8 ms·

Difficult math is about recognizing patterns

116 points by klevertree 5y ago

wombatmobile 5y ago
Don't most experts in most domains work the same way - recognising patterns they've seen before? That's why an expert can charge so much for 1 hour of time. It is more valuable than days or weeks or months of a non-expert's time who doesn't have the library and can't recognise the pattern.
- jstx1 5y ago
  I mean "recognising patterns" is a such broad term that you can use it to describe pretty much any mental skill.
csours 5y ago
Math is about symbol manipulation. Some of those symbols are numbers, and some of the manipulations are arithmetic.
- anyfoo 5y ago
  I'd personally say the symbols and their manipulation are a good representation of what's going on in math, but math is overall more general. (By the way, arithmetic is about numbers, maybe you meant algebraic.) Sometimes manipulating symbols towards an answer is great, but sometimes taking a step back, and looking at a problem through a different lens (e.g. at what you are trying to do intuitively) is vastly superior, and the symbol-manipulating, rigorous formalization (and verification) part comes afterwards. A few examples (out of very many): * In signal processing (both digital and analog) it can often be much more insightful to play with visualizations of time domain, spectra, and convolution and multiplication thereof. * Related but more general: Thinking about the complex exponential as spinning in a circle, or tracing out a corkscrew in 3 dimensions, is a way easier method to grasp it than to look at the equations, which for someone getting into it will look like abstract nonsense[1]. * Topology is about "shape" and "deformation" of objects. * Discrete Structures is about trees, graphs, and so on. In all of those, you can hit paths where an intuitive understanding may stay out of reach, and symbolic manipulation through e.g. algebra might remain the only way to work with it, but that is often not generally true for the whole field. [1] Funnily, that's an actual term used by mathematicians, but usually in another field: https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense
- howtofly 5y ago
  I used to believe this, and failed miserably when studying algebraic topology. Intuition, the so called "feeling", is way more important than pure mathematical logic.
mkl 5y ago
Mathematics is the study of patterns, any kind of pattern, in anything. "Difficulty" of maths problems is a kind of measure of how well you know the patterns involved (which is related to how good our notation, terminology, and visualisations for them are). That means research developing brand new maths or applying it to new problems is often difficult, because no one knows the patterns yet, or has good ways of describing them.
bob1029 5y ago
> What’s important to recognize is that these same attributes apply across all levels of math. Functional/Relational programming models are just a trivial layer on top of math. Everything is pattern recognition at the end of the day. Domain modeling is the logical extension of building standardized "patterns" that can be leveraged for rapidly building & replicating similar ideas. Using good modeling techniques is the most important thing for managing complex systems. If you aren't sure, you can always start modeling at 6th normal form, then walk it back to 3NF as the various pieces start to make sense together. If you have your domain in 6NF and are using purely functional/relational programming, there are mountains of mathematical guarantees you can make about the correctness of your software. For instance, 6NF gets rid of null. It forces you to deal with the notion of optional facts using 0..1-1 relations and applicable query constraints.
- gyam 5y ago
  I want to learn these. How do I start learning good modelling techniques? Are the NFs you’re referring to is same as in DB design? If so, how does that translate to general code/system design. Any pointers you can provide for learning material, concepts, or books will be greatly appreciated. Thanks!
  - Ericson2314 5y ago
    The Codd 1971 stuff on Wikipedia looks like a mess, and I think the original is propbably a bit crufty too (though I don't want to disrespect the old masters, SQL was a high water mark of business thinking about computing in many ways). I would follow the citations of https://ncatlab.org/nlab/show/lens+%28in+computer+science%29 https://ncatlab.org/nlab/show/lens+%28in+computer+science%29 instead. Whatever Spivack can say about this stuff I think is going to be much more worth your while. Looking at https://arxiv.org/pdf/1602.03501.pdf https://arxiv.org/pdf/1602.03501.pdf now.
    - gyam 5y ago
      Thanks for these!
mkl 5y ago
I like the teaching idea, but I feel like there is a step missing in here: > When my students encounter a math problem they can’t answer, I have them put it in the error log with an explanation of how they did and how they knew how to do it. If they can't answer it, where does the "how they knew how to do it" come from? Their teacher/tutor?
tarxzvf 5y ago
Everything is pattern matching (or memorization). You can use this approach to half-automate the solution to a known existing class of problems, but how do you come up with anything new? How did Paul Cohen came up with the forcing technique? Who figured out probabilistic proofs as a possible vector of attack? "Both these properties, predictability and stability, are special to integrable systems... Since classical mechanics has dealt exclusively with integrable systems for so many years, we have been left with wrong ideas about causality. The mathematical truth, coming from non-integrable systems, is that everything is the cause of everything else: to predict what will happen tomorrow, we must take into account everything that is happening today. Except in very special cases, there is no clear-cut "causality chain," relating successive events, where each one is the (only) cause of the next in line. Integrable systems are such special cases, and they have led to a view of the world as a juxtaposition of causal chains, running parallel to each other with little or no interference." - Ivar Ekeland
