The Suicide of Thought (Part One)
Part One: the Murder of Euclid
Imagine you are a detective called to a crime scene. You unlock the room where the corpse, dead under suspicious circumstances, hanged, but also stabbed and shot. Outside the window, at the same time, you see skyscrapers dark, crumbling, cracked and empty, the streetlamps unlit, and corpses heaped in the silent streets. Churches are burned. All monuments and courthouses are coated with graffiti. The public parks are filled with smoldering rubbish, the rivers with floundering wrecks.
As a detective, your mission is not only to solve the murder mystery of the locked room, but discover what unknown force has destroyed the whole city.
A detective works partly by deduction but mostly by induction. You look for clues that form patterns, and seek additional clues as confirmation when they seem to fit the pattern. While it is not impossible for a man to stab, shoot and hang himself, it is rare. When seeking the cause of the downfall of the city, seeing bloated corpses no dogs approach suggests a different conclusion than seeing shattered windows and smoking craters.
I am in something of the same position as that detective, attempting to puzzle out for myself a smaller mystery in the middle of a larger.
Like out hypothetical detective, the task here is one of induction rather than deduction, seeking patterns rather than finding a proof beyond all doubt. The effort is an empirical one, not a deductive certainty as we find in proofs of mathematics and geometry.
The smaller mystery touches on a few well educated men of my casual acquaintance with whom I have had the opportunity to discuss one certain philosophical topic in depth: call this the mystery of science-worship.
The larger question concerns the whole matter of Western thought for over a century: call this the mystery of the suicide of thought.
Let us look at the smaller first.
The smaller mystery is based on this statement: geometry is empirical.
That is the statement, but not the mystery. This statement came up, in one form or another, with well-educated men with whom I had been discussing philosophy. Geometry was not the main topic of conversation, merely an example of something we all agreed was real knowledge, fact and not opinion. Each said that geometry was an empirical science.
The mystery was why they said it.
To say geometry is empirical is an absurd claim, one that can easily and unambiguously be shown false. It is not one of those questions about which there is a sincere argument on both sides well worth pondering. The only way to maintain the claim is by never defining one’s terms, that is, by never discussing it clearly, or perhaps never discussing it at all.
Here is the smaller mystery: Not one or two, but all the educated determinists with whom I spoke, also happen to be convinced of the doctrine that geometry is empirical and, as such, is a branch of natural science like physics or chemistry.
These were men who clearly knew something about science and math, and the should have known the difference between the two: and yet they all affirmed something men in their fields should have been trained to know and show to be false. So why say something so clearly contrary to their own expertise?
We have three clues.
In three of the cases (I cannot speak to the others) the men said that they were determinists because they were radical materialists. That is the first clue.
Some of these men had degrees and professions requiring rigorous and logical thinking. None were dull-witted. One was (or, at least, claimed to be) a nuclear physicist, and yet he had no idea of what the empirical method was, how it worked, nor its limitations. I add this caveat of doubt because this was like meeting an accredited expert ornithologist who cannot tell whether birds are fish.
Questions of scientific methodology, the basis on which scientific models are accepted or rejected, questions of what science could and could not do, or even the question of what a scientific question was, all left them unable to answer.
The second clue is that they are educated, some highly, but none knew the basics of their field.
As near as I can tell from their words, none of these educated gentlemen were playing devil’s advocate nor making a sly joke. None, when questioned, were able to state clearly their opinion, to give any argument in its favor, or to give any reason why to believe it. All seemed taken by complete surprise. None, indeed, seemed to entertain the least suspicion that any opinion to the contrary even existed, or ever had.
Ponder this, for it is rather remarkable.
Imagine being taught astronomy, but with no mention ever made of anyone earlier than Copernicus, so that you are under the impression that all men through all history thought the sun the center of the solar system. You have never heard of Ptolemy.
This is a poor example, since heliocentrism is actually sound. A better example would be to imagine being taught Marxism, but with no mention of any economist before him, or any economic system he criticized or opposed. You have never heard of Adam Smith. You have never heard an argument favoring capitalism because you have never heard of capitalism.
The third clue is that they apparently never had heard any serious contrary opinion.
