శాస్త్ర విజ్ఞానము

7 comments

1. తార Says:
   Posted on July 1, 2013 at 12:26 PM

2. >>అంతేకాక ఈ సంఖ్య మనకి తెలిసిన ఏ ప్రధాన సంఖ్యతోను విభజింపబడలేదని కూడా సులభంగా గుర్తించొచ్చు.
   
   Are you following any Science Literature Book? The problem with Science books is, they are very misleading, and they often leave the heart of the theory may be due to insufficient understanding or to make simplicity. Maths is very rigorous subject, even you change a single word, it will lead to lot of contradictions. More over, most Science writers are not Mathematicians, so they don't have sound understanding of the subject. Finally the end reader gets misguided and he will interpret the whole subject in wrong way.
   
   Here heart of the proof is, UFD (Unique factorization domain), in a UFD you can express every element as "finite" product of irreducibles.
   
   In Set of Natural Numbers, the irreducibles are prime "numbers" (because of the same irreducibility you discussed in previous post). So, essentially, here in this proof, you are assuming that every Natural number can be expressed as "finite" product of Prime "Numbers". Before stating this proof, you should prove that or atleast state a line about this assumption.
   
   With out this assumption, may be there are infinite prime numbers, but your proof is not valid.

    
3. V Srinivasa Chakravarthy Says:
   Posted on July 1, 2013 at 9:58 PM

4. This is a translation of George Gamov's "One, two, three...infinity."
   But what is wrong with the following statement?
   >>అంతేకాక ఈ సంఖ్య మనకి తెలిసిన ఏ ప్రధాన సంఖ్యతోను విభజింపబడలేదని కూడా సులభంగా గుర్తించొచ్చు.
   
   

    
5. తార Says:
   Posted on July 2, 2013 at 12:04 AM

6. Let me explain the Proof.
   
   Lemma: In the set of Natural Numbers, every number can be expressed as finite product powers of irreducibles (called prime numbers).
   Proof: I am leaving it out here.
   
   Theorem: There exists infinite number of Prime Numbers in the set of Natural Numbers.
   
   Proof: Let us assume the otherwise, say there exists only "n" number of prime numbers only. say p1, p2,...pn. Consider the number M = p1*p2*...pn + 1. By Archimedean Principle, this is also a Natural Number. As by previous lemma, M can also be expressed as product of powers of some subset of p1,...pn. But none of the primes divides the number M, (i.e., అంతేకాక ఈ సంఖ్య మనకి తెలిసిన ఏ ప్రధాన సంఖ్యతోను విభజింపబడలేదని కూడా సులభంగా గుర్తించొచ్చు). Hence it satisfies the property of irreducibility, and hence M is a prime number, not one of p1,..pn, which is a contradiction.
   
   ........
   
   Here, that line which is heart of the theorem is based on Lemma. With out that lemma, that line no longer gives you any result. That is just another fact, but significance came to that line just because of previous lemma. So, I quoted that line.
   
   
   
   

    
7. V Srinivasa Chakravarthy Says:
   Posted on July 2, 2013 at 8:23 AM

8. Ok. I understand the issue. Here's my resp.
   
   The lemma you mentioned above is obvious in case of natural numbers. If you are dealing with more abstract entities, you may have to explicitly state it.
   
   Or if you are writing math textbook you will have to explicitly state it for the sake of rigor.
   
   This is a popular science book and pitched at layperson level. So talking about prime numbers as "irreducibles" etc would seem like hairsplitting and might distract the reader.
   
   Another thought:
   
   Time and again I have been noticing that several bloggers are so well informed in various branches of science. It will be great if they used their talent in writing some popular science in Telugu themselves, instead of stopping short at criticizing what others do. If they have concrete ideas about how popular science must be written in Telugu, it is only fair that they reveal them to the world by an actual demonstration...
   
   
   It is not just this blog, or Telugu science popularization... Isnt it a very common feature (malaise?) in our country? In our highly populated country, for every national problem there are a vast number of "commentators" who will illumine us about the best ways to solve those problems. These are the people who discuss, critique, analyze, look at the problem from a thousand angles, and give scholarly lectures on the matter. But there are very few... very, very few... who have the good sense to get off their whatever-it-is and actually do something about it!!!
   
   Personally I'd rather belong to the latter class of people, since there is hardly any competition :-)
   

    
9. తార Says:
   Posted on July 2, 2013 at 12:10 PM

10. >>
    The lemma you mentioned above is obvious in case of natural numbers. If you are dealing with more abstract entities, you may have to explicitly state it.
    >>
    
    Yes quite obvious, but it took thousands of years to prove that lemma, only after "induction" it was proved.
    
    In Maths, many things look obvious, but when you try to prove it, most of the times either you might be wrong or you need to change the whole world :-). Many things look quite obvious because our knowledge is very limited, I mean the examples we can construct are very obvious, so they seem very true, but once you start proving general case, you might need to borrow a couple of thousand years.
    
    Check the abc conjecture, it looks quite obvious, but still the proof is unknown (there are much more harder open problems in number theory, but the most easiest been open from many centuries).
    
    >>
    Or if you are writing math textbook you will have to explicitly state it for the sake of rigor.
    >>
    
    Nope, if you writing Maths text book, that is completely different case. Here you should mention the lemma since, because of that reason only we are studying prime numbers, if that lemma is false, Mathematicians are least bothered about Primes. Prime will be just another number.
    
    Just that simple lemma instigated a whole lot of research in Maths, take localization, why do you localize? you localize because, you study the behavior at individual primes and you glue all the primes together you will get the whole behavior. (this is Geometry, and the thought is from this Lemma only).
    
    That Lemma (also called fundamental theorem of Arithmetic), is much more significant.
    
    For two reasons, here the author should mention that before the line I quoted in my first para.
    
    1. Because of that Lemma only we are interested in primes.
    2. Because of that Lemma only the proof of the theorem is true. (I mean here a serious reader would definitely get a doubt that, why should a prime must divide (1 X 2 X 3 X 5 X 7 X 11…X N) + 1, what is none of the primes divide this number? whats the big deal?)
    
    Again, Fermat also interested in Primes because, he wanted to find a way (an algorithm) for Factorization.
    
    >>Another thought:
    
    Yes I agree, but I want my readers to think and change their attitudes. Give me such readers I will write. (I tried, I wrote some articles with intentional serious flaws, I waited and waited, but none found them, either people knew that either it is beyond science or believed true because of their political ideology.
    
    Even in my college (you might know standard of our Maths dept.), they consult Vaastu experts before making changes in buildings. The whole nation looks at these people and even these people are like that. Instead I prefer to invest same time in learning a new subject. [ Serious readers can find very good English books, like Men of Mathematics, or DFS Macroeconomics]. So, there is no serious loss to a serious reader.

     
11. తార Says:
    Posted on July 2, 2013 at 12:13 PM

12. *I quoted in my first para.
    
    Typo first "comment".

     
13. తార Says:
    Posted on July 2, 2013 at 1:25 PM

14. To be more precise, the proof is not self contained, it requires some advanced level Maths to understand that sentence correctly, certainly the book is not written for such advanced readers.
    
    So, it leaves the reader either assume some thing on his own, this is how a book shouldn't be.