Mathematics Message Board

Yay for Introduction to Electrodynamics by Griffiths. I became strangely attached to that book while it held me by the neck for Fall 05.

Anyone know how LT does the weighting for the "Most commonly shared books" list? My guess was increasing order of P(out of N randomly selected LT members, X have this book) where N is the number of group members and X is the number in the group having the specified book, and only truly "shared" books are included (X > 1).

I figured this group would be the place to get a definitive answer... :)

There are two questions about the "most commonly shared books" algorithm. The first is how Tim does it, the second how should he do it.

Clearly the ratio of (copies owned by the group) / (copies in LT) is no good as all the books only owned by one person, who happens to be in the group rise to the top.

Howerever the ratios for the top ten books as of 10 minutes ago was 4/12 4/15 14/980 5/39 4/21 3/6 4/28 9/494 6/129 3/13.

This shows the some correct trends. The lower the number owned in the group, the higher the ratio needs to be for the same place in the list. And for the same number owned by the group, the ratios are highest first.

However looking at the 100 list on the Group Zeitgeist page it start to go wrong. Entries from 42 to 100 are only owned by two group members, but are not in inverse order of total copies in LT. The first is 2/10 lower down there are 2/2 books (e.g. Carnivorous Plants at number 88).

The algorithm seems to ignore duplicates, it does not count my secod copy of Flatland.

An anecdote perhaps only math-minded people will find humorous:

When I came home from school, the Borders in my town had rearranged itself (to the point that I could no longer find the maths section). I asked one of the people that worked there if they could point me in the right direction, and I heard him over his little earpiece saying,
G(uy) - "Jim, can you show this lady where the maths section is?"
J - "*sneeze* Maps?"
G - "No, *MATH*. M-A-T-H."
(Now, my dad (a sarcastic mathematician), is standing over by Jim)
J - "Like 1+2?"
Dad under his breath - "That's arithmetic, you asshole..."

Well, it amuses me, when I think about it.

In reference to the "most commonly shared books" algorithm, I just posted a similar question in the Recommended Site Improvements Group.

I asked Tim if he could give us (the Math Group) the exact formula used to calculate the "weighted sharing" order and offered up our tweaking services.

Douglas

"In the end, only kindness (eigenvectors between persons?) matters."

Dydo: Thanks for that joke! I loved it!

For years, when struggling to figure a tip on a meal or when trying to recalculate the balance in my checkbook, I've always had to whine "But my degree is in Mathematics, not Arithmetic!"

Douglas

"In the end, only kindness matters."

But arithmetic is subsumed under mathematics, not separate from it! My A Course in Arithmetic by Jean-Pierre Serre is not on how to add up, but is a post-graduate work on analytic number theory, and definitely not elementary!

Now reading: Inconsistent Mathematics, by Chris Mortenson. Also portions of the enormous Paraconsistent Logic, edited by Graham Priest and Richard Routley. I hope I don't drop it on my foot!

Here's a summary of the inconsistent logic stuff that I wrote up for a mailing list I'm on; I hope ou find it interesting:

Godel showed that all formal axiomatic systems are either

a. trivial,
b. incomplete, or
c. inconsistent.

Inconsistent systems, in classical logic, are useless, because in an inconsistent classical system, every sentence is a theorem.

However, as many people have pointed out, this is a poor model for how reasoning is actually done. Certainly most people have some inconsistent beliefs. However, it is probably not the case that these people believe that absolutely everything is true.

For example, I believe that I have some false beliefs, although I don't know what they are. But also, for each of my beliefs B, I believe that B is a true belief. These positions are inconsistent. It does not follow from this, however, that I believe absolutely everything.

The principle that inconsistent theorems imply absolutely everything, that

A, ~A |- B

for all B, is called the "explosion principle". Logic can be weakened so that the explosion principle does not hold. The weakened systems of "paraconsistent logic" are in a certain sense dual to intuitionistic logic: intuitionistic logic rules out the theorem A v ~A; paraconsistent logics allow A & ~A.

I first heard about this yesterday, and I find it amazingly compelling; I've ordered a whole bunch of stuff from the library about it. In the meantime, here is the article that got me started on it:

http://plato.stanford.edu/entries/mathematics-inconsistent/

Paraconsistent logic allows a novel way out of Godel's theorem. Instead of resigning ourselves to incompleteness, we can resign ourselves to inconsistency. But inconsistecy doesn't have to be the disaster we thought it was. For example, the article cited above says:

Robert K. Meyer (1976) seems to have been the first to think of an inconsistent arithmetical theory. . . . Hilbert's program was widely held to have been seriously damaged by Godel's Second Incompleteness Theorem, according to which the consistency of arithmetic was unprovable within arithmetic itself. But a consequence of Meyer's construction was that within his arithmetic R# it was demonstrable by simple finitary means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations.

I liked the logic riff dominus. FWIW you might find Tertium Organum(especially chapter XXI) an interesting contrast/comparison to paraconsistent logic. Also, Fuzzy Logic might be worth a peek too. Cheerio....

Some of y'all might enjoy this discussion about irrational numbers over in the Christianity group.

#10: Wow. Just, wow.

A powerful demonstration of what happens when your axioms are junk.