EvoMath 0: An Introduction

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Occasionally a creationist or an aideeist will make the wild assertion that biologists do not understand math/statistics and that math/statistics actually disproves evolution. This is followed by some random math argument based on ignorance of biology. The irony is that biologists probably understand math better than mathematicians understand biology, for the simple fact that biologists use math in their work more than mathematicians use biology in their work.

Between undergraduate work and graduate work, biologists are usually required to take multiple mathematics and statistics courses, whereas mathematicians will probably only be require to take freshman biology and maybe one higher level course. This is not to say that mathematicians are inherently ignorant of biology, just to point out the irony of someone accusing biologists of being inherently ignorant of biology, while thinking that mathematicians are fully knowledgeable in biology.

Of all of biology, evolution is probably the best characterized mathematically. There is so much theoretical work, that the math often precedes the empirical work. Instead of encountering a phenomenon that cannot be explained, evolutionary biology often has competing theoretical hypotheses just waiting for the experiments and data to sort them out.

Evolutionary biology has a mathematical legacy dating back over a century, the most prominent of which is the work of R.A. Fisher, J.B.S. Haldane, and Sewall Wright. In fact, R.A. Fisher is not only one of the fathers of modern evolutionary biology but is also considered one of the fathers of modern statistics. He invented some of the most powerful tools in statistics for the simple fact that he needed them to study evolution.

With this in mind, I plan on covering much of the classical evolutionary theory for Panda's Thumb in a series of posts.

Index

  1. The Hardy-Weinberg Principle
  2. Testing for Hardy-Weinberg Equilibrium

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The Hardy-Weinberg Principle states that a population satisfying certain primary conditions will not evolve. This result is very important because any departure from these conditions will result in an evolving population. Three scientists in the early ... Read More

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Like biology but unlike creation science, statistics is a real subject that grows and changes. Indeed, while it isn’t the kind of topic much covered by the evening news, we’re in the advanced stages of a revolution in statistical thinking, the Bayesian revolution. The relevance of this development for evolution is that the old, frequentist versions of stat tended to grossly understate the evidence for natural selection because they did not take into account our prior knowledge. Creationists picked up on the frequentist methodology, much as they’ve fastened on to the ideas of Karl Popper, not because they know or care much about either, but because of their ad hoc polemical utility.

I’m hardly the guy to explain statistical ideas. I hope Mr. Carwright deals with the Bayesian angle, however.

Bayesian statistics is not without controversy, however. First, an intro. Bayes’s theorem states:

P(H if D) = P(D if H)*P(H)/P(D)

where P means a probability, H is a hypothesis, D is some data, and P(D) can be calculated by summing P(D if H)*P(H) over all possible H in the problem.

This can be used to calculate the probability of various hypotheses H given some observed data D if one can calculate P(D if H) – which is often much easier than calculating P(H if D) directly.

However, using this theorem requires having values of the prior probabilities P(H) – and the meaning of those probabilities is where the controversy comes in. Are they simply an extra hypothesis or are they something else? I’m not very acquainted with this issue outside of noting that there has been some controversy over this point.

The Bayesian approach to statistics goes far beyond Bayes’ Theorem. It is a significantly different way of practicing as well as thinking about statistics; and it has begun to change the way that research is done, especially in drug trials.

I’ve been struggling through Probability Theory: the Logic of Science, a posthumously published work by E.T. Jaynes that has already had a huge influence in manuscript. As I previously admitted, however, I’m not the right guy to introduce mathematical ideas to this or any other forum. My comments were intended to lure somebody more knowledgeable into explaining the issues involved.

“The irony is that biologists probably understand math better than mathematicians understand biology, for the simple fact that biologists use math in their work more than mathematicians use biology in their work.”

I’m not sure how one would measure understanding, but here’s one possible metric: how many bad books on biology have been written by mathematicians, compared to the number of bad books on mathematics written by biologists? I haven’t done an exhaustive search, but I suspect that biologists come out looking somewhat better.

“The first post in this “EvoMath” series will be on Hardy-Weinberg equilibrium and should be available soon.”

Good, I’m delighted about this. This site/blog’s been publicised by many sites, but I’ve not enjoyed it much. I thought (hoped) it would ‘teach’, and link/explain new discoveries. Instead it seems to be a lot of bickering and name-calling.

I’ll remained subscribed in the hopes of seeing more substance, like the EvoMath series.

With the first post now available, I’ve replaced the sentance “The first post in this ‘EvoMath’ series will be on Hardy-Weinberg equilibrium and should be available soon” with an index.

someone Wrote:

Between undergraduate work and graduate work, biologists are usually required to take multiple mathematics and statistics courses, whereas mathematicians will probably only be require to take freshman biology and maybe one higher level course. This is not to say that mathematicians are inherently ignorant of biology, just to point out the irony of someone accusing biologists of being inherently ignorant of biology, while thinking that mathematicians are fully knowledgeable in biology.

I am a probability/statistics grad student, so I know that more often than not, biologists have an inadequate knowledge of mathematics and statistics, and even though some mathematics and statistics courses are usually required, they are usually the most elementary or special, watered-down courses. But more fundamentally than that, biology is not even remotely on the same epistemological or intellectual footing as mathematics. Indeed, biologists are, if anything, are poor cousins, and only behavioral “scientists” are lower on the totem pole.

