7 ms·

“A Course of Pure Mathematics” – G. H. Hardy (1921) [pdf]

317 points by bikenaga 2y ago

ninalanyon 2y ago
It would be more useful to link to the Gutenberg page that shows links to the various formats: https://www.gutenberg.org/ebooks/38769 https://www.gutenberg.org/ebooks/38769
- bikenaga 2y ago
  Yeah, my bad - I'm so accustomed to pdfs that I forget people often like other formats. Thanks for the link!
  - damhsa 2y ago
    the tex file is huge because its got rasterized images in it, still nice to have though
- deleted 2y ago
  [deleted]
bell-cot 2y ago
A bit of background on Hardy, and his choice of title: https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics
vouaobrasil 2y ago
I read this book as a first-year undergrad. His style inspired me to go after rigour and proof and was a good start to serious mathematics. I always loved Hardy's work and Hardy and Wright's number theory text was also very nice through my PhD in algebra/number theory. I found Hardy's book much nicer than the contemporary calculus texts with irrelevant pictures and modern-day examples. Just straight math! Not for everyone, but it has classical, austere appeal for those who enjoy such things.
- analog31 2y ago
  Classical and austere, but not stilted. For instance... >>> We can state this more precisely as follows: if we take any segment BC on Λ, we can find as many rational points as we please on BC. reads as a normal English sentence. As a student, I also preferred straight math. Proofs were what made math come alive for me. For applications of math, I had plenty of other sources such as physics, electronics, and programming, where the examples weren't forced.
  - vouaobrasil 2y ago
    > As a student, I also preferred straight math. Proofs were what made math come alive for me. For applications of math, I had plenty of other sources such as physics, electronics, and programming, where the examples weren't forced. I guess the difference between us then is that I didn't care about applications.
    - 867-5309 2y ago
      that's like studying medicine then not becoming a doctor
      - AnimalMuppet 2y ago
        Not becoming a patient-treating doctor. Research doctors still matter a great deal in the field of medicine.
        orochimaaru 2y ago
        I doubt you have pure research doctors. Medicine is a field that is so dependent on treatment outcomes. There will be doctors more focused on research. However, I doubt they will stop seeing patients. I know for a fact that pediatric oncology and hematology is entirely driven out of a research hospital or university. But doctors there publish but also treat.
      - analog31 2y ago
curiouscavalier 2y ago
One of my favourite texts. One of those that I found influential early in academics as well as when re-reading later in my career. Even for younger students I think it can be great introduction to more formal approaches, as well as a taste for the austere.
profsummergig 2y ago
IMHO, it has the best ELI5 explanation of integration, still not bested in 100 years.
wannabebarista 2y ago
In A Mathematician's Apology (1940), Hardy has lots of fun musings on math. I don't have the quote handy, but he argues that pure math is closer to reality than applied math since it deals with actual mathematical objects rather than mathematical models of physical objects.
- gjm11 2y ago
  "There is another remark which suggests itself here and which physicists may find paradoxical, though the paradox will probably seem a good deal less than it did eighteen years ago. I will express it in much the same words which I used in 1922 in an address to Section A of the British Association. [...] I began by saying that there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, and that the most important seems to me to be this, that the mathematician is in much more direct contact with reality. This may seem a paradox [...] but a very little reflection is enough to show that the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God; each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense. [...] A mathematician, on the other hand, is working with his own mathematical reality. Of this reality, as I explained in section 22, I take a 'realistic' and not an 'idealistic' view. At any rate (and this was my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy -- I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way." That's not quite the argument you describe -- his point is more that mathematical objects as understood by the mathematician are more like mathematical objects as we encounter them casually, than physical objects as understood by the physicist are like physical objects as we encounter them casually -- the physicist will insist that the chair you're sitting on is really a sort of configuration of fluctuations in quantum fields, but if you count up to 23 then the mathematician will agree that what you just did really does reflect the underlying nature of the number 23. (If you build all mathematics on top of set theory, then you will most likely treat the number 23 as some much more complicated thing. But you'll see that as an "implementation detail" that could be done in lots of different ways, rather than saying that really, deep down 23 is this complicated thing with lots of weird internal structure.)
bira 2y ago
Alternative link: https://archive.org/download/in.ernet.dli.2015.239784/2015.239784.A-Course.pdf https://archive.org/download/in.ernet.dli.2015.239784/2015.2...
dboreham 2y ago
Hmm...that's familiar. Blue cover iirc.
ipnon 2y ago
The preface left a deep impression on me because Hardy says only exceptionally talented and intelligent students will be able to finish and understand this book. It was a revelation because I grew up in schools that claimed anyone can learn maths to any level so long as they had enough interest and support. It was healthy as a young man to know I was beginning to approach my limits as a mathematician, and this book ultimately led me to focus more on computer science and programming.
- psychoslave 2y ago
