[HN Gopher] "A Course of Pure Mathematics" - G. H. Hardy (1921) ...
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"A Course of Pure Mathematics" - G. H. Hardy (1921) [pdf]
Author : bikenaga
Score : 281 points
Date : 2024-12-30 21:20 UTC (1 days ago)
(HTM) web link (www.gutenberg.org)
(TXT) w3m dump (www.gutenberg.org)
| ninalanyon wrote:
| It would be more useful to link to the Gutenberg page that shows
| links to the various formats:
|
| https://www.gutenberg.org/ebooks/38769
| bikenaga wrote:
| Yeah, my bad - I'm so accustomed to pdfs that I forget people
| often like other formats. Thanks for the link!
| damhsa wrote:
| the tex file is huge because its got rasterized images in it,
| still nice to have though
| bell-cot wrote:
| A bit of background on Hardy, and his choice of title:
|
| https://en.wikipedia.org/wiki/G._H._Hardy#Pure_mathematics
| vouaobrasil wrote:
| 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 wrote:
| Classical and austere, but not stilted. For instance...
|
| >>> We can state this more precisely as follows: if we take any
| segment BC on L, 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 wrote:
| > 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 wrote:
| that's like studying medicine then not becoming a doctor
| AnimalMuppet wrote:
| Not becoming a _patient-treating_ doctor. Research
| doctors still matter a great deal in the field of
| medicine.
| orochimaaru wrote:
| 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 wrote:
| Math can be an end unto itself. This can come as a bit of
| a surprise in our prevailing culture, which needs to
| justify the usefulness of everything. Also, it's possible
| for someone to study math as a liberal art, and develop
| the ability to do useful things with it on their own. My
| observation is that the people who grudgingly learned
| math as a means to an end, tend to forget most of it soon
| after graduating. This explains the widespread but
| paradoxical aversion to math among engineers.
| chris_wot wrote:
| Which is exactly what Hardy said himself.
| tikhonj wrote:
| so like studying (human) biology and becoming a biologist
| LPisGood wrote:
| There are lots of medical researchers who don't treat
| patients.
| liontwist wrote:
| Utility is not the ultimate goal of math, understanding
| is.
| vouaobrasil wrote:
| I think that would also be rewarding. I have no desire to
| become a doctor but I'm interested in how medicine works.
| OJFord wrote:
| On the contrary, it's more like studying medicine and
| then remaining in medical research in a purely academic
| (non-clinical) setting.
| lsharkey602 wrote:
| John Keats, Osamu Tezuka, Somerset Maugham, Hector
| Berlioz.
|
| Studied medicine but did not practice (Keats did for a
| little while). Just for interest.
| woopsn wrote:
| That's fair. Hardy himself was a zealot and in fact
| despised applications, writing that he hoped his work would
| never be put to extrinsic use, for then its value would
| become contingent on a particular stage of technological
| development.
| Koshkin wrote:
| Mathematics has, and has always had, applications within
| itself.
| vouaobrasil wrote:
| Obviously. I only meant applications to the real world.
| Don't care about those.
| OJFord wrote:
| I read somewhere that this was Turing's preparation for the
| Cambridge entrance exam; so I read through it in sixth form
| before sitting the STEP exams (modern equivalent for
| mathematics or CS with, and perhaps other programmes depending
| on college). I failed them, but that's a review of my naivete,
| not the work!
| curiouscavalier wrote:
| 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 wrote:
| IMHO, it has the best ELI5 explanation of integration, still not
| bested in 100 years.
| wannabebarista wrote:
| 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 wrote:
| "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.)
| wannabebarista wrote:
| Thanks for finding the quote. Memory can be sketchy
| sometimes!
| ngcc_hk wrote:
| For the number 23, I wonder there must at least 2 schools :
|
| ( ... there are many ...
|
| One can think that there is really a number 23! It was
| discovered and somehow we human has accessed to it. I am not
| sure.
