Tabak J. Mathematics and the Laws of Nature: Developing the Language of Science

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180 MATHEMATICS AND THE LAWS OF NATURE

advantage not just to the individual who carries one copy of the

gene but to the offspring of that individual as well. The fact that a

gene is rare means that the individual with just one sickle-cell gene

probably reproduces with an individual with two normal genes. As

a consequence they can expect that on average 50 percent of their

children will have one copy of the sickle-cell gene and one normal

gene; on average 50 percent of their children will be resistant to

the disease malaria. Under the circumstances we should expect

to see the frequency of the gene for sickle-cell anemia increase

in populations exposed to malaria because those with two normal

genes are less likely to reproduce successfully.

The complication arises as the gene becomes more and more

frequent in the population. In this case it becomes increasingly

likely that two individuals, each with one copy of the normal gene

and one copy of the sickle-cell gene, will reproduce. Under these

circumstances, 50 percent of their children are, on average, still

resistant to malaria, but on average 25 percent of their children

have sickle-cell anemia, and sickle-cell anemia, like malaria, is a

significant cause of death.

The gene for sickle cell is especially common in areas of Africa

where malaria rates are high. Many people in North America,

South America, and the Caribbean can trace their ancestry to

these areas of Africa. They still sometimes carry the gene for sickle

cell, but now there is no advantage in carrying the gene; they live

in an environment where malaria never occurs.

If scientists had not uncovered the role of the gene in the envi-

ronment in Africa where it arose, the presence of the gene would

be impossible to explain. In Canada, for example, where there

has never been any malaria, the gene is clearly disadvantageous.

There the gene is present for historical reasons. In recent times

many Africans have immigrated to Canada, and historically there

were many slaves who escaped their mistreatment in the United

States by escaping to Canada, where slavery was illegal. Sickle-cell

anemia is one example of a genetic disease, the existence of which

can be explained only through science and history. There are other

diseases whose presence can be explained similarly. Scientists can-

not, however, account for all genetic diseases by using a sickle-cell

Natural Laws and Randomness 181

model. This may be because there are other factors at work or

because the historical reasons that might account for the presence

of the disease have yet to be uncovered. The Hardy-Weinberg law

does not offer us any guidance about the specifics, but it does offer

us a way of framing our questions about population genetics.

The Limits of Predictability

A great deal of mathematical modeling has been done to try to

quantify how changes in gene frequencies are related to each of

the hypotheses of the Hardy-Weinberg law. Although the Hardy-

Weinberg conditions seem straightforward, they are not. They

can be violated in various ways that are not especially obvious.

For example, scientists have sought to answer the question, How

large a population is so large that the gene frequencies are not

affected by random fluctuations? Although this question seems as

if it should have a single number for an answer, it does not. For

example, 500 individuals may, under some circumstances, be suf-

ficient to even out random fluctuations for five generations, but

that same population under the same circumstances is probably

not large enough to ensure stability over 500 generations. Over

the course of 500 generations there are many more opportunities

for unlikely events to occur.

One more example of a situation in which a hypothesis of the

Hardy-Weinberg law is violated: Mutations are random changes

in genes, and so far we have described them as if they occur at a

uniform rate. Sometimes this is just what happens. (Mutation rates

can be measured by the number of spontaneous changes occur-

ring in the DNA of an organism per unit of time.) Scientists have

discovered, however, that certain species of organisms have genes

that affect the mutation rate itself. This means that individual

organisms can carry a “mutation gene” that significantly accel-

erates the rate at which changes occur in the organism’s DNA.

Predicting the effect or even the existence of such genes can be

very difficult indeed.

For many years population geneticists had to content them-

selves with studying extremely simple model problems. “Real-life”

182 MATHEMATICS AND THE LAWS OF NATURE

problems involving actual populations were far too difficult to

understand. More complicated problems were often mathemati-

cally intractable, but even when this was not the case, the measure-

ments on which the solutions depended were too difficult to make

accurately. As a consequence, the connection between theory and

practice was weak. This is changing.

