Assumptions in Science

by Jeffrey Sax
November 3, 2003

“Concepts which have been proved to be useful in ordering things easily acquire such an authority over us that we forget their human origin and accept them as invariable.”
– Albert Einstein

Assumptions form an integral part of the scientific method. Every scientific theory is based on a collection of assumptions. These assumptions demarcate the domain where the theory is valid. The theory can’t be applied reliably to a situation if one or more of its assumptions are violated.

The fact that a theory 'works' says nothing about the universal validity of its assumptions. It says nothing about what will happen if one of the assumptions does not apply. All too often, however, as the theories that rely on them become more universally established, the assumptions themselves become self-evident truths that are validated by the success of those theories.

Every assumption sets a boundary on the kind of pursuits that are considered valuable within a field. When an assumption becomes a truth, this boundary turns into a solid wall. This has the beneficial effect that the implications of the theory and its assumptions are very thoroughly investigated. Even within the walls, there are vast regions of unexplored terrain. Limiting our explorations to the region inside the wall allows us to give these unexplored regions the attention they deserve.

The downside is that highly original ideas aren’t really given a chance. Anything that lies beyond the wall of truth is considered a waste of resources. Ideas are dismissed before their implications can be studied in any detail. Progress is inherently limited to what lies inside the wall. For all intents and purposes, there is nothing outside the walls.

When the subject of the theory is the nature of reality, the implication is more far-reaching. In this case, the reality we live in is bounded by those same walls. Nothing can exist beyond them. Any evidence that something beyond those walls might exist is discarded. Anyone who believes there is something beyond those walls is delusional and is unworthy of attention.

Yet history has shown time and again that the most significant breakthroughs come when someone dares to look over the established walls. These scientists have had the courage and insight to investigate how different or less stringent assumptions may cover a wider area and still match the terrain. Eventually the impenetrable walls are torn down and replaced with fences that simply mark the area where the assumption is valid.

The scientific method

According to Karl Popper’s concept of falsifiability, a theory can only be called scientific if there is a possibility to prove it is false. This means the theory must be based on assumptions that can be verified. If a theory cannot be falsified, then it is of little value to science.

We will now look at two examples of assumptions made in the physical sciences that proved to be false. Each of these examples will teach us an important lesson about the role of assumptions in the development of science.

Euclidian geometry

In the 4^th century B.C., the Greek mathematician Euclid of Alexandria wrote one of the pivotal works in the history of science. His Elements, which spanned 13 volumes, not only set out the foundations of geometry for the next two millennia. It also presented very clearly the axiomatic and deductive structure that to this day characterizes mathematics and the exact sciences in general. Euclid started out with a set of postulates or axioms. The axioms contained some common notions, such as a point and a straight line, which were considered self-evident. The axioms themselves were also believed to be obvious, and were given without proof. From these, he deduced dozens of theorems and relationships between geometrical objects.

The most famous of these postulates is the fifth postulate, also known as the parallel postulate:

“That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less than two right angles.”[1]

In its more familiar form, the postulate states that, given a straight line and any point not on it, there exists one and only one straight line that passes through this point and never intersects the first line, no matter how far they are extended.

Many mathematicians felt uneasy about this postulate. It felt less intuitive and obvious than the first four of Euclid’s postulates. However, it was necessary to prove important theorems such as Pythagoras’ theorem. For centuries, many mathematicians believed this postulate could be derived from the first four postulates. Countless attempts were made, and many proofs were published, but all of them were flawed. Nevertheless, the truth of the postulate was never questioned.

Until the year 1823. In that year, Janos Bolyai and Nicolai Lobachevsky both independently realized that, if they relinquished the parallel postulate and replaced it with a different postulate, they could create a consistent and entirely new kind of geometry. Where the fifth postulate produced what would later be called Euclidian geometry, the alternative postulates produced non-Euclidian geometries.

In non-Euclidian geometry, there are either zero or infinitely many straight lines through a point that don’t intersect another line. The sum of the angles of a triangle can be larger or smaller than 180 degrees. Pythagoras’ theorem is no longer valid, which means that distances between points are measured differently.

Less than a century after Bolyai and Lobachevsky’s discovery, non-Euclidian geometry turned out to be one of the mathematical corner stones of one of the major breakthroughs in 20^th century physics: Einstein’s special theory of relativity. Einstein proposed that space-time is non-Euclidian. As a result, if it was forced to fit into a Euclidian world, it would appear to be curved.

