Comments on: What might an expository mathematical wiki be like?

By: Tiddly Formal Math « Formalized Mathematics

Tiddly Formal Math « Formalized Mathematics — Sun, 09 Mar 2008 15:13:57 +0000

[...] is an old discussion on Gower’s Weblog where Gowers says “Terry’s discussion of skeleton proofs is closely [...]

By: Frederick Ross

Frederick Ross — Wed, 26 Dec 2007 14:43:48 +0000

I think there’s a clean solution to the naming problem: don’t name the tricks. Instead, just rewrite them into Dijkstra’s calculational style of mathematics (predicate calculus is probably plenty for most cases). Then you can give the tricks rigorous definitions, and point out their use in various places.

Any trick takes the form of an implication or identity, so you should be able to easily build a search system for them based on putting some rough set of structures on both sides of an equality into a search box.

For instance a search ‘P = Q and R’ would return all tricks consisting of decomposing something into two properties, or ‘P = ’ is basically any embedding.

Such an approach is also valuable because then you can treat tricks as theorems, and make them reflexive manipulations of symbols.

Someone pointed out the design patterns stuff in software design. The last thing you want to do is go this route, because it makes it almost impossible to reason about these structures.

By: General news — December 2007 « Gowers’s Weblog

General news — December 2007 « Gowers’s Weblog — Tue, 18 Dec 2007 23:12:05 +0000

[...] to theorem-proving techniques—will almost certainly exist in the near future. Remarkably, my earlier post on this idea led to an offer of technical help that will be enough to turn it from a fantasy into a reality. And [...]

By: Julio

Julio — Wed, 28 Nov 2007 21:15:30 +0000

Terence
referring to your comment
“A mathematical-tricks wiki would be wonderful! But it is going to be more difficult to set up than other mathematics-oriented wikis. One of the first problems is that of nomenclature: mathematical theorems and objects tend to have standardised names, but mathematical tricks usually don’t; indeed, even mathematicians who use a trick routinely may not even be aware that they are doing so, except at a very non-verbal level.”

This was the same problem faced by people in computer science a while back until a book came out by Erich Gamma et al “Design Patterns: Elements of Reusable Object-Oriented Software”.

Naming software patterns, so what you describe as mathematical tricks could be name on the same token as Mathematical Patterns and be define in similar way as in the Gamma book.
Naming the patterns in a very engaging way that is easy to remember. This will make the demonstrations of theorems once ones knows the mathematical pattern language at another level.

see this

http://en.wikipedia.org/wiki/Software_pattern

for some explanation as to what a software pattern is.
If a way of naming the mathematical pattern is selected it will probably be useful to select names that remind us of the solution similar as it is done in the software pattern book.

By: davidspeyer

davidspeyer — Mon, 17 Sep 2007 18:11:59 +0000

A bit of self promotion: I have written a post on “That trick where you embed the free group into a Lie group” Comments are welcome.

By: That trick where you embed the free group into a Lie group « Secret Blogging Seminar

Mon, 17 Sep 2007 18:03:56 +0000

[...] a comment on Tim Gowers’ blog, Terry Tao suggests that embedding the free group into has a number of applications. I’m afraid that I [...]

By: Adam

Adam — Thu, 13 Sep 2007 20:15:26 +0000

Dear Tim,
I find your observation that in algebraic geometry “problem-solving
techniques seem to be easier to formalize as lemmas” very interesting.
I think that research areas are heavily influenced by their founders.
Modern algebraic geometry was given rigorous foundations by Serre
and Grothendieck. The philosophy of Grothendieck is well essayed in
NAMS 2004, issues 9-10. E.g., page 1197:
One thing Grothendieck said was that one should never try to prove anything that is not almost obvious. … ”if you don’t see that what you are working on is almost obvious, then you are not ready to work on that yet, …”
So with an ocean of definitions, I guess it is possible to find some more or less obvious relationships between them and call them lemmas. An example of what such intellectual endevours can lead to is on page 1196:
This can be seen in the legend of the so-called “Grothendieck prime”. In
a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
It is interesting how algebraic geometry is being promoted nowadays.
E.g., an article in Wikipedia says about it: “… some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.”
However, the limits of “technique” that the human brain can handle
are now well known. So, chess can be considered a very simple math game, more technical than conceptual. No human can win anymore with
a desktop chess engine. I watched the last loss (2-4) of the world champion Kramnik with Deep Fritz (2006), a Sicilian. At some moment during the game, grandmaster commentators were hailing Kramnik’s position. A few moves later, he resigned. No analysis was given after the game. The computer’s play was just beyond the grandmasters’ imagination
and book knowledge.

By: Adam

Adam — Thu, 13 Sep 2007 20:13:32 +0000

By: Web 2.0 Science: Rise of the Wiki, Part I « An American Physics Student in England

Thu, 13 Sep 2007 18:07:16 +0000

[...] 13 Sept 07: I appear to have been partially-scooped by a fields medalist. Professor Gowers has initiated a nice discussion on the value of a wiki-based math-0-pedia. (I [...]

By: Jason Dyer

Jason Dyer — Thu, 13 Sep 2007 17:27:14 +0000

gowers, for your “no magic” style of proof, I am reminded of the Fermat’s Last Theorem blog. Some proofs are hyperlinked back at least five layers.