  Note that both statements can be compatible. With exceptional interest and support, certainly anyone can absorb all these concepts. Of course if we take the case of someone about to die in 24h max due to brain cancer, then sure we just don't have the knowledge and resources to successfully make someone acquire that kind of knowledge. But in the general case, people not learning advanced math notions is everywhere in the intersection between individual having no interest and society not pushing them in that direction through resource allocations. Also since it's about Hardy we can not withdraw the case of Ramanujan. Yes, there are people whose brain is wired in very uncommon way that push them toward exploring uncharted territories where few have interest and even less have the ability to go through and survive. That said, once the path is paved and everything is in place to accommodate the accessibility of the place, there is no longer the same level of struggle to be expected. More often, we lake the great teaching resources, rather than the sufficiently apt learners.
  - n4r9 2y ago
    "Exceptionally talented and intelligent" has a very different meaning to "with exceptional interest and support". When people say talent, they invariably mean a natural endowment or inherent skill. Something distinct from what id possible with support and interest. It's arguable that no such thing as talent exists. That when we say "talent" we are really using a short-hand for a kind of knowledge and experience. I suspect that Hardy would strongly reject such a claim, based on reading his Apology.
- 392 2y ago
  You should read David Bessis's Mathematica.
OldGuyInTheClub 2y ago
From the title page. No Ph.D. necessary to be a legend in them days. ;-) G. H. HARDY, M.A., F.R.S. FELLOW OF NEW COLLEGE SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE
- scapp 2y ago
  M.A. was the highest degree available in the UK at the time [1]. The closest equivalent to a PhD program might be the Prize Fellowship that Hardy had from 1900 to 1906, though it's certainly not one to one. Don't get the impression that Hardy was done with his education when he got his masters degree. [1] https://www.economics.soton.ac.uk/staff/aldrich/Doc1.htm https://www.economics.soton.ac.uk/staff/aldrich/Doc1.htm [2] https://royalsocietypublishing.org/doi/10.1098/rsbm.1949.0007 https://royalsocietypublishing.org/doi/10.1098/rsbm.1949.000...
  - OldGuyInTheClub 2y ago
    Interesting. The Wikipedia page for his contemporary Littlewood lists a Doctoral Advisor. https://en.wikipedia.org/wiki/John_Edensor_Littlewood https://en.wikipedia.org/wiki/John_Edensor_Littlewood Edit: The Mathematics Genealogy Project lists Littlewood as an M.A. as well. https://www.mathgenealogy.org/id.php?id=10463 https://www.mathgenealogy.org/id.php?id=10463
- griffzhowl 2y ago
  Freeman Dyson famously never got a PhD either, he just made seminal contributions to quantum field theory instead
  - OldGuyInTheClub 2y ago
    Yeah... he hated the Ph.D. system, called it evil. He also said a great advantage of the Institute for Advanced Study was that he could work with postdocs - recent products of that same evil system. Source: Why I don't like the PhD system (95/157) https://www.youtube.com/watch?v=DzC1IRYN_Ps https://www.youtube.com/watch?v=DzC1IRYN_Ps
nephanth 2y ago
103 years old. Pretty recent at the scale of Mathematics, although that does make it older than Bourbaki. And than category theory
adalacelove 2y ago
Nice book. For another old but excellent math book I recommend Geometry and the Imagination by David Hilbert. No gutenberg remake I'm afraid, maybe because of the numerous (and incredibly high quality) illustrations.
- mayd 2y ago
  David Hilbert and Stefan Cohn-Vossen.
  - adalacelove 2y ago
    True. I was citing from memory and only recalled the most famous author.
mayd 2y ago
Hardy's "A course of Pure Mathematics" has been highly regarded since it was first published in 1908 because it was an innovative text: rigorous, modern, well-written. Its intended readership was always first year "honours" mathematics students. This book inspired innovation in subsequent generations of textbook writers. However, in the 21st century, this book really can no longer be recommended for its original teaching purpose. As a textbook it is outdated (a term I hate, but it is true). It is now an historical curiosity - although one which I am pleased to own, and the exercises in the book are still worth a look. Calculus teaching has progressed considerably since 1908. The construction of the real number system in Hardy's book, using the Dedekind Cut method is overly complicated - the use of the of Least Upper Bound is much simpler and clearer. Hardy defines the concept of integral solely as the anti-derivative; there is no discussion of Riemann sums, or Darboux sums, etc. I am sure I would not want to take Hardy's approach today. I think we are better off recommending books are more modern. I will start by recommending "Calculus" by Michael Spivak.
- chris_wot 2y ago
  I’m currently reading through Martin to teach myself Calculus - can really recommend it!
- housecarpenter 2y ago
  What do you mean by "the use of the Least Upper Bound"?
  - denotational 2y ago
    Seconded, the “least upper bound” method for constructing the reals that I know about is… …Dedekind cuts.
    - davidgrenier 2y ago
      I haven't looked at Hardy's but the presentation in Spivak is also Dedekind cuts. Perhaps Hardy uses a different approach and OP misnamed it? Rudin's chapter 1 annex also use Dedekind's cuts.
globalnode 2y ago
Ive been nerd sniped by point 158 on page 353. I cant believe I slipped through so many calc classes without understanding Leibniz's rule for taking the derivative of a definite integral. I didn't actually follow through with the calculation in Hardy's book but I bet it haunts me until I do :(.
JPC21 2y ago
I remember having a hard time understanding limsup/liminf and only actually understood the concept after reading this book. I am sure it is not for everyone due to its age, but I think this book is a much better introduction than baby Rudin.
nthingtohide 2y ago
I think in the book, Hardy mentions that Mathematics is one big tautology. Intial axioms setup a graph structure of theorems and the task of mathematicians is to find shortcuts in that graph.
TheRealNGenius 2y ago
[dead]
deleted 2y ago
[deleted]

