Tag Archives: paradox

Hilbert Effect

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Human beings tend to measure the influence of a person(s) on a particular field of study by associating their name to cornerstones. For example: Urysohn lemma, Tychonoff theorem, Gauss Lemma, Eisenstein Criterion, Chinese Remainder Theorem, Hensel Lemma, Langlands program, Diophantine Analysis, Clifford algebra, Lie Algebra, Riemann Surface, Zariski Topology, Banach–Tarski paradox, Russell paradox, Bernstein polynomial, Bernoulli Number ……

In mathematics their have been some fights about naming the cornerstones, which ended up creating a compund-name. For example, Lobachevsky-Bolyai-Gauss geometry (in textbooks it is generally referred as hyperbolic geometry), Bolzano–Weierstrass theorem (Bolzano prove it in 1817, later Wierstrass proved it again rigorously and popularized it), Schönemann–Eisenstein theorem (in textbooks it is generally referred as Eisenstein Criterion), ……

But, David Hilbert influenced mathematics at a whole new level. Apart from terms like Hilbert Cube (and many more..) named after him, he introduced exotic words in mathematics which are very popular in (research-level) mathematics. Following are some of the terms:

  • Eigen: This word troubled me a lot when I came across the term “eigen-vector” and “eigen-values” a couple of years ago. Hilbert used the German word “eigen”, which means “own”, to denote eigenvalues and eigenvectors of integral operators by viewing the operators as infinite matrices. You can find more information about the history of introduction of this term in mathematics in this web-page by Jeff Miller.
  • Entscheidungsproblem: It is german word for “decision problem”, but still mathematicians tend to use this particular term. For example, the famous paper by Alan Turing titled “On computable numbers, with an application to the Entscheidungsproblem“.
  • Syzygy: Interestingly, “syzygy” is greek word used in astronomy to refer to the nearly straight-line configuration of three celestial bodies in a gravitational system. In Hilbert’s terminology,  “syzygies” are the relations between the generators of an ideal, or, more generally, a module. For more details refer to this article by Roger Wiegand titled “WHAT IS…a Syzygy?“.
  • Nullstellensatz: It is german for “Set of zeros” (according to google translate). But today, just like syzygy, it has whole new meaning in mathematics. For more details, refer to this MathOverflow discussion: What makes a theorem *a* “nullstellensatz.”

Apart from the terms used in mathematics, Hilbert popularized the term “ignorabimus” in philosophy during his famous radio address. For more details read this short Wikipedia article.

It appears that mathematicians (sometimes) tend to use their creativity in naming theorems like Snake Lemma

 

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Integration & Summation

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A few months ago I wrote a series of blog posts on “rigorous”  definitions of integration [Part 1, Part 2]. Last week I identified an interesting flaw in my “imagination” of integration in terms of “limiting summation” and it lead to an interesting investigation.

The Paradox

While defining integration as area under curve, we consider rectangles of equal width and let that width approach zero. Hence I used to imagine integration as summation of individual heights, since width approaches zero in limiting case. It was just like extending summation over integers to summation over real numbers.

integration (1)

My Thought Process..

But as per my above imagination, since width of line segment is zero,  I am considering rectangles of zero width. Then each rectangle is of zero area (I proved it recently). So the area under curve will be zero! Paradox!

I realized that, just like ancient greeks, I am using very bad imagination of limiting process!

The Insight

But, as it turns out, my imagination is NOT completely wrong.  I googled and stumbled upon this stack exchange post:

There is the answer by Jonathan to this question which captures my imagination:

The idea is that since \int_0^n f(x)dx can be approximated by the Riemann sum, thus \displaystyle{\sum_{i=0}^n f(i) = \int_{0}^n f(x)dx + \text{higher order corrections}}

The generalization of above idea gives us the Euler–Maclaurin formula 

\displaystyle{\sum_{i=m+1}^n f(i) = \int^n_m f(x)\,dx + B_1 \left(f(n) - f(m)\right) + \sum_{k=1}^p\frac{B_{2k}}{(2k)!}\left(f^{(2k - 1)}(n) - f^{(2k - 1)}(m)\right) + R}

where m,n,p are natural numbers, f (x) is a real valued continuous function, B_k are the Bernoulli numbers and R is an error term which is normally small for suitable values of p (depends on n, m, p and f).

Proof of above formula is by principle of mathematical induction. For more details, see this beautiful paper: Apostol, T. M.. (1999). An Elementary View of Euler’s Summation Formula. The American Mathematical Monthly, 106(5), 409–418. http://doi.org/10.2307/2589145 

Little Things Matter

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This is a very famous puzzle.  Here is motivation:

ezgif.com-video-to-gif

Infinite Chocolate (gif from 9gag.com)

Now consider following variation.

puzzle

Where does the hole in second triangle come from? (image credit: Rookie1Ja , http://brainden.com/forum/index.php?/topic/139-64-65-geometry-paradox/)

The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is

\arctan(\frac{2}{3}) - \arctan( \frac{3}{8}) = \arctan (\frac{1}{46})

which is less than 1 degree 15′ . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the “hidden” lozenge is equal to that of a small square of the chessboard.

p2

It looks like a triangle, because a thick line was used. Hypotenuse of the composite triangle is actually not a straight line – it is made of two lines. Forth cusps are where the arrows point (c9, l6).

Also there is an interesting video illustrating this in real life: