[HN Gopher] A Mathematician's Apology
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A Mathematician's Apology
Author : benbreen
Score : 120 points
Date : 2020-02-02 04:22 UTC (1 days ago)
(HTM) web link (en.wikipedia.org)
(TXT) w3m dump (en.wikipedia.org)
| valgor wrote:
| While Hardy's book is still relevant, there is a more modern
| attempt at answering the same questions for those interested:
| Mathematics without Apologies: Portrait of a Problematic Vocation
| by Michael Harris.
| nik61 wrote:
| Many people, mathematicians and aspirants, would find
| "Mathematics made Difficult" by Carl E Linderholm (pub. 1972)
| entertaining and possibly instructive. PDFs are available to
| those without scruples.
| Ohn0 wrote:
| How about those of us with scruples?
| Anon84 wrote:
| In that case, Amazon is your friend:
| https://www.amazon.com/Mathematics-made-difficult-Carl-
| Linde...
| ducaale wrote:
| > mathematics is a "young man's game"
|
| Why is that the case?
| ggggtez wrote:
| Are you asking for a summary of the book? I'm sure you can
| google that.
| nautilus12 wrote:
| Reminds me of how when designing software systems for companies
| you always start with the user access patterns otherwise you end
| up building some elegant stream based system for something that
| only needs batch access. Pure math can be the same way in alot of
| ways, why invest the precious resource of human innovation
| getting lost in the woods? It's possible alot of pure math is
| just us convincing ourselves of it's value unchecked by any
| actual means of producing value from it. Context: used to be a
| pure mathematician, now an engineer.
| awild wrote:
| > He justifies the pursuit of pure mathematics with the argument
| that its very "uselessness" on the whole meant that it could not
| be misused to cause harm.
|
| honestly, why not do something good and apply yourself to a cause
| that you believe in instead of doing something you intently don't
| believe in. It honestly reads like something out of badly written
| marxist satire of bourgeoisie.
| spanxx wrote:
| Would you recommend that book to a software engineer with an
| interest in Maths?
| jlarcombe wrote:
| Definitely, it's a brilliant read.
| solar_perplexus wrote:
| I would suggest picking up a textbook in mathematics for a
| specific area relevant to your interest.
|
| I found that studying specific topics (currently probability
| theory and statistics) helped me comprehend the field better
| and in ways that make it practical for my planned career.
| sfsylvester wrote:
| It's a great read. I have also found a great reading list of
| from the University of Cambridge[0] that has a whole different
| range of books and websites for those who have a cursory
| interest in Mathematics.
|
| [0]
| https://www.maths.cam.ac.uk/sites/www.maths.cam.ac.uk/files/...
| math123 wrote:
| Slightly updated version of the list at
| https://www.maths.cam.ac.uk/documents/reading-list.pdf/
| FabHK wrote:
| No. It's been a while since I read it, but I found it an
| unapologetic, tedious, arrogant, self-serving screed lionising
| pure mathematics (and its practitioners, but only the very
| best), denigrating anything applied or even applicable, yet
| basically asking the hoi polloi doing that nether pedestrian
| work of actually working to put food on his table.
|
| And, it's not even particularly good in instilling some
| appreciation of the beauty of mathematics.
|
| Save your time.
| vector_spaces wrote:
| I think A Mathematician's Apology is a good read, but if you're
| looking to learn mathematics there are probably better places
| to start.
|
| If you want a cursory view of various parts of mathematics, you
| might prefer Courant's book "What is Mathematics?". Depending
| on your background and interest, there is a volume of books
| (available as a consolidated cheap Dover paperback) called
| Mathematics: Its Content, Meaning, and Methods.
|
| I recently came across A Programmer's Book of Mathematics[0] --
| I haven't read it, but the author is a developer and the
| content might be more appropriate if you're just starting out
| -- both of the other books I mentioned are older, and are
| really wonderful texts, but might possibly be overwhelming
| depending on your appetite and background.
|
| Finally, if you're more interested in math that's relevant to
| software engineers, there's Knuth's book "Concrete
| Mathematics".
| synthmeat wrote:
| I'd second the "What is Mathematics?" and "Mathematics: Its
| Content, Methods and Meaning" recommendations, in that order.
| Both are very cheap, comprehensive, and in ascending rigour.
| After those, it's really dealer's choice, with "Concrete
| Mathematics" being of particular interest to computer
| scientist.
|
| Additionally:
|
| - "Princeton Companion to Mathematics" is really fun to have
| around for exploration
|
| - if you're really really rusty with math, take a week or two
| with "Mathematical Handbook - Elementary Mathematics" by
| Vygodsky
|
| I literally have all these on my desk at this very moment,
| what a fun coincidence.
| jonjacky wrote:
| It's not much about maths, it's a memoir by a particular
| mathematician written at an unhappy point in his life,
| reflecting back on what he has done. It belongs more to the
| literature about artists and creativity, than to books of
| mathematical content. For that, see instead Hardy's Course of
| Pure Mathematics [1]
|
| If you do read the Apology, be sure to get an edition with C.P.
| Snow's forward, which gives the back story that puts Hardy's
| memoir in context. It also includes the wonderful story of
| Hardy and Ramanujan.
