Irrational Pi Irrational Physicists
From my youth up, I have noticed professional physicists adding amateur metaphysical speculations to their theories, where, as far as I could tell, their theories required no such assumptions.
Moreover, they regarded metaphysics as contemptable, as mere word spinning, even while they indulged in schoolboy errors in the field.
Theories about knowledge, called epistemology, or theories about the nature of being, called ontology, are likewise non-empirical.
Frequently I encounter propositions taught by physicists as physics, including in textbooks on the topic, as if the proposition was one that could be proved or disproved by the empirical methods to which physics is confined, or had been proved, but which, upon examination, are non-empirical hence entirely outside the orbit of physics, modern or no.
I cannot emphasize the simplicity of the errors being made: they all had the same peculiar irony as when an idiot gardener is sitting in a tree on the very branch he is sawing off.
None seemed to realize, nor did I have the power to explain to them, that empiricism cannot logically support any conclusion which implies that empiricism itself is baseless.
My sole contention here is hardly controversial. It is that an empirical test cannot disprove a proposition that does not rest on empiricism for its proof.
This is not a theory of physics; it is a theory of epistemology. Physics questions how the world works. Epistemology questions how knowledge works.
I submit that physics is an empirical discipline, and hence no properly formed theory of physics contains matter than an observation could not, in theory, disprove.
Let us take the irrationality of pi as an example.
I submit that the statement “pi is an irrational number” is true. It is true in all times and places and under all conditions, hence it is a universal and necessary statement, not a conditional.
Hence, the ratio between the diameter and the circumference of any ideal circle of any radius cannot be expressed as two whole numbers. This is true for any ideal circle whatsoever.
Yet no measurement of the physical distances of the rims and spokes of circle-shaped objects in the physical world, such as wagon wheels, confirms this statement or can ever.
Contemplate a thought experiment.
Place a loop of beads on a flat surface such that all beads are equidistant from the center. Place a shorter string of beads in a straight line running through the center to bisect the area enclosed by the first.
The string represents the diameter of a circle; the loop represents the circumference.
Let the beads be as small as can be, given the physical limits of our universe. In this universe, matter is not continuous. There will be some point at which beads cannot be made smaller, and still be made of any observable physical substance.
No matter the number of beads, it will be a whole number. No bead is cut in half, for example.
The whole number of the string representing the diameter and the whole number of the loop representing the circumference are both whole numbers. The ratio between them is rational.
But the ratio of the diameter of a circle to its circumference, called pi, is irrational. At least, so Johann Heinrich Lambert tells us.
If Lambert is correct, the knowledge that the ratio of a diameter to a circumference of a circle is irrational cannot be observed from any observation of any representation of any circle whatsoever. No lens, no wheel, no arc of a rainbow, nothing in the physical world has this property. All such representations of circles seen or measured by any means in this universe will be non-continuous.
This means the representation is not the thing being represented. This means the thing being represented is not a physical thing. It cannot possibly exist in any universe where matter is non-continuous.
It means we have never seen a circle with our eye. We have only seen representations and approximations.
But we know what circles are, and we know their properties.
We know pi is an irrational number. It is not an approximation, not an hypothesis, not a theory, not a guess. It is a certainty, or nothing is.
If all knowledge comes through observation, on the other hand, we do not know anything about circles, or about numbers, or about pi.
Hence the knowledge that pi is irrational is not empirical knowledge.
I recently had a famous physicist make the unqualified statement to me that the category of metaphysics has no content. This, apparently without realizing that this statement is, itself, a metaphysical one.
In the same letter, he assured me that there were no valid metaphysical arguments, on any subjects, that are not just arguments from physics. One again, he seemed not to notice that is was itself a metaphysical argument.
Again, he decreed that “all proofs are empirical” — but if all proofs are empirical, and if this statement is not empirical, then it is itself unproven. It cannot pass its own test.
