SCIENCE MATTERS - Mathematics: Road map of science

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Most scientific endeavors require the application of some kind of mathematics. Foremost, it is important for modeling a wide variety of phenomena. And, where observations are quantifiable (measurable), it is used to characterize data.

Of central importance to science is repeatability of experimental results. However, no matter how hard one tries, repeated measurements rarely come out exactly the same. Statistical methods were developed to clarify the validity of measurements.

Suppose six people each having mechanical stop watches measure the time of a sprinter in a 100 meter dash. You are unlikely to get six identical results.

In addition to variances in reaction time of the timers, stop watches have inherent limits to their accuracy. Suppose three timers measure 11.5 seconds, and other timers measure 11.4, 11.8 and 11.9 seconds. Should the average of the six times (11.6) be recorded? Or, should special consideration be given the three identical readings of 11.5?

If the watches are started with the sound of a starter's pistol, running times will be shorted by the time it takes the sound of the pistol to travel 100 meters from the starter to the timers.

This is an inherent bias. It can be removed by using a pistol that emits a puff of smoke on firing. The speed of light (the visual cue) is nearly instantaneous relative to the speed of sound (auditory cue).

Better yet, electronic timers with stop-action video can reduce biases significantly. But, in a tight photo finish, it may come down to fractions of an inch that cannot be distinguished because of limited resolution of the video monitor.

The point is, all measurements have the potential for bias and are limited in accuracy by the instruments and methods being employed.

Mathematical tools help us recognize those limitations and account for them in drawing conclusions. They make it possible for an experimenter to determine if a single measurement is so out of line with other measurements it is likely to be the result of something completely different from what is being studied.

The presentation of scientific data should always be accompanied by determinations of the uncertainties associated with the values given. Results must be given in a way that the range of results is evident to a reviewer. One wants to know how tightly repeated measurements cluster around a given data point.

A reviewer must be able to tell that results come from a plentiful data base and they are truly representative of the phenomena. In some instances, large uncertainties suggest an experiment was not done with sufficient rigor. It could also mean that inherently the phenomena being studied gives rise to widely scattered results.

All of this reflects on the confidence one can assign to results. Mathematical tools have been developed that quantify the extent to which data is widely or tightly distributed and confidence levels we can assign to results.

Another essential use of mathematics is to model physical phenomena. Symbols are used to represent a wide range of properties. Mathematical methods demonstrate the relationships between those properties.

Properties can be almost anything you imagine. They might include things as simple as locations, times, electrical charges, and masses. Or, they might be slightly more complex things involving ratios such as density (mass per unit volume), speed (distance per unit time) or pressure (force per unit area).

Finally, symbols can represent "macroscopic" properties which characterize the state of a system. Examples of this involve more abstract concepts. They include things we easily relate to, such as temperature, or, something unfamiliar such as entropy or the "wavefunction" of a subatomic, quantum mechanical system.

The give and take between theoretical physicists and experimental physicists is a wonder to behold. The theoretician's wavefunctions are intellectual constructs. They represent what is known about the physical world in abstract, unfamiliar terms. Yet, they must predict results measurable by instruments.

Many of the equations of classical physics and chemistry are not so far removed from everyday experience. An "equation" is intuitive in the sense it suggests a balancing. It expresses a relationship as an equivalence of one set of conditions or circumstances with another.

Relationships imply change. It is often the case that the rate at which one thing changes in relation to another is not simple. Calculus, invented by Isaac Newton (also, independently by Gottfried Leibniz), is a beautiful and powerful mathematical tool for dealing with such relationships.

The invention of calculus was pivotal to advances in all the sciences. It makes it possible to calculate outcomes for processes that involve rates of change. When changes in one property occur in relation to another property, calculus is used to determine a result at any point along the way.

Examples of such dynamic relationships are everywhere. Chemical concentrations may vary with changes in temperature. The strength of a chemical signal can be determined as it diffuses through a barrier. The age of an artifact can be calculated from principles of radioactive decay. The effects on animal populations can be determined in response to changes in their food supply, fertility, and predation.

Science could not function without mathematics. It is how we demonstrate regularities in nature. It is how we come to understand universal truths. It is how we avoid misleading ourselves.

Steve Luckstead is a medical physicist in the radiation oncology department at St. Mary Medical Center. He can be reached at steveluckstead@charter.net.

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