317 points by bikenaga 2y ago

It would be more useful to link to the Gutenberg page that shows links to the various formats: https://www.gutenberg.org/ebooks/38769 https://www.gutenberg.org/ebooks/38769

bikenaga 2y ago
Yeah, my bad - I'm so accustomed to pdfs that I forget people often like other formats. Thanks for the link!
- damhsa 2y ago
  the tex file is huge because its got rasterized images in it, still nice to have though
deleted 2y ago
[deleted]

bell-cot 2y ago

A bit of background on Hardy, and his choice of title: https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics

vouaobrasil 2y ago

I read this book as a first-year undergrad. His style inspired me to go after rigour and proof and was a good start to serious mathematics. I always loved Hardy's work and Hardy and Wright's number theory text was also very nice through my PhD in algebra/number theory. I found Hardy's book much nicer than the contemporary calculus texts with irrelevant pictures and modern-day examples. Just straight math! Not for everyone, but it has classical, austere appeal for those who enjoy such things.

analog31 2y ago
Classical and austere, but not stilted. For instance... >>> We can state this more precisely as follows: if we take any segment BC on Λ, we can find as many rational points as we please on BC. reads as a normal English sentence. As a student, I also preferred straight math. Proofs were what made math come alive for me. For applications of math, I had plenty of other sources such as physics, electronics, and programming, where the examples weren't forced.
- vouaobrasil 2y ago
  > As a student, I also preferred straight math. Proofs were what made math come alive for me. For applications of math, I had plenty of other sources such as physics, electronics, and programming, where the examples weren't forced. I guess the difference between us then is that I didn't care about applications.
  - 867-5309 2y ago
    that's like studying medicine then not becoming a doctor
    - AnimalMuppet 2y ago
      Not becoming a patient-treating doctor. Research doctors still matter a great deal in the field of medicine.
      - orochimaaru 2y ago
        I doubt you have pure research doctors. Medicine is a field that is so dependent on treatment outcomes. There will be doctors more focused on research. However, I doubt they will stop seeing patients. I know for a fact that pediatric oncology and hematology is entirely driven out of a research hospital or university. But doctors there publish but also treat.
    - analog31 2y ago

curiouscavalier 2y ago

One of my favourite texts. One of those that I found influential early in academics as well as when re-reading later in my career. Even for younger students I think it can be great introduction to more formal approaches, as well as a taste for the austere.