This is a freshman error in philosophy. I have empirical proof of this, because I began to read the letter from the physicist to a freshman, and, even before, I had finished the sentence, the schoolboy had caught the error.
What separates physics and metaphysics is nature of subject hence method of proof. Physics deals with particular and contingent facts; metaphysics deals with universal and necessary categories. We use those categories to categorize those facts, for without this, facts have no meaning and make no sense.
Physics is empirical. Metaphysics is non-empirical.
Physics is observational. Metaphysics is non-observational.
Theories of physics are disprovable. Theoretically, the case could go either way. Theories of metaphysics are non-disprovable. No other case is possible or imaginable, even in theory.
Statements about physical objects are and must be conditional and particular, if-then statements resting on a qualification that all variables are being held as equal. Metaphysical statements are unconditional and universal, concerning categories of being and things that must be true, as it were, by nature, by definition.
Because statements about physical objects are and must be conditional and particular, any general statement or theory of physics can and must be disprovable:
We cannot know, before we observe it, whether two cannonballs of different weight but the same volume dropped at the same time from the top of the Leaning Tower of Pisa will strike the ground at the same time, as Galileo proposes, or if the heavier falls faster and strikes first, as Aristotle proposes.
Hence, the Galilean proposition that his two weights strike the ground at the same time rests for it confirmation on observation. The result could have been different. We could have found ourselves living in a universe governed by Aristotelian principles.
For an accurate test, of course, we must take air resistance into account, and find some way to mitigate or control for what influence, if any, it may have on our experiment. A balloon of helium and a ball of lead will strike at the same time if dropped from an airless tower on the moon through the silent vacuum to the cratered lunar surface, but on Earth and the weight of the air and friction of air resistance might drive a different result.
This is empirical knowledge. It is contingent knowledge, that is, it is true but not necessarily true. It is that way, but it did not have to be that way. No principle of logic would be violated had the result been different.
That the interior angles of any triangle sum to two right angles, on the other hand, is necessary. The results cannot be different, given the axioms, common notions, and definitions of Euclidean geometry.
Note the conclusions of Riemannian or Lobachevskian geometry are different, because the axioms are different, namely, Playfair’s Postulate is not accepted. Nonetheless, no statement of geometry, Euclidean or Noneuclidean, is disprovable by observation, since no statement of geometry rests on observation for confirmation, but they are contingent upon their axioms.
That a statement cannot be true and not true at the same time or in the same sense is necessarily the case in all cases. This is not contingent upon any axiom, nor is it disprovable. It is true by definition. It is true by the nature of the subject matter. It is necessarily true.
Hence, there is no point in hauling two weights to the top of the Leaning Tower of Pisa in order to show that a given cannonball cannot be both a ball and not a ball in the same sense of the word and at the same moment in time, under the same conditions. This is not because of something about the particular cannonball that may or may not be true; this is because of the nature of sameness.
Likewise again, one need not bring anything to the top of the tower to show that two things equal to a third thing are equal to each other. It is a statement that cannot be unambiguous and untrue.
And yet there is many a modern thinker, including at least one teacher of physics, and one highly respected leader in the field, who neither can refute the argument given above, nor answer it, but who nonetheless heap scorn on it.
It frankly baffles me. How can men be so accomplished in their field, without knowing the foundations and assumptions on which those fields rest? How can they not see that natural philosophy is a part of philosophy, not the other way around?
It seems more like a mental illness than a philosophical proposition to make the philosophical argument that philosophy does not exist, or utter the metaphysical statement that metaphysics has no content.
Or, rather, it seems like a heresy, a dogma of the faith called modernism or nihilism, which, unfortunately, even scholars of proper Christian conviction seem to have absorbed at such a fundamental level, that, to them, the matter is not open to question or doubt.
About John C Wright
John C. Wright is a practicing philosopher, a retired attorney, newspaperman, and newspaper editor, and a published author of science fiction. Once a Houyhnhnm, he was expelled from the august ranks of purely rational beings when he fell in love; but retains an honorary title.
