Can someone please help me understand an argument in Hardy's A Course of Pure Mathematics

[ColabDog]

But that is not the argument that Hardy was giving, nor is this the question currently under discussion.

Are you a spammer?

I am just trying to help him by providing a visual aid so he can better understand his initial question - in particular the section below. If this is not what I am meant to do - please direct me to the posting guidelines and I can re-visit this. I have helped a few people in other forums as well improve their logical reasoning and through visual aids already.

The difficulty I have is --- it appears to me that the proof concludes with the sentence "Thus m=p^2, n=q^2, as was to be proved".

 

[mmm4444bot]

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[imath]\;[/imath]

 

[NotJeffM]

I am getting confused by where you are getting confused.

It is quite obvious that Hardy uses notation like BC to refer to either a line segment or to the measure of the line segment’s length. (See the bottom of page 2 for example.) Moreover, he has made it very clear in Section 2 that the geometrical discussion is merely a visualizing device to help with the reasoning about unspecified numbers. Whenever the geometry is not helpful, you must take pencil and paper to do the underlying algebra. Formally, B and C are numbers and the length of BC is a visualization of C - B (or technically [imath]| \ C - B \ |[/imath]).

I am somewhat perplexed by your concern over the lack of definition of irrational numbers at the start of Section 3. For the first few pages of that section he is discussing properties of rational numbers. So it is irrelevant that there has not yet been any definition of irrational numbers or real numbers. When he says, “BC includes infinitely many rational points,” it is quite clear that he is asserting something about rational numbers. Admittedly, he has not explicitly specified that B and C are rational numbers, but he also has not specified that they are irrational numbers. So, I do not see where you think he is making unjustified assumptions about irrational numbers.

It might help if a single post provided just two or three sentences where you are getting hung up. That way it becomes clear where we are in Hardy’s exposition and where exactly you do not get his intent.

By the way, it may not be obvious why at the beginning of Section 3, the interval [B, C] is embedded in the interval [1, 2]. Nothing about the logic demands [1, 2]. It is strictly speaking unnecessary and seemingly arbitrary. But that interval includes the square root of 2, which will be Hardy’s introductory irrational. He is already setting up the idea that irrationals will be defined by sets of rationals.

 

[MaxMath]

I am getting confused by where you are getting confused.

I feel sorry about the fact that my second question indeed appears confusing, and it appears that the more I expand the more confusing it becomes. Apparently, that's only because by expansion, there become more problems for me to deal with, i.e. more problems (to me, not necessarily to mathematicians) open up.

It is quite obvious that Hardy uses notation like BC to refer to either a line segment or to the measure of the line segment’s length. (See the bottom of page 2 for example.) Moreover, he has made it very clear in Section 2 that the geometrical discussion is merely a visualizing device to help with the reasoning about unspecified numbers. Whenever the geometry is not helpful, you must take pencil and paper to do the underlying algebra. Formally, B and C are numbers and the length of BC is a visualization of C - B (or technically [imath]| \ C - B \ |[/imath]).

I fully understand this. That's why I said now I take BC as a "magnitude" in what appears to me to be the Archimedes Axiom (or Property, or Lemma).

I am somewhat perplexed by your concern over the lack of definition of irrational numbers at the start of Section 3. For the first few pages of that section he is discussing properties of rational numbers. So it is irrelevant that there has not yet been any definition of irrational numbers or real numbers. When he says, “BC includes infinitely many rational points,” it is quite clear that he is asserting something about rational numbers. Admittedly, he has not explicitly specified that B and C are rational numbers, but he also has not specified that they are irrational numbers. So, I do not see where you think he is making unjustified assumptions about irrational numbers.

You are probably right. But I'm still unable to throw away my way of looking at this based on its position in the whole exposition of Hardy about pure mathematics. Again, this is probably my problem. I will reflect on this for a bit more time to see if I can get through.

It might help if a single post provided just two or three sentences where you are getting hung up. That way it becomes clear where we are in Hardy’s exposition and where exactly you do not get his intent.

Good advice. I already received advice from other people that I should start another thread with my 2nd question, which I will definitely follow next time. I'm still here just because people who are interested and willing to help seem to be able to follow without too much difficulty so far.

I have realised I should do a bit more homework myself first.

By the way, it may not be obvious why at the beginning of Section 3, the interval [B, C] is embedded in the interval [1, 2]. Nothing about the logic demands [1, 2]. It is strictly speaking unnecessary and seemingly arbitrary. But that interval includes the square root of 2, which will be Hardy’s introductory irrational. He is already setting up the idea that irrationals will be defined by sets of rationals.

I'm not sure if I'm following this. But anyway, I now kind of understand what I raised as a follow-up question (the 2nd point in my post #14). That is, according to the Axiom of Archimedes, one version of which reads (as far as I understand) "given two positive numbers [imath]x[/imath] and [imath]y[/imath], there is an integer [imath]n[/imath] such that [imath]n x>y[/imath]", we can always find an integer [imath]k[/imath] that meets [imath]k\cdotp BC > 1[/imath].

