Tag: Zeno of Elea

“Fog Everywhere” Continued

[read the previous essay on this topic]

Consider mathematics and the aspect of fog. Nietzsche argued that the world is not knowable to us as a whole. This includes trying to express it with numbers. A recent example is Max Tegmark’s assertion that the universe is not something explained by mathematics; rather, it is itself mathematics. To contrast, Karl Jaspers, summarizing Nietzsche, wrote:

We cannot say what the world as a whole is. It is false to change all processes into a familiar world of our own, and then say: “All is will (everything wills); all is pleasure or pain (everything endures); all is motion (everything flows); all is tone (everything sounds); all is spirit (everything thinks); all is number (everything reckons).” Nietzsche warns us against all notions of the whole: “Let us guard against thinking that the world is a living being…or that the universe is a machine…Let us guard against saying that there are laws of nature…Let us guard against thinking that the world eternally creates novelties.” All these “shadows of God” darken actuality. We are within the world, and the whole of the world is, as a whole, not accessible to us.

Karl Jaspers, Nietzsche: An Introduction to the Understanding of His Philosophical Activity, Henry Regnery Company, 1966, p. 293

Notice in the paragraph quoted above that Nietzsche specifically excludes reducing everything to numbers, in opposition to Tegmark and the simulation hypothesis.

Robert Kanigel describes Srinivasa Ramanujan as “the man who knew infinity” in his biography of the same title, but mathematicians disagree on the ontological status and utility of infinity. The discussion of applying Ramanujan’s sum to arrive at -1/12 is very intricate and complex.

To make things even more puzzling, math students are confronted with the Gaussian integral, with its positive and negative infinities.

You can gain additional insight into infinity with:

Cauchy and Weierstrass had eliminated infinitely small and infinitely great numbers from analysis and replaced them by limits. But the theory of limits that thereby became so central required a clearer theory of the real line, that is to say, a theory of the irrational numbers. And that theory promptly reintroduced the infinite into analysis. The old infinity of infinitesimal and infinite numbers was simply replaced by the new infinity of infinitely large collections.12

In 1831 Carl Friedrich Gauss said [Kli72, p. 994], “I protest against the use of an infinite quantity as an actual entity; this is never allowed in mathematics. The infinite is only a manner of speaking, in which one properly speaks of limits to which certain ratios can come as near as desired, while others are permitted to increase without bound.” But only 52 years later, we find this in Cantor’s Grundlagen [Can76, p. 75]: “The idea of considering the infinitely large not only in the form of the unlimitedly increasing magnitude and in the closely related form of convergent infinite series…but to also fix it mathematically by numbers in the definite form of the completed infinite was logically forced upon me, almost against my will since it was contrary to traditions which I had come to cherish in the course of many years of scientific effort and investigations.”

12 See Russell’s Principles of Mathematics [Rus03, p. 304] for a related sentiment.

Shaughan Lavine, Understanding the Infinite, Harvard University Press, 1994, pgs. 38-39

In 1961, the great Russian physicist George Gamow’s popular math and physics overview was republished to great acclaim among science and math aficionados. Intriguingly titled One, Two, Three…Infinity, the book gives you a hint that the relationship of numbers and integers to the concept of infinity is still intriguing even after the intricate analyses of Cauchy, Weierstrass et al.

Let us conclude by considering Zeno’s dichotomy paradox of motion, in which “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” As the goal is divided into halves, one must complete an infinite number of tasks, which Zeno maintains is an impossibility.

[read the previous essay on this topic]

Education and the Need for Enchantment

Max Weber (1864-1920) and Émile Durkheim (1858-1917) are considered the two fathers of modern sociology at the highest level.

Weber sees the modern world as the zone of “Ent-zauber-ung:” where ent means removal of, Zauber means magic or enchantment and ung means the process of.

He sees our world as “dis-enchanted.” Everything is scientific or profitable or unwelcome. This makes modern life a productive engine of sorts but extremely desiccated and arid and leads to what Durkheim calls “anomie” (the sense of being adrift, directionless).

We argue in this book that education should be seen as the “last exit” to enchantment before the “grind of life” comes down on the student after the “moratorium” of college.

What is enchantment? Enchantment is that special feeling about something, some topic, field, math problem, painting exhibit, novel, movie, debate, that there’s something there that “makes it all worthwhile” and like a great piece of music, “gets to you” and flies under all cynical radar. The best kind of enchantment can last from age 19-95, if you live that long.

Think of a math or physics problem or novel or painting that gives the student “permanent uplift.”

The pedagogical dimension of enchantment works like this: the student encounters a puzzle or conjecture or story or depiction that constitutes a “healthy obsession.”

After interaction with this phenomenon, he or she can “walk backwards” to the 900-page textbook and go to those pages that are relevant, this making the textbook more like a dictionary that serves as a handy reference book and not as a daunting, exhausting endless “Mt. Everest” of names and equations or faces or dates. The student can “conquer” textbooks by enchantment and only enchantment. Without that engine or motor for the mind and will, one is weighed down and demoralized in advance.

Let’s do two quick examples:

Heraclitus is supposed to have said, “you can’t step into the same river twice.” Zeno says you can’t really cross the street because first you have to reach the midpoint, then the next midnight, and so on forever. You never complete your crossing (see Joseph Mazur’s book, Zeno’s Paradox, from 2008).

Such ancient paradoxes are still perplexing. Great thinkers like Bertrand Russell, Whitehead, Frege, et al wrestled with them many decades.

It’s also puzzling that certain math or logic questions open up “oceans” of analysis. Why might that be? Is that enchanting or depressing?

The last chapter of Tolstoy’s War and Peace masterpiece is a set of reflections on history itself. It’s very enchanting as he wrestles with this “caprice machine” called history.

Enchantment gives you the first steps towards what we call “pre-understanding,” a prerequisite for all deep study.