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 As a textbook it is outdated (a term I hate, but it is true).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.
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 Bézier curves/surfaces --- could you suggest texts on those subjects?
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://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).
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.)
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:
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
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.
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.
If one only took the method of teaching that is most common in US university lecture halls, and applied it to a small class of pre-teens or teenagers, it probably wouldn't be very effective.
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.
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.
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...
This can be addressed practically enough by introducing the notion of a 'modulus of convergence'.
What's the misleading part of this supposed to be?
The equivalence classes of integers: pairs of naturals with (a, b) ~ (c, d) := (a + d) = (b + c).
The equivalence classes of rationals: pairs of integers with (a, b) ~ (c, d) := ad = bc.
It’s “easy” to tell whether two integers/rationals are equivalent, because the equivalence rule only requires you to determine whether one pair is a translation/multiple resp. of the other (proof is left to the reader).
Cauchy sequences, on the other hand, require you to consider the limit of an infinite sequence; as the GP points out, two sequences with the same limit may differ by an arbitrarily large prefix, which makes them “hard” to compare.
We can formalise this notion by pointing out that equality of integers and rationals is decidable, whereas equality of Cauchy reals is not. On the other hand, equality of Dedekind reals isn’t decidable either, so it’s not that Cauchy reals are necessarily easier than Dedekind reals, but more that they might lull one into a false sense of security because one might naively believe that it’s easy to tell if two sequences have the same limit.
That won't help you much if you don't know what you're working with, but the same is true of rationals.
I'm missing something as to this:
> equality of Dedekind reals isn’t decidable either
Two Dedekind reals (A, B) and (A', B') are equal if and only if they have identical representations. [Which is to say, A = A' and B = B'.] This is about as simple as equality gets, and is the normal rule of equality for ordered pairs. Can you elaborate on how you're thinking about decidability?
Direct:
Make one of the sets uncomputable, at which point the equality of the sets cannot be decided. This happens when the real defined by the Dedekind cut is itself uncomputable. BB(764) is an integer (!) that I know is uncomputable off the top of my head. The same idea (defining an object in terms of some halting property) is used in the next proof.
Via undecidability of Cauchy reals:
Equality of Cauchy reals is also undecidable. The proof is by negation: consider a procedure that decides whether a real is equal to zero; consider a sequence (a_n) with a_n = 1 if Turing machine A halts within n steps on all inputs, 0 otherwise; this is clearly Cauchy, but if we can decide whether it’s equal to 0, then we can decide HALT.
Cauchy reals and Dedekind reals are isomorphic, so equality of Dedekinds must also be undecidable.
Hopefully those two sketches show what I mean by decidable; caveat that I’m not infallible and haven’t been in academia for a while, so some/all of this may be wrong!
I meant BB(748) apparently.
To elaborate on this point a bit, I specifically mean uncomputable in ZFC. There may be other foundations in which it is computable, but we can just find another n for which BB(n) is uncomputable in that framework since BB is an uncomputable function.
But you're arguing that equality of Dedekind reals is undecidable based on a problem that occurs when you define a particular "Dedekind real" only by reference to some property that it has. If you had a representation of the values as Dedekind reals, it would be trivial to determine whether they were or weren't equal. You're holding them to a different standard than you're using for the integers and rationals. Why?
Let's decide a question about the integers. Is BB(800) equal to BB(801)?
It sure seems like it isn't. How sure are you?
In particular, it is quite possible to prove a theorem that a sequence is Cauchy, but that there is no way to explicitly figure out N for a given epsilon. The sequence is effectively useless. This presumably is possible, and common, with using the Axiom of Choice. One can even imagine an algorithm for such a sequence that produces numbers and eventually converges, but the convergence is not knowable. Again, if this is just approximating something, then we can simply say it is a useless approximation scheme. But defining real numbers as the equivalence class of Cauchy sequences suggests taking such a sequence seriously in some sense and is the answer.
In contrast, consider integer and rational number versions, it is quite immediate how to reduce them to their canonical form, assuming unlimited finite arithmetic ability. For example, 200/300 ~ 2/3 and one recognizes that 200/300 and 2/3 are different forms of what we take to be the same object for most of our purposes. There is no canonical Cauchy sequence to reduce to and concluding two sequences are equivalent could take a potentially infinite number of computations /comparisons. While that is somewhat inherent to the complexity of real numbers, it feels particularly acute when it is something that must be done in defining the object.
Dedekind cuts have the opposite problem. There is only one of them for an irrational number, but it is not entirely clear what we would be computing out as an approximation, particularly if the lower cut viewpoint is adopted.
Intervals, on the other hand, inherently contain the approximation information. By dividing them and picking out the next subinterval, one also has a method to computing out a sequence of ever better approximations. I suppose one could prove the existence of the family of containment intervals without explicitly being able to produce them, but at least the emptiness of the statement would be quite clear (nothing is produced) in contrast to the sequences that could produce essentially meaningless numbers for an arbitrarily large number of terms.
Uh, least upper bound of what? most subsets of Q have no extrema in Q.
>>> 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.
I guess the difference between us then is that I didn't care about applications.
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.
Studied medicine but did not practice (Keats did for a little while). Just for interest.
It is quite dense but at least personally I find that style of textbook much more useful than the American style enormous textbooks which takes a chapter to explain what could be said in a paragraph. You just need to know to expect it will take you quite a while to read each page.
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.
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.
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.
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.)
( … 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.
…
(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.)
"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.
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.
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[1] https://www.economics.soton.ac.uk/staff/aldrich/Doc1.htm
[2] https://royalsocietypublishing.org/doi/10.1098/rsbm.1949.000...
https://en.wikipedia.org/wiki/John_Edensor_Littlewood
Edit: The Mathematics Genealogy Project lists Littlewood as an M.A. as well.
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.
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.
Source: Why I don't like the PhD system (95/157)
Intial axioms setup a graph structure of theorems and the task of mathematicians is to find shortcuts in that graph.