The perfect timeless world of our greatest logicians: At this point, it's important to remember who we're talking about.
We're speaking here about Kurt Godel, "who has often been called the greatest logician since Aristotle." So says the widely praised science/math writer Jim Holt in the opening pages of his new book, When Einstein Walked With Godel: Excursions to the Edge of Thought.
Is it true? Was Godel really the greatest logician since Aristotle? Since Godel (1906-1978) lived in the twentieth century, that would make him the greatest logician in something like 2400 years!
You'd almost think that a person like that would have made some great, identifiable contribution to human thought. You'd almost think his name would be well known.
With Godel, the story is different. In Holt's telling, this second greatest logician seemed to struggle with mental illness since perhaps the age of 5—mental illness which became so extreme that he ended up dying of self-starvation.
Then too, there were the crazy ideas. Holt discusses them early on, contrasting Godel with his friend, Albert Einstein:
HOLT (page 4): Although Einstein’s private life was not without its complications, outwardly he was jolly and at home in the world. Gödel, by contrast, had a tendency toward paranoia. He believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases; he refused to go out when certain distinguished mathematicians were in town, apparently out of concern that they might try to kill him. “Every chaos is a wrong appearance,” he insisted—the paranoiac’s first axiom.Our greatest logician believed in ghosts. He had a morbid fear of gases from the fridge.
Holt essay first appeared in 2005 as a somewhat disguised review of Rebecca Goldstein's book, Incompleteness: The Proof and Paradox of Kurt Godel. As such, almost all Holt's material seems to be drawn from Goldstein's book, in which she presented an amusing list of Godel's apparently crazy ideas—crazy ideas which didn't seem to spring directly from paranoia or other such illness.
Does it seem strange to think that our greatest logician was perhaps best known, in his adult years, for his various crazy ideas? We'll let you wrestle with that one. For ourselves, we'd be inclined to see this syndrome as perhaps being instructive, illustrative of the vast intellectual dysfunction at the heart of the human experience.
Godel became known as "the greatest logician since" because of his incompleteness theorems, which Holt and Goldstein struggle to explain in their respective texts. We'll examine those struggles next week, marveling at the kind of journalistic/academic work which is reflexively praised for its clarity by long lines of scripted nimrods within the upper-end press.
This week, we're puzzling about something different; we're puzzling about Godel's alleged "Platonism." At great length and in flowery language, Goldstein describes the murky "doctrine" as the enduring, rapturous love of Godel's life—as a doctrine with which he "fell in love" when he was only 19.
Holt's essay condenses this portrait of Godel's affair—but what the heck is "Platonism" even supposed to be? As we've noted, the following passage includes Holt's first bite at that worm-infested apple. As he continues (see text below), some comical themes will emerge:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians...It's just as we told you last week! According to Holt by way of Goldstein, the young Godel was "seduced by...the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind."
So says Holt, almost seeming to assume that this puzzling definition of Platonism makes some sort of earthly sense.
Does Holt go on to explicate this "doctrine?" We'll consider his fleeting effort below. First, let's consider the puzzling places to which we humans can be taken in explorations of modern "philosophy" at its highest levels.
Holt is halfway through a lengthy paragraph at the point where we've left off. From the rest of his graf, we can extract a minor attempt to flesh out the concept of "Platonism"—but we're also taken to a peculiar place, a place where we ponder 2 + 2 and the mysterious way in which 2 + 2 can be known to equal 4.
Are we humans the rational animal, or was that widely bruited assertion Aristotle's error? Is it possible that our highest intellectual elites have more often turned out ruminations which more closely resemble clown shows?
Have our highest academic elites sometimes resembled harlequins, clowns? We cut-and-paste, you decide:
HOLT (continuing directly): In the philosophical world of nineteen-twenties Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.A significant name pops up in that passage—the name of Ludwig Wittgenstein. Before our post-philosophical explorations are done, we'll consider his admittedly murky, but conceptually simple, later work at some length.
Setting that matter aside for the present, let's look at Holt's attempt to explicate Platonism. Also, let's consider the fact that our greatest minds have struggled over the nature of the famous schoolboy proposition, 2 + 2 = 4.
In that extended passage, Holt seems to contrast young Godel's Platonism—his fervent belief that the number 3 lives a perfect timeless existence—with the hard-boiled views of a group of thinkers called the Vienna Circle.