- gregorygoc 5y ago
  I think the key to innovation is to first know what’s out there. Then you’re able to combine, twist and augment known ideas. Special theory of relativity did not come out of nowhere. Neither did the geometry, nor algebra. It was all about humans’ curious mind and joy of exploring what’s “possible” out there.
- planet-and-halo 5y ago
  This is such a great quote. There is a similar line in the book "Creation: Life and How to Make It." The author says that causality is a web, not a chain.
  - peterburkimsher 5y ago
    Yes, my thoughts follow a web pattern, not only a chain! There are chains of thought, but they jump all over the place, even in loops. And it all ends in philosophy [0]. Hyperlinks on the web are one-directional. But links are much stronger if they're bidirectional. That's possible using backlinks, or in real life, by saying "thank you". Thank you planet-and-halo for reminding us of the web analogy. Thank you zR0x for relating the abstract maths to tangible reality. Thank you tarxzvf for suggesting that everything is pattern matching (I agree, matter & energy are finite, it's only the connections between them that we can create). I believe that these connections hold true for dad jokes, social situations, software, maths, physics, chemistry, biology... every created thing. Let's thank our creator, and all the teachers who helped us grow. Are there under 6 degrees of separation between everything in the universe? Or is it as few as 3.5 degrees? [1] [0] https://en.wikipedia.org/wiki/Wikipedia:Getting_to_Philosophy https://en.wikipedia.org/wiki/Wikipedia:Getting_to_Philosoph... [1] https://research.fb.com/blog/2016/02/three-and-a-half-degrees-of-separation/#:~:text=The%20majority%20of%20the%20people,and%204.2%20degrees%20of%20separation.&text=Estimated%20average%20degrees%20of%20separation,between%203%20and%204%20steps https://research.fb.com/blog/2016/02/three-and-a-half-degree....
JacobiX 5y ago
In discrete mathematics (combinatorics) there are definitely some tools and techniques but seemingly every problem is unique, I’m not quite sure that pattern matching is very useful in this subfield of mathematics ?
- Ericson2314 5y ago
  That's why combinatorics is a bit of black sheep :D
- paulpauper 5y ago
  Combinatorics is just counting but can be so hard finding the shortcuts
- contravariant 5y ago
  Yeah this view that it's all pattern matching isn't too helpful. I find it more helpful to consider mathematics the art of generalization. Discrete math is frequently about things with very little structure (e.g. graph theory, where you basically just have any binary relation), so inevitably ends up trying to prove things that are way too general. The flipside is that those theorems do tend to crop up everywhere.
resters 5y ago
A lot of math is taught in a sloppy way, which thwarts the pattern recognition progress in brains trying to learn it. Programming is significantly easier than math (for something equivalently complex) because of things like syntax checking and compiler/interpreter errors. This speeds up the pattern recognition process in the human brain. People who are identified as being skilled at math or programming at a relatively early age are usually those who understood it in spite of the teacher/curriculum, so the ability comes as a surprise. But many such people do not go on to distinguish themselves in either field in any way. There are always things that come easily to one person vs another, but in math and programming, the early birds are typically the only ones whose interest in the subject isn't destroyed by the teaching methods (because the learning happened in spite of them).
- creata 5y ago
  > Programming is significantly easier... because of things like syntax checking and compiler/interpreter errors. People[0][1] are using theorem provers to help teach students the general structure of a proof. And this is a bit of a tangent, but if you want to mess around with very simple proofs in first-order logic, the Open Logic Project[2] has an online proof editor and a textbook. [0]: https://link.springer.com/content/pdf/10.1007/s40753-021-00140-1.pdf https://link.springer.com/content/pdf/10.1007/s40753-021-001... [1]: https://xenaproject.wordpress.com/category/learning-lean/page/2/ https://xenaproject.wordpress.com/category/learning-lean/pag... [2]: https://proofs.openlogicproject.org/ https://proofs.openlogicproject.org/
TrackerFF 5y ago
Same goes for problems in data structures and algorithms (read: hackerrank / leetcode problems) Some people will recognize patterns pretty fast, while others must solve literally hundreds of different problems, before becoming comfortable with the concepts. People always seem amazed and baffled that some candidates can practically walk into white-board interviews unprepared, other than what they learned / did in their DS&A classes in college, and nail the interviews, while others have to basically prep 6-12 months before passing the same interview.
- sjg007 5y ago
  The big issue is understanding what the problem is asking. Once you get that the pattern is easy.
  - foxes 5y ago
    I am not sure that is always true, there are plenty of problems that are easy to state, but finding the "pattern" can be out of reach. Eg Collatz conjecture? However there are obviously a subset of problems I guess that can be solved once you realise what they are asking.
    - sjg007 5y ago
      Are you suggesting that the Collatz conjecture is a valid interview question?
      - rejectedandsad 5y ago
