Title: The Mystery of the Aleph
Subtitled: Mathematics, the Kabbalah, and the Search for Infinity
Categories: Mathematics, Number Theory, Biography, Philosophy, Jewish Mysticism
Author: Amir D. Aczel
Copyright 2000, published by Washington Square Press

This wonderful work has stimulated my mind in such a way that I haven't experienced in years. In its very essence, it is a journey into the concept of "actual infinity," rather than the "potential infinity" explored by mathematics such as Calculus. The book's central character is the Russian-born, but primarily Jewish mathematician Georg Cantor. Cantor received his doctorate in mathematics at the University of Berlin, but spent the majority of his life in Halle, Germany, where he patiently awaited the invitation to move to Berlin or Göttingen, two of the most mathematically active cities in the world at that time. He tried his best to find work in either city, but every opportunity he had fell through, which led him to fits of furious rage. He had somewhat of an explosive personality, and experienced several bouts of insanity and depression throughout much of his life, especially during periods in which he focused on the so-called "continuum problem."

After presenting biographical information in the chapter "aleph null," the book delves into ancient Greek mathematics and logic. We are first introduced to Zeno's Paradox, considered to be the first paradox ever conceived by man, or at the very least, the first one that was formally documented and preserved throughout history. His two famous paradoxes, which are really one in the same, are discussed; the one involving the race between Achilles and the Tortoise, and the so-called dichotomy paradox, which says that in order to traverse a distance, one must traverse half that distance, then half of that distance, and so on, making motion impossible. As silly as these paradoxes might seem, they presented a difficult problem in logic and at at the same time ensapsulated the notion that series involving infinite sums of numbers can produce finite quantities. For example, the sum of 1/(2^n) as n starts at 0 and approaches infinity produces a sum approaching 2.

Aczel then discusses Pythagoras and his famous theorem that brought to surface the concept of irrational numbers. He provides an excellent history of the cult of Pythagoras, which ran into the so-called irrational numbers upon the discovery of the Pythagorean theorem. Evidently up until this point no one had really thought of the concept of "square root of 2." The Pythagoreans were well aware of so called "square numbers" such as 1,2,4,16 and so on, but hadn't really bumped into the need to find square roots of numbers that aren't "even squares." The Pythagoreans believed that "god is number" and that the highest and most spiritual numbers were the natural numbers, that is, the non-negative integers starting from 0. They were able to explain away the existence of fractional numbers by expressing them as ratios of whole numbers. For example, 0.6666 repeating can be expressed as the ratio of 2 divided by 3. So, despite the fact that there were fractional numbers that meandered off into infinity, up until this point they could all be expressed as ratios of "god's numbers."

Upon discovery of the Pythagorean theorem, that is, the theorem that tells us given a right triangle with sides A and B and hypoteneuse C, the lengths of the sides can be expressed by the equation A² + B² = C², the Pythagoreans were confronted with the notion of irrational numbers. Consider a square of one foot by one foot. Divide the square in half along the diagonal. How long is the diagonal? Well, it's clear that we've created two perfect right triangles with sides "A=1" and "B=1." So what is C? Well, if 1² + 1² = C², then 2 = C² and C = Square Root(2). And what is the square root of two? Can it be expressed numerically? Can it be expressed as a ratio of whole numbers? Nope. The only way to truly represent the number is to express it as "square root 2," or as the root of an algebraic equation.

This presented an interesting dilemma to the Pythagoreans, who worshipped the whole numbers and believed up until this point that anything could be expressed as a ratio of "god's numbers." So, they kept this discovery a secret for as long as they possibly could, but it was only a matter of time before somebody let the meme loose. The society member Hippasus is believed to have leaked the knowledge and according to legend was either expelled from the society, was buried alive, killed by the head honcho himself (the mighty Pythagoras), or was set sail by other members and then sunk to his death. Shortly thereafter, Greek geometry was born, which presented us with the somewhat "obvious" fact that on any given number line, there is an infinitude of irrational numbers just waiting to be discovered.

