
A Mathematician’s Apology - benbreen
https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology
======
enriquto
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](https://www.uni-
muenster.de/Physik.TP/~munsteg/arnold.html)

[2]
[http://math.ucr.edu/home/baez/Polymath.pdf](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](http://www.pdmi.ras.ru/~arnsem/Arnold/arn-papers.html)

~~~
oarabbus_
>"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.

~~~
enriquto
> 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.

------
hjorthjort
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
Sounds like the beginning of excluding women from stem.

~~~
generationP
The _beginning_?

------
LeanderK
> 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
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
[https://www.smbc-comics.com/index.php?id=4130](https://www.smbc-
comics.com/index.php?id=4130)

------
valgor
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.

------
karlicoss
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."

~~~
jesuslop
What about elliptic curve cryptography?

~~~
EthanHeilman
Yep, or RSA and most modern cryptography.

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

------
nik61
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
How about those of us with scruples?

~~~
Anon84
In that case, Amazon is your friend: [https://www.amazon.com/Mathematics-made-
difficult-Carl-Linde...](https://www.amazon.com/Mathematics-made-difficult-
Carl-
Linderholm/dp/0529045524/ref=sr_1_1?keywords=%22Mathematics+made+Difficult%22&qid=1580748854&sr=8-1)

------
nautilus12
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.

------
spanxx
Would you recommend that book to a software engineer with an interest in
Maths?

~~~
vector_spaces
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
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.

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ducaale
> mathematics is a "young man's game"

Why is that the case?

~~~
ggggtez
Are you asking for a summary of the book? I'm sure you can google that.

~~~
ducaale
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

------
awild
> 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.

