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Friday, November 4, 2011

Thoughts about number theory (1)

Has it ever happened to you that you skipped a proof because no matter how hard you tried you simply "didn't get it"? - Or worse, that you had to learn an algorithm but the 'why' wasn't given, let alone a proof. - Mathematics may be hard and difficult but at the end of the day you should 'own' the theorems. Where ownership stands for the notion that you could have created the theorem, in principle, yourself. Or look at ownership at this way: browse through a mathematics book that you studied two or three years ago, or even longer. Everything in it looks really simple because slowly over the years, you took ownership of that particular subject of mathematics.

When I am stalled on a particular topic or proof, I simply accept the proposition knowing that somehow my brain is working on it. It is a better alternative than remaining stalled. It is possible though that you are stalled because the author decided to leave out 'a few details'. Some topics in number theory, for example, are simple if you look at it from a group theory perspective. The author then has to decide if his proposed audience has already studied group theory or not. And even then his publisher may decide otherwise because from his perspective the audience should be made as large as possible.

Number Theory may be called the Queen of Mathematics the Queen needs a lot of help from the 'people'. There is analytical, algebraic, combinatorial and computational number theory and I wouldn't be surprised if there are a few more. The theory of prime numbers and group theory are strongly interconnected for example. Numbers are among the first mathematical objects that man studied but is a number an object? Unlike graphs, sets and geometric or topological shapes numbers don't really exist. Even groups exist, not just as sets, but they are part of nature itself in the form of symmetries everywhere. What -is- one? Like the one in 'one apple'? I argue that 'number' is a property like color, and the rest is just physics.

What about 'God invented the integers', 'Number theory is beautiful' and so on? The truth is that man invented a God that supposedly invented the integers. I have been studying number theory for a while now, and I haven't found its beauty yet. Just open problems, a lot of open problems everywhere. - Inspector Columbo would call that loose ends. And for him that is proof of human error. - That these open problems are a challenge is another issue.


  1. That is an important thing to keep in mind. The subconscious will continue to work even if you are not consciously working on the problem. I am getting used to the fact I will not and sometimes cannot understand everything I am learning in my undergrad career but I am relying on the honesty and reliability that math and mathematicians bring to the table when they publish material. (ie. I am sure I am not studying a theory not founded on rigorous proof). But with the goal that I will be able to fill in the blanks with time. Maybe not all of the blanks but as many as possible.

  2. Here is the Prime Algorithm:

    the website:

  3. @WarriorClass - Thanks for posting.
    Will reply more after reading.


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