Understanding how they thought of math proofs in the first place

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In summary, the conversation is about the philosophy of learning real analysis from Rudin's book. The speaker is struggling with understanding how the proofs in the book were thought of and is wondering if they need to delve deeper into understanding the proofs or if filling in the gaps is enough. They also discuss different approaches to finding a rational number q to prove that there exists a rational number q > p such that q^2 < 2. They question the role of irrational numbers and whether they can be avoided in the proof. They also mention other books and their own preference for knowing the thought process behind a proof.
  • #1
Constructivist
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Hi,I am learning pure math say real analysis from Rudin. Now, I am getting confused on philosophy of learning. Well, if I read Rudin, somehow with struggle, I am able to fill the gaps in the proof which he gives. Though I can completely understand the proof, my question is how was the proof thought of in the first place. To give an example , as shown in the screenshot,Rudin takes q as p - ((p^2 - 2)/(p+2)) to prove that for every rational number p (where p^2 < 2), there exists a rational number q >p such that q^2 <2. Well, I can fill in the gap in the proofs mostly by myself or ask someone for help say in PF. What I am asking over here i, s how was this particular q = p - ((p^2 - 2)/(p+2)) thought of in the first place? So, I was pondering on this. let's say we take q = (sqrt(2) + p)/2 which is a reasonable guess but we cannot take this number as sqrt(2) is irrational and q will turn an irrational. So, next guess is q = (1.4+p)/2. In that case, q is rational but what if p itself is 1.4? Clearly, there lie number between 1.4 and sqrt(2) and we do not know if they are rational or not? So, q has to be a function of p alone and let q = f(p). Also, the properties of the function are:

1. q = f(p) which is greater than p
2. (f(p))^2 < 2.
3. if p is rational number tending to square root of 2, then q also tends to a rational number tending to square root of 2.

So, intuition is to have q = p + (r- p)/m . Where r is rational number closer to square root of 2 and m is a rational number (Here , (r- p)/m is the small value if m is big ). But, this cannot be the case. As we have discussed before, the proof fails if p tends to fixed r. The fact the square root of 2 is irrational is a pain. So, how can we avoid it? Discuss the variables in-terms of squares. Example is let p be rational and p^2 < 2. The trick is to take a q such that 2>q^2>p^2 and have this q as rational. Since p^2 -2 <0, we need to have a q^2-2 <0.

So, let q^2-2 = (p^2 -2)/m where m should be positive. The one concern we have for m is: Is it a rational number say 5 or 10 or 1000 or is it a function of p. If m were a fixed number, then q^2 = ((p^2 -2)+2m)/m and q = sqrt((p^2 -2)+2m)/m). There are some problems with this m. Can (p^2 -2)+2m)/m be rational ? ----a first question. Even, if it is rational, the problem is ' Is q = sqrt((p^2 -2)+2m)/m) rational ?' (Example: 2 is rational but sqrt(2) is not). So, why cannot m be perfect square like 4, it is the same issue (q becomes sqrt(p^2 +6)/2 . Then is sqrt(p^2 +6)/2 is rational? Let m be Ap+B where A nd B are rational. In that case, m is rational. But wait, we never know if sqrt(Ap+B) is rational. So, the better solution is to go with m = (Ap+B)^2. In that case, q becomes sqrt(p^2 -2 + 2 (Ap+B))/(Ap+B). So, if (p^2 -2 + 2 (Ap+B)) is a perfect square, then q is also rational. So, our task is to find A relation between A and B and make this term a perfect square. The equation I got is 2A^2 -B^ + 1 = 0 . A is 1/sqrt(2) and B is sqrt(2). And hence we get q = p - ((p^2 - 2)/(p+2)).My question is do I need to go this depth or understanding the proofs just enough? Do we need to think on how they investigated to get the proof as we read along Rudin or can I just fill in the holes in proofs. I feel such an approach will give some experience in deriving newer formulae in future. Your suggestions are welcome.
1609278406074.png
 
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  • #2
Like you, I benefit from knowing how a proof was thought of. For that reason I never recommend Rudin, which lacks that side of things more than maybe any other book. I usually like books by Simmons or Berberian but it has been a long time since I studied analysis. I might recommend something by Apostol.
 
