Philosophers and Mathematicians

(So, “What is a Philosopher, Anyhow?” The second post in that series. Scroll back a few to find post one. Thanks!)

A famous contemporary mathematician

It may well be true that Philosophy started with Mathematics; well, at least here in ‘the western world.’ Nobody knows for sure. The ancient Greeks had their Drama, their Mythology, their Religious Rituals, their Political Oratory, and each of these contributed to the rise of Abstract Thinking; but they also had their math. Even today, the most abstract math, “pure math,” exists and makes contributions to our life and our thought in ways hard to precisely capture.

(The Chorus of a Greek Drama [left], often commented and emphasized as if an all-knowing observer. [Top Right] The Pythia is the Oracle of Delphi, some believe her ‘powers’ were enhanced by the inhalation of hydrocarbon gasses; illustration’s author unknown. [Bottom] Depiction of Pericles’ Funeral Oration in the agora of Athens—Franz Foltz, 1853—honoring those lost in the Peloponnesian war versus Sparta, circa 431 BCE. The speech is relayed to us by, probably actually the work of, Thucydides in his History of that war. This book is one of the earliest “histories” of all time and the speech is considered by some to be the greatest ever given!)

Mathematician Gregory Chaitin says he has a very practical side too, “but there’s also, in me, a side that likes beautiful mathematical arguments and these fantasy worlds of pure mathematics…We do need some people who do pure mathematics with no applications, at least maybe not for a hundred years. People who think philosophically...”

Chaitin is one of today’s greatest mathematicians. Born of Argentinian parents, he is the discoverer (or inventor: “I invented it,” he says) of The Omega Number (also called Chaitin’s number). “Omega is sort of a unicorn or a flying horse, — a mathematical fantasy in the Platonic world of ideas…which has this strange number glowing there,” he says (my emphases). (See the following Link.) Adding to the amazement, he did the initial work on Omega while a senior in High School! (Hey, as a senior, I was too busy chasing after a girl named Beverly Bushel to mess with any mathematical stuff!)

The Omega Number: is known to be between zero and one , and to go on forever with no repeating pattern (it is an irrational number).

I have been trying to develop a short and shallow summary of this strange creature–this number–but honestly, without much success. It has something to do with the “elegance” of a theory or a computer program. Elegance meaning spare, to the point, and yet complete and perfectly expressive of the phenomena being described or replicated. Its essence captured “beautifully.” Chaitin proves, apparently, that there is no way to be certain that any program accomplishes this , that there may always be a better way to do it lurking somewhere or somehow.

There is an important set of problems in mathematics that involve conjectures that cannot be proven but seem accurate. Conjectures about the character of all prime numbers was given as an example. To test them, all that can be done is look at every prime number, for which there is no end, of course. So, the issue has now turned to speculation about The Tree of All Possible Computer Programs and their evaluation in terms of the number of bits of information each would use.

The Omega Number provides us with the answer to whether a program or programs, among this tree, will eventually find its answer and come to a halt—find a prime number that defies the conjecture, for example—or go on forever looking, and this without running the program but just from the program’s form.

The Omega relies on The Turing Halting Problem and its theoretically possible solution with a Turing Halting Oracle program. The oracle discovers the number of programs that will halt. But the famous mathematician and founder of the computer, Alan Turing, also proved (apparently) that there is no practical solution to The Halting Problem, it is incomputable, though theoretically true. Therefore, The Omega Number, though real and true, is—also—practically beyond our calculating powers, i.e. unknowable to us.

Now that is a philosophical and a mathematical puzzle! I hope my summary was remotely accurate.

Many Philosophers Have Been Mathematicians

Russell in 1959. He was from a noted British aristocratic family. He and his father were early and outspoken atheists and Bertrand refused to participate in WWI.

The famous modern British philosopher, Bertrand Russell, along with his mentor Alfred North Whitehead—also a philosopher and mathematician—wrote Principia Mathematica, a three volume work published from 1910 to 1913. It attempted to base all math in symbolic logic and to reduce the fundamental principles of math into a set as small as possible. It contained “hundreds of pages of numbers, symbols, and equations,” and “later in life (Russell) claimed he knew of only six people who had read it from beginning to end” (from the book Wittgenstein’s Poker).

In philosophy, Russell led the revolt against Idealism, for which Whitehead was a prominent proponent, and started the modern Analytic Tradition. Its goal is to reduce any theory or tradition of speech (such as ethical language) to its basic assumptions and components.

Portrait of Rene Descartes, by Frans Hal.

But it was a famous Frenchman who wrote, “I think, therefore I am.” (cogito, ergo sum). The story goes that as a young soldier in the Thirty Year War between Protestant and Catholic countries in central Europe starting in about 1620, Rene Descartes huddled inside an old abandoned stove to seek warmth and there began to write the first volume of his famous Philosophical Meditations. He is most known for his explicit division of reality into two very contrasting realms—the material and the ideal, or, in other words, the mechanical (the Body) and the mental (the Mind). This is called Philosophical Dualism.