After this short mathematics history lesson, Aczel delves into the mystical world of the Kabbalah. He tells of the establishment of the Jewish priesthood by the Israelites and the mysterious "Urim veTumim," a necklace consisting of a gold chain containing an array of twelve squares of precious metals, each corresponding to a tribe of Israel. This priesthood and its sacred necklace marked the beginning of so-called "Jewish Mysticism." We learn of the first-century Rabbi Joseph ben Akiva, who penned the beginnings of what would later be collectively known as the Kabbalah (or Qaballah, or however you'd like to transliterate it). His writings were meant to create vivid mental images that lead one to divine meditation and thus oneness with god. Evidently, the good Rabbi himself was capable of falling into the deepest of meditations, in which he experienced powerful hallucinations, which he warned his subjects not to obsess over, lest they drive themselves deep into madness. The legend goes that along with three other Rabbis, Akiva entered a palace of meditation and attempted to see the "chaluk," that is, the infinitely bright robe which god wore upon meeting Moses on Mount Sinai. One participant, Ben Azai, was said to have died upon experiencing the infinite brightness of god, for he could not go on living after experiencing such tremendous beauty and "enlightenment," pun not intended. The second, Ben Abuya, is said to have seen two gods instead of one, and became an apostate. The third, Ben Zoma, is said to have fallen into utter insanity, for he could not comprehend what he saw and relate it back to ordinary life.

We are told that Akiva, and only him, truly survived the experience. We learn much more about the Kabbalah in this chapter, although I'd have to re-read the book quite a few times in order to feel fully capable of explaining exactly what it consists of, and how it relates to the whole of the work. Basically, the point I gathered here is that the mystical conception of god is expressed as a function of ten interconnected, hyperdimensional worlds. God is both these ten worlds, and the emptiness in which they exist, at the same time. God is in the worlds, and god is beyond the worlds. This is interesting because the Pythagoreans also believed in a "ten-ness" and an "infinity-ness" of god. The ultimate, far reaching point I gathered from this chapter on the Kabbalah is that god is finite, infinite, nothing, one, and all, and that attempting to gaze into this seeming paradox will drive all but the most spiritually prepared person into mind-shattering psychosis. On might say that Akiva possessed an almost Zen-like quality that allowed him to experience this blissful wondrousness without losing his mind.

After this mind-altering chapter on mysticism we are once again immersed in the world of hard science, where we learn about Galileo Galilei. We are taught that with the discovery of Calculus, the concept of "potential infinity" was realized and exploited to many extremes in order to deal with infinity on a finite level. The aforementioned sum series "1/(2^n)" is one example of potential infinity. We don't need to actually comprehend infinity itself to realize that this sum converges on two. Run n from 0 to one thousand and you've got a pretty clear idea on what's going on here. But actual infinity is something else altogether. We are given a nice historical run down of Galileo's persecution and subsequent house arrest, and how in this seclusion he was given the opportunity to deal with infinity itself as something more than a mere mathematical abstraction. Galileo delved into set theory, and found that within any infinite set, infinite subsets could be derived. One example is the set of all natural numbers starting from 1, that is {1,2,3,...}. Now consider the set of all natural numbers which are even squares -- {1,4,16,25,...) Despite the fact that the second set is more "restrictive" and each number is far more distant from the next, both sets contain an equal amount of numbers -- an infinitely equal amount.

We then learn of Bernhard Bolzano, who took Galileo's reasoning a step further and applied it to the notion of the number continuum, that is, the infinitely dense number line. For example, if we consider all of the numbers between 0 and 1, there lies an infinity of quantities within a finite distance. The line is only one unit in length, yet it can be divided forever. Yet the same thing could be said for a line defined by the endpoints 0 and 2. While one line is twice as long as the other, they both contain an "equal" number of elements within. This corresponds to Galileo's discovery that a more strictly defined subset of numbers still contains the same number of elements as its less restrictive parent set.

