Tag Archives: David Hilbert

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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Revision 1: Inquisitive Mathematical Thinking

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In this post I wish to expand my understanding about, “asking Why?“.

In 1930, David Hilbert gave radio address lecture. I want to discuss following paragraph from that lecture (when translated to English):

With astonishing sharpness, the great mathematician POINCARÉ once attacked TOLSTOY, who had suggested that pursuing “science for science’s sake” is foolish. The achievements of industry, for example, would never have seen the light of day had the practical-minded existed alone and had not these advances been pursued by disinterested fools.

Science exists because we (human beings) want to find reason for everything happening around us (like how air molecules interact, which bacteria is harmful…) . We claim that this will enrich our understanding of the nature thus enabling us to make rational decisions (like when should I invest my money in stock market, from how much hight I can jump without hurting myself…).

Let me illustrate the point I want to make: Mathematicians make observations about real/abstract objects (shape of universe/klein bottle) and try to explain them using logical arguments based on some accepted truths (axioms/postulates). But today we have “science” for almost every academic discipline possible. Therefore, we (human beings) have become so much obsessed with finding reasons for everything that we even want to know why the things happened a moment ago so that we are able to predict what will happen in a moment from now. So the question is:

Should there be a reason for everything?

Can’t some thing just be happening around us for no reason. Why we try to model everything using psedo-randomness and try to extract a meaning from it? In case you are thinking that probability helps us understanding purely random events, you are wrong. We assume events to be purely random, we are never sure of their randomness and based on this assumption we determine chances of that event to happen which infact tells nothing about future (like an event with 85% chances of happening may not happen in next trial).

In same spirit, I can ask: “Should there be reason for you being victim of a terrorist attack?” We can surely track down a chain of past events (and even the bio-chemical pathways) leading to the attack and you being a victim of it.

Why we try to give “luck” as reason for some events? Is this our way of acknowledging randomness or our inability to find reason?

Moreover, David Hilbert ends his lecture with following slogan (in German):

Wir müssen wissen, Wir werden wissen.

which  when translated to English means: “We must know, we will know.”.