        It is, it’s a fairly standard warmup interview problem. https://leetcode.com/discuss/interview-question/489169/Google-or-Phone-or-Decode-String-and-Collatz-Conjecture https://leetcode.com/discuss/interview-question/489169/Googl...
      - foxes 5y ago
        Proving it no one has any ideas, but generating the sequence given the input is definitely an interview style question.
paulpauper 5y ago
It is more than just having the toolbox. You need to know how to use the tools and know how to combine them . Anything harder than the basics is going to require a lot of outside the box thinking and epiphanies rather than just pattern recognition r. Very subtle and hard most of the time
Ericson2314 5y ago
Getting into sort of pattern matching I think requires a certain optimism that things are "nice and symmetrical" after enough analysts. Of course, that can get us into trouble with e.g. "supersymmetry in physics", but usually I think that optimsim is a feature not a bug and necessary in any case. I think instilling this optomism in students --- following their curiosity won't lead deeper in a bottomless pit, if something doesn't make sense it's might be them but a lack of information, etc. --- is the essentially hard part, and requires undoing a lot alienation people experience. Conversely, I think messing around with block boxes like machine learning we don't understand is giving into the alienation. (Studying it to understand it rather than do things is fine.) I worry more use of machine learning like things will be a another nail in liberalism's coffin as do the equivalent of regressing back to alchemy from chemistry. Now, looking for patterns is what machine learning does, but while Rorschach-test-style grappling in the dark might be the basal "reptiling" instinct that lead to more high level theory-based pattern renegotiation, they should not be conflated.
adamnemecek 5y ago
Literally all math is about adjoints, norms and fixed points https://github.com/adamnemecek/adjoint/ https://github.com/adamnemecek/adjoint/
hyperpallium2 5y ago
> When my students encounter a math problem they can’t answer, I have them put it in the error log with an explanation of how they did and how they knew how to do it. I'm gonna assume a step where they learned how to do it? TFA's method is for incremental discovery expertise. Feynman talks about an inverse, where he maintained a list of interesting problems, and when he learnt a new technique, tried it on each one. But Feynman's actual breakthroughs came from playfully looking at phenomena. I think the incremental skills are basics like reading, writing and arithmetic - it's harder to really get to grips with something you've noticed without them. I mean, Einstein famously didn't have adequate math for special relativity and sought help. He was however the one to notice something. A library of techniques is a poor substitute for actual thought.
- mhh__ 5y ago
  > Einstein famously didn't have adequate math for special relativity and sought help. General relativity. Special Relativity is an extremely subtle insight into relatively simple mathematics, general relativity is basically a chasm of rich detail that requires advanced mathematics to express and use. Beyond examining the mathematics for yourself, you can see evidence for this in that there are very little texts considering "mathematical" (i.e. for mathematicians) on special relativity but many on general relativity. (For what it's worth, I find some of these mathematics-first one's to be remarkably poor pedagogically and far enough away from any useful physics that I sometimes question why some of them exist, although I am a very long way from being a mathematician).
hyperpallium2 5y ago
Let's be clear: this is an ad for his app.
- MonkeyClub 5y ago
  You've just noticed a (dark) pattern
ackbar03 5y ago
I guess its kind of obvious now that practice always helps, but for a while, mostly during high school, I used to think that being good at math because you've seen similar problems hundreds of before was kind of "cheating" and you weren't really that smart. Instead, you were smart if you managed to do a test/competition really well without doing tons of practice questions. Consequently during math classes I used to sit at the back of the class and play counter strike all day on my laptop. Nobody seemed to care since I'd ace all the tests and still compete for my school in math competitions and stuff. However I completely wrecked my math education, and come university (I skipped last year of high school for uni, there's a standard program for it in my country) I had completely forgotten how to prepare for a math exam and was systematically left further behind with every year Lol. Looking back I still kind of regret my perspective on doing practice problems. In hindsight it was kind of stupid but it was mostly because I thought it was kind of lame that I did well sometimes because I practiced more than other people, whereas some other students seemed to do pretty good without (seemingly) having practiced at all. On the plus side I do feel I learn things a lot faster than much people and am pretty descent at a wider variety of things
linspace 5y ago
A math professor of mine said that the first step to solve a math problem was to know the answer. I remember during my first year buying a book called something like 1000 limit problems. I "just" did about 300. It was definitely pattern matching and nothing Mathematica wouldn't do better than me.
- iamcreasy 5y ago
  > A math professor of mine said that the first step to solve a math problem was to know the answer. Yes, I've learned the same thing in my MS as well. Before I put pencil on paper I need to have a rough idea why something is true. The books sounds fascinating. Can you provide a link to this book?
  - linspace 5y ago
    Unfortunately I cannot find a link. It was a fat blue soft cover book from what I recall from about 20 years ago

