This post is a reply to Alex and Harv, but filed as a new topic to better facilitate easy access to the numerous responses I’m hoping to receive. It goes without saying (well, it doesn’t now) that everyone is encouraged to respond.
The assertion has been made that numbers are an inherent part of reality. That is to say, numbers are prescriptive of reality. I disagree. It is my position that mathematics is merely a descriptive device.
In order for a representative device to be prescriptive, everything it identifies must be unwaveringly consistent with everything else it likewise identifies. Indeed, an effective discipline must have its own, self-contained prescriptions. Linguists throw prescriptive rules around to better facilitate proper use of a language (e.g., “any noun beginning with a vowel sound must be preceded by ‘an’ instead of ‘a’ ”). Attorneys define legal maneuvers based within a prescriptive set of legal standards. And yes, scientists use prescriptive mathematics to better predict and understand data.
But such prescriptions are only valid within their respective disciplines. Axioms, functions, and the like are only prescriptive *within* the boundaries of mathematics; to say that mathematics is wholly prescriptive beyond itself is a fallacy I have labeled “hyperpolation.”
Hyperpolation: From forensics to astrophysics, we use various methods of extrapolation and interpolation to discern what cannot be directly observed. Additionally, extra- and interpolation are often used in debate. But it is a mistake to apply one set of variables to an entirely different type of variable set.
In hyperpolation, variables from two very different domains are incorrectly juxtaposed. When Nicomachus asserted that the universe was a huge musical “scale,” he was hyperpolating. When someone opposes evolution with the “if I threw computer pieces into a washing machine, ran the machine for 15 billion years, and hoped the pieces would somehow evolve into a computer” argument, he or she is hyperpolating.
Similarly, to mistake the success of a discipline’s internal prescriptions as an indication that the discipline itself is externally prescriptive is hyperpolation.
Viewing mathematics at its most fundamental as a numbering system, you might be inclined to assert that, since we can draw hash marks to represent just about anything, and since these hash marks would be understandable to anyone, numbers are something inherent to reality.
As an illustration, I could tally up the number of planets in our solar system thus: IIIIIIIII. I could remind everyone there is no greater common sense than that which accompanies ‘self-evident’ truths like ‘nine planets are nine planets are nine planets.’ I could further stress that there were this many planets before we humans discovered there are this many planets, and even before we humans came to be!
Common sense, sure, but is my ‘self-evident’ truth an indication that our conceptual & descriptive hash marks are prescriptive as well? Remember: a prescriptive “1” would require anything it identifies to be *precisely* the same as all other phenomena identified by “1.” Do commonsensical illustrations like mine indicate that numbers are more than just a language we invented, but rather a reality we’ve discovered?
No. Mercury does not *explicitly* equal Jupiter (1 ≠ 1). Simply using these numbers to establish quantity falls short of the requirements a prescriptive device necessitates. Any admission that these numbers are merely expressing a singular quantitative value among a sea of deeper qualifications (atmosphere, mass, distance from the sun, etc.) is an admission that the "nine planets" illustration is only descriptive.
Fine, but aren’t I oversimplifying the issue by using the “nine planets” illustration? Am I not guilty of begging the question in order to effect my conclusion via *reductio ad absurdum*?
No, because I can reduce the entire premise down to its most absurd conclusion without evoking specific examples: no particle is absolutely equivalent to any other particle. Every entity has a different history and location. If you wish to identify two entities as having an equivalent numerical value, you’d have to remove *all* non-similar quantities (i.e., qualifications) before ascribing to your numbering system the power of absolute inherence.
Indeed, if we wish to attribute numbers with the responsibility of prescription, we’d be forced to ultimately conclude that “1” is meaningless, and there is no such thing as “=”.
Mathematics is a beautiful language, but our best language is no more inherent to reality than our worst. We no more ‘discovered’ numbers than we did music, colors, or cartography.
Thanks for a great topic, Alex.