# Category Archives: Mathematics

## Infinity and Beyond! Math for Primates Episode 14

I’ve just uploaded our new episode on Infinity and Fractals at the Math for Primates website.

In this episode:

• Zeno’s beach bungalow and why he can’t come out to play
• Do train tracks actually touch at infinity?  How do YOU know? Have you been there?
• Watch out for the Hordes!!
• Welcome to the Hilbert Hotel … you can check in any time you like, blah, blah, blah …
• What do you do with an infinite number of mathematicians? Run!
• Explaining jokes at the Unit Interval, a great neighborhood bar.
• Is one type of infinity BIGGER than another kind?
• Counting ALL the Real Numbers, “… 2, ah ah … 3, ah ah … pi, ah ah!”
• The Power Set:  the enemy of the X-Men.
• Fractals, Hippies, and gettin’ trippy
• Is Nick a quantum human?

## Coordinated Punishment Leads to Cooperation

A new study suggests that cooperation is maintained by punishment

Humans are incredibly cooperative, but why do people cooperate and how is cooperation maintained? A new research study by UCLA anthropology professor Robert Boyd and his colleagues from the Santa Fe Institute in New Mexico suggests cooperation in large groups is maintained by punishment.

This study looks to be attempting a solution to the “free-rider” problem in human behavior research.  A free-rider is someone who, in a large enough society to support such things, refuses to cooperate, or play by the rules, but still benefits from the cooperation of others.  Think of someone who doesn’t pay their taxes in Canada, but still gets to use their health care system.

The “problem” we face when dealing with the free-rider issue is not that it exists, but that it doesn’t exist MORE.  It should by any good Game Theoretic reading.  It’s in the best interest of the individual to NOT pay their taxes, and to still get the health care.  That’s a win-win. But, the reality is that the vast majority of Canadians DO pay their taxes.  Why?

But it turns out that most members of large groups cooperate. Why? Boyd and his colleagues suggest cooperation is maintained by punishment, which reduces the benefits to free riding. There are tribes, for example, that punish free-riders who do not participate in warfare by not allowing them to take a bride. Thus, there is the threat of losing societal benefits if a member does not cooperate, which leads to increased group cooperation.

The results of their model look a lot like what is seen in most human societies, where individuals meet and decide whether and how to punish group members who are not cooperating. This is coordinated punishment where group members signal their intent to punish, only punish when a threshold has been met and share the costs of punishing.

The important point being that the cost of punishment (to the society) is not higher than the cost of allowing the person to free-ride.

Since theft and other forms of (non-violent) crime can be seen as free-riding, one has to ask, if the current system of punishment in the US has not resulted in lowered crime rates (think of marijuana related cases, for instance) then might it be because the cost of the punishments outweighs that of the crime to us as a whole?  Just a thought.

## Natufian Villages, Wisdom Teeth, and Probability

Some of the earliest villages ever discovered were from the Fertile Crescent (modern day middle east).  The culture that inhabited these villages has been dubbed Natufian, and dates to around 12,500 BC.  Two of these villages, Hayonim and ‘Ain Mallaha were relatively close to one another.  At first it would have been reasonable to presume that these villagers bred with one another.  They were close and each one had relatively small populations.  But, it turns out they didn’t, and the reason has much to do with simple probability.

Well, that and wisdom teeth, or the third-molars. The Hayonim people had ‘agenesis’ of the third molar – that is, they never grew in.  But, the ‘Ain Mallaha group nearly all had their third molars.

If it had been true that the two groups interbred, then we’d get a mixing of types – those that had wisdom teeth, and those that didn’t – in about equal proportions.   Therefore, since we don’t have that, there must not have been significant interbreeding.

Sounds reasonable enough.  But, how can I be so sure that IF there was interbreeding THEN the proportion of third molars would in fact be about equal in both groups (or at least spread around a bit more)?

To answer that question we need to go over what an “allele” is.

## Alleles as Soda-Pop

Genes are complicated little buggers.  It isn’t so simple as “this gene codes for that trait”.  Any single gene comes in a number of variations. We call those variants ‘alleles’.

