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There’s more to mathematics than rigour and proofs (2007)
- ffhhj 4y agoWhich are the newest developments in math? Someone told me Geometric Algebra has been around for a long time but wasn't really useful until some recent theorems.
- peterhalburt33 4y agoIt really depends on which field you are talking about. I’d say it is very hard to find an area of math that’s completely new , but you will often find existing areas where novel perspectives are driving math forward. Geometric algebra may be a hot topic in some areas, but the ideas of exterior algebra go back more than a century at this point, so is it really new?? Just to humor you though, I think Deep Operator learning is a vastly exciting new field which combines ideas from functional analysis and deep learning in order to do things like solving PDEs.
- devnulll 4y agoOne day I'll retire and go back to school. The idea of learning Math - really learning & understanding Math - as a fun pastime is so appealing. What's stopping me now? That sweet overpaid SDE salary and the endless obligations that come from being an adult. I suspect I am not alone...
- paulpauper 4y agokeep the salary. if the goal it to change the world, to understand reality, to have an impactful life, have a good standard of living, etc. math is one of the hardest ways of achieving that. It's such a saturated field. Almost everything you can imagine has been done to the highest possible degree of abstraction. Every stone overturned except for things which may take a lifetime to even try to understand. Writing a blog post is probably way more fulfilling and also a doable challenge. A top mathematician may spend years working on a result that maybe if he is lucky be worthy of a footnote somewhere. As a field I think math is well past its diminishing returns imho. It's like 'what was the last big philosophical discovery'...yeah...hard to think of one. Maybe the P zombie concept or the simulation hypothesis. But new and important books, fiction and non fiction, are being written all the time.
- bombcar 4y agoI think the whole point would be to retire and do math for fun, not desiring anything more than the joy of discovery, no footnotes, no recognition, just math. 99.99% of everyone won't be remembered for their "contributions" so why not do something you enjoy?
- lupire 4y agoEven if no one remembers you, your work is a contribution. Learning for its own sake is a hobby for yourself, like watching TV or reading a book.
- eyelidlessness 4y agoLet me be remembered for watching TV. I would be utterly delighted if that’s my long term legacy. “They finally rested and enjoyed the most banal show imaginable. It was relaxing. Any other lasting contributions will be in other records should you care.” There I’ve written my eulogy.
- formerkrogemp 4y agoMy grandfather's last 10 years were spent developing diabetes, watching those three shows, and playing Microsoft solitaire until death. If that's what you want, then, sure, you can be remembered that way.
- andi999 4y agoAbout the simulation hypothesia, how is that different from Descartes evil demon? https://en.m.wikipedia.org/wiki/Evil_demon https://en.m.wikipedia.org/wiki/Evil_demon (apart from saying that the evil demon is a future type computer)
- postingposts 4y agoThe simulation stuff is just “Plato’s Cave”:Reloaded for people who never understood the concept of the cave in the first place. At a basic level you can interpret it from Gödel’s incompleteness theorems, which state that systems of logic need some form of observer to function. That observer concept reaches way back across different philosophical domains and authors as well.
- foobarian 4y agoI'm still grumpy that I accepted I knew what real numbers were just because I could recite back the definition given by the teacher. There is so much depth there if you go looking...
- gxs 4y agoThe way I explain this super simply to people is absolute value. To most people, it just means that when you take the absolute value of a negative number it becomes positive, and the absolute value of a positive number stays positive. Now there is more to it, but how you might think of absolute value instead is as a distance function, particularly how far away from zero you are on a number line. This is way over simplified, but an example of how there can be a little more buried beneath the surface.
- l33t2328 4y agoI contend no one understands real numbers if they can’t explain (without resorting to essentially purely symbolic rigor) why you can cover the rationals with intervals of arbitrarily small total length, but you can’t do the same with R.
- Koshkin 4y agoWell, to understand an explanation one must already know something, otherwise first you have to explain those other things, e.g. the simple fact that unlike the reals the set of rational numbers is countable…
- qsort 4y agoOne of my biggest regrets is not having studied (more) math. But would I regret not studying CS had I studied math? You really can't win :/
- qiskit 4y agoCS is math...
- dumpsterlid 4y agoIn a practical sense, not really.
- l33t2328 4y agoI’d argue math is actually (a proper subset of) CS.
- Koshkin 4y agoBut the real numbers cannot be faithfully represented in a computer!
