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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-w

by 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 As a textbook it is outdated (a term I hate, but it is true).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.
    - fweimer 2y ago
      It looks like Hardy used Dedekind cuts from starting with the second edition (1914), but not in the first edition (1908). What's the advantage of Dedekind cuts over say equivalence classes of Cauchy sequences of rational numbers? Particularly if you start out by introducing the integers and rational numbers as equivalence classes as well.
      - zozbot234 2y ago
        Cauchy sequences can be made constructive (providing a nice foundation for numerical analysis); Dedekind cuts cannot.
      - jostylr 2y ago
        The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. For example, a sequence of 0,0,0,...,0 where the number of dots is the count of all the atoms in the universe and then followed by the decimal approximations of pi: 3, 3.1, 3.14, 3.141, ... The key component is the error clause of getting close, but that can vary greatly from sequence to sequence as to when that happens. The cute idea of being able to look at a sequence and see roughly where it is converging just is not captured well in the reality of the equivalence classes. More or less, one can think of a Cauchy sequence of generating intervals that contain the real number, but it can be arbitrarily long before the sequence gets to "small" intervals. So comparing two Cauchy sequences could be quite difficult. Contrast that with the rational numbers where a/b ~ c/d if and only if ad = bc. This is a relatively simple thing to check if a, b, c, and d are comfortably within the realm of human computation. Dedekind cuts avoid this as there is just one object and it is assumed to be completed. This is unrealistic in general though the n-roots are wonderful examples to think it is all okay and explicit. But if one considers e, it becomes clear that one has to do an approximation to get bounds on what is in the lower cut. The (lower) Dedekind cut can be thought of as being the set of lower endpoints of intervals that contain the real number. My preference is to define real numbers as the set of inclusive rational intervals that contain the real number. That is a bit circular, of course, so one has to come up with properties that say when a set of intervals satisfies being a real number. The key property is based on the idea behind the intermediate value theorem, namely, given an interval containing the real number, any number in the interval divides the interval in two pieces, one which is in the set and the other is not (if the number chosen "is" the real number, then both pieces are in the set). There is a version of this idea which is theoretically complete and uses Dedekind cuts to establish its correctness[1] and there is a version of this idea which uses what I call oracles that gets into the practical messiness of not being able to fully present a real number in practice[2]. 1: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-dedekind-cuts.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl... 2: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-oracles.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
- davidgrenier 2y ago
  Where we define the real numbers as the least upper bounds of special sets. There is a bijection between these sets and the set of real numbers which we commonly think of and that bijection is the least upper bound of such sets.
- griffzhowl 2y ago
  Probably what Abbott in Understanding Analysis calls the axiom of completeness: every set that is bounded above has a least upper bound. Making this stipulation distinguishes the reals from the rationals, as e.g. the set of numbers whose square is less than 2 is bounded above by any number whose square is greater than or equal to two, but among the rationals there is no least upper bound: given any rational number whose square is greater than or equal to two we can always find a smaller such rational. Assuming the axiom of completeness, we define the square root of two as the least upper bound of the set of numbers whose square is less than two
  - red_trumpet 2y ago
    But that is an axiom, not a construction! The point of Dedekind cuts is that they give a construction of the real numbers, and one can prove that this satisfies the Axiom of Least Upper Bounds.
    - itchyjunk 2y ago
      The axiom is used to give an alternative construction of real. Everything starts with some axioms somewhere.
    - wbl 2y ago
      You don't need a construction for a calculus class. If you do need one Cauchy sequence completion is more generalizable and somewhat easier to work with.
      - denotational 2y ago
        I don’t really know what a “calculus” class is since here (the UK) that term isn’t really used for university-level mathematics; we’d usually say “analysis” instead, but I know that “analysis” is a class in the US too, so I don’t know if calculus is closer to what we would do just prior to university (a bit of limits, differentiation, Riemann integrals, a bit of vector calculus). Virtually every first year UK undergraduate analysis course will start with a construction of the reals via Dedekind cuts, and this is about the level that this book is pitched at. The original commenter suggested that “least upper bounds” is a simpler approach, and that Hardy’s book is outdated by using Dedekind cuts; it may be that constructing the reals is not something that would be done at “calculus”-level in the US, but clearly the book isn’t aimed at that level. Dedekind cuts (or Cauchy sequences) are totally standard, and I don’t think it’s fair to criticise their use at all.
