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The Anti-Pigeonhole Conjecture

April 26, 2015


A conjecture about faculty behavior

rdkibzwang
“Dr. Kibzwang” source

Colin Potts is Vice Provost for Undergraduate Education at Georgia Tech. His job includes being a member of the President’s Cabinet—our president, not the real one—and he is charged with academic policies and changes to such policies. He is also a College of Computing colleague and fellow chess fan.

Today I want to state a conjecture about the behavior of faculty that arose when Tech tried to change a policy.

I am currently at Georgia Tech, but this conjecture applies I believe to all institutions, all faculty. Ken mostly agrees. Potts recently supplied a wonderful example of the conjecture in action—I will get to that after I formally state it. Perhaps we should call it Potts’s Conjecture?

The Conjecture

The conjecture is easy to state:

Conjecture 1 Let {X} be any issue and let {A_{1},\dots,A_{n}} be any collection of distinct faculty members. Then during a long enough period of email exchanges among the above faculty on {X} at least {n+1} opinions will be voiced.

You can see why I refer to it as an anti-pigeonhole principle. Right?

I have tried to prove the conjecture—I view it as a kind of Arrow’s Paradox. I have failed so far to obtain a formal proof of it. The conjecture does have the interesting corollary:

Corollary 2 Let {X} be any issue and let {A_{1},\dots,A_{n}} be any collection of distinct faculty members. Then during a long enough period of email exchanges on the issue {X} some faculty member {A_{i}} will voice at least two different opinions.

A weaker version that we will cleverly call The Weak Conjecture is the following:

Conjecture 3 Let {X} be any issue and let {A_{1},\dots,A_{n}} be any collection of distinct faculty members. Then during a long enough period of email exchanges on the issue {X} at least {\sqrt{n}} opinions will be voiced.

The point is that the total number of opinions is unbounded. Strong or weak, we can call it the Conjecture.

The Example

Of course, being mathematicians we want proofs not examples. But as in areas like number theory, one is often led to good conjectures by observations. In any event simple tests of conjectures are useful to see if they are plausible enough to try to prove.

Here is the policy change that has been suggested. You are free to skip this or go here for even more detail. The point is that this is the issue {X}.

Per the proposal, starting in fall 2015, classes would not meet on the Wednesday before Thanksgiving, giving students an additional day for their break. A change implemented as a pilot this spring will continue to stand, which eliminated finals being held during the last exam session on the Friday before Commencement to prevent finals overlapping with graduation festivities. Starting the next academic year, it was approved to extend the individual course withdrawal deadline by two weeks, allowing students more time to evaluate whether to drop a class.

In Spring 2016, the current Dead Week would be replaced with Final Instructional Class Days and Reading Periods. The new schedule would designate Monday and Tuesday of the penultimate week of the semester as Final Instructional Class Days, followed by a day and a half of reading period, and administering the first final on Thursday afternoon. Finals would be broken up by that weekend and resume Monday, with an additional reading period the next Tuesday morning. Finals would finish that Thursday, allowing Friday for conflict periods and a day between exams and Commencement.

{\dots}

The final recommendation would extend the length of Monday/Wednesday/Friday classes during spring and fall semesters from 50 to 55 minutes. Breaks between classes would extend from to 10 to 15 minutes.

Pretty exciting, no? No.

The result of Potts announcing the above was a storm of emails from our faculty members. As you would expect, given the Conjecture, this quickly led to a vast number of opinions. The number of opinions seems easily to follow the Conjecture.

No-Gradient Hypothesis

Ken analogizes this kind of policy tuning for a university {U} to finding a regional optimum in the landscape of a multivariable function {f_U}. A proposal like Potts’s, with so many little changed parts, resembles a step in simulated annealing where one periodically jumps out of a well to test for better conditions in another. He is not surprised that such a ‘jump’ would bring multiple reactions from faculty.

Even so, however, one would expect there to be a gradient in the new region so that opinions could converge to the bottom of the new well. This is a different matter: a helpful gradient should be in force after a jump.