116 points by klevertree 5y ago

Don't most experts in most domains work the same way - recognising patterns they've seen before? That's why an expert can charge so much for 1 hour of time. It is more valuable than days or weeks or months of a non-expert's time who doesn't have the library and can't recognise the pattern.

jstx1 5y ago
I mean "recognising patterns" is a such broad term that you can use it to describe pretty much any mental skill.

csours 5y ago

Math is about symbol manipulation. Some of those symbols are numbers, and some of the manipulations are arithmetic.

anyfoo 5y ago
I'd personally say the symbols and their manipulation are a good representation of what's going on in math, but math is overall more general. (By the way, arithmetic is about numbers, maybe you meant algebraic.) Sometimes manipulating symbols towards an answer is great, but sometimes taking a step back, and looking at a problem through a different lens (e.g. at what you are trying to do intuitively) is vastly superior, and the symbol-manipulating, rigorous formalization (and verification) part comes afterwards. A few examples (out of very many): * In signal processing (both digital and analog) it can often be much more insightful to play with visualizations of time domain, spectra, and convolution and multiplication thereof. * Related but more general: Thinking about the complex exponential as spinning in a circle, or tracing out a corkscrew in 3 dimensions, is a way easier method to grasp it than to look at the equations, which for someone getting into it will look like abstract nonsense[1]. * Topology is about "shape" and "deformation" of objects. * Discrete Structures is about trees, graphs, and so on. In all of those, you can hit paths where an intuitive understanding may stay out of reach, and symbolic manipulation through e.g. algebra might remain the only way to work with it, but that is often not generally true for the whole field. [1] Funnily, that's an actual term used by mathematicians, but usually in another field: https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense
howtofly 5y ago
I used to believe this, and failed miserably when studying algebraic topology. Intuition, the so called "feeling", is way more important than pure mathematical logic.