|
| Or one can think about it. This is the mapping of empty set
| to 0, set of empty set to 1, set of empty set of empty set to
| 2 ... ...)
|
| one can think of 23-ish item is a set with all 23 elements
| whose combination of any 2 elements does not reduce. You need
| a thousand page to prove 1 + 1 = 2, with the reason that the
| first 1 is not the same as the second 1 to avoid this
| operation collapse back to 1. Our counting always assume
| different object, but to be rigorous there is nothing in the
| first 1 is explicitly said in that equation is not the same
| as the second one, as pointed out by a previous Hn refer to
| latest article .
|
| ...
|
| Or my beloved : there is no 23. Only 0 and an operation +1
| exist. You can say 23 as the result of a marker after 23 +1
| operation on 0. It is 0 +1 -> 1 ... 1 +1 -> 0 +1 +1 -> 2,
| Qed. If you have 23 stones/... with you, you do a counting by
| doing a mapping to this 0 obj +1 ops in your head-compute
| somehow.
|
| ...
| mellosouls wrote:
| In 1922 this claim will have seemed increasingly resonant as
| the contemporary quantum revolution upended the classical
| view of the world.
| lasermike026 wrote:
| And the engineering takes their crude tools striking
| inanimate matter while the mathematician and the physics
| argue over the nature of reality.
| gjm11 wrote:
| Mentioned in a footnote in that book is the following, which I
| have always rather liked: "A science is said to be useful if
| its development tends to accentuate the existing inequalities
| in the distribution of wealth, or more directly promotes the
| destruction of human life."
|
| (He mentions it in a footnote mostly to clarify that he doesn't
| really quite believe it -- it was a "conscious rhetorical
| flourish" in something he wrote in 1915, and in the main body
| of the text he gives a less cynical account of what it means
| for something to be useful.)
| globalnode wrote:
| Maybe that cynicism was one of the reasons he appeared to be
| against applications, in as much as they would be a form of
| monetisation or exploitation of his work. edit: Just read
| some more about him and it seems this may be well documented
| already heh.
| bikenaga wrote:
| You might be thinking of: "One rather curious conclusion
| emerges, that pure mathematics is on the whole distinctly more
| useful than applied." I wonder what he would think if he could
| see the ways in which number theory, once often regarded as the
| purest of the branches of math, is now used in things like
| cryptography.
|
| "Apology" is definitely worth reading. Some of his opinions can
| seem rather elitist:
|
| "Statesmen, despise publicits, painters despise art-critics,
| and physiologists, physicists, or mathematicians usually have
| similar feelings; there is no scorn more profound, or on the
| whole more justifiable, than that of the men who make for the
| men who explain. Exposition, criticism, appreciation is work
| for second-rate minds."
|
| At the same time, he is very honest about himself - in fact, he
| seems to have been suffering from depression over what he
| perceived as a decline in his ability to do math at the level
| he was accustomed to:
|
| "If then I find myself writing, not mathematics but 'about'
| mathematics, it is a confession of weakness, for which I may
| rightly be scorned or pitied by younger and more vigorous
| mathematicians. I write about mathematics because, like any
| other mathematician who has passed sixty, I have no longer the
| freshness of mind, the energy, or the patience to carry on
| effectively with my proper job."
|
| Or:
|
| "A mathematician may still be competent enough at sixty, but it
| is useless to expect him to have original ideas."
|
| Or more sadly, but with some serenity:
|
| "It is plain now that my life, for what it is worth, is
| finished, and that nothing I can do can perceptibly increase or
| diminish its value. It is very diffciult to be dispassionate,
| but I count it a 'success'; I have had more reward and not less
| than was due to a man of my particular grade of ability."
|
| "If I had been offered a life neither better nor worse when I
| was twenty, I would have accepted without hesitation."
|
| The personal reflections bookend a central portion where he
| illustrates with several examples (e.g. Euclid's proof of the
| infinitude of the primes) his feelings about the "importance"
| of math, its "usefulness", and the distinction between pure and
| applied math.