|
| [1] https://en.wikipedia.org/wiki/A_Course_of_Pure_Mathematics
| LeanderK wrote:
| > On the other hand, Hardy denigrates much of the applied
| mathematics as either being "trivial", "ugly", or "dull", and
| contrasts it with "real mathematics", which is how he ranks the
| higher, pure mathematics.
|
| Not sure if this surprising, since he was a vocal pure
| mathematician. But since I don't agree (and it sometimes looks
| like the pure mathematicians look down on the applied), I wonder
| whether there are some texts from famous applied mathematicians
| defending their branch.
| contravariant wrote:
| History's greatest example of irony is that Hardy's field of
| interest ended up being one of the most widely cited examples
| of the practical applications of abstract mathematics.
| zen_of_prog wrote:
| https://www.smbc-comics.com/index.php?id=4130
| generationP wrote:
| Tom Korner's "The Pleasures of Counting" (
| https://www.amazon.com/Pleasures-Counting-T-W-K%C3%B6rner/dp...
| ) is an amusingly written yet intellectually challenging
| introduction to applied maths for the general audience. (His
| website https://www.dpmms.cam.ac.uk/~twk/ will give you an idea
| of the writing style.)
| LeanderK wrote:
| but does it contain a philosophical argument like Hardy?
| galangalalgol wrote:
| Philosophical arguments are for pure math navel gazers.
| Applied folks are too busy actually doing things that make
| people's lives better.
|
| That rant does not actually represent my beliefs, it just
| seemed like what the ggp wanted so, tada!
| LeanderK wrote:
| Not sure what you're getting at. I think it's often
| fascinating to read texts about the instrinsic motivation
| of these different crafts, as an example "the beauty of
| programming" from linus torvalds:
| https://www.brynmawr.edu/cs/resources/beauty-of-
| programming
|
| So I wonder, since this is more or less direct attack on
| the beauty of applied mathematics how they would respond.
| What makes applied math beautiful? I can certainly say
| what I find beautiful about CS and Machine Learning! It
| strikes me as obvious, but that's probably why I like it.
| chongli wrote:
| _make people 's lives better_
|
| But arguably also making people's lives worse. Applied
| mathematicians and physicists have contributed to the
| development of weapons of war since the time of
| Archimedes. As a committed and outspoken pacifist, Hardy
| wanted no part in warfare. This informed a good part of
| his philosophy, along with aesthetics.
|
| I would hardly call that navel gazing.
| generationP wrote:
| It's not a philosophic pamphlet; Chapters 18 and 19 have
| some amount of philosophizing, but the rest of the book is
| heavily "show, don't tell". Which is the logical way to
| write a book on applied(!) maths.
| bonoboTP wrote:
| Look at this interview with Cornelius Lanczos:
| https://youtu.be/avSHHi9QCjA?t=141
|
| At the linked timestamp he talks about how he doesn't like the
| split of "pure" and "applied" math and how it's a very new
| concept of the 20th century.
| enriquto wrote:
| This is a beautiful read. It should not be taken too much
| seriously, though.
|
| For a nice complement, see the writings of another, and arguably
| much greater, mathematician V.I. Arnold. His contrasting view on
| the nature of mathematics is that "mathematics is the part of
| physics were experiments are cheap".
|
| EDIT: I add quotes and links to some of Arnold's writing.
|
| " Mathematics is a part of physics. Physics is an experimental
| science, a part of natural science. Mathematics is the part of
| physics where experiments are cheap.
|
| The Jacobi identity (which forces the heights of a triangle to
| cross at one point) is an experimental fact in the same way as
| that the Earth is round (that is, homeomorphic to a ball). But it
| can be discovered with less expense. "
|
| --[1] _On Teaching Mathematics_
|
| "All mathematics is divided into three parts: cryptography (paid
| for by CIA, KGB and the like), hydrodynamics (supported by
| manufacturers of atomic submarines) and celestial mechanics
| (financed by military and by other institutions dealing with
| missiles, such as NASA.).
|
| Cryptography has generated number theory, algebraic geometry over
| finite fields, algebra, combinatorics and computers.
|
| Hydrodynamics procreated complex analysis, partial derivative
| equations, Lie groups and algebra theory, cohomology theory and
| scientific computing.
|
| Celestial mechanics is the origin of dynamical systems, linear
| algebra, topology, variational calculus and symplectic geometry.
|
| The existence of mysterious relations between all these different
| domains is the most striking and delightful feature of
| mathematics (having no rational explanation)."
|
| --[2] _Polymathematics: Is mathematics a single science or a set
| of arts?_
|
| [1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
|
| [2] http://math.ucr.edu/home/baez/Polymath.pdf
|
| [3] Link to many Arnold's writings:
| http://www.pdmi.ras.ru/~arnsem/Arnold/arn-papers.html
| oefrha wrote:
| As a physicist who studied mathematics as an undergrad and was
| admitted into some top math graduate programs (but didn't go
| because I pursued theoretical physics instead) -- no, only part
| of mathematics has something to do with physics at all, let
| alone being inspired by physics.
