eon

Posted by tpc at February 22nd, 2005

Two quotes on logic by Lewis Carroll, author of Alice In Wonderland, British logician.

‘Contrariwise,’ continued Tweedledee, ‘if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’

from Through the Looking-Glass

Then Logic would take you by the throat, and FORCE you to do it!

Extracted from an article entitled “What the Tortoise said to Achilles”

If you cannot get enough of Achilles and the Tortoise, the book to read is Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter.

Posted by tpc at February 20th, 2005

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The proof of the fascinating result can be found in Peter Walker’s Elliptic Functions - A Constructive Approach.

Posted by tpc at February 20th, 2005

In a local forum discussion about numb3rs, someone mentioned an old tv show called Square One TV. If you have time, download and watch the clip “Mathematics of Love”. It’s a funny skit about Roman numerals.

Incidentally, I quote from Gilles Godefroy’s book The Adventure of Numbers.

Archimedes is said to have been killed in 212 B.C. by a (Roman) soldier who stepped on his geometric figures, after a last protest: “Don’t disturb my circles …” We don’t want to disparage the grandeur of Roman civilization by following in the tracks of a famous contemporary thinker (”Ils sont fous ces Romains”)*, and will only point out the Romans’ disinterest in mathematics … We have already mentioned the archaic nature of their number system … The most amazing thing is that they were satisfied with it throughout the thousand years of their history.

I forgot where I heard this.

“The only thing the Roman numerals is good for is telling which Rocky movie came before the other.”

* “The Romans are crazy.” - Asterix the Gaul

Posted by tpc at February 8th, 2005

No one shall expel us from the paradise that Calculus has created.
- tpc

The fact that an analytic solution to number theoretical problems exist is bewildering. And often, the analytic proof is much simpler than the elementary proof. The Prime Number Theorem is the most famous example. Another is the evaluation of and in terms of double integrals.

The following is strictly not a number theory problem:
A rectangle is divided into smaller rectangles. Each of the smaller rectangles has the property that at least one of the sides has integer length. Show that the large rectangle has the same property.

I first encountered the problem at kuro5hin via Chapterzero, but apparently it dates back to de Bruijin in the sixties. I couldn’t solve it but as luck would have it, saw a solution in Paul Zeitz’s The Art and Craft of Problem Solving, which provided a reference to an article in American Math Monthly (vol 94) by Stan Wagon which gave 14 proofs, including a one line calculus proof.

Posted by tpc at February 7th, 2005

A little number theory problem posted by higherpi.
Let be an integer such is prime.
Prove that .

(more…)

Posted by tpc at February 7th, 2005

I quote verbatim from the preface in Simmons’ book.

It seems to me that a worthwhile distinction can be drawn between two types of pure mathematics. The first - which unfortunately is somewhat out of style at present - centers attention on particular functions and theorems which are rich in meaning and history, like the gamma function and the prime number theorem, or on juicy individual facts like Euler’s wonderful formula

The second is concerned primarily with form and structure …

40 years later, my view is that “form and structure” is still very much the fashion.

Posted by tpc at February 7th, 2005

No one shall expel us from the paradise that Cantor has created.
-David Hilbert (1862-1943).

Hilbert probably wrote in German, so the above quote may not be word perfect, but the meaning is clear. I first learnt set theory from the first chapter of “Introduction to Topology and Modern Analysis” by G.F. Simmons. I was hooked and wanted to jump into the continuum hypothesis. (The book was published in 1963, and so did not mention Cohen.) Then I took a course on formal set theory, which included among other things, the axiomatic development of the real numbers, yes Dedekind cuts and all that jazz. Even more exasperating was cardinal arithmetic and transfinite induction. So formal Set theory became to me one of those things that you should see once in your life, but not ever again.

Posted by tpc at February 6th, 2005

I’ve never seen the proof of this theorem, although it is mentioned in every advanced calculus and geometry/topology course that I’ve taken. What I’ve learnt recently is that it cannot be generalised to higher dimensions, and the Alexander’s Horned Sphere is a counter example. I particularly like the picture of Conway.
See the following link for more details.

Posted by tpc at February 2nd, 2005

This is classic. Check out the main site simpsonsmath.com for a comprehensive list of all things mathematical that appeared in the series.