mkl 5y ago

Mathematics is the study of patterns, any kind of pattern, in anything. "Difficulty" of maths problems is a kind of measure of how well you know the patterns involved (which is related to how good our notation, terminology, and visualisations for them are). That means research developing brand new maths or applying it to new problems is often difficult, because no one knows the patterns yet, or has good ways of describing them.

bob1029 5y ago

> What’s important to recognize is that these same attributes apply across all levels of math. Functional/Relational programming models are just a trivial layer on top of math. Everything is pattern recognition at the end of the day. Domain modeling is the logical extension of building standardized "patterns" that can be leveraged for rapidly building & replicating similar ideas. Using good modeling techniques is the most important thing for managing complex systems. If you aren't sure, you can always start modeling at 6th normal form, then walk it back to 3NF as the various pieces start to make sense together. If you have your domain in 6NF and are using purely functional/relational programming, there are mountains of mathematical guarantees you can make about the correctness of your software. For instance, 6NF gets rid of null. It forces you to deal with the notion of optional facts using 0..1-1 relations and applicable query constraints.

gyam 5y ago
I want to learn these. How do I start learning good modelling techniques? Are the NFs you’re referring to is same as in DB design? If so, how does that translate to general code/system design. Any pointers you can provide for learning material, concepts, or books will be greatly appreciated. Thanks!
- Ericson2314 5y ago
  The Codd 1971 stuff on Wikipedia looks like a mess, and I think the original is propbably a bit crufty too (though I don't want to disrespect the old masters, SQL was a high water mark of business thinking about computing in many ways). I would follow the citations of https://ncatlab.org/nlab/show/lens+%28in+computer+science%29 https://ncatlab.org/nlab/show/lens+%28in+computer+science%29 instead. Whatever Spivack can say about this stuff I think is going to be much more worth your while. Looking at https://arxiv.org/pdf/1602.03501.pdf https://arxiv.org/pdf/1602.03501.pdf now.
  - gyam 5y ago
    Thanks for these!

mkl 5y ago

I like the teaching idea, but I feel like there is a step missing in here: > When my students encounter a math problem they can’t answer, I have them put it in the error log with an explanation of how they did and how they knew how to do it. If they can't answer it, where does the "how they knew how to do it" come from? Their teacher/tutor?

tarxzvf 5y ago

Everything is pattern matching (or memorization). You can use this approach to half-automate the solution to a known existing class of problems, but how do you come up with anything new? How did Paul Cohen came up with the forcing technique? Who figured out probabilistic proofs as a possible vector of attack? "Both these properties, predictability and stability, are special to integrable systems... Since classical mechanics has dealt exclusively with integrable systems for so many years, we have been left with wrong ideas about causality. The mathematical truth, coming from non-integrable systems, is that everything is the cause of everything else: to predict what will happen tomorrow, we must take into account everything that is happening today. Except in very special cases, there is no clear-cut "causality chain," relating successive events, where each one is the (only) cause of the next in line. Integrable systems are such special cases, and they have led to a view of the world as a juxtaposition of causal chains, running parallel to each other with little or no interference." - Ivar Ekeland

gregorygoc 5y ago
I think the key to innovation is to first know what’s out there. Then you’re able to combine, twist and augment known ideas. Special theory of relativity did not come out of nowhere. Neither did the geometry, nor algebra. It was all about humans’ curious mind and joy of exploring what’s “possible” out there.
planet-and-halo 5y ago
This is such a great quote. There is a similar line in the book "Creation: Life and How to Make It." The author says that causality is a web, not a chain.
- peterburkimsher 5y ago
  Yes, my thoughts follow a web pattern, not only a chain! There are chains of thought, but they jump all over the place, even in loops. And it all ends in philosophy [0]. Hyperlinks on the web are one-directional. But links are much stronger if they're bidirectional. That's possible using backlinks, or in real life, by saying "thank you". Thank you planet-and-halo for reminding us of the web analogy. Thank you zR0x for relating the abstract maths to tangible reality. Thank you tarxzvf for suggesting that everything is pattern matching (I agree, matter & energy are finite, it's only the connections between them that we can create). I believe that these connections hold true for dad jokes, social situations, software, maths, physics, chemistry, biology... every created thing. Let's thank our creator, and all the teachers who helped us grow. Are there under 6 degrees of separation between everything in the universe? Or is it as few as 3.5 degrees? [1] [0] https://en.wikipedia.org/wiki/Wikipedia:Getting_to_Philosophy https://en.wikipedia.org/wiki/Wikipedia:Getting_to_Philosoph... [1] https://research.fb.com/blog/2016/02/three-and-a-half-degrees-of-separation/#:~:text=The%20majority%20of%20the%20people,and%204.2%20degrees%20of%20separation.&text=Estimated%20average%20degrees%20of%20separation,between%203%20and%204%20steps https://research.fb.com/blog/2016/02/three-and-a-half-degree....