|
| > Cryptography has generated number theory, algebraic geometry
| over finite fields, algebra, combinatorics and computers.
|
| > Hydrodynamics procreated complex analysis, partial derivative
| equations, Lie groups and algebra theory, cohomology theory and
| scientific computing.
|
| > Celestial mechanics is the origin of dynamical systems,
| linear algebra, topology, variational calculus and symplectic
| geometry.
|
| Now those are just backwards.
| enriquto wrote:
| > Now those are just backwards.
|
| Of course, it is a rhetorical device. If you read the rest of
| the linked article you'll see that the author has quite a
| sense of humor. The content is rather serious though (about
| unexpected links between seemingly unrelated parts of math).
| [deleted]
| oarabbus_ wrote:
| >"All mathematics is divided into three parts: cryptography
| (paid for by CIA, KGB and the like), hydrodynamics (supported
| by manufacturers of atomic submarines) and celestial mechanics
| (financed by military and by other institutions dealing with
| missiles, such as NASA.).
|
| This seems like an odd classification or backwards. The
| mathematics existed, long before humans existed. Humans simply
| _discovered_ it when studying some of the disciplines mentioned
| above.
|
| The statement "all mathematics is divided into cryptography,
| hydrodynamics, and celestial mechanics" seems untrue. I'd
| personally disagree that for example, topology can be entirely
| attributed to celestial mechanics, scientific computing to
| hydrodynamics, algebra to cryptography, etc.
|
| Also, it seems fair to call Arnold a greater mathematician than
| Hardy, rather than just "arguably" so based on their direct
| work. Hardy's greatest contribution to mathematics was
| discovering and nurturing Ramanujan, who was a top 10
| mathematical talent of all time.
| cma wrote:
| Pretty sure it was intentionally facetious.
| jjtheblunt wrote:
| I wonder if he was alluding to Caesar asserting "All Gaul is
| divided into three parts:..." (albeit in Latin).
| domnomnom wrote:
| >This seems like an odd classification or backwards. The
| mathematics existed, long before humans existed. Humans
| simply _discovered_ it when studying some of the disciplines
| mentioned above.
|
| Nope. When you get into real analysis you see that this is
| not true. It's hard to even get the Real Numbers down.
| enriquto wrote:
| > The statement "all mathematics is divided into
| cryptography, hydrodynamics, and celestial mechanics" seems
| untrue.
|
| He's obviously using an over-the-top generalization to be
| provocative and funny. I guess we are not supposed to
| understand these words literally.
|
| The adscription of topology to mechanics is not entirely
| casual in his case. He's essentially the father of
| topological methods in dynamics, and he proved (with
| Kolmogorov and Moser) the famous "KAM" theorem about the long
| term stability of the solar system with probability one.
|
| Notice that Hardy also uses exaggeration to state some of his
| finest claims, and he's probably a better writer than Arnold
| because he manages to do so without the reader noticing.
| karlicoss wrote:
| Great read! The bit that I found the most curious is actually
| listed on Wikipedia (this was written in 1940):
|
| > "No one has yet discovered any warlike purpose to be served by
| the theory of numbers or relativity, and it seems unlikely that
| anyone will do so for many years."
| Craighead wrote:
| But isn't that actually a false statement? Intelligence
| satellites rely on relativity to work out gravity formulas to
| stay afloat.
| jesuslop wrote:
| What about elliptic curve cryptography?
| EthanHeilman wrote:
| Yep, or RSA and most modern cryptography.
| madcaptenor wrote:
| But not the cryptography that was in use at the time - for
| example Enigma is more combinatorial than number-theoretic.
| hjorthjort wrote:
| What about pretty much all of cryptography?
| madaxe_again wrote:
| Relativity, particularly the mass-energy equivalence principle
| of special relativity, lead to the thinking that lead to The
| Bomb less than five years later.
| Armisael16 wrote:
| Atomic weapons were far more an outgrowth of quantum research
| than relativity. The mass-energy equivalence is a pretty
| minor part of the energy release of the atomic bomb at ~10%.
| Most of the energy is simple atomic binding energy; energy
| before and energy after.
|
| Einstein's politics had more to do with the bomb than his
| physics.
| mikorym wrote:
| >> "No one has yet discovered any warlike purpose to be served
| by the theory of numbers or relativity, and it seems unlikely
| that anyone will do so for many years."
|
| > The GPS project was started by the U.S. Department of Defense
| in 1973 [1]
|
| [1] https://en.wikipedia.org/wiki/Global_Positioning_System
| whatshisface wrote:
| > _From 1942 to 1946, the [Manhattan] project was under the
| direction of Major General Leslie Groves of the U.S. Army
| Corps of Engineers._
|
| https://en.wikipedia.org/wiki/Manhattan_Project
| hjorthjort wrote:
| It almost sounds like this book is the reason we so often hear
| and talk about the imposed hierarchy between "pure" and
| "applied", and that "mathematics is a young man's game". Did it
| have a huge impact? Or were these ideas commonplace before it was
| published?
| Ohn0 wrote:
| Sounds like the beginning of excluding women from stem.
| generationP wrote:
| The _beginning_?
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