- stocknoob 4y agoAim for financial independence on that SDE and you can retire a few decades ahead of schedule. And you’ll have plenty of time for that sweet math learning.
- hebrox 4y agoI'm actually looking into this. Just an hour ago I mailed the local university that I won't be doing any courses there. My initial plan was to do a bachelor at around 50% speed. But working and having to girls (1,5 years and 3 weeks) makes that quite impossible. And looking at photos of myself at 17 makes me feel rather out of place at a university at age 42. The Open University has an AI master that I'm thinking about right now. It has about 25% of the math that I want to learn, so that would be a good start. I did some prep work (an official high school math certificate) last few months and I noticed that I need a schedule to keep me going. One thing that I'm quite certain about, is that doing math is the most important thing. And doing math leads to more doing math.
- ivansavz 4y agoRE learning math: you might want to check out some of my (non-free) books on MECH+CALC https://minireference.com/static/excerpts/noBSmathphys_v5_preview.pdf https://minireference.com/static/excerpts/noBSmathphys_v5_pr... and LINEAR ALGEBRA https://minireference.com/static/excerpts/noBSLA_v2_preview.pdf https://minireference.com/static/excerpts/noBSLA_v2_preview.... They are written especially with adult readers in mind. RE schedule to keep you going: I've had some students use the concept maps as a "world map" (like the stage map in Mario World) and check off boxes as you progress through them (each concept corresponds to roughly one section). See https://minireference.com/static/conceptmaps/math_and_physics_concepts.pdf https://minireference.com/static/conceptmaps/math_and_physic... and https://minireference.com/static/conceptmaps/linear_algebra_concepts.pdf https://minireference.com/static/conceptmaps/linear_algebra_... I guess you could time-box these and do N of sections each week to make this into a schedule. Make sure you dedicate lots of time for the exercises/problems, because that's when the real learning happens...
- markus_zhang 4y agoIn the same boat. One of my dreams is to go back to school, learn vector analysis, differential equation, differential geometry, classic mechanics, electromagnetic, special relativity and finally general relativity. Actually quite doable as long as one can grit through the Math, some of which do not need a back to back read.
- lordnacho 4y agoWhy would you need to go back to school to learn those things? At best school is a syllabus telling you what people think you should know, fairly easy to get a hold of, and maybe a bit of accountability to make sure you actually learn it.
- markus_zhang 4y agoYeah I agree with that. I can also buy a few books and go from there. But essentially O cannot do that now because of X, Y and Z.
- stult 4y agoI’ve had a very similar experience and my solution was to incrementally move into more and more math intensive jobs, starting from regular old SWE working on a web app to working on an aerospace-related web app that involved lots of physics and geospatial calculations and then moving toward MLE/DS jobs. All without a STEM degree, teaching myself the math as I go. It hasn’t been easy but I enjoy what I do more and more over time.
- jtaft 4y agoImpressive :)
- darksensei 4y agoThis is a bit tangential, but was the lack of a STEM degree ever an obstacle in getting those jobs? Just asking because I'm currently in a similar position, regular SWE job, but I'd like to eventually move to something more mathy.
- Koshkin 4y agoBe sure not to retire too late. At an older age, with all the experience you will have under your belt, picking up new concepts and methods will not be a problem, but retaining details in memory will. You have no idea how incredibly hard it will be. So start early.
- agumonkey 4y agonot alone, or more like the vast majority I'm still trying to find a part time dev gig so I can just focus on graphs and advanced combinatorics
- lupire 4y agoYes, you can't do something if you chose to do something else instead. Clearly you find earning/spending money more appealing than math.
- pontus 4y agoWhat a ridiculous oversimplification of life...
- postingposts 4y agoYou can’t really learn Math at school. Not really from the perspective of understanding mathematical beauty deeply. I wish there was an outlet or means to do so. I always found schooling to be woefully inept at assisting in learning the craft.
- mbrodersen 4y agohttps://arxiv.org/pdf/1910.09336.pdf https://arxiv.org/pdf/1910.09336.pdf
- eyelidlessness 4y agoMy lofty future academic pursuit for fun when I can relax from my professional programming career would probably be compsci and then research if I’m still interested. Gonna get really old and tired working in the semicolon mines, then turn around and learn all the stuff I was supposed to learn when I started. At least that’s my picture of my casual retirement interests.