profsummergig 2y ago

IMHO, it has the best ELI5 explanation of integration, still not bested in 100 years.

wannabebarista 2y ago

In A Mathematician's Apology (1940), Hardy has lots of fun musings on math. I don't have the quote handy, but he argues that pure math is closer to reality than applied math since it deals with actual mathematical objects rather than mathematical models of physical objects.

gjm11 2y ago
"There is another remark which suggests itself here and which physicists may find paradoxical, though the paradox will probably seem a good deal less than it did eighteen years ago. I will express it in much the same words which I used in 1922 in an address to Section A of the British Association. [...] I began by saying that there is probably less difference between the positions of a mathematician and of a physicist than is generally supposed, and that the most important seems to me to be this, that the mathematician is in much more direct contact with reality. This may seem a paradox [...] but a very little reflection is enough to show that the physicist's reality, whatever it may be, has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons, or an idea in the mind of God; each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense. [...] A mathematician, on the other hand, is working with his own mathematical reality. Of this reality, as I explained in section 22, I take a 'realistic' and not an 'idealistic' view. At any rate (and this was my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but '2' or '317' has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy -- I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way." That's not quite the argument you describe -- his point is more that mathematical objects as understood by the mathematician are more like mathematical objects as we encounter them casually, than physical objects as understood by the physicist are like physical objects as we encounter them casually -- the physicist will insist that the chair you're sitting on is really a sort of configuration of fluctuations in quantum fields, but if you count up to 23 then the mathematician will agree that what you just did really does reflect the underlying nature of the number 23. (If you build all mathematics on top of set theory, then you will most likely treat the number 23 as some much more complicated thing. But you'll see that as an "implementation detail" that could be done in lots of different ways, rather than saying that really, deep down 23 is this complicated thing with lots of weird internal structure.)

bira 2y ago

Alternative link: https://archive.org/download/in.ernet.dli.2015.239784/2015.239784.A-Course.pdf https://archive.org/download/in.ernet.dli.2015.239784/2015.2...

dboreham 2y ago

Hmm...that's familiar. Blue cover iirc.

ipnon 2y ago

The preface left a deep impression on me because Hardy says only exceptionally talented and intelligent students will be able to finish and understand this book. It was a revelation because I grew up in schools that claimed anyone can learn maths to any level so long as they had enough interest and support. It was healthy as a young man to know I was beginning to approach my limits as a mathematician, and this book ultimately led me to focus more on computer science and programming.

psychoslave 2y ago
Note that both statements can be compatible. With exceptional interest and support, certainly anyone can absorb all these concepts. Of course if we take the case of someone about to die in 24h max due to brain cancer, then sure we just don't have the knowledge and resources to successfully make someone acquire that kind of knowledge. But in the general case, people not learning advanced math notions is everywhere in the intersection between individual having no interest and society not pushing them in that direction through resource allocations. Also since it's about Hardy we can not withdraw the case of Ramanujan. Yes, there are people whose brain is wired in very uncommon way that push them toward exploring uncharted territories where few have interest and even less have the ability to go through and survive. That said, once the path is paved and everything is in place to accommodate the accessibility of the place, there is no longer the same level of struggle to be expected. More often, we lake the great teaching resources, rather than the sufficiently apt learners.
- n4r9 2y ago
  "Exceptionally talented and intelligent" has a very different meaning to "with exceptional interest and support". When people say talent, they invariably mean a natural endowment or inherent skill. Something distinct from what id possible with support and interest. It's arguable that no such thing as talent exists. That when we say "talent" we are really using a short-hand for a kind of knowledge and experience. I suspect that Hardy would strongly reject such a claim, based on reading his Apology.
392 2y ago
You should read David Bessis's Mathematica.