Think of 2 soda cans.  One of them is flavored cherry, the other is flavored grape.  What they have in common is that they are both soda.  The gene is the soda.  But, this gene comes in one of two flavors – or we can say it has two alleles – grape and cherry.

So, an allele is like a flavor.  Sometimes, as is the case with eye color, a change in alleles is no big deal.  It really is much like a flavor in that it causes you no harm to have blue vs. brown eyes.  But, there are times when having a different flavor (allele) does matter, like in the case of polio.

Humans are diploids.  That means that we all have two chromosomes.  We get one from our Father and one from our Mother.  Each of them gave us one allele of each gene, and our ‘genotype’ for that gene (that is, what type of that gene we got) is a pairing of the alleles.  But, here’s the rub.  Our parents each had two alleles of the gene, and we end up with two.  How many possible choices were there for what we could have ended up with?  That all depends.

Let’s say the two allele variants for gene X are A and B.  If Dad had an AA pairing and Mom had a BB pairing, then our only possibility was an AB pairing.   However, if our Dad was AB and our Mom was BB, then we’d have 2 possible choices:  AB or BB.  And it would be a 50-50 chance that we got either one.

Now, if both our parents were AB, then we’d have 3 choices (note that $AB=BA$) but they are not all equally likely.   There’s only 1 AA and only 1 BB, but there are 2 AB’s.  So there is a 25% chance of AA, a 25% chance of BB, and a 50% chance of AB.

If it turns out that the A allele is dominant, then from the outside we’d not be able to tell the difference between someone with AA vs. someone with AB.  Only the BB’s would show any signs.  Again, this is no biggy with something as silly as eye color or handedness, but when it comes to diseases or something life threatening, it certainly is.

## Times Have Changed

Back in ye-olden days, sometime after Darwin did the damage, it was thought that a dominant gene variant, or allele, would, given enough time, simply wipe out a recessive one.  It seemed reasonable at the time.  If there is a negative selection value for a particular trait, evolution should select it out.  That’s how Natural Selection works, right?

Well, no.  As the above calculations show, life ain’t that simple.  If two members of a population have that AB type, where B is a deleterious (or “bad for you”) allele.  Then, there is a 1/4 probability that they will have a child with the negative BB variation.

In other words, no matter how bad a trait can be, it can never be wiped out unless you killed off all of the AB’s along with the BB’s.   But, since the AB’s don’t exhibit the negative trait, nature won’t “know” to select them out.  They pass the test.

## Back to Wisdom Teeth

Given this new information, we can answer the question posed at the beginning with some confidence.  Even though it is totally true that having wisdom teeth grow in in a prehistoric village can spell death (via infection and impaction), it would not do so until after the person reached breeding age.  In fact, wisdom teeth often don’t show up until the early twenties or later.  By then, the person could have had multiple children.

So, if there was breeding going on between these populations (the one without wisdom teeth, and the one with them) then there would have been a lot of mixing.

Even if the first was all AA’s which coded for NO wisdom teeth, and the other was all BB’s which coded for them.  And we allowed the A to be dominant, then we’d still see a ton of variation in both villages.  AA x BB = AB at least half the time.  And then AB x AB = BB 25% of the time.  In just 2 generations you’d have signs of wisdom teeth.  In a few more it would grow since AB x BB = BB 50% of the time.

## Disclaimer

I want to just make one last point.  When I’ve been using the word gene, I mean that almost metaphorically.  It could be that there’s a single gene for eye color (there isn’t), but that isn’t the point.  The point is that there is a gene, or a collection of genes, that code for a trait.  And that there are variants of that gene, or collection of genes, called alleles.  These different alleles combine to give us an array of possibilities.  Whether the trait we’re looking at is coded for by a single gene or a collection is often glossed over because it doesn’t actually make much of a difference from that standpoint.  But, it can seem confusing sometimes to hear “gene” but really be thinking of a collection of genes that work together.