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  • #3
mathwonk said:
Like you, I benefit from knowing how a proof was thought of. For that reason I never recommend Rudin, which lacks that side of things more than maybe any other book. I usually like books by Simmons or Berberian but it has been a long time since I studied analysis. I might recommend something by Apostol.
Thanks Mathwonk. So, is thinking in such a way, good in the long run? Will I lose a lot of time as a result? I see that people who just fill the gaps in proofs move through the material faster rather than knowing how it was thought about. Sometimes, I feel the easier way is faster
 
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  • #5
"Thanks Mathwonk. So, is thinking in such a way, good in the long run? Will I lose a lot of time as a result?"

yes it is good in the long run. and yes, you will spend a lot of time, but it will not be lost.
 
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  • #6
mathwonk said:
Like you, I benefit from knowing how a proof was thought of. For that reason I never recommend Rudin, which lacks that side of things more than maybe any other book.

I have a mathematician friend who often teaches real analysis, and he says similar things about Rudin.
 
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  • #7
Constructivist said:
Hi,I am learning pure math say real analysis from Rudin. Now, I am getting confused on philosophy of learning. Well, if I read Rudin, somehow with struggle, I am able to fill the gaps in the proof which he gives. Though I can completely understand the proof, my question is how was the proof thought of in the first place. To give an example , as shown in the screenshot,Rudin takes q as p - ((p^2 - 2)/(p+2)) to prove that for every rational number p (where p^2 < 2), there exists a rational number q >p such that q^2 <2. Well, I can fill in the gap in the proofs mostly by myself or ask someone for help say in PF. What I am asking over here i, s how was this particular q = p - ((p^2 - 2)/(p+2)) thought of in the first place? So, I was pondering on this. let's say we take q = (sqrt(2) + p)/2 which is a reasonable guess but we cannot take this number as sqrt(2) is irrational and q will turn an irrational. So, next guess is q = (1.4+p)/2. In that case, q is rational but what if p itself is 1.4? Clearly, there lie number between 1.4 and sqrt(2) and we do not know if they are rational or not? So, q has to be a function of p alone and let q = f(p). Also, the properties of the function are:

1. q = f(p) which is greater than p
2. (f(p))^2 < 2.
3. if p is rational number tending to square root of 2, then q also tends to a rational number tending to square root of 2.

So, intuition is to have q = p + (r- p)/m . Where r is rational number closer to square root of 2 and m is a rational number (Here , (r- p)/m is the small value if m is big ). But, this cannot be the case. As we have discussed before, the proof fails if p tends to fixed r. The fact the square root of 2 is irrational is a pain. So, how can we avoid it? Discuss the variables in-terms of squares. Example is let p be rational and p^2 < 2. The trick is to take a q such that 2>q^2>p^2 and have this q as rational. Since p^2 -2 <0, we need to have a q^2-2 <0.

So, let q^2-2 = (p^2 -2)/m where m should be positive. The one concern we have for m is: Is it a rational number say 5 or 10 or 1000 or is it a function of p. If m were a fixed number, then q^2 = ((p^2 -2)+2m)/m and q = sqrt((p^2 -2)+2m)/m). There are some problems with this m. Can (p^2 -2)+2m)/m be rational ? ----a first question. Even, if it is rational, the problem is ' Is q = sqrt((p^2 -2)+2m)/m) rational ?' (Example: 2 is rational but sqrt(2) is not). So, why cannot m be perfect square like 4, it is the same issue (q becomes sqrt(p^2 +6)/2 . Then is sqrt(p^2 +6)/2 is rational? Let m be Ap+B where A nd B are rational. In that case, m is rational. But wait, we never know if sqrt(Ap+B) is rational. So, the better solution is to go with m = (Ap+B)^2. In that case, q becomes sqrt(p^2 -2 + 2 (Ap+B))/(Ap+B). So, if (p^2 -2 + 2 (Ap+B)) is a perfect square, then q is also rational. So, our task is to find A relation between A and B and make this term a perfect square. The equation I got is 2A^2 -B^ + 1 = 0 . A is 1/sqrt(2) and B is sqrt(2). And hence we get q = p - ((p^2 - 2)/(p+2)).My question is do I need to go this depth or understanding the proofs just enough? Do we need to think on how they investigated to get the proof as we read along Rudin or can I just fill in the holes in proofs. I feel such an approach will give some experience in deriving newer formulae in future. Your suggestions are welcome.
View attachment 275340
Yes, its a good approach to think of the ideas that the author has discussed before arriving at the theorem in question, and try to think of what he was building towards. Try to do it first without looking, a good honest approach, then compare your reasoning to the authors. With more practice, you can offer valid and alternative proofs.