The smaller mystery, hence, is this:
How is how is it possible for educated men (1) to believe something which is effortlessly easy to prove untrue (2) and which is openly contradicts the methodology that is absolutely foundational to their chosen (3) and yet to be utterly unaware of the contradiction?
The larger mystery is what turn of philosophy made modern man constitutionally unable to link even the simplest of logical arguments together in his mind? What is the philosophy that killed philosophy? What is the thought that causes the suicide of thought?
The larger mystery much wait for the answer to the smaller, for the two are linked. Let us start small:
None of these educated men actually, that I recall, made the claim that all knowledge is empirical, but, if he had, it would at least made a kind of thematic sense that a radical materialist would also be a radical empiricist.
Radical materialism, also called panphysicalism, holds that no substance aside from material substance does or can exist; everything that seems to be non-physical can be reduced to a physical phenomenon. Radical empiricism holds that no knowledge aside from empirical knowledge can or does exist; everything that seems to be non-empirical knowledge is either induction from empirical examples, or mere arbitrary opinion.
Had any of them been able to put together an argument for their position (none did) he would have argued something like this: 1. All things are matter. 2. Empirical observation is the sole means to gather knowledge about matter. 3. Therefore empirical observation is the sole means to gather knowledge about any thing; therefore all knowledge is empirical. 4. Geometry is a branch of knowledge. 5. Like all knowledge, geometry is gathered solely by empirical observation: therefore geometry is empirical.
That argument is valid, that is to say, the logical form is followed. This means that if the premises and definitions are true, then the conclusion must be true.
But the hackles of our candid suspicions should be raised by two points:
First, the syllogism contradicts itself by holding all things known to be true to be known by empirical means, that is, based on an observation.
For the statement 3 ” all knowledge is empirical” cannot be based on any observation unless an omniscience observer has observed all things and gathered all knowledge about them, and seen, not deduced but seen, that all knowledge is empirical.
But empiricism, by definition, is limited to what we can observe.
What of thing is observed? There could be one object on Mars, smaller than a mouse, or object larger than the moon orbiting a dead star in some dwarf galaxy Horologium supercluster, which is both true and, for the space of four minutes and a half, was entirely non-empirical in how it was known. Until and unless the observer goes to Mars, and to Horologium, and to all other planets and stars, and every square inch of space in the entire macrocosmic universe, and every four minute interval of time, then the statement “All” cannot honestly be uttered.
One must say “all things known heretofore to humans I happen to have met are empirical” or say some other limiting qualification. In which case the syllogism fails, because if only some knowledge is empirical, from the statement the geometry is knowledge the conclusion that it is empirical does not follow.
In sum, the syllogism given above has both universal and unconditional axioms and thus has universal and unconditional conclusions, which empirical observation, by definition, cannot confirm.
If the statement “All knowledge is empirical” is true, then the statement “no universals are known” must also be true, since no empirical knowledge confirms a universal statement.
But if “no universals are known” is true, and if I know it to be true, then I know a universal, in which case the statement is false. Which is a contradiction in terms.
Second, it is evident that geometry, which consists of planes with zero volume, lines of zero thickness, and points of zero size, consists of nothing but objects that could never be seen by the eye. Photons cannot bounce off am imaginary point with no extension, no location, and no duration.
Nor could things of two, one, or no dimensions exist in the material universe, nor things occupying no extension and having no location, nor things that are timeless, infinite, eternal, and incapable of change or decay. But points, lines, figures and solids are precisely such things.
The empirical method is based on observation. Observation cannot possibly study something which presents nothing, directly or indirectly, to the senses.
Perhaps, in counterargument, one might say that geometers like Euclid and Pythagoras first suspected the properties of these imaginary and mathematical objects by the inductive reasoning from examples of physical objects. The problem with this counterargument is that such things are hunches, daydreams, suspicions, or idle thoughts concerning the subject matter of geometry, but themselves are not part of the science and nothing the student of geometry need learn. Asking where scientists or mathematicians get their inspirations is like asking poets where they get theirs: it is a non-rigorous mental process with no power to convince a skeptic, and no ability to prove a conclusion is true. The dreams of geometers are not geometry, and not part of the method of geometry: only their definitions, common notions, postulates, proofs and conclusions.
