Hurdles for mathematical thinking

From my years as both a math professor and observer of people, I’ve come up with a list of  hurdles for mathematical thinking.  These are what I believe to be the essential set of skills  a  person must have if they want to understand and do mathematics.  They don’t need to have all these skills to use mathematics but would need most of them if they want to progress far in mathematics.  Identifying what sorts of conceptual barriers people may have could help in improving mathematics education.

I’ll first give the list and then explain what I mean by them.

1. Context dependent rules

2. Equivalence classes

3. Limits and infinitesimals

4. Formal  logic

5. Abstraction

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Darts and Diophantine equations

Darts is a popular spectator sport in the UK. I had access to cable television recently so I was able to watch a few games.  What I find interesting about professional darts is that the players must solve a Diophantine equation to win.  For those who know nothing of the game, it involves throwing a small pointed projectile at an enumerated target board that looks like this:


A dart that lands on a given sector on the board obtains that score.  The center circle of the board called the bulls eye is worth 50 points.  The ring around the bulls eye is worth 25 points.  The wedges are worth the score ascribed by the number on the perimeter.  However, if you land in the inner ring then you get triple the score of the wedge and if you land in the outer ring you get double the score.  Hence, the maximum number of points for one dart is the triple twenty worth 60 points.

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The NP economy

I used to believe that one day all human labour would be automated (e.g. see here).  Upon further reflection, I realize that I am wrong.  The question of whether or not machines will someday replace all humans depends crucially on whether or not P is equal to NP.   Jobs that will eventually be automated will be the ones that can be solved easily with an algorithm.  In computer science parlance, these are problems in the computational complexity class P (solvable in polynomial time).   For example, traditional travel agents have disappeared faster than bluefin tuna because their task is pretty simple to automate.  However, not all travel agents will disappear.  The ones that survive will be more like concierges that put together complex travel arrangements or require negotiating with many parties.

Eventually, the jobs that humans will hold (barring a collapse of civilization as we know it) will involve solving problems in the complexity class NP (or harder).  That is not to say that machines won’t be doing some of these jobs, only that the advantage of machines over humans will not be as clear cut.  While it is true that if we could fully reproduce a human and make it faster and bigger then it could do everything that a human could do better but as I blogged about before, I think it will be difficult to exactly reproduce humans.  Additionally, for some very hard problems that don’t even have any good approximation schemes, blind luck will play an important role in coming up with solutions.  Balancing different human centric priorities will also be important and that may be best left for humans to do.   Even if it turns out that P=NP there could still be some jobs that humans can do like working on undecidable problems.

So what are some jobs that will be around in the NP economy?  Well, I think mathematicians will still be employed. Theorems can be verified in polynomial time but there are no known algorithms in P to generate them.   That is not to say that there won’t be robot mathematicians and mathematicians will certainly use automated theorem proving programs to help them (e.g. see here). However, I think the human touch will always have some use.  Artists and comedians will also have jobs in the future.  These are professions that require intimate knowledge of  what it is like to be human .  Again, there will be machine comics and artists but they won’t fully replace humans.  I also think that craftsmen like carpenters, stone masons, basket weavers and so forth could also make a comeback.  They will have to exhibit some exceptional artistry to survive but the demand for them could increase since some people will always long for the human touch in their furniture and houses.

The question then is whether or not there will be enough NP jobs to go around and whether or not everyone is able and willing to hold one.  To some, an NP economy will be almost Utopian – everyone will have interesting jobs.    However, there may be some people who simply don’t want or can’t do an NP job.   What will happen to them?  I think that will be a big (probably undecidable) problem that will face society in the not too distant future, provided we make it that far.

Arts and Crafts

There is an opinion piece by  Denis Dutton in the New York Times today on Conceptual Art, which presents some views that I am very sympathetic to.   All creative endeavours involve some inspiration and perspiration – There is the idea and then there is the execution of that idea.  Conceptual art essentially removes the execution aspect of art and makes it a pure exercise in cleverness.  In some sense it does crystallize the essence of art but I’ve always found it lacking.   I just can’t get that inspired by a medicine cabinet.  I’ve always found that the craft of a work of art to be as compelling (if not more) as the idea itself.  In many cases the two are inseparable.  Dutton argues that the craft aspect of art will never disappear because people intrinsically enjoy witnessing virtuosity.  I’m inclined to agree.  So while Vermeer or Caravaggio will remain timeless Damien Hirst may just fade away in time.


