[HN Gopher] Difficult math is about recognizing patterns ___________________________________________________________________ Difficult math is about recognizing patterns Author : klevertree Score : 26 points Date : 2021-08-09 20:49 UTC (2 hours ago) (HTM) web link (get21stnight.com) (TXT) w3m dump (get21stnight.com) | wombatmobile wrote: | 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 wrote: | I mean "recognising patterns" is a such broad term that you can | use it to describe pretty much any mental skill. | resters wrote: | 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). | JacobiX wrote: | 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 wrote: | That's why combinatorics is a bit of black sheep :D | paulpauper wrote: | Combinatorics is just counting but can be so hard finding the | shortcuts | contravariant wrote: | 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. | tarxzvf wrote: | 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 | mkl wrote: | 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 wrote: | > 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. | saeranv wrote: | Is there some high-level mathematical way to interpret these | normal forms? | | I am not familiar with relational databases and just googled | database normalization to get a kind of crude idea of what you | are saying, but all the examples I find don't seem to give some | high-level mathematical interpretation of database | normalization. And it seems like there should be one (I could | be way off here) based on how you are talking about forms here. | Ericson2314 wrote: | My guess is that stuff is kinda iffy from a math standpoint. | Relational databases are good but I've long suspected the | theory needs to be overhauled a bit as we've gotten a lot | better at connecting CS to math foundations than in the | 1970s. (And indeed math foundations is itself getting better | from that process.) | gyam wrote: | 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 wrote: | 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 | 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 now. | TrackerFF wrote: | 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. | mkl wrote: | 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? | csours wrote: | Math is about symbol manipulation. Some of those symbols are | numbers, and some of the manipulations are arithmetic. | anyfoo wrote: | 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 ___________________________________________________________________ (page generated 2021-08-09 23:00 UTC)