SCIENCE MATTERS - Math holds keys to greater understanding


A school science project designed to illustrate the solar system might include an assemblage of Styrofoam balls held together by wires. This is, quite literally, a model of the actual solar system in an easily visualized representation.

Scientists develop models for similar purposes. They serve to explain phenomena that are unfamiliar to everyday experience. They provide a shortcut means for discussing abstract ideas.

A model strips away nonessential aspects of a larger process. It focuses on the features one wants to illustrate. It ignores things that are irrelevant but obscure the main process.

The intention of a Styrofoam solar system model may only be to illustrate the relational positions and sizes of the planets. More representative models, with proportionate sizes and distances, would be prohibitively large and awkward to build and display.

Scientific models come in a variety of forms, but only occasionally do they involve handicrafts with material objects.

Many models are not visual at all. Science is often a very mathematical enterprise. Equations used to characterize the motions, states and compositions of natural processes are actually abstract models.

Often more than one model may be used to represent something in different settings. Take for instance the atomic model of matter. The iconic representation of an atom is a grouping of little ball-like objects in a nucleus surrounded by smaller balls orbiting in well defined paths.

This gives a very crude visual picture of an atom that serves for an introductory discussion. Atomic scientists have better models than those familiar to the general public. Some deal primarily with the structure of atomic electron shells. Others deal explicitly with nuclear interactions in the nuclear core.

The most useful atomic models are mathematical. They account, in a precise way, for natural laws that give atoms their characteristic properties. Those natural laws are generally referred to as quantum mechanics.

Subatomic objects don't behave as we expect from experience with large objects. There are natural limitations to measurements in the atomic realm that we don't have with objects such as baseballs and planets.

Though the quantum world is not intuitive, it is well understood from mathematical models. From a practical point of view, the math makes possible the bulk of modern electronic devices. Additionally, it provides investigational tools across a wide range of scientific fields.

In a stricter representation of an atom, electrons do not occupy discrete orbital paths in the way our iconic model depicts. Measurement of the exact position of an electron is inherently confounded by quantum mechanics.

Repeated measurements of the positions of subatomic objects that are seemingly identical yield a range of results. Some results cluster around an average or yield different values at predictable frequencies. However, the exact outcome of any single measurement is unpredictable.

The simple model of distinct electron orbits in atoms can be thought of as an approximation. Here, an orbit represents the most likely path of an electron.

More accurate models depict electrons smeared out in cloud-like configurations resembling shells. The higher densities in the cloud represent the more likely location of the electron were it to be localized in some way.

Describing an orbital electron as being spread out in a region of space around the nucleus of an atom has real advantages. The properties of the entire atom, of which an electron is but one part, are dictated by the distribution of electrons in the atom's outer shells.

This physical model is useful. It gives chemists ways to think about how assemblages of atoms in molecules take the forms that they do. It gives physicists a shortcut way to talk about light emission from atoms that gives rise to characteristic spectral lines on passing through a prism.

I say shortcut because not every discussion needs to encompass all the nuances imbedded in more accurate depictions. In this case the model glosses over conceptual details of the nucleus since they are incidental to electron behavior in shells.

Ultimately, to a purist, a mathematical characterization is the most accurate. Equations are sought to account for every known, relevant characteristic. These models are least likely to lead us astray and most likely to lead to new insights, and so, are the most satisfying.

What a given model glosses over depends on the application it is used for. While using a simpler model, one is more likely to forget its limitations. With any model, one must be careful not to exceed the limitations of its explanatory scope.

Mathematical models come in many forms. At their foundation some symbols represent a physical thing. The rules governing the particular math being applied in a given situation must mimic the physical process the math is meant to represent. The fact that mathematical formalisms can be found or invented to represent such a vast array of natural phenomena is truly profound.

Saying "found or invented" reveals a fundamental conundrum of the scientific endeavor. Is the universe inherently mathematical? Are we just inventing math to do bookkeeping for outcomes from our observations and experiments? Or, is there a deeper significance to the math?

These are not easy questions. The foundations of some of math are entirely abstract. Group theory, at some level, simply deals with symmetries. The rules are entirely derived from logic without reference to anything physical.

Yet, when that formalism is applied to operations relating to physical space, a whole host of consequences becomes evident. As more knowledge is accumulated, especially about the subatomic realm, those consequences are seen to be manifested throughout the physical world.

Group theory can be used to formalize what is implicit about translating or rotating coordinates in space. It's not limited to the ordinary space we live in, but can be applied to the abstract spaces conceived of by physicists and mathematicians.

It would all seem like an empty exercise if these ideas where not subject to testing. The theorists tell the experimentalists what is suggested by their mathematical models. Theorists tell what new forms of matter exist, under what conditions and what properties they possess.

At giant laboratories like the Large Hadron Collider elaborate experiments are designed to create the conditions the theorist say are necessary. Theories survive or die when the data comes in.

When a mathematical model is confirmed we have learned about more than the existence of a previously unknown elementary particle. Factoids outside of their context have limited value. We have learned fundamental truths about the fabric of the universe. We have learned that fundamental properties of the universe can be characterized by mathematics derived wholly from abstract logic.

The distinction between model and reality becomes blurred in such heady endeavors. I never cease to marvel that humans have developed such profound insights from tools they have invented. Or, is it discovered?

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


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