Or, again, perhaps, in counterargument, one might say the geometrical concepts can be perceived indirectly, in the same way, from the motions of objects in the air, the changes of chemical weights, or the actions of friction and erosion, we deduce atoms exist, albeit we cannot see them.
One might say that things like the dot above the letter iota, or grains of sand; or walking sticks or the horizon seen at sea; or the shapes of tripods, frame tents, and the pillars of public buildings; and all fashion of other visible things from boxes to billiards, suggest points and lines and figures and solids to the imagination, and act as visible agents of an invisible reality, much as an emissary represents a king without himself being the distinct hence unseen king. Deductions from the audible words of the visible emissary is a sound basis to deduce the intentions and state of mind of the unseen king.
The problem with the counterargument is that no one who has read Euclid could possibly be deceived or confused.
There is no way to explain, see, or observe this link between the visible examples of things that are approximately straight or approximately triangular and the real, perfect, timeless and infinite mathematically straight line or mathematically perfect triangle.
Without this link of representation being visible to observation, it cannot be confirmed that is it trustworthy, or even that it exists at all.
Since by definition we cannot see the infinitesimal and indivisible mathematical point, the mathematically straight and infinite line, the mathematically perfect triangle, then how do we know if any alleged representation of them are accurate? If we do know they are accurate, is this knowledge based on observation? If so, observation of what?
If the emissary is an ambassador to only an imaginary king, he is not an ambassador at all.
The theorem of Pythagoras, Euclid 1.47, does not consist of a request to build a model of a right triangle out of three walking sticks sawed to differing lengths, and then count the square inches of the square erected on the hypotenuse using a convenient postage stamp, then to do the same for squares built on the remaining two sides, and to compare the numbers thereof: and to do that for a sufficient number of models, each one with legs and hypotenuse of different size, perhaps made out of matchsticks or soda straws in one case, or, later, with lines ploughed very straightly in a small field or a large one, or, as mathematicians grow more ambitious, with highways paved in marble running perfectly straight commanded to be built by a mad king exactly for that purpose; and then inductively to conclude whether there is a repeating pattern in all the ratios of all the examples, and, if so, what it is, and to what cases it applies.
This is not the method Euclid uses to convince the student of geometry that the theorem of Pythagoras is true in all cases for all right triangles.
Indeed, had a mad king build such an extensive road system and sent his slaves for hundreds of years placing postage stamps over each square inch of terrain, presumably levels of trees, houses, and populations, the answer received would invalidate the theorem, for the insensible curvature of the earth would make the marble-paved highways representing the right triangle a false representation. A three sided figure inscribed on the surface of a sphere does not have the properties of a triangle in a flat plane, for the sum of the three angles in a spherical figure do not equal two right angles in a plane.
Euclid, in his proof, makes no mention of the distance of his line segments, nor the size of the angles opposite the right angle. No matter what these values are, the argument is the same and the conclusion is the same. Hence, his proof is necessarily true universally, absolutely, for all time, and everywhere.
Had the proof been gathered by frantic students duct tapping walking sticks together, or frightened slaves counting the square inches of provinces enclosed by the mad king’s triangular highway system, it would not and could not be necessarily true universally, absolutely, for all time, and everywhere.
An empirical proof of the theorem of Pythagoras would be true only under the conditions that all trials of observation happened to hold in common: true for us, here on earth, during the span covered by historical records, but with no particular assurances that it will be true tomorrow, or on Mars, or for triangles too small to see or too large to measure. We would no more know what a Martian triangle was like than we know the scent of the thin and freezing Martian wind at dawn. As of the time of this writing, no man as smelled it.
Likewise, no real circular object you see with your eye will ever actually have an irrational ratio called pi between its radius and its circumference, simple because all objects seen by your eye will be composed of some sort of particle, atomic or photonic or what-have-you, and particles alter their mass or location when placed under observation and one value or the other is determined, and this indeterminacy is quantized, that is, the particles exist in a non-continuous state. They are either here or there is no halfway point in between. So likewise the ratio of the number of all particles occupying on strand of the radius compared to the number of all particles occupying the circumference will be a rational number.