JacobiX 5y ago

In discrete mathematics (combinatorics) there are definitely some tools and techniques but seemingly every problem is unique, I’m not quite sure that pattern matching is very useful in this subfield of mathematics ?

Ericson2314 5y ago
That's why combinatorics is a bit of black sheep :D
paulpauper 5y ago
Combinatorics is just counting but can be so hard finding the shortcuts
contravariant 5y ago
Yeah this view that it's all pattern matching isn't too helpful. I find it more helpful to consider mathematics the art of generalization. Discrete math is frequently about things with very little structure (e.g. graph theory, where you basically just have any binary relation), so inevitably ends up trying to prove things that are way too general. The flipside is that those theorems do tend to crop up everywhere.

resters 5y ago

A lot of math is taught in a sloppy way, which thwarts the pattern recognition progress in brains trying to learn it. Programming is significantly easier than math (for something equivalently complex) because of things like syntax checking and compiler/interpreter errors. This speeds up the pattern recognition process in the human brain. People who are identified as being skilled at math or programming at a relatively early age are usually those who understood it in spite of the teacher/curriculum, so the ability comes as a surprise. But many such people do not go on to distinguish themselves in either field in any way. There are always things that come easily to one person vs another, but in math and programming, the early birds are typically the only ones whose interest in the subject isn't destroyed by the teaching methods (because the learning happened in spite of them).

creata 5y ago
> Programming is significantly easier... because of things like syntax checking and compiler/interpreter errors. People[0][1] are using theorem provers to help teach students the general structure of a proof. And this is a bit of a tangent, but if you want to mess around with very simple proofs in first-order logic, the Open Logic Project[2] has an online proof editor and a textbook. [0]: https://link.springer.com/content/pdf/10.1007/s40753-021-00140-1.pdf https://link.springer.com/content/pdf/10.1007/s40753-021-001... [1]: https://xenaproject.wordpress.com/category/learning-lean/page/2/ https://xenaproject.wordpress.com/category/learning-lean/pag... [2]: https://proofs.openlogicproject.org/ https://proofs.openlogicproject.org/

TrackerFF 5y ago

Same goes for problems in data structures and algorithms (read: hackerrank / leetcode problems) Some people will recognize patterns pretty fast, while others must solve literally hundreds of different problems, before becoming comfortable with the concepts. People always seem amazed and baffled that some candidates can practically walk into white-board interviews unprepared, other than what they learned / did in their DS&A classes in college, and nail the interviews, while others have to basically prep 6-12 months before passing the same interview.

sjg007 5y ago
The big issue is understanding what the problem is asking. Once you get that the pattern is easy.
- foxes 5y ago
  I am not sure that is always true, there are plenty of problems that are easy to state, but finding the "pattern" can be out of reach. Eg Collatz conjecture? However there are obviously a subset of problems I guess that can be solved once you realise what they are asking.
  - sjg007 5y ago
    Are you suggesting that the Collatz conjecture is a valid interview question?
    - rejectedandsad 5y ago
      It is, it’s a fairly standard warmup interview problem. https://leetcode.com/discuss/interview-question/489169/Google-or-Phone-or-Decode-String-and-Collatz-Conjecture https://leetcode.com/discuss/interview-question/489169/Googl...
    - foxes 5y ago
      Proving it no one has any ideas, but generating the sequence given the input is definitely an interview style question.

paulpauper 5y ago

It is more than just having the toolbox. You need to know how to use the tools and know how to combine them . Anything harder than the basics is going to require a lot of outside the box thinking and epiphanies rather than just pattern recognition r. Very subtle and hard most of the time

