A Mathematician’s Apology
en.wikipedia.org
en.wikipedia.org
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
> 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.
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).
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.
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.
Nope. When you get into real analysis you see that this is not true. It's hard to even get the Real Numbers down.
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.
That rant does not actually represent my beliefs, it just seemed like what the ggp wanted so, tada!
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.
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.
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.
The closest I can think of is Brady Haran's Numberphile or perhaps Lex Fridman's AI interviews (for example the Stroustup interview did go into technicalities). But these are still much more palatable and worried about losing the general viewership.
> "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."
Einstein’s politics had more to do with the bomb than his physics.
> The GPS project was started by the U.S. Department of Defense in 1973 [1]
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".
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.
[0] https://www.maths.cam.ac.uk/sites/www.maths.cam.ac.uk/files/...
And, it's not even particularly good in instilling some appreciation of the beauty of mathematics.
Save your time.
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
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.
Why is that the case?
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.