Baseball Families and Math Families

(This article had help from Clyde Kruskal, Joe Kruskal, Bill Kahn, Hans Courant, and Lucy Moser.)

Bill James, the baseball statistician, once had an article on measuring Baseball Families. Who was the best baseball family? Would it be ...

1. The Alous: Felipe, Jesus, Matty were brothers. Moses was Felipe's son. All made the major leagues, and some of them were pretty good.
2. The Bonds: Bobby and Barry Bonds. Father/Son- both excellent. While you might rather have them then the four Alou's on your team, having four in a family just seems like a stronger family.
3. The Aarons: Hank and Tommie Aaron. Hank was a great player who hit 755 home runs (without steroids). Tommie, his brother, had a very short career. Even though its a real family, still doesn't seem like the right answer.
4. Babe Ruth alone: Bill James him rates as the best player of all time. Hence the "Ruth family" would seem to be a very good baseball family.

So, how to go about the question of who is the best baseball family ever? First we need a way to assign to a player some number saying how good he is. Bill James has already done this via a method called win shares. The idea is that you assign to a player how much he helped the team win. This is very complicated and I won't explain it here. However I will point out that it can't be negative and it varies quite a bit. For example Hank Aaron had 643 win shares, while his brother Tommie had 15. For some examples of players with high win shares see this.
So, you could just take the sum of the family members win shares. But this makes the family of one, the winner. We want a real family to get some credit for being a real family. Bill James' solution: Say the best person has a₁ win shares, the second a₂, etc. a₁ > a₂ > ... > a_n. Rate the family via a₁ + 2a₂ + ... + na_n. Under this metric the Alou brothers were the best baseball family as of 2003. See here for the numbers.

But I am still bothered. Why the combination a₁ + 2a₂ + ... + na_n? Is there some other way that is more mathematically sound or that can be derived? I doubt it since the question is somewhat subjective.

Clyde Kruskal has brought up another point. What if you had a longer link between relatives? What if you had a great-grandfather, grandfather, father, son. If the father is the best baseball player, start there. People who share half his genes (son and grandfather) count fully. But the great-grandfather counts less. You could even do this if someone in the family does not play baseball. We will see an example below under (what else) the Kruskal Math Family.

Who are the best Math families of all time? Here are some, not in order. I am sure there are more. Corrections and additions welcome! (I may make a website out of it.) I only count a family if it has at least three members and all of the people are related by blood (sorry Blums). Aside from that, I am fairly informal. This list is not meant to be the final word.

1. The Bernoulli Bunch Link. There were eight of them. Jakob-Bernoulli numbers, Nickolaus-Prob theory and Geometry (NOT Bernoulli Dist), Johann- Brachistochrone problem and possibly L'Hopital's rule, Daniel-Bernoulli's principal (also physics and probability), Wikipedia does not have info on the other four, but says they were math folks.
2. The Kruskal Kin: William (Kruskal-Wallace test in statistics), Martin (Solitons), and Joseph (Min Spanning tree, Kruskal Tree Theorem (set of all trees under embedding is a well quasi order), Kruskal-Katona Theorem). They are all brothers. Martin's son is Clyde (Parallel Computation, Coloring the Plane). Clyde's son is Justin (Ramsey Theory, though he's still in High School, so perhaps shouldn't count). William's son is Vincent (Computer Science-IBM research). Rosaly and Molly Kruskal are sisters of Martin/William/Joseph. They do not do math, but Molly has two sons who do math: William Kahn (statistician), and Ted Kahn (Statistical Software); and Rosalie's son is Jeremy Evnine (OR, Math Finance). (Using Clyde's theory these people would count some.) The following does not count as he is not a blood relative: Joe Kruskal's son-in-law is Neal Madras (Math prof at York Univ in Canada).
3. A Nest of Noethers: Father: Max Noether one of the finest mathematicians of the nineteenth century according to Auguste Dick who wrote a book on Emmy Noether. Max's son was Fritz Noether. Fritz Noether. Max's daughter was Emmy Noether the most important woman in the history of mathematics according to Einstein and others. (That was meant as a compliment but sounds so odd nowadays.) A great mathematician independent of her gender. The only father-son-daughter combination that I know of.
4. The Markov Chain: Andrey Markov (Markov Chains), his brother Vladmir Markov (Markov's inequality co-authored with his brother), and Andrey Markov Jr. (logic) son of Andrey Markov.
5. The Browder Brothers: Felix (PDE's), William (Topology and Geometry), and Andrew Browder (Analysis).
6. A Litter of Lenstras: Hendrik (Computational Number Theory), Arjen (Crypto), and Jan Karel.

Interesting misses:

1. A Research of Rabin's: Michael and Tal. A Father-Daughter both in Crypto. Probably the only such.
2. A Manifold of Millars: Terry and Jessica. A Father-Daughter both in Recursive Model Theory. Definitely the only such.
3. A Troop of Tardos': Eva and Gabor. The only other Brother-Sister combo I know of is the Noethers.
4. The Courant Clan: Richard Courant (Math), Hans Courant (son of Richard, Physics), Ernst Courant (son of Richard, Physics) Ted Courant (son of Hans, Math), Jurgen Moser (son-in-law of Richard, Math) Lucy Moser-Jauslin (Jurgen Moser's daughter, Math), Jerry Berkowitz (son-in-law of Richard, Math), Peter Lax (son-in-law of Richard, Math), Carl Runge (Father-in-law of Richard, Physics and Numerical Analysis). Note that Jurgen and Lucy Moser are a father-daughter combination.
5. A Band of Blums: Manuel and Lenore (married) and their son Avrim. All do TCS at CMU. Lenore-Avrim is the only mother-son combo I know.
6. A Geek of Gauss' or A Nerd of Newtons or An Egghead of Eulers or An Ark of Archimedes' are competitive with any of these families. But one person does not a family make.