Analytic Geometry: beloved by many a high schooler (or not!).

In mathematics, Descartes was quite accomplished. He invented (or discovered) Coordinate Geometry, which was then highly significant to Isaac Newton’s work. Among his other contributions is the convention of using a superscript to denote powers or exponents (x2).

Descartes was also a scientist and attempted to study animals as Mechanisms, but contended Humans were both machines (body) and soul (Mind, non-mechanism). Though he contended he was a devout Catholic, he fought for the Dutch Protestant armies (as a mercenary, and military engineer–where he studied the flight of cannon balls and the craft of aiming a cannon). Other philosophers of his time accused him of atheism because in his thinking God played no further role after getting the mechanisms of the world started.

Calculus provides the description of an orderly curved line, like the flight of a cannon ball or the orbit of the Moon. Its flight is formed by the pull of gravity of the Earth and its compromise with the tangential momentum of the Moon’s, or ball’s, forward velocity. The moon is “falling around the Earth;” it can be said, while the cannon ball is in an orderly fall back to Earth.

Isaac Newton is thought of primarily as a scientist, though in the 17th century the line between these more abstract pursuits was blurry. Newton wrote extensive biblical interpretations and speculations, along with his creation of classical physics.

For his mathematical description of the movements of the planets, he invented Differential Calculus which was a development upon Descartes’ Analytic Geometry. Curiously, a philosopher living at the same time, in Germany, Gottfried Leibniz, also made this ‘discovery’ of calculus. The two engaged in a bitter dispute over its authorship lasting the remainder of their careers. Today, it is generally acknowledged as a simultaneous invention/discovery.

Back to The Greeks for The Finish

Pythagoras depicted in a detail from Raphael’s giant fresco from a wall in the Vatican, painted from 1509-1513.

Pythagoras is one of the western world’s earliest known mathematicians and philosophers. It is questionable how much math and philosophy attributed to him was actually his work.

For example, the famous Pythagorean Theorem, A2+B2=C2, was not known theoretically–as in this form, I guess–but was known practically by the Babylonians and Egyptians.

Line C is, of course, the hypotenuse. Every Right Triangle is composed of this orderly relationship between the Area of three squares–A,B, and C– that can be aligned with its sides.

I was once working with a friend of mine, a great craftsman who is short on formal education but long on intelligence. We were “finishing” his basement, framing the walls. We got to the first corner and he wanted the angle just right, 90 degrees. I thought we had it lined up pretty well just using a T square but he wanted it perfect (as always!) and said we will use the 3-4-5 rule. He measured out from the temporary corner 3 ft. on one wall, marked it, then 4 ft. on the other wall. If the diagonal line connecting these two marks was then 5 ft., the corner was true, 90 degrees. I realized, “that is the Pythagorean theorem,” I announced: 32 + 42 = 52 . Well, Pythagoras, or one of his associates, at least formalized this practical knowledge.

There is a legend that one day Pythagoras was walking by a group of men working metal, “blacksmiths.” They were pounding out a piece on an anvil using various hammers. He heard the sharp clanging sounds created by the repeated blows and began to think of their relation to music. He realized there was an orderly relationship between the size of the hammer, the force of the blow, and the sound that it made.

“A series of fifths generated can give one seven notes: a diatonic major scale on C in Pythagorean tuning.” Thanks to Wiki for diagram and explanation.

He then began experimenting with the four strings of a lyre and discovered that the sounds most often used musically, the Notes, the one’s most pleasing to our ear, could be organized into sets of four (a tetrachord) and then a set of eight (an octave) with a steady interval between them. These notes or pitches then coincided with a ratio of length of the lyre’s string used to create them. The ratio he found or decided upon as closest to those pleasing notes was 3:2, or what became known as “The Perfect Fifth” and this became the basis of the Pythagorean Tuning system used in western music up to the early 1500s. Our modern tuning system is very close to Pythagoras’, but differs just slightly in the frequency of vibration for each note.

(God, do I feel stupid as I try to write this post! I am not much of a mathematician or musician, so once again, I hope the above description of sound, music theory, acoustics is roughly accurate. Please, Comment [Bob, or anyone] if it is not, so that I may learn!)

Finally, dear old Pythagoras had another idea worth mentioning: The Harmony of The Spheres. To this man, and his followers, Numbers and Mathematics were godly, religious in character. All things were essentially numbers, or something like that. So, he speculated The Heavens (what we would now call stars, planets, and moons) were also numerically organized. That was a good idea, but since numerical relations were also Musical, he contended that the orderly movements of the sky also created A Harmonious Sound, A Celestial Symphony, but we could not hear it!

Atlas holding up the sky, which is A HARMONY OF SPHERES

IF ONLY IT WERE TRUE! What a pleasant idea! I’m not so sure that this last idea helps prove my case, THAT PHOLOSOPHERS KNOW SOMETHING. Maybe sometimes they just get carried away! But Pythagoras’ idea did influence Johannes Kepler!

Salvador Dali: The Harmony of The Spheres (1978)

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