## Combinatorics: Let My People Count – Math for Primates Episode 011

Tom and I have gone all fancy and started working with a producer for our math/comedy program “Math for Primates”.  We enlisted the help of Keith Schreiner of Auditory Sculpture.

The format is more fun, there are musical interludes etc.  Check out the new episode here.

In This Episode:

• All about Combinatorics – the Mathematicians fancy way of “counting”.
• The magical musical and production styling’s of Keith Schreiner make Tom and Nick seem “almost” respectable.
• Tom invents the natural numbers in under 3 minutes.
• 5 primates are standing in a line … can you come up with a punch line?
• Is the factorial [5! = 5x4x3x2x1] just a way for mathematicians to justify yelling?
• How many ways can you cage 2 primates out of 5?  And is this illegal?
• The importance of watching Stargate SG1.
• If you went far enough on a space ship, would your toothpaste taste like rye bread?
• Apparently Nick ‘Horton’ has an alter-ego, Nick ‘Lion’.  It does sound more manly.

The Stargate SG1 picture above will make more sense once you’ve heard the episode.

## Clash of the Titans: Mature vs New Science

John Hawks takes a paragraph of a new book by William Burroughs, “Climate Change in Prehistory,” and runs with it.   It has to do with the clash between mature sciences and emerging new sciences.  Here’s the paragraph Hawks refers to:

It is often easier to write with confidence on fast-developing and relatively new areas of research, such as climate change and genetic mapping, than to review the implications of such new developments for a mature discipline like archaeology. Because the latter consists of an immensely complicated edifice that has been built up over a long time by the painstaking accumulation of fragmentary evidence from a vast array of sources, it is hard to define those aspects of the subject that are most affected by results obtained in a completely different discipline. Furthermore, when it comes to many aspects of prehistory, the field is full of controversy, into which the new data are not easily introduced. As a consequence, there is an inevitable tendency to gloss over these pitfalls and rely on secondary or even tertiary literature to provide an accessible backdrop against which new developments can be more easily projected (Burroughs 2005:10).

Hawks makes the point that this paragraph’s suggestion that a new science in facing resistance from an entrenched mature science can lead to one of two possible conclusions

1. … and therefore the simple conclusions of the immature sciences may be wrong.

or

2. … and therefore those wishy-washy archaeologists had better get their act together.

He comes to the defense of (what he calls) the mature science of archaeology.  In this defense he points out …

What marks a "mature" discipline is the emergence of informed critiques focused on the limits of methods of analysis. When archaeology was immature, before the 1950s or so, almost all archaeologists were simple (some say "naive") positivists. They excavated and found the traces of ancient people, just as today’s archaeologists do. And what they found was what there must have been. Find a handaxe, you know people made handaxes; find a temple, you know they worshipped gods of some kind. Dig in a mound, find a grave, you know that the people had rituals associated with death that required substantial non-subsistence directed labor.

Notice his definition of “mature”:  An emergence of informed critiques, focused on the limits of methods of analysis.  This isn’t a horrible definition (I’ll argue for a different one below).  He goes on:

Of course, today’s archaeologists tend to be positivists, too. There’s no sense twiddling around with hypotheses that will never be testable. The religion of Neandertals? Well, it’s one thing to speculate about it, but the fact is that it’s devilishly hard to test hypotheses about religion from the material remains of any pre-monumental culture. In the absence of information, we may as well stick to the facts.

But there’s a deeper sense in which archaeologists have a much more complicated view of their evidence. Archaeology has gone through many periods where different researchers developed and applied distinctive analytical techniques. These techniques have often been incommensurable. Sometimes they settle debates. For example, the systematic study of skeletal element representation and cutmark taphonomy has gone far toward testing (and verifying) the occurrence of hunting in some Early Pleistocene contexts. The hunting versus scavenging debate still goes on, with renewed emphasis on active or confrontational scavenging. But knowledge advanced by means of analytical critique.

## What is a “Mature” Science?

Now, I don’t want to sound like a curmudgeon, but I would never call Archeology a mature science.  At least not by my definition (which I’ll outline in a second).  It is an adolescent science, albeit an exciting one on the verge of maturity.