WillAdams 2y ago
I need to brush up on math for my current project: https://github.com/WillAdams/gcodepreview https://github.com/WillAdams/gcodepreview and found the book series: - _Make:Geometry_ https://www.goodreads.com/book/show/58059196-make https://www.goodreads.com/book/show/58059196-make - _Make:Trigonometry_ https://www.goodreads.com/book/show/123127774-make https://www.goodreads.com/book/show/123127774-make - _Make:Calculus_ https://www.goodreads.com/book/show/61739368-make https://www.goodreads.com/book/show/61739368-make a helpful review and extension of my slipshod math education (remember how Feynman once critiqued some math books, esp. calling out one for using made-up associations of colors and star temperatures? guess which one the school system I attended was using...). Next step is I need to work with conic sections and after that Bézier curves/surfaces --- could you suggest texts on those subjects?
- dalton01 2y ago
  These look great, thanks for sharing :)
  - WillAdams 2y ago
    You're welcome! Hopefully someone can make recommendations for books (or other references) which address the next two aspects of math I'll need, conic sections and Bézier curves/surfaces. I have found: - _Practical Conic Sections_ by J.W. Downs - https://pomax.github.io/bezierinfo/ https://pomax.github.io/bezierinfo/ - https://www.youtube.com/watch?v=aVwxzDHniEw https://www.youtube.com/watch?v=aVwxzDHniEw and I'm reading through _METAFONT: The Program_ but if someone could provide a list of math texts which build from the Make: <foo> series (the coverage of conic sections in them was sparser than I was hoping for) I'd be glad of it. Links to helpful Github repositories would be welcome as well (for reviewing once I've digested the books).
noqc 2y ago
>the use of the of Least Upper Bound is much simpler and clearer Uh, least upper bound of what? most subsets of Q have no extrema in Q.
- deleted 2y ago
  [deleted]
thaumasiotes 2y ago
> 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. A Dedekind cut is a partition of the rational numbers into two sets, A and B, where every number that belongs to A is less than every number that belongs to B. The cut represents a rational number if B has a least element, and an irrational number if it doesn't. (In full generality, it's a rational number if either (1) A has a greatest element, or (2) B has a least element, but in the case where A has a greatest element, we transfer that element into B, where it's the least element.) The real number represented by a Dedekind cut is always the least upper bound of A (and the greatest lower bound of B). How does "the use of the Least Upper Bound" differ from Dedekind cuts? They aren't just the same thing in some arcane abstract sense where they both map onto the real numbers - they're the same thing in the most direct sense possible. (For comparison, my analysis class defined real numbers as Cauchy sequences of rationals. The limit of such a sequence is a real number, but that real number need not be an upper or lower bound to anything.)
kccqzy 2y ago
I recently looked at the syllabus of a course by my alma matar targeted at the same audience of ambitious first-year mathematics students with an emphasis on rigor. It is quite different from Hardy's book and the changes are, I think, for the better. Mostly, it introduces a lot more discrete mathematics, with a little bit of even graph theory and a little bit of enumerative combinatorics (generating functions and the like). In contrast this book is mostly focused on calculus and analysis, which are already separate courses. Freshman students should be exposed to a wider variety of topics in college-level mathematics and not just harder versions of what they already studied in high school.
teleforce 2y ago
Really wished modern mathematics textbook cover geometric algebra not only complex algebra, what a shame and a lost of great opportunity [1]. It seems the contents of pure math textbooks do not changed and improved much after more than a century and when I did my pure math A level many years ago the contents are not much different than Hardy's textbook contents. Apparently and ironically when the textbook was written when Hardy was the Savilian Chair of Geometry at Oxford University. If he had introduced GA into the contents back then it will be a game changing and I'd probably had studied GA in my pure math class back then. [1] Projective Geometric Algebra: https://projectivegeometricalgebra.org/ https://projectivegeometricalgebra.org/
1vuio0pswjnm7 2y ago
Why not read both
lanstin 2y ago
What do you think of Hardy's book in number theory? I have never done number theory but having to study it a bit while I work on sets of traps that are approximable by algebraic numbers and it is really a nice read.
- lanstin 2y ago
  sets of reals, not traps.

