**nhaliday + heavyweights + reflection**
21

The Future of Mathematics? [video] | Hacker News

8 days ago by nhaliday

https://news.ycombinator.com/item?id=20909404

Kevin Buzzard (the Lean guy)

- general reflection on proof asssistants/theorem provers

- Kevin Hale's formal abstracts project, etc

- thinks of available theorem provers, Lean is "[the only one currently available that may be capable of formalizing all of mathematics eventually]" (goes into more detail right at he end)

hn
commentary
discussion
video
talks
presentation
math
formal-methods
expert-experience
msr
frontier
state-of-art
proofs
rigor
education
higher-ed
optimism
prediction
lens
search
meta:research
speculation
exocortex
skunkworks
automation
research
math.NT
big-surf
software
parsimony
cost-benefit
intricacy
correctness
programming
pls
python
functional
haskell
heavyweights
research-program
review
reflection
multi
pdf
slides
oly
experiment
span-cover
git
vcs
teaching
impetus
academia
composition-decomposition
coupling-cohesion
database
trust
types
plt
lifts-projections
induction
critique
beauty
truth
elegance
aesthetics
Kevin Buzzard (the Lean guy)

- general reflection on proof asssistants/theorem provers

- Kevin Hale's formal abstracts project, etc

- thinks of available theorem provers, Lean is "[the only one currently available that may be capable of formalizing all of mathematics eventually]" (goes into more detail right at he end)

8 days ago by nhaliday

Making of Byrne’s Euclid - C82: Works of Nicholas Rougeux

12 days ago by nhaliday

https://www.math.ubc.ca/~cass/Euclid/byrne.html

Tufte: https://www.gwern.net/docs/statistics/1990-tufte-envisioninginformation-ch5-bryneseuclid.pdf

https://habr.com/ru/post/452520/

techtariat
reflection
project
summary
design
web
visuo
visual-understanding
math
geometry
worrydream
beauty
books
history
early-modern
classic
the-classics
britain
anglo
writing
technical-writing
gwern
elegance
virtu
:)
notation
the-great-west-whale
explanation
form-design
heavyweights
dataviz
org:junk
org:edu
pdf
essay
art
multi
latex
Tufte: https://www.gwern.net/docs/statistics/1990-tufte-envisioninginformation-ch5-bryneseuclid.pdf

https://habr.com/ru/post/452520/

12 days ago by nhaliday

big list - Are there proofs that you feel you did not "understand" for a long time? - MathOverflow

nibble q-n-a overflow soft-question big-list math proofs expert-experience heavyweights gowers mathtariat reflection learning intricacy grokkability intuition algebra math.GR motivation math.GN topology synthesis math.CT computation tcs logic iteration-recursion math.CA extrema smoothness span-cover grokkability-clarity

10 weeks ago by nhaliday

nibble q-n-a overflow soft-question big-list math proofs expert-experience heavyweights gowers mathtariat reflection learning intricacy grokkability intuition algebra math.GR motivation math.GN topology synthesis math.CT computation tcs logic iteration-recursion math.CA extrema smoothness span-cover grokkability-clarity

10 weeks ago by nhaliday

One week of bugs

may 2019 by nhaliday

If I had to guess, I'd say I probably work around hundreds of bugs in an average week, and thousands in a bad week. It's not unusual for me to run into a hundred new bugs in a single week. But I often get skepticism when I mention that I run into multiple new (to me) bugs per day, and that this is inevitable if we don't change how we write tests. Well, here's a log of one week of bugs, limited to bugs that were new to me that week. After a brief description of the bugs, I'll talk about what we can do to improve the situation. The obvious answer to spend more effort on testing, but everyone already knows we should do that and no one does it. That doesn't mean it's hopeless, though.

...

Here's where I'm supposed to write an appeal to take testing more seriously and put real effort into it. But we all know that's not going to work. It would take 90k LOC of tests to get Julia to be as well tested as a poorly tested prototype (falsely assuming linear complexity in size). That's two person-years of work, not even including time to debug and fix bugs (which probably brings it closer to four of five years). Who's going to do that? No one. Writing tests is like writing documentation. Everyone already knows you should do it. Telling people they should do it adds zero information1.

Given that people aren't going to put any effort into testing, what's the best way to do it?

Property-based testing. Generative testing. Random testing. Concolic Testing (which was done long before the term was coined). Static analysis. Fuzzing. Statistical bug finding. There are lots of options. Some of them are actually the same thing because the terminology we use is inconsistent and buggy. I'm going to arbitrarily pick one to talk about, but they're all worth looking into.

...

There are a lot of great resources out there, but if you're just getting started, I found this description of types of fuzzers to be one of those most helpful (and simplest) things I've read.

John Regehr has a udacity course on software testing. I haven't worked through it yet (Pablo Torres just pointed to it), but given the quality of Dr. Regehr's writing, I expect the course to be good.

