Following on from today's session, here is a version of the proof of the infinity of primes:
1.Each number is either prime or not prime (ie composite)
2.All composite numbers can be broken down into prime factors (eg 154 = 2 × 7 ×11) – this is the fundamental theorem of arithmetic
3.Let’s suppose that there is only a finite number of primes
4.Let’s call the members of this set: a, b, c, d, and so on, up to z, which would be the largest prime
5.Multiply all the members of the set of all primes together: a × b × c × d × …. × z – let’s call this number P, the product of all primes
6.Add 1 to make P + 1
7.P + 1 must be composite since it is larger than the supposedly largest prime
8.So, according to statement 2, there must be a prime number – let’s call it p – that divides P + 1
9.But for p to be a member of the set of primes a, b, c, d… z it would have to be also a divisor of P, and no number larger than 1 can divide two consecutive natural numbers
10.Hence for any set of prime numbers there will always be at least one more prime outside the set; therefore the number of primes must be infinite – QED
Do you see how some people can sense elegance or beauty in such a construction? More broadly, does mathematics have anything in common with the arts? Historically, mathematics has sometimes been regarded as an art...
Please also note that the powerpoint presentation that I showed you today can be found on the local network in the folder at K:\Staff_To_Students\IB Subjects\Core\TOK.
Tuesday, March 24, 2009
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4 comments:
I agree that Euclid's proof of the infinitude of primes is very beautiful and elegant.
I think it is important to note that in mathematics when we say a result is beautiful or elegant we simply mean that it simple or does not involve the use other complex results.
For example, Andrew Wiley's 200 page proof of Fermat's Last Theorem is not beautiful because it is very complex and long and few number theorists are capable of understanding it. The theorem itself is very simply stated and can be described as elegant or beautiful.
I hope this helps to clarify what we mean by beauty in mathematics.
This is not meant to be conclusive or bring the debate to an end because this is a bone of contention in mathematics, because some ask," Who decides what is simple?"
For instance, if i state Riemann's Hypothesis which says that the zeros of the Riemann Zeta function occur at the poles where the real part is half, this is simply stated and does not involve any complex math (depending upon your mathematical prowess) but i bet almost no one understands it. So is this beautiful or not?
hmm.. I think I disagreewith Gartaa's definition of mathematical beauty, That is unless there is some global agreement that this is its definition. Even then...
As far as I know, beauty is subjective and what i may perceive as beautiful is not necessarily what you may call it; its not as simple as 1+1=2.
Personally, I find the idea of a 200 page proof of a theorem quite fascinating. Granted, i may not(more like will not) comprehend half of what is in the pages but id do find the idea truly fascinating And beautiful. Plus, if you did understand what was in those pages would you not feel some sense of aesthetic satisfaction.
I find structure and logic beautiful regardless of length or complexity. Coming to think of it wouldn't you say that its complexity would make it more beautiful to the understander(Gartaa, I ask you; you would know)?
Gartaa and anteye504,
Nice discussion going on here. Some very insightful observations too. The concept of beauty in mathematics is very intriguing to many people. I think we will discuss the topic properly in class. But it is good to have some ideas about it in the mean time. It is good to use some examples to discuss what we mean by mathematical beauty, because surely mathematical beauty is not visual but rather conceptual.Here is an equation that many mathematicians consider the most beautiful equation ever. What makes it beautiful? To what extent is this kind of beauty subjective?
πi
e + 1 = 0
(e to the power of π times i plus 1 = 0)
Where,
e is the base of the natural logarithm,
i is the imaginary number, one of the two complex numbers whose square is negative one,
π is pi, the ratio of the circumference of a circle to its diamete,r
0 marks the beginning of whole number and
1 the beginning of natural number
So, wherein lies the beauty?
I write this in reponse to Mr. Kidane's submission about Euler's equation and to touch on what antye504 said about mathematical beauty.
But first to antye504. I agree that we have diferent standards when it comes to beauty (in the ordinary sense). You refer to the 200 page proof as fascinating (granted) but beautiful (i am not sure). I am sure many a mathematician would disagree with you. I say this because even in the mathematical community, only an elite few number theorists understand it because it requires knowledge of very complex properties. This is not an attempt to make mathematics look any more rigid than it already does but mathematical beauty is basically simplicity.
I couldn't be more grateful to Mr. Kidane for his wonderful example about Leonhard Euler's formula involving complex numbers. This was not the original statement of the result. The original involved many more complex properties of complex numbers and natural logarithms. In its initial form, people did not regard it as beautiful. It was further simplified and is now one of the most famous results in all of mathematics for its beauty (due to its simplicity)
Having said this, I dare say that this "definition" (note the inverted commas) of mathematical beauty is based on the views of the statistical majority of mathematicians but of course even among mathematicians there are some qualms as to if Euler's equation is simple enough to be called beautiful because one still requires some knowledge of complex numbers and the argand diagram and/or infinite series to understand it. Again, i think it depends on how much of the result you are able to comprehend.
Now to the personal question as to whether i find complex mathematical concepts and processes beautiful, i say, i find them fascinating and intriguing and it gives me a lot of joy when i understand them and am able to apply them to solve many other problems.
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