Tag: irrational number

“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 “Intuition Pumps”

Professor Daniel Dennett of Tufts uses the word “intuition pumps” in discussing intuitive understanding and its tweaking.

Let’s do a simple example, avoiding as always “rocket science,” where the intricacies weigh you down in advance. We make a U-turn and go back by choice to elementary notions and examples.

Think of the basic statistics curve. It’s called the Bell Curve, the Gaussian, the Normal Curve.

The first name is sort of intuitive based on appearance unless of course it’s shifted or squeezed and then it’s less obvious. The second name must be based on either the discoverer or the “name-giver” or both, if the same person. The third is a bit vague.

Already one’s intuitions and hunches are not fool-proof.

The formula for the Bell Curve is:

\begin{equation} y = \frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}} \end{equation}

We immediately see the two key constants: π (pi) and e. These are: 22/7 and 2.71823 (base of natural logs).

The first captures something about circularity, the second continuous growth as in continuous compounding of interest.

You would not necessarily anticipate seeing these two “irrational numbers” (they “go on” forever) in a statistics graph. Does that mean your intuition is poor or untutored or does it mean that “mathworld” is surprising?

It’s far from obvious.

For openers, why should π (pi) be everywhere in math and physics?

Remember Euler’s identity: eiπ + 1 = 0

That the two key integers (1 and 0) should relate to π (pi), e, and i (-1) is completely unexpected and exotic.

Our relationship to “mathworld” is quite enigmatic and this raises the question whether Professor Max Tegmark of MIT who proposes to explain “ultimate reality” through the “math fabric” of all reality might be combining undoubted brilliance with quixotism. We don’t know.