Esoteric Dissertations from a One-Track Mind

July 29, 2008

Bertrand Russell: Face to Face Interview

Filed under: culture, math — Tags: — codesmithy @ 9:12 am

Bertrand Russell was an anti-war activist, philosopher and mathematician.  I think he is correct in the conclusion that scientific man can not be at war with itself.  We are much too clever.  We wield great power, but haven’t the sense not to use, nor can we contemplate all the problems we are sure to create.

In The Systems Bible by John Gall, it states

Destiny is largely a set of unquestioned assumptions.

Gall gives this piece of wisdom with respect to problem avoidance.

If you’re not there, the accident can happen without you.

Avoiding catastrophe can be a lot easier than dealing with it.

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July 13, 2008

Ramanujan and 1729

Filed under: books, math — Tags: , , — codesmithy @ 11:32 am

I’ve been reading The Oxford Book of Modern Science Writing an anthology put together by Richard Dawkins.  It provides another good reason why life is far more interesting than the human imagination.  In it, there is a piece by C. P. Snow that speaks of relationship between Ramanujan and G. H. Hardy.  Ramanujan died at the age of 32. I can’t help but wonder what discoveries he would have made if his life had not been tragically cut short.

G. H. Hardy deserves credit for being able to recognize genius when it confronted him, especially since some of his colleagues apparently didn’t.  Hardy is said to have made the following comment about Ramanujan’s work: “must be true, because, if they were not true, no one would have the imagination to invent them.”  This dynamic is related by Snow in a anecdote he relays when Hardy visits Ramanujan in the hospital.

Hardy used to visit him, as he lay dying in hospital at Putney.  It was on one of those visits that there happened the incident of the taxicab number.  Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance.  He went into the room where Ramanujan was lying.  Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729.  It seemed to me rather a dull number.’  To which Ramanujan replied: ‘No, Hardy!  No, Hardy!  It is a very interesting number.  It is the smallest number expressible as the sum of two cubes in two different ways.’

That is the exchange as Hardy recorded it.  It must be substantially accurate.  He was the most honest of men; and further, no one could possibly have invented it.

How someone just knows that, I will never know.  But from the informal glancing at Ramanujan’s work, he saw numbers differently.  How much of this was a product of not being substantially formally educated, I don’t know.  I do think that learning in a more open-ended fashion, as Ramanujan did, has benefits.  Hardy made some remarks in a similar vein.

Regardless, I double-checked Ramanujan assertion, although there are a few caveats.  Since we are dealing with cubes, we will limit ourselves to whole numbers.  The sum of cubes that one arrives at 1729 are 1^3 + 12^3 = 1729 and 9^3 + 10^3 = 1729.  1729 turns out to be smallest.  The two distinct sum sequence is as follows.

1 12 9 10 1729
2 16 9 15 4104
2 24 18 20 13832
10 27 19 24 20683

In the context of the book, it is meant to show how people of a scientific persuasion see the world differently.  Where one person sees something bland, another sees it as a source of wonder.  In a certain sense, every number has something special about it.  Ramanujan’s gift was seeing what that special thing was.

July 15, 2007

Incorrect Mathematical Proofs

Filed under: math, random — codesmithy @ 2:37 am

Digg ran a story that showed an fallacious proof that 1 = 2 using complex numbers. This is a rehash of the 0 = 1 fallacy. Which goes something like this.

  1. a = b
  2. a – b = 0 (subtract both sides by b)
  3. (a – b)/(a – b) = 0 / (a-b) (divide both sides by a – b)
  4. 1 = 0 (simplify)

Although there tends to be some steps in between to mask the a – b step. Since a and b are equal, we are clearly dividing by zero producing the nonsensical result. The divide by 0 is a common caveat that might be forgotten in the factor and divide rule, especially when it is masked by unknowns.

In more advanced math, more seeming oddities can sometimes arise. Such as the following proof, posted in the thread discussion.

  1. e^(pi * i) + 1 = 0
  2. e^(pi * i) = -1 (subtract 1)
  3. e^(pi * i) * e^(pi * i) = 1 (square both sides)
  4. ln e^(2 * pi * i) = ln 1 (take the ln of both sides)
  5. 2 * pi * i = 0 (simplify)

Every thing is fine up to step 5. Although, it looks fairly innocuous. Because ln is usually defined as ln y = x where y = e^x, so it would seem that ln e^x would be x, by definition. Although, the way we evaluate e^(y) changes slightly when y is complex. Namely, e^(a + b * i) = e^a * cos(b) + e^a * sin(b) * i. Therefore, the ln must take this into account, the proper definition for complex numbers is ln(e^a * cos(b) + e^a * sin(b) * i) = a + b * i. In short, we can’t leave the i in the exponent when dealing with complex numbers and taking the ln.

Using the correct definition, we evaluate e^(2 * pi * i) = 1 * cos(2*pi) + 1 * sin(2 * pi) * i = 1 + 0i = 1. ln 1 = 0, where we reach the correct solution.

This particular fallacy is possibly because of the nature of the sin and cos evaluation, it is fairly trivial to prove e^(a + b * i) = e^(a + (2 * pi * k + b) * i) where k is an integer, which this particular fallacious proof was exploiting in a hidden way. Since, everyone would recognize e^(2 * pi * i) = e^0, ln e^0 = 0.

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