If you do not believe me, try it. Take a number of pingpong balls or round beads, as many as you like, and place as many as fit on the radius and the circumference. There will be no half, quarter, eighth, or other fractional pingpong balls. The larger you make the circle or the small you make the beads, the close your derived ratio will approach pi, but in the empirical world, you can never reach this value.
Pi does not exist in the empirical world. It is a geometrical entity only.
Empirical knowledge is always provisional, never universal, and never absolute. Empirical conclusions are accepted as given only until an observation contradicts one of them. When, for example, Newtonian mechanics inaccurately predicts the precession of Mercury, or predicts wrongly the behavior of light moving through the luminiferous aether, or fails to predict the bending of light seen around Sol during an eclipse, the whole of Newtonian mechanics is revised and dethroned, and held to apply only to limited cases, as approximates.
Newtonian mechanics certainly still holds true, but only for macroscopic cases, over non-astronomical distances, near the Earth’s surface, where the increments of time are rounded to the nearest second, and all this for objects not moving near lightspeed.
In the same way the Ptolemaic model is useful, indeed, necessary for stellar navigation from the surface of the sea at night, so, too, Newtonian mechanics is useful for ballistics and billiards and all other non-extreme cases involving non-relativistic motion. But Newton cannot explain the ignition of an atomic bomb.
This is because empirical axioms are not accepted intuitively or logically or metaphysically, but provisionally. They are if-then statements. If you accept Kepler’s three laws of motion or Maxwell’s four laws of electromagnetics, then certain motions in planets or magnets are predicted: when and where and if real observation does not contradict the predictions, the assumptions are assumed, until further notice, to be laws of nature. When observation does contradict, the laws is taken to be limited to certain cases but not to apply to other cases, for which a more general law is then sought.
The scientific method, which even at least one highly educated and accredited scientists with whom I had the misfortune to argue did not seem to know, plainly stated, is this: an hypothesis intended to explain a wide number of cases of matter in motion all belonging to some one category is imagined. Motion here means any fashion of physical change. The explanation eschews any mention of final cause: only efficient causes are regarded. The forms of the motion, where they can be, are reduced to a formula, equation, ratio, or simple set of rules.
These rules are called a model. If the model is inelegant, that is, it multiplies entities unnecessarily, it is ruled out in favor of a less inelegant model. If the model is not robust, that is, it only covers certain cases and not others, it is ruled out in favor of a more robust model.
An example of an inelegant model is Ptolemy or Copernicus, whose circular orbits requires scores of epicycles to save the appearances. It was rejected in favor of Kepler, who postulated elliptical orbits that followed three simple rules of celestial motion.
An example of a non-robust model is Kepler, whose three laws explained celestial motions only, and took no account of inertia, mass, or gravity, and could not be used to explain any terrestrial motions, such as the collisions of billiard balls or the parabolic flightpath of cannon balls. Newton, with his three simple laws, explained both celestial and terrestrial motions.
In neither case was the model of Ptolemy yielded to Kepler or Kepler to Newton due to inaccuracies of prediction. It is not as if Ptolemy predicted a solar eclipse would happen on a Tuesday, Kepler predicted it for a day earlier, and when the eclipse happened on Monday, Kepler was vindicated. Nothing like that happened.
The methodology is humble, therefore invulnerable. Any skeptic unconvinced need only examine the evidence himself, take his own observations, make his own measurements, and draw his own conclusions. Nature volunteers no information, but she tells no falsehoods.
So empirical science is limited to cases precisely like these: theoretical models making falsifiable predictions which have not yet been proved wrong, and such models are not ad hoc, but make minimal assumptions (elegant), and apply to maximum possible cases (robust).
A falsifiable prediction is one that some possible, imaginable sense impression can prove wrong. If I predict a solar eclipse on Tuesday, and it happens on Monday, my prediction is wrong. That is an empirical prediction based on a scientific model. If I predict the sun will always and forever rise in the East, but I do not take into account that tidal friction slows the rotation of the Earth one day to a standstill, my prediction is false, but my model is a scientific one. I am making a statement that empirical science can disprove.