This means my question has now been reduced to its original form -- why this argument still holds (for BC) when B is a rational number but C is an irrational number (not a rational number).

I now have a sneaky feeling that there may not be a proper proof for this but rather can only be based on the fact that the Axiom applies to both rational and irrational numbers (i.e. all real numbers). This may be ok, but based on my very limited exposure to pure mathematics so far, this looks pretty shaky as a foundation to move on.

 

[pka]

I am getting confused by where you are getting confused.

It is quite obvious that Hardy uses notation like BC to refer to either a line segment or to the measure of the line segment’s length. (See the bottom of page 2 for example.) Moreover, he has made it very clear in Section 2 that the geometrical discussion is merely a visualizing device to help with the reasoning about unspecified numbers. Whenever the geometry is not helpful, you must take pencil and paper to do the underlying algebra. Formally, B and C are numbers and the length of BC is a visualization of C - B (or technically [imath]| \ C - B \ |[/imath]).

In reading GH Hardy it is most important to remember that he died in 1947. So a great deal of mathematics has been done since.
See this LINK. But it is fair to say that Hardy had more influence on twenty century mathematics that any other British mathematician.
From that link we see that he never "took" a PhD but yet had an oversized contribution to American mathematics.
One example of change. The question here about [imath]\sqrt{2~}[/imath] being rational is disposed quickly as:
if [imath]\rho[/imath] is a non-square positive integer then [imath]\sqrt{\rho~}[/imath] is irrational.
That is proven using the floor or ceiling function.
Moreover, every [imath]x\in\mathbb{R}[/imath] is either rational or it is irrational.

 

[MaxMath]

This does not directly answer my 2nd question (in post #9). But it appears to me that I cannot get away from the Axiom of Archimedes.

After a bit of digging, I'm surprised by how loosely terms are being thrown around; it's indeed very confusing to outsiders like me. So far I consider this (Archimedean property) the most accurate, proper, yet concise, exposition of this subject.

What's said in Hardy's book as the axiom is described here as "Corollary 1", as far as I can see.

This page is as dense as, if not denser than, Hardy's book. So I will need to read it multiple times to really grasp all of it.

Also, I think I have managed to escape the 'prison' I put myself in, as an obstacle to applying the argument in section 3 (all about rational numbers only) to section 17 (including irrational numbers). I now look at it this way—

Indeed the book of Hardy develops incrementally from the very basic to more complex ideas, such as from rational to real numbers. In this sense, when reading it, we should be cautious by assuming what has not been discussed as non-existent but only derive more complex ideas by deduction and definition step-by-step very carefully. This is in line with the essence of pure mathematics in my opinion.

We must keep in mind, however, that the order in which the theories/concepts/theorems were discovered or invented is not necessarily the same as the order in which they are explained in this book. For example, while the Axiom of Archimedes is introduced before the realisation of irrational numbers in Hardy's book, the latter was discovered before the former. The important implication of this is that, while the discussion of section 3 is indeed strictly limited only to rational numbers – so we should not take it for granted that the same holds for irrational numbers, the Axiom of Archimedes, by itself and as how it was first discovered, applies to all real numbers (that is, including irrational numbers).

Because of this, and specifically because the 'axiom' of Archimedes can be proved as true (this is why I put quotes around "axiom"), which applies to real numbers, the argument of section 3 indeed can be extended to section 17 for the case where B/C are no longer only rational numbers, but rather one rational the other irrational number.

I think this is the proper answer to my own question (by "proper", I mean compatible with how my rusty brain is wired). Thanks for the help of everyone!

 

[NotJeffM]

I repeat that I do truly understand where you are having difficulty.

Here is how I follow Hatdy’s argument.

Section 1: We assume as true the basic facts about integers and rational numbers. (I presume he means the axioms of an ordered field and certain well known theorems such as unique factorization of an integer > 1. He is not talking about some theorem known only to professional mathematicians. The book is intended for undergraduates.)

Section 2: He discusses the geometry of the straight line as a useful analogy to thinking about rational numbers. (This means to me that, (1) any time you find the geometric argument confusing, resort to algebra and (2) only algebra will deliver a proof.)

Section 3: Initially this section involves some general propositions on rational numbers. He sketches one proof with the simplifying assumption that an interval has a length of 1, but if you turn the proposition into a completely general algebraic proposition about two numeric intervals [A_1, A_2) and (B, C) such that A_1, A_2, B, and C are rational numbers such that
0 < | C - B | [imath]\le[/imath] | A_2 - A_1 |, you should be able prove this to be a valid theorem. He then interrupts his discussion of rational numbers to address the meaning of of “infinity.” He then moves on to presenting without proof the key propositions he wants about rational numbers in the paragraph starting “The reader will easily convince himself …” I suggest you do the algebra. The balance of this section is devoted to showing the rational numbers are not sufficient to do pure mathematics (they are all that you can use for applied math). Nothing is proved in a mathematical sense except the need for a more inclusive definition of “number” than the rational numbers.