Corrected the spelling of Damien Hirst’s name on May 15,2012

Retire the Nobel Prize

I’ve felt for sometime now that perhaps we should retire the Nobel Prize.  The money could be used to fund grants, set up an institute for peace and science, or even have a Nobel conference like TED.  The prize puts too much emphasis on individual achievement and in many instances misplaced emphasis.  The old view of science involving the lone explorer seeking truth in the wilderness needs to be updated to a new metaphor of the sandpile, as used to described self-organized criticality by Per Bak, Chao Tang, and Kurt Wiesenfeld.  In the sandpile model, individual grains of sand are dropped on the pile and every once in awhile there are “avalanches” where a bunch of grains cascade down.  The distribution of avalanche sizes is a power law.  Hence, there is no scale to avalanches and there is no grain that is more special than any other.

This is just like science.  The contributions of scientists and nonscientists are like grains of sand dropping on the sandpile of knowledge and every once in awhile a big scientific avalanche is triggered.  The answer to the question of who triggered the avalanche is that everyone contributed to it.  The Nobel Prize rewards a few of the grains of sand that happened to be proximally located to some specific avalanche (and sometimes not) but the rewarded work always depended on something else.

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Talk at MBI

I’m currently at the Mathematical Biosciences Institute for a workshop on Computational challenges in integrative biological modeling.  The slides for my talk on using Bayesian methods for parameter estimation and model comparison are here.

Title: Bayesian approaches for parameter estimation and model evaluation of dynamical systems

Abstract: Differential equation models are often used to model biological systems. An important and difficult problem is how to estimate parameters and decide which model among possible models is the best. I will argue that Bayesian inference provides a self-consistent framework to do both tasks. In particular, Bayesian parameter estimation provides a natural measure of parameter sensitivity and Bayesian model comparison automatically evaluates models by rewarding fit to the data while penalizing the number of parameters. I will give examples of employing these approaches on ODE and PDE models.

Human scale

I’ve always been intrigued by how long we live compared to the age of the universe.  At 14 billion years, the universe is only a factor of 10^8 older than a long-lived human.  In contrast, it is immensely bigger than us.  The nearest star is 4 light years away, which is a factor of 10^{16} larger than a human, and the observable universe is about 25 billion times bigger than that.    The size scale of the universe is partly dictated by the speed of light which at 3 \times 10^8 m/s is coincidentally (or not) the same order of magnitude faster than we can move as the universe is older than we live.

Although we are small compared to the universe, we are also exceedingly big compared to our constituents. We are comprised of about 10^{13} cells, each of which are about 10^{-5} m in diameter.  If we assume that the density of the cell is about that of water (1 {\rm g/ cm}^3) then that roughly amounts to 10^{14} molecules.  So a human is comprised of something like 10^{27} molecules, most of it being water which has an atomic weight of 18.  Given that proteins and organic molecules can be much larger than that a lower bound on the number of atoms in the body is 10^{28}.

The speed at which we can move is governed by the reaction rates of metabolism.  Neurons fire at an average of approximately 10 Hz, so that is why awareness operates on a time scale of a few hundred milliseconds.  You could think of a human moment as being one tenth of a second.  There are 86,400 seconds in a day so we have close to a million moments in a day although we are a sleep for about a third of them.  That leads to about 20 billion moments in a lifetime. Neural activity also sets the scale for how fast we can move our muscles, which is a few metres per second.  If we consider a movement every second then that implies about a billion twitches per lifetime.  Our hearts beat about once a second so that is also the number of heart beats in a lifetime.

The average thermal energy at body temperature is about 10^{-19} Joules, which is not too far below the binding energies of protein-DNA and protein-protein interactions required for life.   Each of our  cells can translate about 5 amino acids per second, which is a lot of proteins in our lifetime.  I find it completely amazing that  a bag of 10^{28} or more things, incessantly buffeted by noise, can stay coherent for a hundred years.  There is no question that evolution is the world’s greatest engineer.  However, for those that are interested in artificial life this huge expanse of scale does pose a question –  What is the minimal computational requirement to simulate life and in particular something as complex as a mammal?  Even if you could do a simulation with say 10^{32} or more objects,  how would you even know that there was something living in it?

The numbers came from Wolfram Alpha and Bionumbers.