Courage and humility

...my dear hope is that political leaders will have the courage and the humility as well to overcome political sensitivity and concerns and doctrines, which are perfectly legitimate, for the sake of the entire country and for the sake of the global economy.   

IMF Managing Director Christine Lagarde in a PBS News Hour interview,  on the U.S. debt ceiling crisis.

Courage and humility.  Without these qualities, leaders simply can't be counted on.  Ms. Lagarde later says,

Political courage is required. And I'm not - I'm not suggesting that it is lacking, but it has to be demonstrated in moments of crisis. I was last week in Brussels, and there was a moment of courage and solidarity amongst the European leaders, members of the eurozone area. It comes in times in crisis. And when it does, it's quite extraordinary.

Well, we are all dearly hoping to see this real soon now from our US political leaders. 

The combination of these two qualities is what makes leaders who can be counted on.  To be humble, to acknowledge mistakes, to genuinely listen, to take accountability - all of these things require courage and all are necessary to gain confidence, followership and support in difficult times.  A great leader once said, "the role of a leader is to define reality and to give hope."  In difficult times, the "defining reality" part often requires accepting responsibility, acknowledging mistakes and sometimes being clear about uncertainty and personal shortcomings.  It also means asking for help and being open and humble in accepting it.  What some leaders don't understand is that showing this kind of courage and humility in defining reality provides a solid foundation for hope and when you give people that kind of hope, amazing things can happen.  

Ms. Lagarde's gentle but firm admonishment to the US political leadership really applies to all of us.  We have all enjoyed the "exorbitant privilege" that she (quoting Giscard d'Estaing) ascribes to our currency.  That "privilege" was earned by generations of Americans who built homes, families, careers, businesses and institutions showing these key qualities of leadership.  We need to get back to that.  Now.  Not just in politics, not just in government, but all of us, every day.

Those of you who are parents, teachers or leaders, think about the example that you are setting.  Think about what behaviors you are rewarding and who you are choosing to promote.  Will your children/students/proteges respond with courage and humility or CYA, obfuscation, finger-pointing or "positioning" when real problems arise?  We can only reverse this slide into weak, self-centered, myopic leadership by calling it out when we see it and stopping the cycle of rewarding it.

Thank you, Ms. Lagarde for calling us out!

Saving the phenomenon of OSS

I just reread Dan Pink's Drive, a book that I recommend highly for anyone who leads people or is interested in how human motivation works.  The basic premise of the book is that there is a growing gap between "what science knows and what business does" to motivate people.  He calls the traditional setup that still dominates the mainstream corporate world "Motivation 2.0" and makes a compelling case that this throwback to the industrial age is in need of a major version bump - Motivation 3.0.    Drawing on a substantial body of behavioral science research, Pink demonstrates that the "If-then" reward structure that dominates how we try to motivate people today is not well-suited to the post-industrial workplace where "work" is no longer primarily made up of boring, repetitive tasks.  Instead, he argues that intrinsic rewards are far more effective in today's world.  He calls out three primary drivers of intrinsic motivation: mastery, autonomy and purpose.

What I found very interesting, and at the same time a little troubling, after reading this book, is how well it saves the phenomenon of open source contribution.  Pink mentions OSS as an example of the beginnings of  Motivation 3.0.  At least looking at my own involvement in OSS, the drivers above do a pretty good job explaining how it is that I am willing to spend so much time volunteering.  Much better than what other "anthropologists" have come up with, which generally devolves to Motivation 2.0-speak (we do it because it will enable us to make more money somehow).  Working with great developers on hard problems can provide a sense of mastery that is hard to attain otherwise.  Autonomy is also key.  We "scratch itches" that we have as - and when - we have them.  And finally, there is a sense of contributing to something larger than ourselves or our code - a purpose.

The troubling bit is that we stand at the crossroads now in open source.  We got where we are as a result of really forward-thinking community engagement - being on the vanguard of Pink's Motivation 3.0.   People are now, in large numbers, being paid to work on OSS projects and commercial software companies are "open sourcing" their products faster than established OSS communities can absorb them.   Will we be able to maintain the powerful intrinsic reward structure that has gotten us to this point and use our elevated position to drive positive change in the Motivation 2.0-dominated establishment?  Or will the "OSS phenomenon" gradually vanish as we collectively "grow up?"

With all hats that I may wear, I will be trying to lead the change toward environments that provide the kinds of intrinsic rewards that I have enjoyed in OSS and tried my best to bring to the corporate world.