One factor that made progress possible was better information

about genetics and the mechanisms involved in heredity. As scien-

tists acquired more information, they were better able to predict

mutation rates and the way that individual genes function as part

of the complement of genes that make up the individual and the

species. The second factor that affected progress was computer

technology. Computers enabled scientists and mathematicians

to solve mathematical problems that had previously proved too

difficult. Scientists also developed technology that enables them

to directly sample the DNA of the species of interest. This gives

them better information about current gene frequencies and it

also makes it possible to measure changes in gene frequency more

accurately. All of this is important, but it is a measure of the dif-

ficulty of the problems involved that in many ways progress has

remained incremental. Today scientists are sometimes able to

make reasonably accurate short-term predictions about changes

in the genetic variability of a population of organisms. However,

because of the enormous uncertainties involved, the accuracy of

these predictions quickly decays as the period under consideration

increases.

With all of these qualifications about the difficulty of applying

the laws of population genetics, it may seem that the physical

sciences have made far more progress in their ability to predict

nature than have population geneticists. The situation is, however,

somewhat more complicated. To be sure, there are many situa-

tions in which physical scientists have now become accustomed to

making many highly accurate predictions about the state of some

physical system far in the future. No one, for example, is especially

surprised when a space vehicle is launched toward some distant

planet and arrives just as it was predicted to arrive months or even

years earlier. In fact, we tend to be surprised when this does not

Natural Laws and Randomness 183

happen. There is, as we have already pointed out, no theory or

technique in the field of population genetics that enables scientists

to make the same sorts of long-range predictions.

A major difference between population genetics and classical

physics is related to the role of chance. Newton, Lavoisier, and

Clausius were able to develop very satisfying theories—theo-

ries that can be used to make accurate predictions about future

events—without any appeal to ideas of chance and randomness.

Mendel, by contrast, was unable to form even an elementary

theory of heredity without incorporating ideas of randomness in

his theory. It would be incorrect to assume, however, that the laws

of probability have not also found a home in the physical sciences.

There seem to be inherent limits on predictability even in the

physical sciences. Today there are aspects of the physical sciences

that cannot be expressed without concepts taken from the field of

probability. Sometimes probability is required because the phe-

nomenon to be studied seems to be fundamentally random. This is

the case with turbulent fluids. Fluids that are turbulent—rushing

water and rushing air are two examples—currently defy precise

prediction, and some scientists believe that they always will. It

is certainly true that the best that can now be done is to predict

“most probable” values. One can, for example, predict the most

probable range of velocities that would appear at some point at

some time in a turbulent fluid moving under a given set of condi-

tions. Accuracy in this sense means predicting a narrow range of

velocities with a high probability. Sometimes scientists have been

successful in this regard; often they have not.

The study of turbulence is only one of a number of disciplines in

which physical scientists use probability theory in an essential way.

Another situation arises when scientists cannot accurately measure

the state of the system in which they have an interest. They may

feel confident that they have the right equations—equations that

allow them to predict the future state of the system—but the equa-

tions themselves are never enough. To solve a differential equation

fully, they must also know the initial conditions. (See the section

Models and Reality in chapter 7.) Initial conditions involve speci-

fying the state of the system at some instant of time.

184 MATHEMATICS AND THE LAWS OF NATURE

But it is sometimes the case that scientists are unsure about the

precise state of the system at any time and that they have no meth-

od for eliminating this uncertainty. Scientists have, for example,

developed a wide variety of predictions about motions occurring

deep within the Earth. (These predictions are generally made with

differential equations and random processes

As a matter of definition, it is not possible to accurately predict individual

random events. (If they could be predicted, they would not be random.)

Instead of individual outcomes, researchers predict the frequencies with

which various outcomes will occur. Experience has shown that frequen-

cies can be accurately predicted.