by mayd 2y ago

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.
  - fweimer 2y ago
    It looks like Hardy used Dedekind cuts from starting with the second edition (1914), but not in the first edition (1908). What's the advantage of Dedekind cuts over say equivalence classes of Cauchy sequences of rational numbers? Particularly if you start out by introducing the integers and rational numbers as equivalence classes as well.
    - zozbot234 2y ago
      Cauchy sequences can be made constructive (providing a nice foundation for numerical analysis); Dedekind cuts cannot.
    - jostylr 2y ago
      The equivalence class of Cauchy sequences is vastly larger and misleading compared to those of integers and rational numbers. You can take any finite sequence and prepend it to a Cauchy sequence and it will represent the same real number. For example, a sequence of 0,0,0,...,0 where the number of dots is the count of all the atoms in the universe and then followed by the decimal approximations of pi: 3, 3.1, 3.14, 3.141, ... The key component is the error clause of getting close, but that can vary greatly from sequence to sequence as to when that happens. The cute idea of being able to look at a sequence and see roughly where it is converging just is not captured well in the reality of the equivalence classes. More or less, one can think of a Cauchy sequence of generating intervals that contain the real number, but it can be arbitrarily long before the sequence gets to "small" intervals. So comparing two Cauchy sequences could be quite difficult. Contrast that with the rational numbers where a/b ~ c/d if and only if ad = bc. This is a relatively simple thing to check if a, b, c, and d are comfortably within the realm of human computation. Dedekind cuts avoid this as there is just one object and it is assumed to be completed. This is unrealistic in general though the n-roots are wonderful examples to think it is all okay and explicit. But if one considers e, it becomes clear that one has to do an approximation to get bounds on what is in the lower cut. The (lower) Dedekind cut can be thought of as being the set of lower endpoints of intervals that contain the real number. My preference is to define real numbers as the set of inclusive rational intervals that contain the real number. That is a bit circular, of course, so one has to come up with properties that say when a set of intervals satisfies being a real number. The key property is based on the idea behind the intermediate value theorem, namely, given an interval containing the real number, any number in the interval divides the interval in two pieces, one which is in the set and the other is not (if the number chosen "is" the real number, then both pieces are in the set). There is a version of this idea which is theoretically complete and uses Dedekind cuts to establish its correctness[1] and there is a version of this idea which uses what I call oracles that gets into the practical messiness of not being able to fully present a real number in practice[2]. 1: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-dedekind-cuts.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl... 2: https://github.com/jostylr/Reals-as-Oracles/blob/main/articles/rational-betweenness-relations-oracles.pdf https://github.com/jostylr/Reals-as-Oracles/blob/main/articl...
davidgrenier 2y ago
Where we define the real numbers as the least upper bounds of special sets. There is a bijection between these sets and the set of real numbers which we commonly think of and that bijection is the least upper bound of such sets.
griffzhowl 2y ago
Probably what Abbott in Understanding Analysis calls the axiom of completeness: every set that is bounded above has a least upper bound. Making this stipulation distinguishes the reals from the rationals, as e.g. the set of numbers whose square is less than 2 is bounded above by any number whose square is greater than or equal to two, but among the rationals there is no least upper bound: given any rational number whose square is greater than or equal to two we can always find a smaller such rational. Assuming the axiom of completeness, we define the square root of two as the least upper bound of the set of numbers whose square is less than two
- red_trumpet 2y ago
  But that is an axiom, not a construction! The point of Dedekind cuts is that they give a construction of the real numbers, and one can prove that this satisfies the Axiom of Least Upper Bounds.
  - itchyjunk 2y ago
    The axiom is used to give an alternative construction of real. Everything starts with some axioms somewhere.
  - wbl 2y ago
    You don't need a construction for a calculus class. If you do need one Cauchy sequence completion is more generalizable and somewhat easier to work with.
    - denotational 2y ago
      I don’t really know what a “calculus” class is since here (the UK) that term isn’t really used for university-level mathematics; we’d usually say “analysis” instead, but I know that “analysis” is a class in the US too, so I don’t know if calculus is closer to what we would do just prior to university (a bit of limits, differentiation, Riemann integrals, a bit of vector calculus). Virtually every first year UK undergraduate analysis course will start with a construction of the reals via Dedekind cuts, and this is about the level that this book is pitched at. The original commenter suggested that “least upper bounds” is a simpler approach, and that Hardy’s book is outdated by using Dedekind cuts; it may be that constructing the reals is not something that would be done at “calculus”-level in the US, but clearly the book isn’t aimed at that level. Dedekind cuts (or Cauchy sequences) are totally standard, and I don’t think it’s fair to criticise their use at all.