Ericson2314 5y ago

Getting into sort of pattern matching I think requires a certain optimism that things are "nice and symmetrical" after enough analysts. Of course, that can get us into trouble with e.g. "supersymmetry in physics", but usually I think that optimsim is a feature not a bug and necessary in any case. I think instilling this optomism in students --- following their curiosity won't lead deeper in a bottomless pit, if something doesn't make sense it's might be them but a lack of information, etc. --- is the essentially hard part, and requires undoing a lot alienation people experience. Conversely, I think messing around with block boxes like machine learning we don't understand is giving into the alienation. (Studying it to understand it rather than do things is fine.) I worry more use of machine learning like things will be a another nail in liberalism's coffin as do the equivalent of regressing back to alchemy from chemistry. Now, looking for patterns is what machine learning does, but while Rorschach-test-style grappling in the dark might be the basal "reptiling" instinct that lead to more high level theory-based pattern renegotiation, they should not be conflated.

adamnemecek 5y ago

Literally all math is about adjoints, norms and fixed points https://github.com/adamnemecek/adjoint/ https://github.com/adamnemecek/adjoint/

hyperpallium2 5y ago

> When my students encounter a math problem they can’t answer, I have them put it in the error log with an explanation of how they did and how they knew how to do it. I'm gonna assume a step where they learned how to do it? TFA's method is for incremental discovery expertise. Feynman talks about an inverse, where he maintained a list of interesting problems, and when he learnt a new technique, tried it on each one. But Feynman's actual breakthroughs came from playfully looking at phenomena. I think the incremental skills are basics like reading, writing and arithmetic - it's harder to really get to grips with something you've noticed without them. I mean, Einstein famously didn't have adequate math for special relativity and sought help. He was however the one to notice something. A library of techniques is a poor substitute for actual thought.

mhh__ 5y ago
> Einstein famously didn't have adequate math for special relativity and sought help. General relativity. Special Relativity is an extremely subtle insight into relatively simple mathematics, general relativity is basically a chasm of rich detail that requires advanced mathematics to express and use. Beyond examining the mathematics for yourself, you can see evidence for this in that there are very little texts considering "mathematical" (i.e. for mathematicians) on special relativity but many on general relativity. (For what it's worth, I find some of these mathematics-first one's to be remarkably poor pedagogically and far enough away from any useful physics that I sometimes question why some of them exist, although I am a very long way from being a mathematician).

hyperpallium2 5y ago

Let's be clear: this is an ad for his app.

MonkeyClub 5y ago
You've just noticed a (dark) pattern

ackbar03 5y ago

I guess its kind of obvious now that practice always helps, but for a while, mostly during high school, I used to think that being good at math because you've seen similar problems hundreds of before was kind of "cheating" and you weren't really that smart. Instead, you were smart if you managed to do a test/competition really well without doing tons of practice questions. Consequently during math classes I used to sit at the back of the class and play counter strike all day on my laptop. Nobody seemed to care since I'd ace all the tests and still compete for my school in math competitions and stuff. However I completely wrecked my math education, and come university (I skipped last year of high school for uni, there's a standard program for it in my country) I had completely forgotten how to prepare for a math exam and was systematically left further behind with every year Lol. Looking back I still kind of regret my perspective on doing practice problems. In hindsight it was kind of stupid but it was mostly because I thought it was kind of lame that I did well sometimes because I practiced more than other people, whereas some other students seemed to do pretty good without (seemingly) having practiced at all. On the plus side I do feel I learn things a lot faster than much people and am pretty descent at a wider variety of things

linspace 5y ago

A math professor of mine said that the first step to solve a math problem was to know the answer. I remember during my first year buying a book called something like 1000 limit problems. I "just" did about 300. It was definitely pattern matching and nothing Mathematica wouldn't do better than me.

iamcreasy 5y ago
> A math professor of mine said that the first step to solve a math problem was to know the answer. Yes, I've learned the same thing in my MS as well. Before I put pencil on paper I need to have a rough idea why something is true. The books sounds fascinating. Can you provide a link to this book?
- linspace 5y ago
  Unfortunately I cannot find a link. It was a fat blue soft cover book from what I recall from about 20 years ago