Self-organization is "supra-optimal"

Thanks to a post shared by Eric Le Goff on Google+, I ran across a very interesting article describing some fundamental research on neuronal network organization.  The authors have developed a research "platform" that enables them to grow physical neuronal networks on chips.  They are using the platform to observe self-organizing behavior of the networks.  What is becoming clear from this and other research is that the large-scale architecture and macro behavior of our brains is most likely the result of self-organizing collaboration among a hierarchy of network clusters, each "operating" with locally conditioned architecture and control.

After reading the article, I was embarrassed to notice that I had not yet heard of Plos One, where it appeared.  Poking about Plos One, I stumbled on yet another article on self-organizing behavior.  Somebody must be trying to tell me something.   The second article is a nice visceral illustration of how we need to think differently about how to manage complex systems.   The authors look at train scheduling in public transportation systems.  They show that "global" scheduling systems for train arrival and departure times can be beaten by a system that imitates ant colony behavior - trains emitting "anti-pheromones" that signal trains behind them to modify their behavior.   What is interesting here is that independent decision-making based on locally available information can result in better global outcomes than a globally engineered solution.

These ideas are not new.  What these two articles illustrate, however, is how much change is going to be required in how we think about modelling and engineering to take advantage of them.  For example, the whole concept of "control" in the pure engineering sense is going to have to be transformed to engineer the kinds of systems that the second article hints at.  Imagine trying to apply six sigma to that transportation system.   Similarly, the first article indicates that it is likely hopeless to try to get much more than we already have from macro-level decomposition of the brain or functional/reductionist analysis of isolated neuronal networks.

I also notice that Plos.org has opened up their search APIs and (jointly with Mendeley) launched "a contest to build the best apps that make science more open."  Hmm...somebody is really trying to tell me something here.

Remembering how to count

Today, rummaging through the boxes of books that follow me from house to house, I stumbled on a first edition, 1940 hardbound copy of Mathematics and the Imagination by Edward Kasner and James Newman that my mother gave me years ago.  Kasner is probably best known for having come up with the name - or more precisely goading a child into inventing the name - "googol."  The book begins with a delightful account of how the very large, but finite is sometimes confused by adults and "moderns" with the infinite, but small children and the ancient Greeks could see things more clearly.  I like this quote:

The Greeks had very definite ideas about the infinite.  Just as we are indebted to them for much of our wit and learning, so are we indebted to them for much of our sophistication about the infinite.  Indeed, had we always retained their clear-sightedness, many of the problems and paradoxes about the infinite would never have arisen.

The authors do a really nice job motivating what was to me as a student just a definition - that cardinality is based on one-to-one correspondences.  What they bring out really nicely is how this really is just counting.

Another great point that comes out in many places in the book is how mathematicians often choose what turn out to be stupid names for things, because we can name them before we know them.  A great example is "transcendental numbers."  Such a name suggests something exotic or rare.  It turns out that almost every real number is transcendental.

What does that mean, exactly?  I am going to try to explain this in less time than it takes you to get bored - in just a short walk through a truly beautiful neighborhood built by Georg Cantor.

A transcendental number is a number that cannot be expressed as a solution to a polynomial equation with rational coefficients.  The square root of two, for example, is not transcendental because it is a solution to the equation x² - 2 = 0.   It turns out that both e and π are transcendental. Initially mathematicians thought this was a very special property worthy of such an exalted name.

Cantor saw two things: 1) algebraic numbers (the complement of transcedental numbers) can be put into one-one correspondence with the set of natural numbers (i.e., they are countable) 2) the whole set of real numbers is uncountable (cannot be put into 1-1 correspondence with the natural numbers).  That means that all but a comparatively tiny set of real numbers are transcendental!  You can take away all of the algebraic numbers without affecting the size of the set of real numbers (or leaving any "holes" in the real line).

Imagine a spreadsheet with an infinite number of rows and columns:

A0 A1 A2 A3 ...

B0 B1 B2 B3 ...

C0 C1 C2 C4 ...

D0 D1 D2 D3 ...
...
Cantor realized that as long as the row and column indexes were all natural numbers,  the entire set of cells could be put into 1-1 correspondence with the natural numbers (i.e. "counted") by just starting in the top left corner and following a path that snakes outward along the diagonals:  A0, A1, B0, C0, B1, A2, A3, B2, C1, D0, D1, C2, B3, A4, ...