|
| It's interesting to compare "Apology" to "Littlewood's
| Miscellany" (I recommend the Cambridge University Press
| version, which contains the essay "The Mathematician's Art of
| Work" - ISBN 0-521-33702-X). There is more math than in
| "Apology" and many anecdotes. J. E. Littlewood was Hardy's
| long-time collaborator.
| wannabebarista wrote:
| I've not read Littlewood's book. Thanks for the suggestion!
| bryanrasmussen wrote:
| > I wonder what he would think if he could see the ways in
| which number theory, once often regarded as the purest of the
| branches of math, is now used in things like cryptography.
|
| I would say number theory proves his statement, although
| perhaps not his point.
|
| Applied math is useful for the applications that are known at
| the time of its creation, and it is likely that it will
| remain with that level of applicability in the future,
| although if the real world applications that it is used for
| fall out of favor we might find that the applied math
| decreases in importance, given its importance is in its
| applicability, and the applicability of things has an
| importance contingent on the importance of the thing that
| they are applied to.
|
| This is of course not 100% sure, as there can also arise new
| applications of things in the future.
|
| Pure math on the other hand, being not tethered to any
| particular application on the time of its creation, may find
| all sorts of applications in the future, pure math has as
| such infinite potential applicability waiting to be
| discovered and thus infinite potential usefulness, whereas
| applied math has limited known applicability and thus limited
| known usefulness.
| bira wrote:
| Alternative link:
| https://archive.org/download/in.ernet.dli.2015.239784/2015.2...
| dboreham wrote:
| Hmm...that's familiar. Blue cover iirc.
| ipnon wrote:
| 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 wrote:
| 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 wrote:
| "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_.
| OldGuyInTheClub wrote:
| 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 wrote:
| 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
|
| [2]
| https://royalsocietypublishing.org/doi/10.1098/rsbm.1949.000...
| OldGuyInTheClub wrote:
| Interesting. The Wikipedia page for his contemporary
| Littlewood lists a Doctoral Advisor.
|
| 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
| griffzhowl wrote:
| Freeman Dyson famously never got a PhD either, he just made
| seminal contributions to quantum field theory instead
| OldGuyInTheClub wrote:
| 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
| ska wrote:
| PhD, and especially its near ubiquity in academic circles, is a
| relatively new phenomena.
| UniverseHacker wrote:
| You don't now either.... I'm an academic scientist and more
| than a few times I have looked up the CVs of well respected
| researchers whose work I really admired, and was shocked to
| discover they had no PhD, yet were still in a permanent PI
| position leading a well funded research effort of some type.
|
| There are a small number of people in academia that are so good
| that they are effectively exempted from the requirement for
| traditional credentials- because everyone in the field knows
| who they are and will make a custom position for them anywhere
| that bypasses traditional requirements and recruitment.
| jll29 wrote:
| That can still happen today.
|
| One of my Cambridge professors was Mr. Phil Woodland, an
| international authority on automatic speech recognition. When I
| asked him why he had no Ph.D., he shrugged and said he was too
| busy with his research work, so he "never found the time".
| Meanwhile, he got promoted to Professor [1].
|
| Another noteworthy fact about Oxford and Cambridge that people
| from elsewhere may not realize is that they award an M.A. "for
| free" to Bachelor students after 5 years have passed, so you
| effectively get two degrees for the price of one.
|
| [1] https://www.eng.cam.ac.uk/profiles/pw117
| nephanth wrote:
| 103 years old. Pretty recent at the scale of Mathematics,
| although that does make it older than Bourbaki. And than category
| theory
| adalacelove wrote:
| 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 wrote:
| David Hilbert and Stefan Cohn-Vossen.
| adalacelove wrote:
| True. I was citing from memory and only recalled the most
| famous author.
| mayd wrote:
| 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 wrote:
| I'm currently reading through Martin to teach myself Calculus -
| can really recommend it!
| housecarpenter wrote:
| What do you mean by "the use of the Least Upper Bound"?