OldGuyInTheClub 2y ago

From the title page. No Ph.D. necessary to be a legend in them days. ;-) G. H. HARDY, M.A., F.R.S. FELLOW OF NEW COLLEGE SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE

scapp 2y ago
M.A. was the highest degree available in the UK at the time [1]. The closest equivalent to a PhD program might be the Prize Fellowship that Hardy had from 1900 to 1906, though it's certainly not one to one. Don't get the impression that Hardy was done with his education when he got his masters degree. [1] https://www.economics.soton.ac.uk/staff/aldrich/Doc1.htm https://www.economics.soton.ac.uk/staff/aldrich/Doc1.htm [2] https://royalsocietypublishing.org/doi/10.1098/rsbm.1949.0007 https://royalsocietypublishing.org/doi/10.1098/rsbm.1949.000...
- OldGuyInTheClub 2y ago
  Interesting. The Wikipedia page for his contemporary Littlewood lists a Doctoral Advisor. https://en.wikipedia.org/wiki/John_Edensor_Littlewood https://en.wikipedia.org/wiki/John_Edensor_Littlewood Edit: The Mathematics Genealogy Project lists Littlewood as an M.A. as well. https://www.mathgenealogy.org/id.php?id=10463 https://www.mathgenealogy.org/id.php?id=10463
griffzhowl 2y ago
Freeman Dyson famously never got a PhD either, he just made seminal contributions to quantum field theory instead
- OldGuyInTheClub 2y ago
  Yeah... he hated the Ph.D. system, called it evil. He also said a great advantage of the Institute for Advanced Study was that he could work with postdocs - recent products of that same evil system. Source: Why I don't like the PhD system (95/157) https://www.youtube.com/watch?v=DzC1IRYN_Ps https://www.youtube.com/watch?v=DzC1IRYN_Ps

nephanth 2y ago

103 years old. Pretty recent at the scale of Mathematics, although that does make it older than Bourbaki. And than category theory

adalacelove 2y ago

Nice book. For another old but excellent math book I recommend Geometry and the Imagination by David Hilbert. No gutenberg remake I'm afraid, maybe because of the numerous (and incredibly high quality) illustrations.

mayd 2y ago
David Hilbert and Stefan Cohn-Vossen.
- adalacelove 2y ago
  True. I was citing from memory and only recalled the most famous author.

mayd 2y ago

Hardy's "A course of Pure Mathematics" has been highly regarded since it was first published in 1908 because it was an innovative text: rigorous, modern, well-written. Its intended readership was always first year "honours" mathematics students. This book inspired innovation in subsequent generations of textbook writers. However, in the 21st century, this book really can no longer be recommended for its original teaching purpose. As a textbook it is outdated (a term I hate, but it is true). It is now an historical curiosity - although one which I am pleased to own, and the exercises in the book are still worth a look. Calculus teaching has progressed considerably since 1908. The construction of the real number system in Hardy's book, using the Dedekind Cut method is overly complicated - the use of the of Least Upper Bound is much simpler and clearer. Hardy defines the concept of integral solely as the anti-derivative; there is no discussion of Riemann sums, or Darboux sums, etc. I am sure I would not want to take Hardy's approach today. I think we are better off recommending books are more modern. I will start by recommending "Calculus" by Michael Spivak.

chris_wot 2y ago
I’m currently reading through Martin to teach myself Calculus - can really recommend it!
housecarpenter 2y ago
What do you mean by "the use of the Least Upper Bound"?
- denotational 2y ago
  Seconded, the “least upper bound” method for constructing the reals that I know about is… …Dedekind cuts.
  - davidgrenier 2y ago
    I haven't looked at Hardy's but the presentation in Spivak is also Dedekind cuts. Perhaps Hardy uses a different approach and OP misnamed it? Rudin's chapter 1 annex also use Dedekind's cuts.

globalnode 2y ago

Ive been nerd sniped by point 158 on page 353. I cant believe I slipped through so many calc classes without understanding Leibniz's rule for taking the derivative of a definite integral. I didn't actually follow through with the calculation in Hardy's book but I bet it haunts me until I do :(.

JPC21 2y ago

I remember having a hard time understanding limsup/liminf and only actually understood the concept after reading this book. I am sure it is not for everyone due to its age, but I think this book is a much better introduction than baby Rudin.

nthingtohide 2y ago

I think in the book, Hardy mentions that Mathematics is one big tautology. Intial axioms setup a graph structure of theorems and the task of mathematicians is to find shortcuts in that graph.

TheRealNGenius 2y ago

[dead]

deleted 2y ago

[deleted]