[HN Gopher] There's more to mathematics than rigour and proofs (...
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There's more to mathematics than rigour and proofs (2007)
Author : _ttg
Score : 75 points
Date : 2022-04-19 18:30 UTC (4 hours ago)
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| ffhhj wrote:
| Which 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 wrote:
| It 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.
| paulpauper wrote:
| #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.
| devnulll wrote:
| One 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 wrote:
| keep 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.
| lordnacho wrote:
| This is why I'm happy to simply read the well established
| facts of a variety of fields. There's more than enough to
| learn for a lifetime, and from what I read about research
| there's a heck of a lot of BS in the way for small
| incremental gains.
|
| Having said that, if something does come along that interests
| someone, they should try it. A friend of mine is doing his
| 2nd PhD 40 years after his first, having found his way into
| it via a love of jazz music.
| andi999 wrote:
| About the simulation hypothesia, how is that different from
| Descartes evil demon?
|
| https://en.m.wikipedia.org/wiki/Evil_demon
|
| (apart from saying that the evil demon is a future type
| computer)
| ffhhj wrote:
| I couldn't wait that much, started researching stochastic
| processes a few years ago and been developing a theory for
| supertasks.
| bombcar wrote:
| I 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?
| DecentAI wrote:
| What was this utter gibberish I just read? I'm absolutely
| positive you have the least idea of what modern mathematical
| research entails or is even about. Please refrain from
| sharing your opinions on subject matters which you clearly
| lack any understanding of.
| nextos wrote:
| I totally agree with what you say, but there's a _lot_ of low
| hanging fruit in mathematizing biology.
|
| It's not easy, but it's certainly very impactful.
| ratzkewatzke wrote:
| I'm not going to gainsay your experience, but it doesn't
| match mine. Much of what you do in graduate school is to
| discover where the accessible areas of research are--where do
| we have a foothold, and are making progress, and what are
| some achievable results?
|
| There are big and hairy problems that are bad investments for
| a young mathematician. I would steer students clear of the
| Collatz conjecture. But once you get up to speed in your
| research area, you usually find interesting problems thick on
| the ground.
|
| Tenure-track positions are competitive, but I don't think
| there are a lack of interesting things to work on.
| paulpauper wrote:
| when I worked on math I found that no matter what problem I
| was working on, someone had already solved it completely or
| to high level of abstraction than I had. got discouraging
| after while.
| xphos wrote:
| This is what a imagine late stage security after OS mature
| there stacks and such. The advent/widespread use of robost
| memory protections like PAC and Cheri are going to be so
| depressing for those on the offensive.
|
| I wish I just had more time to do math like the OP but the
| saturation of the field especially with people who can deeply
| understand the abstraction is very very intimidating.
| Koshkin wrote:
| Be 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.
| foobarian wrote:
| I'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 wrote:
| The 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.
| markus_zhang wrote:
| In 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 wrote:
| Why 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 wrote:
| Yeah 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.
| agumonkey wrote:
| not 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
| qsort wrote:
| One 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 :/
| turtleyacht wrote:
| Same. But we can keep looking for opportunities to learn math
| in our spare time. For me though, old CS books have a lure
| all their own :) Maybe an algorithms book (plus MathOverflow)
| or TLA+ as a gateway.
| qiskit wrote:
| CS is math...
| dumpsterlid wrote:
| In a practical sense, not really.
| stult wrote:
| I'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.
| stocknoob wrote:
| Aim 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 wrote:
| I'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.
| dang wrote:
| Related:
|
| _There's more to mathematics than rigour and proofs_ -
| 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 - Nov 2012 (36
| comments)
| [deleted]
| vlovich123 wrote:
| I 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.
| [deleted]
| adamnemecek wrote:
| I can't wait for theorem provers to be commonplace.
| Koshkin wrote:
| A 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 wrote:
| It'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.
| peterhalburt33 wrote:
| I 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/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.
| cpp_frog wrote:
| This 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).
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