| denotational wrote:
| Seconded, the "least upper bound" method for constructing the
| reals that I know about is... ...Dedekind cuts.
| davidgrenier wrote:
| 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 wrote:
| 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.
| jostylr wrote:
| 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/articl... 2:
| https://github.com/jostylr/Reals-as-
| Oracles/blob/main/articl...
| zozbot234 wrote:
| > 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. ...
|
| This can be addressed practically enough by introducing
| the notion of a 'modulus of convergence'.
| red_trumpet wrote:
| What you are referring to is also called the Principle of
| Nested Intervals: https://en.wikipedia.org/wiki/Nested_in
| tervals#The_construct...
| zozbot234 wrote:
| Cauchy sequences can be made constructive (providing a
| nice foundation for numerical analysis); Dedekind cuts
| cannot.
| davidgrenier wrote:
| 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 wrote:
| 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 wrote:
| 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.
| wbl wrote:
| 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 wrote:
| 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.
| woopsn wrote:
| I went to a University of California school which had 3
| calculus tracks - one for life/social sciences students
| (eg biology, econ), one for physical sciences (chemistry,
| physics, math, ...), and an honors track.
|
| High school went up through what we call Algebra II.
| Calculus is an Advanced Placement (AP) course that most
| students don't take.
|
| I took physical sciences calc + multivariate calc (1 year
| including summer), an intro to proofs and set theory
| course, and then finally a rigorous construction of reals
| was taught in our upper division real analysis course. So
| somewhere in my second year as a math major. Though I had
| already researched the constructions myself out of
| curiosity.
|
| Apart from the material being extraneous for anyone
| outside the major, I think they were in a sense trying to
| be more rigorous by first requiring set theory which
| included constructions of the integer and rational number
| systems.
| kxyvr wrote:
| In the U.S., there is typically a separation between
| calculus and real analysis. Though, the amount of
| difference between the two depends on the university.
|
| In calculus, there is more emphasis on learning how to
| mechanically manipulate derivatives and integrals and
| their use in science and engineering. While this includes
| some instruction on proving results necessary for
| formally defining derivatives and integrals, it is
| generally not the primary focus. Meaning, things like
| limits will be explained and then used to construct
| derivatives and integrals, but the construction of the
| reals is less common in this course. Commonly, calculus 1
| focuses more on derivatives, 2 on integrals, and 3 on
| multivariable. However, to be clear, there is a huge
| variety in what is taught in calculus and how proof based
| it is. It depends on the department.
|
| Real analysis focuses purely on proving the results used
| in calculus classes and would include a discussion on the
| construction of the reals. A typical book for this would
| be something like Principles of Mathematical Analysis by
| Rudin.
|
| I'm not writing this because I don't think you don't know
| what these topics are, but to help explain some of the
| differences between the U.S. and elsewhere. I've worked
| at universities both in the U.S. and in Europe and it's
| always a bit different. As to why or what's better, no
| idea. But, now you know.
|
| Side note, the U.S. also has a separate degree for math
| education, which I've not seen elsewhere. No idea why,
| but it also surprised me when I found out.
| itchyjunk wrote:
| The axiom is used to give an alternative construction of
| real. Everything starts with some axioms somewhere.
| noqc wrote:
| This is basically exactly a dedekind cut.
| WillAdams wrote:
| I need to brush up on math for my current project:
|
| https://github.com/WillAdams/gcodepreview
|
| and found the book series:
|
| - _Make:Geometry_
| https://www.goodreads.com/book/show/58059196-make
|
| - _Make:Trigonometry_
| https://www.goodreads.com/book/show/123127774-make
|
| - _Make:Calculus_
| 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
| Bezier curves/surfaces --- could you suggest texts on those
| subjects?
| dalton01 wrote:
| These look great, thanks for sharing :)
| noqc wrote:
| >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.
| thaumasiotes wrote:
| > 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.)
| globalnode wrote:
| 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 wrote:
| 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 wrote:
| 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.
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