Mercifully, Holt ascribes no particular "isms" to that particular group. But within that passage, a reader can discern a tiny attempt by Holt to explain Platonism in a bit more detail:
You might be a Platonist if! From that passage by Holt, it sounds like a Platonist believes that a proposition like 2 + 2 = 4 is true because "it correctly describes some abstract world of numbers."
Reader, tell the truth! Do you have even the slightest idea what that proposition might mean? Before you give a defensive answer in which you defer to academic authority, please remember that we've asked you to stick to the truth.
People, what makes a proposition like “2 + 2 = 4” true? According to Holt, our loftiest gangs of intellectuals were debating this chin-scratcher as recently as 1930, with Godel clinging to a belief which he was refusing to reveal:
According to Holt, Godel believed the proposition "2 + 2 = 4" is true because it correctly describes some abstract world of numbers. So believed our second greatest logician as he fell in love with the idea which would drive his work throughout his entire life.
Friend, do you have the slightest idea what any of that might mean? Do you feel sure that you can explain what it means to believe in the existence of "abstract worlds" at all? More specifically, do you know what it means to believe in the existence of the "abstract world of numbers"—which, presumably, is the place where the number 3 lives its perfect timeless existence, surrounded of course by the circles?
Do you have even the slightest idea what it means to believe in such things? We're going to guess that you do not—that you could never explicate, explain or unpack the essence of such alleged beliefs, which represent Holt's only attempts to explain what "Platonism" is.
Luckily, Holt briefly extends his explanation at one additional point. As it turns out, the truth of 2 + 2 = 4 was, for Godel, all about our ESP! For now, let's ignore the technical language and stick to the matter at hand:
HOLT (page 10): [Gödel] believed he had shown that mathematics has a robust reality that transcends any system of logic. But logic, he was convinced, is not the only route to knowledge of this reality; we also have something like an extrasensory perception of it, which he called “mathematical intuition.” It is this faculty of intuition that allows us to see, for example, that the formula saying “I am not provable” must be true, even though it defies proof within the system where it lives...How can we know that 2 + 2 equals 4? According to Godel as told by Holt, we have "something like an extrasensory perception" which lets us discern such realities about the abstract world of numbers where the number 3, and all other numbers, live their perfect timeless existence in the company of circles.
Briefly, a word of praise. In her own book, Goldstein offered a perfect, simple explanation of how we can know that 2 + 2 equals 4.
Goldstein's perfect explanation doesn't involve ESP. You'll rarely see a philosophy professor make such a perfect statement. We'll get to that statement next week.
For today, let's leave matters at this. Back around 1930, the greatest logician in 2400 years was trying to determine how we can possibly know that 2 + 2 equals 4.
At Princeton, during his later years, he was known for his various crazy ideas. Starting around the age of 19, he developed the Platonistic idea that we can know about 2 + 2 because we have some faculty which resembles ESP.
This is the way our highest academic elites were playing during these not-so-distant years. According to Holt, even Lord Russell got dragged into this ridiculous mess. Again, ignore the technical language to focus on the sad absurdity of what is being said:
HOLT (page 9): Gödel’s incompleteness theorems did come as a surprise. In fact, when the fledgling logician presented them at a conference in the German city of Königsberg in 1930, almost no one was able to make any sense of them. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it? The very idea seemed absurd. Even the once great logician Bertrand Russell was baffled; he seems to have been under the misapprehension that Gödel had detected an inconsistency in mathematics. “Are we to think that 2 + 2 is not 4, but 4.001?” Russell asked decades later in dismay, adding that he was “glad [he] was no longer working at mathematical logic.”Lord Russell, "the once great logician," worried that he was being asked to believe that 2 + 2 might not equal 4 after all! This made him glad that we was no longer mired in the world of (mathematical) logic.
Respect for academic authority tells us that we must believe that these ruminations, by these great intellectual figures, simply must have made sense. By the end of the 1940s, Wittgenstein had torn such suppositions to shreds in a self-admittedly poorly-written book, Philosophical Investigations.
Professor Horwich has said that our current professors have stopped teaching Wittgenstein because his demonstrations mean that they would pretty much have to stop teaching everyone else. Down the road, we'll return to that claim, concerning which we'd occasionally joked on the world's greatest comedy stages before first encountering Horwich.
We'll return to Horwich's conjecture at some later date! Tomorrow, we'll return to Professor Goldstein's attempts to explicate "Platonism." We'll start with a comical story mentioned by Holt in passing:
A Platonist was hiding among them! Yes, it actually gets that silly, that puny, that tiny, that dumb.
Tomorrow: What a Platonist believes