Central to the idea of prediction in random systems is the concept

of the ensemble. Here is the basic idea: Suppose we are interested in

describing an experiment that has a random component to the outcome.

(Think, for example, of Mendel and his pea plants.) We begin by imagin-

ing that we have performed the experiment many times. The result is a

very large set of experimental runs, each of which is called a “realiza-

tion” of the experiment. The ensemble is the set of all such realizations.

(From a practical point of view, it is often more convenient to imagine the

ensemble as an infinite set of realizations, although, of course, there is

never enough time or money to run an experiment infinitely often. In that

sense the ensemble is a fiction, but it is a useful fiction.)

The next step is to actually perform the experiment several times and

record the results. If this is done correctly, the researcher obtains a rep-

resentative sample of the ensemble. (Sometimes the word ensemble is

used to refer to the sample as well as the parent population from which it

was drawn.) The sample is used to draw conclusions about the statisti-

cal properties of the ensemble. These conclusions are used to predict

the statistics of future experimental runs.

The idea of using a sample to better understand a large parent

population is not, of course, a new one. Advertising firms, for example,

will assemble “focus groups,” representative samples of a much larger

target market, and test the effectiveness of different advertising cam-

paigns on the focus groups. The campaigns that they choose and the

campaigns that they discard will be determined, in part, by the focus

groups’ reactions, which are supposed to accurately reflect those of the

target market.

Natural Laws and Randomness 185

the help of the law of conservation of mass, the law of conservation

of momentum, and the law of conservation of energy.) The prob-

lem is that their predictions depend on certain assumptions about

what is actually occurring deep inside Earth right now; no one

has access to this information. Sometimes the devices necessary to

The same sampling techniques are often employed by those who

develop mathematical models of random phenomena. They will often

repeatedly solve the same set of differential equations. Each solution

corresponds to a slightly different set of initial conditions. The result

is a sample of the ensemble, the much larger population of solutions

from which the sample is supposedly drawn. If the sample is correctly

constructed, its statistical properties will accurately reflect the statisti-

cal properties of the ensemble, and researchers will be able to make a

number of statistical predictions with reasonable confidence.

Sampling the mathematical ensemble involves art as well as science.

Much of the problem lies with the choice of the perturbations, the small

changes in the initial conditions that determine the elements in the

sample. It is the choice of perturbations that determines whether the

sample is an accurate representation of the ensemble.

Once they have a set of “model runs,” which is another name for the

sample, researchers can ask and answer questions about the ensemble

in which they are interested. They might, for example, ask “Within the

ensemble, how frequently is condition A exceeded?” In the simplest

cases, the answer to such a question is obtained by counting the number

of realizations in the sample for which condition A is exceeded and then

dividing that number by the total number of realizations in the sample. If

all has gone according to plan, this number will be a reliable estimate of

the frequency with which the condition is exceeded within the ensemble,

and the researchers will have learned something new about the ensemble.

Sometimes deterministic and nondeterministic models are perceived

as mutually exclusive. But often deterministic and nondeterministic mod-

els are used in a complementary fashion. Each provides information not

available from the other. A deterministic model produces a single pre-

diction about a process, but there is usually some uncertainty about the

reliability of the prediction. Analysis of the corresponding ensemble may

reveal how much confidence should be placed in the results obtained

from the deterministic model. Finding the best way to model random

phenomena remains an active area of research.

186 MATHEMATICS AND THE LAWS OF NATURE

make the measurements have yet to be constructed. Sometimes it

is not clear even in theory how one might construct such a device.

These uncertainties are reflected in the existence of several com-

peting theories about the dynamics of the processes that occur

deep within our planet.

