Summary:
The development of 20th Century physics was intimately interconnected with the ramifications of what constitutes a physical constant or fundamental parameter. Which was first immortalized in Einstein's special relativity as an invariant constant c = the speed of light. As velocity measures length L divided by time T, c represents a metric or 'dimensional constant,' as is mass M or any combination of these quantities. Yet when the SI metric system was formally adopted (firmly ...

The development of 20th Century physics was intimately interconnected with the ramifications of what constitutes a physical constant or fundamental parameter. Which was first immortalized in Einstein's special relativity as an invariant constant c = the speed of light. As velocity measures length L divided by time T, c represents a metric or 'dimensional constant,' as is mass M or any combination of these quantities. Yet when the SI metric system was formally adopted (firmly replacing the CGS system), c had acquired new meaning as a defined constant equal to a precise integer 299,792,458 represented in meters per second. However, as explained by Wikipedia (browse "physical constant"), the distinction between a 'pure' or 'dimensionless number' in mathematics and physics is that the speed of light had to be measured since no theory can predict it; for even then the value would have to be empirically confirmed. .
And though hardly an integer nor yet a 'defined parameter' by NIST, a much 'purer' number as a dimensionless fundamental physical constant is distinctly embodied by the inverse of the electromagnetic coupling constant 'alpha' or "fine-structure (fs-) number" a ~ 137.036. Which essentially equals Planck's constant h multiplied by c (times 4p times the vacuum permittivity) divided by the square of the electric charge e, though all metric dimensional terms are canceled in this ratio. While even constants such as a fundamental particle's mass can be expressed as a dimensionless relationship, say, to the 'Planck scale' or other mass, the fine structure is uniquely a pure number. Which led to some deep pondering from its inception, culminating in what was chosen as the number one quandary facing physics at the 'Strings 2000' conference. As articulated by David Gross this "Millennium Question" is:
"Are all the (measurable) dimensionless parameters that characterize the physical universe calculable in principle, or are some merely determined by historical or quantum mechanical accident and incalculable."
My first impression upon reading this in a morning paper was utter amazement a numerological issue of invariance merited such distinction by eminent modern authorities. For I'd been obsessed with the fine-structure number in the context of my colleague A. J. Meyer's model for a number of years but had come to accept it's experimental determination in practice, pondering the dimensionless issue periodically to no avail. Gross's question then was the catalyst from my complacency; recognizing a unique position as the only one who could provide an answer in the context of Meyer's main fundamental parameter. But even then my pretentious instincts led to two months of inane intellectual posturing until one day sanely repeating a simple procedure explored a few years earlier. I merely looked, and the solution and more struck with full heuristic force.
For the fine-structure ratio effectively quantizes (by h-bar) electric charge (of e2) as a coupled to light, in the same sense as an integer like 241 is discretely quantized compared to the 'fractional continuum' between it and 240 or 242. As we aren't talking directly about the integer 137 at all, the result definitively answers Gross's question. For it follows that the fs-number exactly equals 137.0359996502301