For more on my perspective on testing, there's this.

Everything's broken and nobody's upset: https://www.hanselman.com/blog/EverythingsBrokenAndNobodysUpset.aspx

https://news.ycombinator.com/item?id=4531549

https://hypothesis.works/articles/the-purpose-of-hypothesis/

From the perspective of a user, the purpose of Hypothesis is to make it easier for you to write better tests.

From my perspective as the primary author, that is of course also a purpose of Hypothesis. I write a lot of code, it needs testing, and the idea of trying to do that without Hypothesis has become nearly unthinkable.

But, on a large scale, the true purpose of Hypothesis is to drag the world kicking and screaming into a new and terrifying age of high quality software.

Software is everywhere. We have built a civilization on it, and it’s only getting more prevalent as more services move online and embedded and “internet of things” devices become cheaper and more common.

Software is also terrible. It’s buggy, it’s insecure, and it’s rarely well thought out.

This combination is clearly a recipe for disaster.

The state of software testing is even worse. It’s uncontroversial at this point that you should be testing your code, but it’s a rare codebase whose authors could honestly claim that they feel its testing is sufficient.

Much of the problem here is that it’s too hard to write good tests. Tests take up a vast quantity of development time, but they mostly just laboriously encode exactly the same assumptions and fallacies that the authors had when they wrote the code, so they miss exactly the same bugs that you missed when they wrote the code.

Preventing the Collapse of Civilization [video]: https://news.ycombinator.com/item?id=19945452

- Jonathan Blow

NB: DevGAMM is a game industry conference

- loss of technological knowledge (Antikythera mechanism, aqueducts, etc.)

- hardware driving most gains, not software

- software's actually less robust, often poorly designed and overengineered these days

- *list of bugs he's encountered recently*:

https://youtu.be/pW-SOdj4Kkk?t=1387

- knowledge of trivia becomes more than general, deep knowledge

- does at least acknowledge value of DRY, reusing code, abstraction saving dev time

techtariat
dan-luu
tech
software
error
list
debugging
linux
github
robust
checking
oss
troll
lol
aphorism
webapp
email
google
facebook
games
julia
pls
compilers
communication
mooc
browser
rust
programming
engineering
random
jargon
formal-methods
expert-experience
prof
c(pp)
course
correctness
hn
commentary
video
presentation
carmack
pragmatic
contrarianism
pessimism
sv
unix
rhetoric
critique
worrydream
hardware
performance
trends
multiplicative
roots
impact
comparison
history
iron-age
the-classics
mediterranean
conquest-empire
gibbon
technology
the-world-is-just-atoms
flux-stasis
increase-decrease
graphics
hmm
idk
systems
os
abstraction
intricacy
worse-is-better/the-right-thing
build-packaging
microsoft
osx
apple
reflection
assembly
things
knowledge
detail-architecture
thick-thin
trivia
info-dynamics
caching
frameworks
generalization
systematic-ad-hoc
universalism-particularism
analytical-holistic
structure
tainter
libraries
tradeoffs
prepping
threat-modeling
network-structure
writing
risk
local-glob
...

Here's where I'm supposed to write an appeal to take testing more seriously and put real effort into it. But we all know that's not going to work. It would take 90k LOC of tests to get Julia to be as well tested as a poorly tested prototype (falsely assuming linear complexity in size). That's two person-years of work, not even including time to debug and fix bugs (which probably brings it closer to four of five years). Who's going to do that? No one. Writing tests is like writing documentation. Everyone already knows you should do it. Telling people they should do it adds zero information1.

Given that people aren't going to put any effort into testing, what's the best way to do it?

Property-based testing. Generative testing. Random testing. Concolic Testing (which was done long before the term was coined). Static analysis. Fuzzing. Statistical bug finding. There are lots of options. Some of them are actually the same thing because the terminology we use is inconsistent and buggy. I'm going to arbitrarily pick one to talk about, but they're all worth looking into.

...

There are a lot of great resources out there, but if you're just getting started, I found this description of types of fuzzers to be one of those most helpful (and simplest) things I've read.

John Regehr has a udacity course on software testing. I haven't worked through it yet (Pablo Torres just pointed to it), but given the quality of Dr. Regehr's writing, I expect the course to be good.

For more on my perspective on testing, there's this.

Everything's broken and nobody's upset: https://www.hanselman.com/blog/EverythingsBrokenAndNobodysUpset.aspx

https://news.ycombinator.com/item?id=4531549

https://hypothesis.works/articles/the-purpose-of-hypothesis/

From the perspective of a user, the purpose of Hypothesis is to make it easier for you to write better tests.

From my perspective as the primary author, that is of course also a purpose of Hypothesis. I write a lot of code, it needs testing, and the idea of trying to do that without Hypothesis has become nearly unthinkable.