April is the month when US undergraduates have been informed of all their college acceptances and in many—fortunate—cases must make a choice. Ken has a front-row seat this year. From comparing various colleges and universities with widely different policies, and noting the market incentive to diversify, he has come to a conjecture of his own:

Conjecture 4 There is no gradient: for any university {U}, {\nabla f_U} is defined only on a set of measure zero.

To all appearances, this conjecture implies the others. Is it capable of being proved? Again you—our readers—are best placed to furnish input for a proof.

Open Problems

Do you believe any of the conjectures? I hope we get lots of opinions…

Ken and I are divided: he thinks we will not get many, I think we will get a lot, and we both think that we may get just a couple. But in my opinion it is possible that …

[added to first paragraph; format fixes]

15 Comments leave one →
  1. April 26, 2015 11:08 pm

    I found tenured faculty members far stranger than students. Once tenured, the feel free to reveal their actual home planet. I only hope to get away from Earth before their Army of Flying Monkeys arrives.

    • April 26, 2015 11:22 pm

      Here is mine—as eldest son of the First Earl of Nussex, it was my unrenounceable inheritance.

  2. April 26, 2015 11:22 pm

    I think this is because some agent or coalition among the n agents will eventually try to formulate a compromise or synthesis that subsumes the original n opinions, hence the n +1. This is related to classical paradoxes that there cannot be a finite number n of basic concepts or Platonic ideas, because the idea of their totality would be one more.

  3. George Watson permalink
    April 27, 2015 12:44 am

    Given how freely Academics comment on everything under the sun and how impractical
    most of them are I am surprised there are not N ! opinions – perhaps that is an upper bound.

  4. MattF permalink
    April 27, 2015 6:29 am

    What about the case when N=1? Or, worse, N=0?

  5. phomer permalink
    April 27, 2015 7:13 am

    I’ve found that the world makes a little more sense if you believe that for any possible opinion (whether it makes sense or not) that there is at least one person out there holding it, including that this particular opinion is incorrect. More interestly, most possible combinations of opinions are out there too, somewhere…

    In that sense every opinion has at least one anti-opinion, and since there are usually less people than combinations of opinions, somebody, somewhere, must be flip-flopping. But of course, that is only my opinion🙂

    Paul.

  6. April 27, 2015 8:33 am

    Reminds me of a famous saying in Hebrew – “Two Jews, three opinions”.
    Google even autocompletes it, which is conclusive evidence for me that it’s true🙂

  7. marcelofinger permalink
    April 27, 2015 9:27 am

    [Tong in cheek] You should charge people for expressing a new opinion. Conjecture: Even a small symbolic amount would limit the number of opinions to O(log n).

  8. April 27, 2015 10:10 am

    There’s a relation among Pigeonhole Principles, Wilhelm von Humboldt’s recognition that “language makes infinite use of finite means”, and recursiveness, since it means that some of our means must be used an infinite number of times.

    If a positron is an electronhole, is a pigeonhole an antipigeon?

    As for Antipigeon Pigeons, I am of two minds about that.

  9. Javaid Aslam permalink
    April 28, 2015 6:22 pm

    I suppose there is a tacit assumption in Conjecture 1– the set Y from which all opinions about X are made is infinite.
    Otherwise the conjecture will fail for any finite Y, whenever n > |Y|.

    Considering the complex behavior of human being, putting a lower bound on the number of opinions would amount to modeling a deterministic behavior.
    Is this possible?

  10. paulb permalink
    April 29, 2015 3:25 am

    I’m sure there is a joke there about using a Pott’s model to explore function minimum. Maybe piecewise no gradient…

  11. Serge permalink
    April 30, 2015 4:54 am

    I’ve changed my mind so often about P=NP… that I accept your conjecture without hesitation!🙂

  12. Bobito permalink
    April 30, 2015 6:07 am

    Don’t discard prematurely the hypothesis that given any issue $X$ and any n distinct faculty members $A_1, \dots, A_n$ there is at least one faculty member $A_i$ who initially has two (possibly contradictory) opinions. This faculty member can be called by the special name of “dean”.

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