The Ancients and Infinity

In Everything and More, David Foster Wallace provides an excellent portrayal of the one sided history held by many transfinite theorists. This view places Zeno of Elea and Plato as the greatest heros of mankind. Aristotle (and his minions Thomas Aquinas, Roger Bacon, Descartes, etc.) were the world's greatest villains.

Everything and More is a fun book to read. The work does an excellent job presenting the metaphysical world view held by many transfinite theorists. The work casts subtle barbs at those in the tradition of Aristotle and logic, while heaping praise on those whose works led to Cantor's invention of the denumerable/nondenumerable dichotomy and the other brave new dialectics of 19th century Germany. I actually owe a great debt of gratitude to Mr. Wallace for presenting, in a clear readable manner, the metaphysics that I dislike. People refuse to believe me when I tell them the things I was taught in my College math classes. To my delight, Wallace actually wrote it down.

Wallace's one-sided history casts Aristotle as the great villain. After reading the work, I thought I would like to portray a different one-sided history. In my one-sided history, Socrates and Aristotle are the heros. Of course, I openly acknowledge the one-sidedness of my history. All philosophers have good and bad points. One dimensional representations in texts such as this should not be taken too terribly seriously.

There is one advantage to one dimensional histories. Such histories allow us to develop a single thread of thought in an entertaining fashion. Wallace pursued the treatment of infinity. I wish to pursue a much more important issue: the foundations of logic and the ability to communicate our ideas. Looking through history, we find that some systems of discourse help people communicate. Others tend to hinder the ability to communicate.

In Ancient Greece, we find a society struggling to develop that ability to develop and communicate ideas. It is a society with both triumphs and set backs. Due largely to the works of Socrates and Aristotle, they ultimately triumphed.

Ancient Greece provides an excellent study in the history of discourse. Ancient Greece was a vibrant, complex society. Ancient Greece had a democracy (along with reactionaries to democracy). The had open philosophical disputations and many different schools of thought competing for the attention of the public. Ancient Greece had a strong merchant class, but also suffered the scourge of slavery. As such, Ancient Greece provides many possible threads for historians to pursue.

If you read this history with Wallace's; you will have a two sided history. Even a two sided history is incomplete. Hopefully, you will always keep an open mind and will always find new and interesting threads of thought to follow.

The Pythagoreans

Most histories of infinity begin with the Pythagoreans. The Pythagorean Society are considered by many to be the first true mathematicians. They developed proofs, had an internally consistent systems of thought that incorporated beliefs on numerology and numbers. They made great strides in number theory, music theory, etc.. The Pythagoreans loved measuring things and sought out all of the hidden numbers in nature and mysticism.

The Pythagoreans had one fatal flaw in their belief system: The Pythagoreans only accepted the existence of the natural numbers. The Pythagoreans allowed ratios; however they thought in terms of ratios of wholes. They did not think of ratio in terms of a portion of a whole or the modern fraction.

The Pythagoreans had tremendous success with their mathematical systems. Unfortunately, success often breeds arrogance. They became more and more dogmatic in their beliefs. According to the myth, the society suffered a spiritual crisis when they realized that hypotenuse of the unit square (the square root of two) could not be expressed as a ratio of two numbers. The legends have the Pythagoreans doing horrible things to the mathematicians who crumbled their world view by discovering an incommensurable. Because the Pythagoreans held a dogmatic view, a simple "irrational" number shattered their faith.

The Pythagoreans had effectively developed methods of mathematics and science that greatly expanded their appreciation and understanding of the world. They had difficulties dealing with arguments that fell outside their own domain. Nothing new here.

Zeno of Elea

On this path there are a multitude of indications that what-is, being ungenerated, is also imperishable, whole, of a single kind, immovable and complete. Nor was it once, nor will it be, since it is, now, all together, one and continuous.

Parmenides (c. 475 B.C.)

Zeno of Elea (c495430 BC) was of the school of Parmenides. The School of Parmenides was one of many schools developed in the wake of discovery of incommensurables that hoped to preserve the good works of the Pythagoreans while dealing with incommensurables. Others schools include atomists, who believed that the universe was composed of small building blocks called atoms.

Parmenides philosophy centered around the unity of things. He argued against multiplicity and motion. This multiplicity that appears before us is an illusion of the one. Parmenides would argue that motion and multiplicity involve absurdities—like empty space (non-being). If "non-being" exists then it is.