But, on a large scale, the true purpose of Hypothesis is to drag the world kicking and screaming into a new and terrifying age of high quality software.

Software is everywhere. We have built a civilization on it, and it’s only getting more prevalent as more services move online and embedded and “internet of things” devices become cheaper and more common.

Software is also terrible. It’s buggy, it’s insecure, and it’s rarely well thought out.

This combination is clearly a recipe for disaster.

The state of software testing is even worse. It’s uncontroversial at this point that you should be testing your code, but it’s a rare codebase whose authors could honestly claim that they feel its testing is sufficient.

Much of the problem here is that it’s too hard to write good tests. Tests take up a vast quantity of development time, but they mostly just laboriously encode exactly the same assumptions and fallacies that the authors had when they wrote the code, so they miss exactly the same bugs that you missed when they wrote the code.

Preventing the Collapse of Civilization [video]: https://news.ycombinator.com/item?id=19945452

- Jonathan Blow

NB: DevGAMM is a game industry conference

- loss of technological knowledge (Antikythera mechanism, aqueducts, etc.)

- hardware driving most gains, not software

- software's actually less robust, often poorly designed and overengineered these days

- *list of bugs he's encountered recently*:

https://youtu.be/pW-SOdj4Kkk?t=1387

- knowledge of trivia becomes more than general, deep knowledge

- does at least acknowledge value of DRY, reusing code, abstraction saving dev time

may 2019 by nhaliday

reference request - Essays and thoughts on mathematics - MathOverflow

q-n-a overflow nibble list big-list math writing essay reflection soft-question links meta:math philosophy big-picture thurston gowers meaningness virtu metameta wisdom p:null heavyweights technical-writing communication

february 2017 by nhaliday

q-n-a overflow nibble list big-list math writing essay reflection soft-question links meta:math philosophy big-picture thurston gowers meaningness virtu metameta wisdom p:null heavyweights technical-writing communication

february 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

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 capacity to be alone | Quomodocumque

september 2016 by nhaliday

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

math
reflection
quotes
scholar
mathtariat
lens
optimate
serene
individualism-collectivism
the-monster
humility
the-trenches
virtu
courage
emotion
extra-introversion
allodium
ascetic
heavyweights
psychiatry
september 2016 by nhaliday

On proof and progress in mathematics

pdf thurston math writing thinking synthesis papers essay unit nibble intuition worrydream communication proofs the-trenches reflection geometry meta:math better-explained stories virtu 🎓 scholar metameta wisdom narrative p:whenever inference cs programming rigor formal-methods meta:research info-dynamics elegance technical-writing heavyweights guessing trust

august 2016 by nhaliday

pdf thurston math writing thinking synthesis papers essay unit nibble intuition worrydream communication proofs the-trenches reflection geometry meta:math better-explained stories virtu 🎓 scholar metameta wisdom narrative p:whenever inference cs programming rigor formal-methods meta:research info-dynamics elegance technical-writing heavyweights guessing trust

august 2016 by nhaliday

ho.history overview - Does any research mathematics involve solving functional equations? - MathOverflow

math reflection expert characterization tidbits q-n-a overflow oly mathtariat gowers motivation nibble expert-experience rec-math heavyweights explanation roots explanans properties

july 2016 by nhaliday

math reflection expert characterization tidbits q-n-a overflow oly mathtariat gowers motivation nibble expert-experience rec-math heavyweights explanation roots explanans properties

july 2016 by nhaliday

soft question - Famous mathematical quotes - MathOverflow

math aphorism reflection list quotes q-n-a overflow soft-question big-list mathtariat stories lens nibble giants von-neumann darwinian old-anglo poetry letters troll lol creative algebra geometry linear-algebra thick-thin moments high-variance elegance heavyweights

june 2016 by nhaliday

math aphorism reflection list quotes q-n-a overflow soft-question big-list mathtariat stories lens nibble giants von-neumann darwinian old-anglo poetry letters troll lol creative algebra geometry linear-algebra thick-thin moments high-variance elegance heavyweights

june 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

On writing | What's new

april 2016 by nhaliday

also: on reading papers

writing
papers
academia
math
tcs
advice
reflection
thinking
expert
gowers
long-term
🎓
checklists
grad-school
scholar
mathtariat
learning
nibble
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april 2016 by nhaliday

Work hard | What's new

april 2016 by nhaliday

Similarly, to be a “professional” mathematician, you need to not only work on your research problem(s), but you should also constantly be working on learning new proofs and techniques, going over important proofs and papers time and again until you’ve mastered them. Don’t stay in your mathematical comfort zone, but expand your horizon by also reading (relevant) papers that are not at the heart of your own field. You should go to seminars to stay current and to challenge yourself to understand math in real time. And so on. All of these elements have to find their way into your daily work routine, because if you neglect any of them it will ultimately affect your research output negatively.

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april 2016 by nhaliday

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