I could go on forever here (the book goes over many other mathematical conundrums such as squaring the circle), but I'd like to get at the matter at hand before I end up re-writing the entire book myself. It's an excellent book and should be purchased by anyone with the slightest interest in number theory. What we eventually learn is that within any finite continuum of numbers, transcendental numbers form the highest order of infinity. That is to say that even though there are infinite rational and irrational numbers between the numbers 0 and 1, there are "more" infinite transcendental numbers. Galileo and Bolzano discovered that in countable yet infinite sets of numbers, smaller and smaller subsets still contain an equal number of elements. But transcendental numbers are not countable. There is no way to move from one transcendental number to the next.

So what's the difference between a run of the mill "irrational" number and a "transcendental number?" Well, all transcendentals are irrational, but not all irrationals are transcendental. We could take this to mean that transcendental numbers are a subset of irrational numbers, yet perplexingly enough, given any number line, if one were to pick any point at random, the likelihood of it being transcendental, not just "irrational," but "transcendental," is 1. Probability one. That's right. It seems insane, but it's true. This is what Georg Cantor discovered -- that there are higher orders of infinity. But back to the original question -- there are two types of irrational numbers. There are the "algebraic" irrationals, and the "transcendental" irrationals. You might recall before when we discussed how the Pythagoreans were blown away by the irrational numbers because they cannot be expressed as simple ratios of whole numbers. Yet they can still be expressed as functions of whole numbers. For example, the square root of two is an irrational number, but we can still arrive at this seemingly infinitely complex (in that the digits after the decimal point go on forever but never repeat in any recognizable pattern) number by considering a function of "two." These are called "algebraic irrational numbers" because while they cannot be represented by simple ratios of whole numbers, they can be represented as roots of polynomial equations involving whole numbers. Consider x² - 2 = 0. This becomes x² = 2, and then "x" becomes the square root of two. So even though the square root of two goes on forever, we can still use a very finite equation involving very simple whole numbers to express this notion in an abstract manner, without having to write out every decimal place.

Transcendental numbers cannot be expressed in such a fashion. What this means is that every other "order" of real numbers can be seen as a function involving whole numbers, with the exception of transcendentals. The only reason we know with certainty that pi and epsilon are transcendental is due to the famous Euler equation, in which we are told that (e^(i*pi))+1 = 0. Without this equation, we'd never really be able to know for sure, because one would have to imagine every comprehensible polynomial equation possible and would never be able to rule out any function entirely, having had to try an infinite number of coefficients. So, even though transcendental numbers seem rare, "few and far between" if you will, they form the richest consituency on the number line. As mentioned before, pick any point at "true" random, and the likelihood of hitting a transcendental is absolutely certain.

Does this not boggle the mind? It certainly boggled Georg Cantor's. Yet he was the first mathematician to truly struggle with actual infinity involving uncountable sets of numbers. By uncountable I mean that there is no way to begin enumerating a set of transcendental numbers. There is no way to arrive at the next number from the previous one, whereas the set {1,2,3,4,...} can be stretched our forever without any confusion as to where it's going. Even algebraic irrational numbers can be counted, by adding a "one" to each decimal place in order to arrive at the next number (which Cantor himself discovered). Cantor essentially discovered that there were orders of infinity, and that the continuum represented the highest possible order. In his efforts to solve the continuum hypothesis, which involved determining whether there were any orders of infinity lying between the lowest order, and the order of the continuum itself, he drove himself mad on many occasions, constantly coming up with one proof, disproving it, then moving on to another. He wound up in the mental hospital on multiple occasions, and then would seem to recover fully, only to relapse upon attempting to reconcile these problems once again.

