Monday, 29 August 2011

Roy's Fortnightly Puzzle: Volume 8

Not really a puzzle, but something to think about:

Okay, so you probably know what an Erdos number is - the number of links - where a link is a co-published paper - between Paul Erdos and yourself. You also probably know that Erdos numbers are used as a tongue-in-cheek measure of mathematical status. Much more irrational, of course, is the fact that logicians and others in more technical areas of philosophy tend to brag about their Erdos numbers (mine is 7, by the way - Erdos-Shelah-Woodin-Welch-Rayo-Uzquiano-Shapiro-Cook).

What is much more interesting than closeness to Erdos, however, is what all of this implies about the connectivity with respect to collaboration of the mathematical community. For example, as of the year 2000, approximately 98.4 percent of authors listed on Math Reviews who have an Erdos number at all have an Erdos number no higher than 7 (99.6% are no higher than 8, and 79.1% are no higher than 5. About 2/3 of authors published in Math reviews have an Erdos number).

Erdos number distribution:

504 have Erdos Number 1.
6593 have Erdos Number 2.
33605 have Erdos Number 3.
83642 have Erdos Number 4.

(it levels off, and then eventually tapers off, after this).

All this data, and much more, can be found here.

Setting Erdos aside, the average distance between any two mathematicians (again, where they are connected at all) is 7.64.

This got me thinking about philosophy, and how connected it is via collaboration in comparison with mathematics (and other disciplines). So I picked a random person (myself) and computed Cook numbers. Here are the results:

4 have Cook Number 1.
19 have Cook Number 2.
39 have Cook Number 3.
~250 have Cook Number 4.

Unlike the Erdos numbers, however, these numbers don't look like they will taper off anytime soon. Instead the numbers seem likely to continue increasing. For example, there are a number of philosophers who have collaborated a good bit, including Barry Smith, Walter Sinnott-Armstrong, and Otavio Beuno, who have Cook number 4.

Part of the issue is that I started with myself, and I have few collaborators (although my recent work in aesthetics did help the list become rather varied speciality-wise rather quickly).

Anyway, Cook numbers are merely suggestive (and silly as well). Nevertheless, the numbers do seem to grow steadily. It would be interesting to know what the collaboration graph for philosophy looks like in comparison to mathematics. Some interesting questions (which I would love to hear people's opinions/guesses on):

(1) What is the average length of the chain between two philosophers (who are connected at all)?

I am going to hazard a guess that the average chain length is similar to the length in math. My guess would be that the difference will be the number of distinct chains between two people. Keep in mind that some of the slower growth suggested by the numbers above will be offset by the smaller population of philosophers as compared to mathematicians.

(2) Within philosophy, is there a single, large 'network' as there is in mathematics?

Probably (see below). For example, as noted above, 2/3 of all authors listed on Math Reviews are in the network connected to Erdos, including a surprising number of people who strictly speaking aren't mathematicians, such as myself. Interestingly, the average number of collaborators for people in the large network is 4.73, while the average for people who aren't (but who have collaborated) is 1.63.

(3) What is the average number of collaborators within philosophy?

For example, on average an author listed in Math Reviews has 3.36.

(4) Is there any way to get access to the Phil Papers data to actually find any of this out?

Anyway, here is a challenge, just to make it interesting: I suspect that the answer to question (2) is "yes", and that the 'big' network includes the majority of philosophers in American and UK philosophy departments (and, obviously, I think that the network is the one I mapped a bit of above). Further, I make the following conjecture:

Any American or UK philosopher who has collaborated with at least four distinct people has a Cook number.

(Note that this is equivalent to the claim that any philosopher who has collaborated with four other people has an Erdos number)

Prove me wrong?

Bonus Question: Do any philosophers have Erdos-Bacon numbers (the sum of one's Erdos number and one's Bacon number, computed in terms of distance from Kevin Bacon via co-starring in a film). I know of one.

1. My Erdos-Cook number is nine. Jeff Roland, Mark Silcox, and Jason Megill are all elevens.

One question- Are mathemeticians more likely to collaborate on work that has more than two authors? This is extremely common in the sciences and almost unheard of in philosophy. If math were somewhere in between it might account for some of the different behaviors of Erdos and Cook numbers.

2. My Erdos number is 6, be jealous everybody! :) It goes through Stephen Read, Graham Priest, and the rest I don't remember (the Australian connection is great for low Erdos numbers).

As for my Cook number, not sure. But via Otavio Bueno, it wouldn't take long to make it to Stephen, presumably (via Graham perhaps?), so it should also be around 6.

3. Actually, there are now 5 people with Cook number 5. My wife (who is really NOT a math person) and I just got a co-written paper on comics and adult education accepted in an anthology on comics and education.

This actually brings up a interesting question on how to compute Erdos numbers (my wife is rather amused by the fact that she now has one). The general consensus seems to be that any academic paper counts. But isn't counting a paper on comics and education stretching the concept a bit?

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