I define a field to be "mature" if and only if it has a reasonably well developed empirical AND theoretical side.  Without both, you are only half a science.

(OK, I used the ambiguous word "reasonably" in my definition.  And this opens the door for questions about what we mean by that.  But, that’s the way laws should be written – with room for interpretation.)

I’ve found most people I talk to about this (in the sciences) to be rather hostile to my definition.  I suspect the reason is that if we take it to be strict, there is only ONE mature science – Physics.  (I include engineering in physics as applied-physics, the way that we include medicine in biology as applied-biology.)  The reason is that it is the only science that has serious mathematics and theoretical work being done “in house”.  They don’t rely on Mathematicians to do the hard labor for them.  There IS great work being done on the theoretical side of a lot of other sciences, but nearly all of it is done by Mathematicians and Physicists.

Let’s go into more detail as to what I mean in my definition:

### Empirical Science

The empirical side of science is what everyone thinks of when they think of science.  That is, when YOU think “science”, I’m guessing that you’re thinking of guys in white lab coats pouring boiling blue liquid into a beaker.  This side of science is well developed in nearly every field save for economics (that’s a whole different discussion – and a strange one at that).

This side of science is all about hypothesis testing, data collection, and statistical and other methods to deal with the vast amount of data that is gathered.  That is, this is the “get your hands dirty” part of science.  It’s why most people who go into science went in to it in the first place.  They loved all that went with it.  Primatologists love to hang out with primates, Chemists love to mix chemicals, Archaeologist love to dig in the dirt.

As I discussed in my article on Karl Popper, a science must have a robust empirical side in order to test hypothesis.  Without it, we have no way to know if we’re just blowing smoke or not.

What most sciences don’t have (and some refuse to take seriously) is a serious theoretical side of their field.

### Theoretical Science

Theoretical Science is all about hypothesis generating.  Darwin’s theory of natural selection is an example of a work of theoretical science.  Einstein was a theoretical physicist, and the theory of relativity is a work of theoretical science also.

As a field gets more developed, theoretical science converges more and more toward mathematical and computational work.  That is, the models become so complicated that only the tools of mathematics and computer science are able to deal with them.

Don’t get this confused with statistics.  We need complex statistical models to deal with the data collected by empirical scientists.  But, theoretical scientists don’t deal with data – at all.  Sure, they may be inspired by data.  But, the point is that they are developing theories about how the world works that are then able to be tested.    They follow lines of implication – if this is true, then this other thing MUST be true.  It is logical philosophy, mathematics, theorem-proof.

No science is totally devoid of theory.  Obviously.  Paleoanthropologist gave us the “out of Africa” theory which has proven to be rather robust.  But, no science other than physics has a dedicated “in house” world of theoreticians who’s ONLY job is to follow lines of implications and thereby generate new and diverse hypothesis.

Theoretical physics predicted Black holes before they were seen on a telescope.  They predict things like an expanding universe.  They predict dark matter, super strings, etc.  All of this is done by physicists who are not passed off by their empirical counterparts as “just” mathematicians, or “arm chair” physicists.

They do their job with very complex mathematics.  Some times the experimental physicists will prove them right … sometimes wrong.   But, the important point is that they are full fledged members of the physics community.

In most other sciences, theoretical (and especially mathematical) work is met with skepticism and sometimes outright disdain.  If you do ONLY theoretical work, then you are not really a member of this science at all … you’re a mathematician.  A real scientist DOES something.  They do field or lab work.  They get their hands dirty.  Blah, blah, blah …

## Why Are Most Sciences So Hostile to Mathematics, and What Can We do About It?

I suspect the reason why most sciences have been traditionally so hostile to treating mathematical modeling as a serious part of their field is simply because most of the members of that science haven’t ever taken any serious math.  Oh, they may have taken a calculus class or two, but let’s get real.  Calculus is a FRESHMAN level class for math, physics, and engineering students.  There is an entire world of mathematics that comes after that that is hard to describe to people who haven’t seen it (imagine explaining what “red” means to a blind man).