Uncertainty in any science has never proved a barrier to the

development of new theories or the making of new predictions. It

is reflected, instead, in a general lack of confidence in the validity

of apparently correct predictions. If a prediction turns out to be

reasonably accurate—and no theory in population genetics or in

the theory of turbulent fluids, for example, has proved to be more

than “reasonably accurate”—there is still the question of whether

the agreement is the result of a coincidence or whether the predic-

tion displays real insight. The answer to this type of question is

usually not clear. When scientists are interested in studying “ran-

dom” phenomena, either they are forced to discuss the most likely

“effect” of a likely “cause,” or they abandon the notion of cause

and effect entirely and instead search for correlations between

one data set and another. Rather than being mutually exclusive,

the philosophy of cause and effect that is so characteristic of clas-

sical physics and the probabilistic analysis that is characteristic of

population genetics are now just two alternative, complementary

ways of perceiving the same phenomenon.

187

conclusion

Nature is to some extent arbitrary. Certain facts are important and

yet inexplicable. This is not to say that an explanation has not yet

been found but to suggest that in some important cases no expla-

nation exists. The Sun, for example, has a particular location with-

in our spiral galaxy, the Milky Way. We could imagine that it is

located somewhere else—perhaps on a different arm of our galaxy

or perhaps in some other galaxy. But the location of our Sun (and

by extension our own location as well) is not amenable to expla-

nation. It simply is. The same can be said for laws of nature. We

can imagine different and contradictory laws—universes where

energy is not conserved, for example—and novelists can explore in

their fiction what might happen in such situations, but engineers

and scientists must use laws that coincide with the world as it is

revealed through observation and experiment. And from the most

fundamental observations and experiments researchers construct

the laws of nature. These laws of nature form the conceptual loca-

tion from which researchers begin to reason.

Reasoning works because in addition to its arbitrary aspects,

there are many ways in which nature is also a highly ordered sys-

tem. Many natural processes evolve in ways that are both compre-

hensible and predictable. For whatever reason—and philosophers

have written extensively and inconclusively about why this is

so—the rules of logic apply to much of nature. Nature is math-

ematical. Using the methods described in the preceding chapters,

researchers have learned to deduce the future from knowledge of

the present. These methods do not seem to apply to every field of

human endeavor—at least not to the same degree—but in those

fields where the methods have been found to be the most appli-

cable, progress has been most evident.

But if those branches of knowledge that depend upon these

methods have evolved rapidly, the same cannot be said for the

methods themselves. It has taken the entire recorded history of

188 MATHEMATICS AND THE LAWS OF NATURE

the human race to develop the small number of concepts and

methodologies that underpin research in physics, chemistry, some

branches of the life sciences, and engineering. For the interested

layperson this is fortunate, because it means that the level of

knowledge required to evaluate advanced research is less than the

level of knowledge required to do the research.

In a democracy, we are often asked to pass judgment on the mer-

its of various scientific and technical policies. Voting provides an

opportunity to express ourselves, of course, but discussions with

friends or family are another important venue. Activist groups pro-

vide other opportunities. Despite the difficulties often associated

with scientific and technical discussions, such discussions need not

be left to the experts or to the loudly ill-informed. Familiarizing

oneself with the ideas described in this book is an important part

of what it means to be well-informed about technical and scien-

tific issues. It is hoped that this book has made a contribution in

spreading awareness of these important ideas.

189

afterword

an interview with

dr. renate hagedorn

on meteorology,

modeling, and the

quantification of

uncertainty

Renate Hagedorn is the education officer for the European Centre for

Medium-Range Weather Forecasts (ECMWF) research department.

She is also a scientist in the probabilistic forecasting division at the

Centre. She earned her Ph.D. at the Institute for Marine Sciences in

Kiel, Germany, where she developed a model of the Baltic Sea that

revealed new information about atmosphere-ocean interactions. At

ECMWF, she helped to develop

a seasonal forecasting system as

part of the DEMETER team

(DEMETER is an acronym

for Development of a European

Multimodel Ensemble system for

seasonal to inTERannual pre-

diction.) This project received

widespread recognition. More

recently, she has worked on

improving the ECMWF ensem-

ble prediction systems with the

goal of producing the most useful

Dr. Renate Hagedorn (Dr. Renate

Hagedorn)