Mathematics Without Apologies Portrait Of A Problematic Vocation Pdf

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

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.

> 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).

>"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.

> 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.

I wonder if he was alluding to Caesar asserting "All Gaul is divided into three parts:..." (albeit in Latin).

>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.

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?

> 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.

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.

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.

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!

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.

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.

wow, this is a fascinating, outstanding interview. I don't know why, but I was immediately hooked. I watched the whole thing.

Agreed. I somehow can't imagine such an interview being made today. Where mathematicians would talk about the math in a serious but fascinated tone,talking about theoretical stuff without flashy animations or cutesy stories.

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.

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.

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."

But not the cryptography that was in use at the time - for example Enigma is more combinatorial than number-theoretic.

Relativity, particularly the mass-energy equivalence principle of special relativity, lead to the thinking that lead to The Bomb less than five years later.

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.

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.

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.

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".

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.

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.

I'm not complaining, just pointing it out because it's an interesting quibble: "hoi" means "the" in greek, so no need to say "the hoi polloi"

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

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.

No, I mean the phrase. This is not the only time I am seeing that as people age, they lose their edge in math (e.g bertrand russell focused more on philosophy as he got older). Is it same in in computer science

> 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.