Of course, this is changing.  Chemistry has always been in second place to Physics as the most mathematical of sciences.  They had to be.  Now Biology is catching up.  Theoretical Biology is (in my opinion), hands down, the most exciting emerging field (it’s been emerging for about 25 years).  But, still most of the work is done by math people, not biologists.

What’s wrong with that?  Why not just let mathematicians do the work, and leave scientists alone to do the dirty stuff?

There are 2 reasons.

1. Mathematicians have their own work to do.
2. Scientists and Mathematicians can’t communicate properly with one another.

First, contrary to popular belief amongst many scientists, Mathematicians are not here to serve you.  Yes, oftentimes they come up with highly useful tools that scientists find they can’t live without.  But, mathematicians generally get into math for its own sake … not because they care so much about furthering some particular science.

Second, even amongst those mathematicians who DO get in on the action of a particular science, it’s often impossible for them to communicate with the members of said science.  This goes both ways.

Mathematicians are frustrated by the total lack of knowledge of even basic mathematical skill by scientists, and scientists are shocked at how little mathematicians know about the basics of their field.

What physicists have figured out is that if you train your own theoreticians, then you can train them from the get-go to be able to communicate with the experimenters. They’ll know the big problems in the field, they’ll know the history, the language, the nuts and bolts.  Similarly, they train ALL physicists up to a threshold level of mathematical maturity, even the ones who become experimenters.  This way, everyone can talk to everyone else.

So far, no other field has ever gotten this right.  They only train empirical scientists.  The only math required is what any advanced high school kid can do.  And as such, the theoretical side of their field is grossly underdeveloped.

Again, this IS changing.  Most of the hard sciences are making strides fast, but it will take a lot more time. But, because of the reasons outlined above, I can’t call Archaeology a “mature” science.

## Pi Day – What’s the Big Deal?

Tom and I, being math podcasters, were forced by a code of mathematical obedience to do a  podcast in honor of pi day (March 14th). Yes, $\pi$, the Greek letter turned mathematical object, number, and spawner of cults across the world.

Both Tom and I approached this topic with hesitation.  You see, while the most common response that math-folk like us get when we tell someone the we are math people is something akin to fear, distrust, disdain, and outright horror.

But, the second is a kind of strange respect normally reserved for guru’s and shaman.  A common refrain that accompanies this type of reaction is something like this, “Wow, you do math. That’s so … like … cool.  I really like pi myself.  It’s part of everything.  It’s, like, at the center of the universe …”

Frightening, to say the least.

But we are hardly the only people to have experienced such weirdness.  The Greeks suffered all manner of cults that worshiped this little unassuming number/letter.  And I’m sure some of them have survived to this day.

Tom and I aimed to dispel some of the myth of pi.  It’s just a number.  A weird number, yes.  Lot’s of applications to lots of cool stuff, yes.  But, mystical, no.  Our relationship with $\pi$ is purely platonic (get it, Platonic … Greek … OK, that was bad).

Here’s the link to the podcast again.

In this podcast:

• Tom and Nick are not as happy with Pi Day as you’d expect.
• Why do Hippies like Pi so much?
• Where does pi come from, and why do we care?
• Is mounting a Ferris Wheel on a Flat-Bed Truck a good idea?
• How many digits of pi can YOU recite?  I’ll bet not 69,000!
• Nick and Tom give you back 23 hours and 40 minutes of your life … ish.

## Math for Primates episode 009 – Partial Orderings and What’s Wrong with the Olympics

We’ve posted our 9th episode over at Math for Primates on Partial Orderings.

Here’s the breakdown of our discussion

• What’s wrong with the “medal count” ordering in the Olympics?
• Is it really all that surprising that Canada is good at Curling?
• quasi ordering vs partial ordering vs total ordering – who cares?
• Nick, leave the carrots out of this!
• Dealing with our Daddy issues with an ancestor semi-lattice
• Are you your own ancestor?
• For that matter, is a sandwich equal to 5 dollars?
• Discovering the quasi-order of joy-points