Since the rational numbers are just numerator/denominator pairs of integers, the technique above can be used to count them - i.e., to show that the rationals are countable.  But really, what it shows is that 1) the set of ordered pairs of any countable set is countable and 2) the union of countably many countable sets is countable.  The second is clear when you think of the rows of the spreadsheet as enumerations of the elements in the sets over which the union is taken.

Now imagine a "spreadcube" - a rubix-cube like thing that has little subcubes for cells.  Again, imagine it as countably infinite in all three dimensions.  The "pick a corner and snake out" game works there too.  So the set of ordered triples (a, b, c) from any countable set is also countable.  To get to the next dimension, you can either just test your 4-dimensional intuition (too hard for me) or just look at (a, b, c, d) as ((a, b, c), d) and use the fact that there are only countably many (a, b, c) (note that we could have done this for n = 3 instead of imagining the spreadcube).  Obviously, there is no stopping us here, so we see that for each n, the set of n-tuples of elements from any countable set is countable.   Since the coefficients of a polynomial determine the polynomial, that means that for each n, there are only countably many polynomials of degree n with rational coefficients.  By 2) above that means that there are only countably many polynomials with rational coefficients.  Each of these has only finitely many roots, so the union of all of the sets of roots of all of the polynomials with rational coefficients is countable - i.e. there are only countably many algebraic numbers.

Now the coup de gras.  Since we seem to be able to count everything, lets suppose that via snakes, pairing or some other ruse we have managed to enumerate the real numbers - i.e., we have put them in a list that goes r0, r1, r2 ... and eventually hits every real number.   Every real number has a decimal expansion.  Some terminate, like 5.2, some repeat, like 0.333333 and some, like π do exotic things that keep computers busy for indefinite amounts of time; but all have decimal representations and two real numbers are equal if and only if they have exactly the same values in their decimal expansions in every digit [1].   Suppose that our enumeration looked like this:

r0 = 12.238769045
r1 =  3.1415923988888
r3 = 23.1783459823999
...

Consider the number r = 0.359 in relation to the three numbers above.  It is obviously not equal to any of them.  It was constructed to be different from any of them as follows.  Put a 3 in the first decimal digit because r0 has a 2 there; put a 5 in the second decimal digit because r1 has a 4 there; put a 9 in the third decimal digit because r3 has an 8 there.   Now imagine the list continuing.  We continue to define the decimal expansion of our "special" number r making it different from the nth element in the list in the nth decimal place.  The result will be a number that differs from every number in the list.  This is a contradiction,  proving that no such enumeration can exist.

This is not a satisfying place to leave Mr. Cantor's neigborhood, since we have not succeeded in counting the real numbers at all.  So even though you are likely, like my dog Casper, getting a little tired of this walk, I need to stop by one more place on the way home.   To count the real numbers, lets start by convincing ourselves that it suffices to count the elements of the open interval (0, 1).  Look at the middle section of the graph below of the tangent function

The middle section of this graph shows that (-π/2,π/2) can be mapped onto the full real line. Every y value in the graph corresponds to some x value in the bounded open interval above.   The graph depicts a one-to-one correspondence between the whole real line (the y axis) and the little π-length interval in the middle of the picture along the x axis. Just squishing the picture a little and moving it to the right shows that (0, 1) could similarly be mapped onto the whole real line.

Now, similarly to decimal representation, every real number has a binary expansion.  The number 1/2 is 0.1 in binary, 0.11 is 1/2 + 1/4 = 3/4, 0.101 is 1/2 + 0/4 + 1/8 = 3/8 and so on.  Like decimal expansions, binary expansions are unique, some terminate and some don't and two numbers between 0 and 1 are equal iff their binary expansions have 1's (or 0's) in all the same places.   Therefore, there is a one-one correspondence between the real numbers between 0 and 1 and the set of subsets of the natural numbers.  Let each real number correspond to the set of natural numbers where its binary expansion has a "1".  So 1/2 corresponds to {0}, 3/4 corresponds to {0, 1}, 3/8 corresponds to {0, 2} and so on.  The set of all subsets of a set is called the power set of the set.  The above shows two important things: 1) the power set of the natural numbers in uncountable and 2) the power set of the natural numbers has the same cardinality as the real numbers.   An interesting question is does there exist a size in between the natural numbers and the real numbers.  We are not going to see an answer to that question in this neighborhood.

[1] Strictly speaking, some numbers have two infinite decimal expansions.  For example 0.9999999 (repeating indefinitely) equals 1.  We are assuming above that all of the representations selected are minimum length (i.e. if there is a finite representation, choose that).