- ifokiedoke 4y agoSame situation, but I've started! I only spend about 30 minutes every other day, and it's extremely slow-going, but so fulfilling. I had the same goal as you -- _really_ learning & understanding math. I finished pre-Algebra last year and I'm halfway through an Algebra text by Gelfand & Shen now. My friends look at me funny when I tell them I'm re-learning Math from the ground up for fun (esp. with a degree in CS) but it has been so rewarding. I probably won't get to finishing Calculus for another couple years but I'm already having so much fun. Stumbled upon deriving some exponent laws last month by accident and truly understanding the sum and difference of squares has been awesome.
- yobbo 4y agoIf your goal of learning is for your own benefit, it's much easier than actually going to school and doing exams, assignments, and getting credits which involves a massive amount of formalities. A good starting point could be MIT OCW 18.01, 18.02, and 18.03. Do all their problem sets and get as much understanding you want. It corresponds to a first year university engineering curriculum.
- dang 4y agoRelated: There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=9517619 https://news.ycombinator.com/item?id=9517619 - May 2015 (32 comments) There’s more to mathematics than rigour and proofs - https://news.ycombinator.com/item?id=4769216 https://news.ycombinator.com/item?id=4769216 - Nov 2012 (36 comments)
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- paulpauper 4y ago#2, #3 means being at the stage where you can look at something and be like "no this cannot work" or "maybe this can work" without having to do all the steps.
- peterhalburt33 4y agoI love Terry Tao’s writing on math. One thing that strikes me about him is that, despite being an absolute technical powerhouse, he writes in a very down to earth style that connects disparate areas of math - e.g. his article on “what is a gauge” https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/amp/ https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/am... where he explains how dimensional analysis might be viewed as a change of coordinates. Too often exposition in math is myopic and fails to impart a unique perspective on the subject, but Tao imbues his writing with a wisdom that I consider the sign of a true genius.
- adamnemecek 4y agoI can't wait for theorem provers to be commonplace.
- Koshkin 4y agoA proof that no one would understand in not a good proof. The ideal approach to proving theorems, at least according to how Grothendieck did it, is to build a beautiful theory in which the proof becomes elementary.
- throwamon 4y agoIt's quite a big assumption to think truth can always be bent so as to satisfy our ridiculously limited cognition. And math has been used instrumentally from the very beginning, so results are often much more important than the process. Theoreticians may still value elegance because that gives them pleasure or whatever, but few other people care about that as long as they can use the results.
- l33t2328 4y agoThe results aren’t often nearly as useful as the techniques used to find them. For example, a completely opaque proof resolving P vs NP is completely useless to theoretical CS.
- mbustamanter 4y agoThis is remarkably accurate and resonates with me a lot. I did mathematical olympiads in high-school, where intuition to crack problems plays a major role. Then went on to college to study an undergraduate degree in maths (concentration in analysis). Analysis requires, at least in its rigorous foundations, to be careful and have a skilled knowledge of logic/quantifiers (more than elementary abstract algebra in my humble and biased opinion), often very scrupulously. Then in my graduate studies intuition along with the maturity of rigor work to produce new theorems. I'm impressed that several times I look at a paper or series of results and can read them "diagonally" to get the motivation without scanning all the text (of course, if the aim is to cite/build on top of/generalize/apply it then close attention to reasoning should be paid).
- vlovich123 4y agoI feel like this is how all domain expertise works, no? Start with intuition which helps you solidify some of the foundation. Flush out the foundation and start building complicated structures. Now that you've built up the experience, go back and use your intuition to figure out new types of buildings to build.
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- mbrodersen 4y agoI understood math much better after following the Software Foundations tutorials: https://softwarefoundations.cis.upenn.edu/ https://softwarefoundations.cis.upenn.edu/ Another good starting point (for mathematicians) would be the Lean community group and the Math lib project: https://arxiv.org/pdf/1910.09336.pdf https://arxiv.org/pdf/1910.09336.pdf https://github.com/leanprover-community/mathlib https://github.com/leanprover-community/mathlib
- kazinator 4y agoMathematics needs imagination to come up with conjectures that need proving; without that there is nothing to apply rigor to.
- auggierose 4y agoYou can infer from this how Artificial Intelligence and Logic should be combined: AI enables a "post-rigorous" mode, and Logic is how you know you are still doing something sensible, and how you expand your sure footing.
- formerkrogemp 4y agoAll you need to get by in life is a solid understanding of a little applied math and statistics. Leave the rest for professors and hobbyists.