Kurt Gödel attempted to do the same thing in his lifetime, and also drove himself mad on multiple occasions in doing so. Though Cantor had a personal history (but no family history) of mental trouble, Gödel did not. Though Cantor had many ideological enemies who opposed what he did and thus had reason for developing a persecution complex, Gödel did not. Yet they both developed such a disorder, feeling as if they were working for god and were being opposed by his enemies, who are bent on hiding the truth. When Cantor reached his breaking point, he became obsessed with a completely non sequitor unsolvable problem. He wanted to prove that Shakespeare never wrote any of his plays; that they were really written by Francis Bacon. A similar thing happened to Gödel -- in attempting to solve this continuum problem, he fell into an obsession with proving that the mathematician Leibniz plagiarized many of his theories. Both men faced the problem of actual infinity, and became paranoid, raving maniacs as a result. The author relates this back to the aforementioned chaluk; the infinite white robe that maimed, killed and otherwise drove mad the three Rabbis who tried to comprehend it.

So what does all of this mean? In the process of attempting to resolve these issues, Gödel ran into his infamous incompleteness theorem, which tells us that within any formal, axiomatic system (of which mathematics is one example), there will be theorems that cannot be proven within the confines of that system. In the book we learn that the natural numbers are ultimately defined by the mathematician Peano through set theory. Zero is the empty set. One is the set containing the empty set. Two is the set containing the set containing the empty set, plus another empty set, and so on. We also learn of Bertrand Russell's paradox, which shines doubt on the whole of set theory itself, despite the fact that much of mathematics relies on it as a foundation. Russell invites us to consider the set of all sets that are not members of themselves. Such an example would be the set of all things that are animals. Since sets aren't animals, the set of all things that are not animals is not a set of itself. But consider the set of all things that aren't animals. Since sets aren't animals, the set itself belongs to itself. But if we consider the set of all sets that are not members of themselves, we are confronted with the question -- is that set itself a member of itself? If it is, then it isn't. If it isn't, then it is. Russell attempted to reconcile this paradox by coming up with a convoluted version of set theory, in which there are lower "orders" of sets. For example, the lowest order of sets cannot contain sets, only numbers. The next order can contain only numbers, and lower-ordered sets. Yet this hopelessly complicates set theory and presents problems of its own. There simply is no way to skirt around the issue using a formal system, which Gödel proved.

Since the definition of natural numbers relies on the foundation created by set theory, this seemingly innocuous paradox tells us that the whole of mathematics relies on foundations that cannot be completely resolved within themselves. Though it seems obvious that 1+1=2, these assumptions rest on the definitions of "1" and "2" being sets of empty sets, and set theory itself is incomplete, so we really don't even know for sure what "2" means. And the rational and algebraic irrational numbers are defined as ratios or functions of equations of involving natural numbers, we cannot be sure that these numbers are "legitimate" or "valid," or "real," as odd as that may seem. But the transcendental numbers such as pi cannot be expressed as functions of sets, and cannot be counted. So while the Pythagoreans believed that the natural numbers were god's numbers, it is the transcendental numbers that exist beyond our most basic assumptions about math. It is the transcendental numbers that "exist" beyond the realm of human consciousness (assuming you aren't a subscriber to the anthropic principle) -- we find a rough approximation of "e," the approximation itself being transcendental as well -- in a reproducing colony of bacteria. We find a rough approximation of "pi," the approximation itself being transcendental as well -- in the beautiful sphere of thermonuclear, gravitational chaos that we call a star. These "numbers exist" beyond our scope of awareness. Without us, they'd still be there, defining the continuum of the heavens themselves. It is the natural numbers that require us to validate their existence. It is the natural numbers which cannot be said to certainly "exist" on their own merits. Aczel himself delves into the question "do numbers really exist?" and discusses the notion that "everythingness" contains "nothingness" as well. It can be said that all sets contain the empty set, which is the most abstract mathematical "proof" of this notion that nothing is inextricably connected to everything. There is no distilling the two apart.

After reading this book, I am possessed with a belief that there is much to be learned about the power of transcendental numbers, and that many of the mysteries of the inner workings of the Universe lie in discovering the properties of these incredibly bizzare and ubiquitous quantities. Upon reading this book, my next recommendation, if you haven't read it already, would be Gödel, Escher, Bach: An Eternal Golden Braid. Have fun...

You might notice that I never really explain what "aleph" means within the context of this book. I did not do this intentionally, but I will leave finding that out as an exercise for the reader. :)