Now if you read the argument differently from that, let’s discuss specifically where and why?

 

[MaxMath]

I repeat that I do truly understand where you are having difficulty.

Here is how I follow Hatdy’s argument.

Section 1: We assume as true the basic facts about integers and rational numbers. (I presume he means the axioms of an ordered field and certain well known theorems such as unique factorization of an integer > 1. He is not talking about some theorem known only to professional mathematicians. The book is intended for undergraduates.)

Section 2: He discusses the geometry of the straight line as a useful analogy to thinking about rational numbers. (This means to me that, (1) any time you find the geometric argument confusing, resort to algebra and (2) only algebra will deliver a proof.)

Section 3: Initially this section involves some general propositions on rational numbers. He sketches one proof with the simplifying assumption that an interval has a length of 1, but if you turn the proposition into a completely general algebraic proposition about two numeric intervals [A_1, A_2) and (B, C) such that A_1, A_2, B, and C are rational numbers such that
0 < | C - B | [imath]\le[/imath] | A_2 - A_1 |, you should be able prove this to be a valid theorem. He then interrupts his discussion of rational numbers to address the meaning of of “infinity.” He then moves on to presenting without proof the key propositions he wants about rational numbers in the paragraph starting “The reader will easily convince himself …” I suggest you do the algebra. The balance of this section is devoted to showing the rational numbers are not sufficient to do pure mathematics (they are all that you can use for applied math). Nothing is proved in a mathematical sense except the need for a more inclusive definition of “number” than the rational numbers.

Now if you read the argument differently from that, let’s discuss specifically where and why?

Thank you. I read it the same way. It does seem to make sense to me now. But to be honest, I may need to park some of my questions, keep notes of them, and keep reading and re-reading. The answer to my questions may be found in later parts of the book or may come up when revisiting the earlier sections. Anyway, I will raise my questions here if I need help!

 

[NotJeffM]

I realize that this thread is probably done, but I read through Section 4 today. I still do not know where the OP was getting hung up because he and I read the first three sections the same way. And it seems quite clear to me that the argument in Section 4 is that the numbers we want, such as the square root of 2, lie BETWEEN two sets of RATIONAL numbers, one consisting of all rational numbers less than the number we want and the other consisting of all rational numbers greater than the number we want.

If I were a mathematician, I might see subtleties, but this seems to my humanities-trained mind a very clever but fundamentally simple argument. (I am not trying to disparage the creativity of Dedekind et al; I am merely saying that once the answer is shown, comprehending it does not require being a mathematician.)

 

[pka]

I realize that this thread is probably done, but I read through Section 4 today. I still do not know where the OP was getting hung up because he and I read the first three sections the same way. And it seems quite clear to me that the argument in Section 4 is that the numbers we want, such as the square root of 2, lie BETWEEN two sets of RATIONAL numbers, one consisting of all rational numbers less than the number we want and the other consisting of all rational numbers greater than the number we want.

If I were a mathematician, I might see subtleties, but this seems to my humanities-trained mind a very clever but fundamentally simple argument. (I am not trying to disparage the creativity of Dedekind et al; I am merely saying that once the answer is shown, comprehending it does not require being a mathematician.)

I am not in any way a historian of mathematics. But I have had extensive experiences in the philosophy of mathematics, (A NIH fellowship 1992) Double major in mathematics & philosophy. I began my university teaching in a philosophy department teaching the logic from Copi's textbooks. That said recommend a most readable book ot you: GEORG CANTOR His Mathematics and Philosophy of the Infinte. By J. W. Dauben see here
Look at the used book market. If you really want to understand this thread read the first five chapters (119 pp)

 

[MaxMath]

I’m digressing, but it’s ok. Thank you @pka for recommending a good math history/philosophy book. I know little about this subject but find it extremely fascinating. And I believe it’s an extremely important subject, much more important than what I can see how it is being taken.
Coincidentally, I’m looking at something about Cantor recently. And I really struggle with the concept of infinite. Will definitely grab a copy of this!

 

[MaxMath]

... And it seems quite clear to me that the argument in Section 4 is that the numbers we want, such as the square root of 2, lie BETWEEN two sets of RATIONAL numbers, one consisting of all rational numbers less than the number we want and the other consisting of all rational numbers greater than the number we want.

That's correct. The method of "sectioning" the rational numbers makes it clear rational numbers are not all the numbers on the 'straight line'—there are gaps. The same method is used for all real numbers, later in section 17 of the book, and that tells us real numbers are complete (as a "continuum").