Euclid’s parallel postulate was unchallenged for over twenty centuries. To question its validity was considered a worthless pursuit. Many disciplines developed and flourished based on Euclidian geometry: mechanics, electro-magnetism, much of chemistry. In the end, however, the postulate turned out to be only an assumption.

However, this fact does not invalidate Euclidian geometry itself or the sciences that have it as one of their corner stones. Euclidian geometry was and still is an enormously successful tool for describing every-day physical phenomena. To a very high degree of accuracy, our world is Euclidian. And so to a very high degree of accuracy, the assumption of the parallel postulate holds true.

No one in their right mind would consider throwing away the achievements of Newton’s mechanics or Maxwell’s electro-magnetic theory because Euclid’s parallel postulate is not true in our physical universe.

The theory of heat

Another example of an assumption that eventually proved false concerns the nature of heat. According to this theory, all mechanical interactions involve some conversion of mechanical energy into heat. Therefore, any self-contained mechanical system must eventually come to rest, as eventually all the available mechanical energy has been converted into heat. The assumption under scrutiny here was worded by James Clerk Maxwell as follows:

“Admitting heat to be a form of energy, the second law asserts that it is impossible, by the unaided action of natural processes, to transform any part of the heat of a body into mechanical work, except by allowing heat to pass from that body into another at a lower temperature.”[2]

The assumption is that heat is a form of energy that is essentially different from mechanical energy.

This classical theory of heat was immensely successful. It has been used to accurately describe heat exchange, efficiency of engines. It is, quite literally, the foundation of rocket science!

Later in the 19^th century, Maxwell, Boltzmann and others developed the modern kinetic theory of heat. According to this new theory, heat is nothing more than kinetic mechanical energy of the molecules that make up the solid objects, liquids, and gases of the classical theory. The molecules move, too. They bounce around and tumble chaotically, colliding with each other like little billiard balls. The temperature of a body is a measure for how much its molecules are moving.

The conversion from mechanical energy to heat is then nothing more than the redistribution of that mechanical energy from the body as a whole to its constituent atoms and molecules.

For example, when a ball is rolling on the floor, its molecules are all moving in an orderly, coherent fashion. During this process, the molecules of the ball collide with molecules in the air and on the floor. Mechanical energy is exchanged between the molecules in the ball and the molecules in its surroundings. This results in a gradual loss of coherence, until finally all the coherent mechanical energy is converted into chaotic movement of molecules of the ball as well as the floor and the air. When the ball has come to a halt, the temperature of the ball, the floor, and the air will have risen slightly.

Won’t the molecules themselves come to rest after a while, just like the balls on a pool table? The answer is no. No energy is lost in the collisions between molecules. The energy has nowhere to go! The molecules will just keep bouncing around happily ever after. This constant motion of molecules is actually visible in a phenomenon called Brownian motion. Named after Robert Brown, a botanist, who first observed the constant motion of particles suspended in water or air. The molecules themselves are too small to see, but they do carry enough energy to show the effect of their collisions with particles that are visible under a microscope.

The kinetic theory of heat went on to become even more successful than its predecessor. It was able to explain everything the previous theory could explain. It was able to explain Brownian motion. It was even able to quantify the heat capacity of substances, which is a measure for the amount of energy needed to raise the temperature by a certain amount.

However, for most applications, including sending astronauts to the Moon, the original theory, based on the assumption that heat was an essentially different form of energy, was good enough. The reason it was good enough is because this assumption has no or very little bearing on the further development of the theory of heat. The kinetic theory of heat provided an alternative foundation for thermodynamics, but the same quantities used in the classical theory can be easily derived in the kinetic theory. The structure of the theory on top of the new foundation remained the same.

Today, very few people still hold on to the original concept of heat as an essentially different form of energy. One of the reasons it was given up so easily is that it was clearly shown that it was not necessary. It was demonstrated that the new kinetic theory of heat led to the same results. More importantly, the new theory stayed well within the bounds of the established scientific framework.

Conclusion

The two examples in this section illustrate two important points:

1. Genuine progress is only possible by thinking outside the walls of established assumptions. Unfortunately, this activity is discouraged for the most well-established and basic assumptions of science.
2. Even if an assumption lies at the base of a theory, it may be possible to replace this assumption while keeping the theory built on top of it intact.