to variable habitat features. Graded changes in relative brain size could also be explained by outsourcing information processing to environmental features. More generally, niche-constructed structures emerge as prime candidates for extending animal cognition, generating the selective pressures that help to shape the evolving cognitive system.

https://www.quantamagazine.org/the-thoughts-of-a-spiderweb-20170523/

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org:sci
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summary
nibble
org:inst
https://www.quantamagazine.org/the-thoughts-of-a-spiderweb-20170523/

april 2017 by nhaliday

Cat Ladies | West Hunter

march 2017 by nhaliday

hmm...:

If toxo naturally can make people like cat piss, it’s already preadapted to become (with suitable genetic engineering) the model system for many kinds of infectious behavior modifiers.

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parasites-microbiome
disease
toxo-gondii
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epidemiology
drugs
pharma
If toxo naturally can make people like cat piss, it’s already preadapted to become (with suitable genetic engineering) the model system for many kinds of infectious behavior modifiers.

march 2017 by nhaliday

Origins of the brain networks for advanced mathematics in expert mathematicians

february 2017 by nhaliday

The origins of human abilities for mathematics are debated: Some theories suggest that they are founded upon evolutionarily ancient brain circuits for number and space and others that they are grounded in language competence. To evaluate what brain systems underlie higher mathematics, we scanned professional mathematicians and mathematically naive subjects of equal academic standing as they evaluated the truth of advanced mathematical and nonmathematical statements. In professional mathematicians only, mathematical statements, whether in algebra, analysis, topology or geometry, activated a reproducible set of bilateral frontal, Intraparietal, and ventrolateral temporal regions. Crucially, these activations spared areas related to language and to general-knowledge semantics. Rather, mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians, with a corresponding reduction in nearby face responses. The evidence suggests that high-level mathematical expertise and basic number sense share common roots in a nonlinguistic brain circuit.

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february 2017 by nhaliday

The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers

february 2017 by nhaliday

The child’s acquisition of language has been suggested to rely on the ability to build hierarchically structured representations from sequential inputs. Does a similar mechanism also underlie the acquisition of geometrical rules? Here, we introduce a learning situation in which human participants had to grasp simple spatial sequences and try to predict the next location. Sequences were generated according to a “geometrical language” endowed with simple primitives of symmetries and rotations, and combinatorial rules. Analyses of error rates of various populations—a group of French educated adults, two groups of 5 years-old French children, and a rare group of teenagers and adults from an Amazonian population, the Mundurukus, who have limited access to formal schooling and a reduced geometrical lexicon—revealed that subjects’ learning indeed rests on internal language-like representations. A theoretical model, based on minimum description length, proved to fit well participants’ behavior, suggesting that human subjects “compress” spatial sequences into a minimal internal rule or program.

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february 2017 by nhaliday

The Learning Brain: Neuronal Recycling and Inhibition: Zeitschrift für Psychologie: Vol 224, No 4

february 2017 by nhaliday

Reading is an example of complex learning specific to human beings. In readers, an area of the brain is dedicated to the visual processing of letters and words, referred to as the visual word form area (VWFA). The existence of this brain area is paradoxical. Reading is too recent to be a phylogenic product of Darwinian evolution. It likely develops with intense school training via a neuroplastic ontogenic process of neuronal recycling: neurons in the lateral occipitotemporal lobe originally tuned to the visual recognition of stimuli, such as faces, objects, and animals, will be recycled for the visual recognition of letters and words. Thus, the VWFA inherits the intrinsic properties of these neurons, notably, mirror generalization, a process (or heuristic) applied to all visual stimuli that enables the recognition of a stimulus irrespective of its left-right orientation. On its own, this inherited property is not adapted to reading because it makes children confuse mirror letters, such as b and d in the Latin alphabet. In this article, we present evidence that inhibitory control is critical to avoid mirror errors inherited from the neuronal recycling process by blocking the mirror generalization heuristic in the context of reading. We subsequently argue that the “neuronal recycling + inhibitory control” law constitutes a general law of the learning brain by demonstrating that it may also account for the development of numeracy.

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february 2017 by nhaliday

cc.complexity theory - What is the complexity class most closely associated with what the human mind can accomplish quickly? - Theoretical Computer Science Stack Exchange

q-n-a overflow tcs neuro psychology cog-psych cool interdisciplinary soft-question complexity neurons rigor vague nibble thinking applications retrofit computation

january 2017 by nhaliday

q-n-a overflow tcs neuro psychology cog-psych cool interdisciplinary soft-question complexity neurons rigor vague nibble thinking applications retrofit computation

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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atoms
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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
👳
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giants
cartoons
gowers
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stories
the-trenches
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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
- 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".

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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

Alon Amit's answer to In an online lecture, a professor mentioned that Einstein could draw or imagine a 4-dimensional figure. How can one possibly do that? - Quora

september 2016 by nhaliday

like the idea of treating color as 4th dimension

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september 2016 by nhaliday

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