But if I predict that a square built on the hypotenuse of a right triangle will embrace a surface area equal to the sum of the areas of the squares built on the two remaining sides, that is not a prediction at all, since it is not an even happening any moment in time nor anywhere in space. There is nothing to look at. There is no prediction because I am making a statement about an entity that exist in a timeless and conceptual aspect of reality the senses cannot reach.
Likewise, if I predict that any statement which both affirms and denies its predicate at the same time and in the same way is not true, this again is not a prediction, but is a universal law about purely formal relations between sign and signified that exist in a timeless and conceptual aspect of reality the senses cannot reach.
Knowledge which is self-evident (that is, merely knowing the meaning of the terms is sufficient to affirm certainty) is called intuitive. It is unfortunate that the same word is used to refer to hunches and strong feelings, but that is the correct term. For example, when Euclid offers the common notion that two things equal to a third thing are equal to each other, this is intuitive. No additional proof is needed as no other possibility is logically coherent. Likewise, when he claims all right angles are equal, this is intuitive.
Interestingly enough, Euclid’s Fifth Postulate, also called Playfair’s axiom, is not intuitive, albeit for centuries men held it to be so. The investigation of the logical deductions of systems where Playfair’s axiom does not hold, by Riemann or Lobachevski, explicated Non-Euclidean geometry, of which Euclidean geometry is a limiting case.
Intuitive knowledge is not empirical, as it concerns matters that cannot be any other way. Deductions from intuitive axioms likewise cannot be empirical, since they necessarily must be necessarily the case, that is, universe truths unlimited to any particulars.
Those things that are true and known to be true because they must be true under all circumstances, that is, universally, are called rational. Again, it is regrettable that this word has other meanings, but that is the technical term in epistemology. Rationalism only concern non-empirical universals.
Those things that are known to be true because they fit into an elegant yet robust model of matter in motion, and the predictions of this model both are falsifiable and have not yet been falsified, are called empirical. Empirical models only concern non-rational particulars.
It is no use using the word empirical for anything else: that is not what it means.
Geometry is the study of abstract mathematical objects, such as points, lines and planes, which do not and cannot exist in the world apprehended by observation, nor can they have an observable, that is, a materially measurable relationship with the approximate objects, such as the dot or penstroke on a page, yardstick or discus or billiard ball, that remind us of points, lines, circles, spheres. There is no way to see of measure this thing called “representation” or “reminder.”
Nor can we be reminded of something we have never seen, nor can we know how well, or even if, a material thing represents or resembles another thing when this second thing is nonmaterial, timeless, infinite, eternal, and disconnected utterly from the material universe.
One way to escape the paradox is to claim that geometry is not knowledge, but merely a word game of logic. Its conclusions hold true only in the imagination of man, not in the real world, and hence any resemblance or representation of real figures and real conclusion is merely a coincidence. This is a very difficult, if not impossible, argument to maintain, since it undermines the use of mathematics in all sciences whatsoever, not just astronomy and ballistics. It becomes merely a matter of happy coincidence that planets move in elliptical orbits seemingly obedient to the laws of Newton, or that a gunner can use simple conic sections to deduce the proper elevation of his gun. All science becomes, not knowledge, provisional or otherwise, but an uninterrupted series of happy coincidences.
But the argument, as I said, is so plain that it cannot be denied, nor is there any wiggle room. Geometry is a rationalistic science in the sense of being deductions from intuited non-observable universals axioms; physics is an empirical science in the sense of being falsifiable elegant and robust models of observable matter in motion.
What is rationalistic is non-empirical; what is empirical is non- rationalistic. What is observational is not non-observational. What is non-observational is not observational. What is falsifiable is not non-falsifiable. What is non-falsifiable is not falsifiable.
Geometry makes no predictions and is subject to no observations. Geometry is not and cannot be called empirical by the very definitions of the terms.
Now then, this is obvious, and, at one time, was known to all educated men. How did these educated men of my acquaintance not know it? One and only one answered when I asked. The others grew offended that I dared to question them, and fell silent. That one said he had been so taught by his teachers.