Robert O'Brien Wrote:

But more fundamentally than that, biology is not even remotely on the same epistemological or intellectual footing as mathematics.

You are comparing apples and oranges. Mathematics is inherently abstract, it is impossible to compare it to science which is required to live in the real world. Yes science isn’t math, but math isn’t science. I could very well declare math the red-headed stepchild of biology because math doesn’t rest on a foundation of biological data. But that would be pointless cockfighting.

Now if you really want to compare mathematics to sciences, then you need to look at the mathematical work in those sciences. But then you will find that they are no different.

I’m not sure how one would measure understanding, but here’s one possible metric: how many bad books on biology have been written by mathematicians, compared to the number of bad books on mathematics written by biologists? I haven’t done an exhaustive search, but I suspect that biologists come out looking somewhat better.

Good point.

Reed Cartwright Wrote:

You are comparing apples and oranges. Mathematics is inherently abstract, it is impossible to compare it to science which is required to live in the real world. Yes science isn’t math, but math isn’t science.

Mathematics is a science.

I could very well declare math the red-headed stepchild of biology because math doesn’t rest on a foundation of biological data. But that would be pointless cockfighting.

Nope. Biology needs mathematics, but mathematics does not need biology. Without mathematics biology would be little more than haruspicy. As Roger Bacon said:

Mathematics is the door and key to the sciences For the things of this world cannot be made known without a knowledge of mathematics. Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world. There are four great sciences … Of these sciences the gate and key is mathematics, which the saints discovered at the beginning of the world. … mathematics is absolutely necessary and useful to the other sciences.

By the way, Dr. Wolpert, whom you and your compatriots of like-mind are wont to quote, asserts (as I do) that not all scientific disciplines are on the same epistemological footing and scientists need to own up to that fact.

Leighton Cowart Wrote:

I’m not sure how one would measure understanding, but here’s one possible metric: how many bad books on biology have been written by mathematicians, compared to the number of bad books on mathematics written by biologists? I haven’t done an exhaustive search, but I suspect that biologists come out looking somewhat better.

That would not be an accurate measure because the average biologist who attempted to write a book on mathematics would not get past the prologue. :)

And as we all know, mathematicians - with their extensive Platonic knowledge of Everything - can write books on biology without actually seeing an organism.

Henry Gee in Nature summed this attitude up nicely:

I believe that unless biologists have dissected real animals or experienced natural diversity for themselves, they are not worthy of the name. It was this same exposure that sowed the seeds of evolution in the mind of the young Darwin, turning him away from the theoretical, typological views of German Naturphilosophie that resonate still in those who argue for the presence of a designing hand. The artificial environment of the lab rat is as rarefied as the thoughts of Plato and Aristotle, the philosophy from which this idea derives. It no coincidence that it is in these very environments that Intelligent Design finds its most willing converts.

Mathematics is - whether O’Brien admits it or not - theoretical and typological, a rarified view of nature.

And lawyers for some reason. Many of the big creationists are lawyers. Perplexing.

O’ Brien quotes Bacon:

Mathematics is the door and key to the sciences

For the things of this world cannot be made known without a knowledge of mathematics.

Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of the world.

There are four great sciences … Of these sciences the gate and key is mathematics, which the saints discovered at the beginning of the world.

… mathematics is absolutely necessary and useful to the other sciences.

Two things of note - (i) outdated reliance on Bacon as the “last word” in philosophy of science (the Baconian model died in the late 1800’s), and (ii) ignorance of the concept that mathematics is a language which allows us to describe nature in succinct ways. It need not - in any way - correspond to “reality”.

If O’Brien actually believes “the things of this world cannot be made known without a knowledge of mathematics,” then, I afraid, there is no way that any dialog can occur here.

O'Brien Wrote:

Mathematics is a science.

Nope. Mathematics is mathematics. It is its own special field of human endeavor. If you feel that mathematics is a science, please demonstrate to this mathematical biologist where mathematicians in general use scientific methods.

Nope. Biology needs mathematics, but mathematics does not need biology. Without mathematics biology would be little more than haruspicy.

As a mathematical biologist, I agree that biology needs mathematics. It is a wonderful tool to simplify the complexities of nature. However, biology needs biology more than it needs mathematics. That is because mathematics is worthless to biology without being anchored to reality. Biologists doing field and lab research provide such an anchor; mathematics does not.

O’Brien wrote

Nope. Biology needs mathematics, but mathematics does not need biology. Without mathematics biology would be little more than haruspicy.

And without biology and the other sciences, mathematics would be nothing more than empty syntax devoid of meaning or utility. Reading the entrails of animals, after all, at least can lead to knowledge of anatomy; but no matter how strenuously they’re manipulated, purely syntactic transformations of semantically empty symbols can lead only to more semantically empty symbols, which is an intrinsically sterile exercise.

RBH

*cough* umm blow your own…eerrr… horn and move on Mr O’brien with your “I’m better then all of you” attitude.

About this Entry

This page contains a single entry by Reed A. Cartwright published on April 2, 2004 4:00 PM.

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