Like Mathwonk suggested. I have a copy of Berberian (Linear Algebra), and Simmons (Topology and a Differential Equations book). Both authors write clearly and build up the material well., ie very lucid in aiding you to connect the dots. So check to see of they have an analysis book.

If you want a similar book on Analysis, although a bit more gentle (It stays in R), I would recommend Abbot: Understanding Analysis. Before Analysis was a mystery to me, and it appeared to my eyes as a bag of tricks and algebraic manipulations to make my epsilon/delta arguments work, but this book helped me understand Analysis. It is very neat for self-study and problems.

If you want something along the lines of Rudin but more explanation, Maybe V. Zorich: Mathematical Analysis Volume 1? I am currently reading it, it may be a bit too hard for a first exposure to Analysis, but working through something like Abbot then work through Zorich.

Maybe Lang undergraduate analysis? I have not read the parts pertaining to single variable analysis, so I am not sure how good it is for an introduction, but the multivariable part is great (just started reading it).

But in summary, give Abbot a gander.
 
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  • #8
If you have access to a library take a look at Strichartz. It has its critics, but at the end of the day it comes down to what works for you.
 
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  • #9
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  • #10
Keith_McClary said:
This short and elementary version of the proof was arrived at after 50 years of effort by Toeplitz, Hausdorff, Stone, Halmos and others:
https://www.ams.org/journals/proc/1...-1970-0262849-9/S0002-9939-1970-0262849-9.pdf
Thanks Keith, so I was building up material of 50 years? Is it realistic for a lay person like me to think on how they thought about proofs?

I think I myself have to ask why I do math:
1. Am I doing math to rigorously build in structure (say analysis, topology)?
2. Or am i doing math to create new ideas from the structure?

Any idea guys what has to be my approach here ?
 
  • #11
This inexpensive book, Real Analysis: A Long Form Mathematics Book by Jay Cummings puts a fair amount of effort into explaining the reasoning behind the various aspects of real analysis.

"This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own."

https://www.amazon.com/dp/1077254547/?tag=pfamazon01-20

And it's actually a fun read. The dedication at the front of the book made me chuckle: "To my students, without whom I would have had to edit this whole damn book myself." :)
 
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Related to Understanding how they thought of math proofs in the first place

1. What is a math proof?

A math proof is a logical argument that shows a statement or theorem is true. It involves using established mathematical principles and logical reasoning to demonstrate the validity of a mathematical statement.

2. How did mathematicians come up with the concept of proofs?

The concept of proofs has been around since ancient times, with the Greeks being credited as the first to develop a formal system of mathematical proofs. Mathematicians throughout history have built upon these foundations and developed new methods for proving mathematical statements.

3. Why are proofs important in mathematics?

Proofs are important in mathematics because they provide a rigorous and logical way to verify the truth of mathematical statements. They also help to build a deeper understanding of mathematical concepts and can lead to new discoveries and advancements in the field.

4. How do mathematicians approach creating a proof?

Mathematicians approach creating a proof by first clearly defining the statement they want to prove. They then use logical reasoning, deductive reasoning, and mathematical principles to construct a step-by-step argument that leads to the desired conclusion.

5. Are there different types of proofs in mathematics?

Yes, there are different types of proofs in mathematics, including direct proofs, indirect proofs, proof by contradiction, and proof by induction. Each type of proof has its own set of rules and techniques, and mathematicians may use different types of proofs depending on the statement they are trying to prove.

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