Similar arguments can be made to show why mathematics or formal logic cannot be reduced to empiricism, to say nothing of metaphysics, ontology, epistemology, ethics, law, aesthetics, or, if I may be bold enough to name the forbidden and forgotten name, theology.
Why would the teachers of science, persons allegedly skilled and knowledgeable in the subject matter, made such a tyro mistake about the matter and methods of science, or conflate empirical with non-empirical hence rationalistic types of knowledge? It is not a mistake anyone conversant in the field would make, any more than an ornithologist would call a penguin a fish. It is certainly not a mistake an expert would make.
We can dismiss the idea that this is an innocent mistake, due merely to ignorance, because persons conversant with the field would know this. Scientists know the scientific method.
Is there a possibility that these teachers, whoever they are, happened to be sincerely convinced for whatever reason that the traditional and logical distinctions between empirical and non-empirical do not apply. Perhaps there is an honest philosophical argument, based on Hume or the writings of logical positivists, which gives a proof these teachers found convincing to support radical empiricism.
But if that were so, a competent teacher, in such a case, would teach both sides of the argument, so that a student would at least be aware that there was a second opinion in the matter, or previous models since overthrown.
This rather strongly suggests a dire conclusion to which all three clues mentioned above point.
When a teacher hides even the existence of contrary models, arguments and opinions, he is in no way attempting to teach his students to understand knowledge, but only to recite, without understanding, points of a dogma held to be beyond discussion.
This does not prove the case, but it does fit the pattern. If the teachers were attempting to indoctrinate rather than educate, this explains our third clue, why these gentlemen were not even aware of any contrary opinion. It explains why none could offer any explanation, much less a logical argument, to support their ideas. It explains our second clue, why their education hindered, rather than aided their ability to think clearly about the topic: they have been trained not to think about it.
This seems to suggest something deliberate, and even conspiratorial: a desire not to educate, but indoctrinate.
Again, no case can be proved, but the remaining clue fits a pattern. Radical materialism is not a sincere philosophical posture, but a cult dogma, something believed by an effort of will, not because there is or can be any proof or argument to show is must be or could be true. It is something that passes by indoctrination, from one incurious, unquestioning mind to another.
One of the many flaws in radical materialism is this: if radical materialism were true, radical empiricism must also be true, on the grounds that if nothing is real but matter, no knowledge is real except for knowledge about matter, and facts about matter can only be known by empiricism. But radical materialism is a universal metaphysical theory, and therefore cannot be known empirically, which means it cannot be known at all. Hence, if radical materialism were true, it is false.
It is a doctrine that refutes itself, something which the mere unambiguous statement of the terms proves false. No further argument is need, no other witnesses need be called.
Hence the final clue also fits the pattern, but also leads to a bigger mystery: the reasons why the teachers do not teach and the students do not learn about the basics of science is because of dogmatic yet illogical beliefs that cannot withstand such scrutiny that swirl about science.
These beliefs and beliefs like them are beliefs that make outrageous claims about the prestige of science, which is inflated to serve as a substitute religion. Such beliefs are called science worship. These beliefs flourish only in a dark age, when the lamp of reason is guttering or extinguished.
Science worshippers are not necessarily partisans of radical materialism and radical empiricism, but these and beliefs like them are friendly to science worship. Such beliefs dull the curiosity, encourage dismissive arrogance, or inspire bellicose narrowmindedness, which, in turn, forms a favorable environment to allow science worship to grow like mold.
Science worshippers do not do science, do not understand science, and are easily duped by junk science.
So, the lesser mystery of who killed Euclid now has have a reasonable theory that fits the facts. Since teaching the truth about geometry and science would necessarily cast doubt on a cult belief about science worship that is prevalent in society, it can only be passed along from one uncurious mind to the next by indoctrination.
Raising the eyes of a detective from the corpse to the window, and seeing a dark and dead city in a landscape of destruction, the greater mystery now demands attention. Science worship is a symptom, but only one, of a deeper sickness that afflicts more than just one field of study, more than just one school of thought or more than just one topic.
But that deeper question must wait for another day.
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.