WillAdams 2y ago

I need to brush up on math for my current project: https://github.com/WillAdams/gcodepreview https://github.com/WillAdams/gcodepreview and found the book series: - _Make:Geometry_ https://www.goodreads.com/book/show/58059196-make https://www.goodreads.com/book/show/58059196-make - _Make:Trigonometry_ https://www.goodreads.com/book/show/123127774-make https://www.goodreads.com/book/show/123127774-make - _Make:Calculus_ https://www.goodreads.com/book/show/61739368-make https://www.goodreads.com/book/show/61739368-make a helpful review and extension of my slipshod math education (remember how Feynman once critiqued some math books, esp. calling out one for using made-up associations of colors and star temperatures? guess which one the school system I attended was using...). Next step is I need to work with conic sections and after that Bézier curves/surfaces --- could you suggest texts on those subjects?

dalton01 2y ago
These look great, thanks for sharing :)
- WillAdams 2y ago
  You're welcome! Hopefully someone can make recommendations for books (or other references) which address the next two aspects of math I'll need, conic sections and Bézier curves/surfaces. I have found: - _Practical Conic Sections_ by J.W. Downs - https://pomax.github.io/bezierinfo/ https://pomax.github.io/bezierinfo/ - https://www.youtube.com/watch?v=aVwxzDHniEw https://www.youtube.com/watch?v=aVwxzDHniEw and I'm reading through _METAFONT: The Program_ but if someone could provide a list of math texts which build from the Make: <foo> series (the coverage of conic sections in them was sparser than I was hoping for) I'd be glad of it. Links to helpful Github repositories would be welcome as well (for reviewing once I've digested the books).

noqc 2y ago

>the use of the of Least Upper Bound is much simpler and clearer Uh, least upper bound of what? most subsets of Q have no extrema in Q.

deleted 2y ago
[deleted]

thaumasiotes 2y ago

> 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. A Dedekind cut is a partition of the rational numbers into two sets, A and B, where every number that belongs to A is less than every number that belongs to B. The cut represents a rational number if B has a least element, and an irrational number if it doesn't. (In full generality, it's a rational number if either (1) A has a greatest element, or (2) B has a least element, but in the case where A has a greatest element, we transfer that element into B, where it's the least element.) The real number represented by a Dedekind cut is always the least upper bound of A (and the greatest lower bound of B). How does "the use of the Least Upper Bound" differ from Dedekind cuts? They aren't just the same thing in some arcane abstract sense where they both map onto the real numbers - they're the same thing in the most direct sense possible. (For comparison, my analysis class defined real numbers as Cauchy sequences of rationals. The limit of such a sequence is a real number, but that real number need not be an upper or lower bound to anything.)

kccqzy 2y ago

I recently looked at the syllabus of a course by my alma matar targeted at the same audience of ambitious first-year mathematics students with an emphasis on rigor. It is quite different from Hardy's book and the changes are, I think, for the better. Mostly, it introduces a lot more discrete mathematics, with a little bit of even graph theory and a little bit of enumerative combinatorics (generating functions and the like). In contrast this book is mostly focused on calculus and analysis, which are already separate courses. Freshman students should be exposed to a wider variety of topics in college-level mathematics and not just harder versions of what they already studied in high school.

teleforce 2y ago

Really wished modern mathematics textbook cover geometric algebra not only complex algebra, what a shame and a lost of great opportunity [1]. It seems the contents of pure math textbooks do not changed and improved much after more than a century and when I did my pure math A level many years ago the contents are not much different than Hardy's textbook contents. Apparently and ironically when the textbook was written when Hardy was the Savilian Chair of Geometry at Oxford University. If he had introduced GA into the contents back then it will be a game changing and I'd probably had studied GA in my pure math class back then. [1] Projective Geometric Algebra: https://projectivegeometricalgebra.org/ https://projectivegeometricalgebra.org/

1vuio0pswjnm7 2y ago

Why not read both

lanstin 2y ago

What do you think of Hardy's book in number theory? I have never done number theory but having to study it a bit while I work on sets of traps that are approximable by algebraic numbers and it is really a nice read.

lanstin 2y ago
sets of reals, not traps.