What’s wrong with the modern engineering view of dimensional homogeneity?

Cause-and-effect processes

Engineering phenomena are cause-and-effect processes. For example,

· Electromotive force causes electric current.

· Temperature difference causes heat flux.

· Stress causes strain.

Cause and effect parameters

Cause and the effect parameters generally have different dimensions. For example,

· Electromotive force and electric current have different dimensions.

· Temperature difference and heat flux have different dimensions.

· Stress and strain have different dimensions.

Engineering phenomena are not dimensionally homogeneous

Because engineering phenomena are generally cause-and-effect processes, and because the cause and the effect generally have different dimensions, engineering phenomena are not dimensionally homogeneous. They are dimensionally heterogeneous.

Why the modern engineering view of dimensional homogeneity is irrational

The modern engineering view of dimensional homogeneity is irrational because it requires that heterogeneous phenomena be described by homogeneous equations.

Historical background

The modern engineering view of dimensional homogeneity is the brainchild of Fourier. He described his revolutionary view in the following words from page 128 of The Analytical Theory of Heat (1822):

. . . every undetermined magnitude or constant has one dimension proper to itself, and the terms of one and the same equation could not be compared if they had not the same exponent of dimensions.

Fourier’s view of homogeneity is now considered “almost self-evident”. But it was revolutionary in 1822 because it required the multiplication and division of dimensions—mathematical operations that had been deemed irrational since the time of Aristotle.

Note that Newton and his colleagues generally described Natural phenomena using proportional expressions that were heterogeneous. For example, Hooke’s law (1676) and Newton’s second law (1686) are heterogeneous.

Stress is proportional to strain.

A change in motion is proportional to the motive force impressed . . .

Fourier did not prove that his view of homogeneity is scientifically rigorous. The only rationale he offered is the following from page 128 of The Analytical Theory of Heat:

(This view of homogeneity) is derived from primary notions on quantities; for which reason, in geometry and mechanics, it is the equivalent of the fundamental (axioms) which the Greeks have left us without proof.

Fourier’s contemporaries accepted his revolutionary view of homogeneity solely because he solved many practical and theoretical problems that had never been solved. He attributed his success to the homogeneity in his equations, and he attributed the failure of his contemporaries to the lack of homogeneity in their equations, as in the following from page 59 of The Analytical Theory of Heat:

If we did not make a complete analysis of the elements of the problem, we should obtain an equation not homogeneous, and, a fortiori, we should not be able to form the equations which express the movement of heat in more complex cases.