To support the arguments of Parmenides, Zeno is said to have created a system of 40 paradoxes. Only five of the paradoxes survive to this day. The five existing paradoxes are enormously clever. Each paradox attacks a different aspect of the idea of infinite space, infinitely divisible time, etc.. The paradoxes have been extraordinarily influential. They seem to be quite easy to refute on an individual basis. However, if you take all five in conjunction, you will find that many of the "easy refutations" are in conflict with each other. For that matter, I don't think they have every fully been resolved. When we look at the world of quantum electrical dynamics, we see a chaotic world filled with wave/particle dualities, and can only describe the location of particles with probability equations.

Zeno's paradoxes are quite clever and worth investigating. However, Zeno's methodology had an unfortunate side effect. By frequent use of reductio ad absurdum, he essentially found a methodology to reduce everything in the world to absurdity. Infinity shares much in common with zero. Dividing by zero, I can show that 3 = 5.

     3*0 = 5*0      | Both sides of the equation equal 0
     3*0/0 = 5*0/0  | Divide by zero
     3 = 5          | The zeros cancel each other leaving 3 = 5

     3*∞ = 5*∞      | Both sides of the equation are infinite
     3*∞/∞ = 5*∞/∞  | Divide by infinity
     3 = 5          | The infinities cancel each other leaving 3 = 5

The above demonstrations are clearly flawed. However, the introduction of infinity brings a host of logical difficulties.

There's a number of other paradoxes that tend to show up unannounced. Any system that allows self reference and the negation will be subject to the liar's paradox: This sentence is false.

When you get down to it, it is possible to find absurdities in any system of beliefs (or systems of non-belief for that matter). One of the common examples given is that it is possible for people in a democracy to elect a leader who intends to destroy the democracy; hence, democracy is absurd.

To make matters worse. Excessive use of reductio ad absurdum often leads to the theory's bastard cousin—argument by ridicule and cynicism. You can reduce anyone to absurdity with a carefully timed farting sound.

Essentially what happened is that by frequent use of reductio ad absurdum arguments Zeno's had created a method where he could wrap his beliefs in an impenetrable shield of argumentation. Wrapping oneself in an impeneterable shield of arguments is seductive; however, such shields block the exchange of information in both directions. So while a dialectician might excel at winning arguments, they end up having an impoverished view of the world. Wrapping one's beliefs in paradox tends to lead to a totalitarian mind set. The person open to true communication will learn the value of speaking clearing and the ability to be refuted.

One of the most important dictates of modern science is that scientific theorems must be written in a way that they can be verified or refuted. Science advances because scientists insist on wording theories in ways that can be tested and verified.


The hero of my story is the mythical Socrates. I added the word mythical because Socrates did not believe in writing things down. We do not have any of Socrates' writings, we only have second hand accounts of his works. These second hand accounts are somewhat suspect as they all say different things.

The reason that Socrates did not like to write things down was that he realized that he often made mistakes and did not want to force any of his mistakes on future generations. Our understanding of the world changes as we learn, question and investigate.

The myth of Socrates is that Socrates was an open thinker. The Socratic method is a method for asking questions to find truth. I imagine Socrates being open to wild speculation, but also possessing enough common sense to understand speculation as such.

My mythical Socrates would rather have me learn the ability to think than to tell me how things are. He would look at the diagonal method to see what it would tell him then turn to the next set of questions. This odd notion that the diagonal holds the secrets of continuity would be absurd to him.

When Socrates was young, he met Zeno and Parmenides. I suspect Socrates saw great merit in their arguments, but saw the school of Parmenides as being one of many potential thought systems. Reportedly Zeno could argue both sides of the monist/atomist debate. My mythical Socrates would have been more interested in the method of the disputation than in the conclusion.

Unfortunately, Socrates did not live in a wonderland. Apparently there were detractors who would stand at the door and ridicule the Socratic School. The politics of the day were wicked. Socrates was forced to drink hemlock by his detractors.


Socrates did not write things down. Plato did.

As one of the top students of Socrates, Plato was in the position to inherit the formidable reputation of the great thinker. As Socrates did not publish his work, Plato was in the position to interpret Socrates in ways that would fit his own world view.

In The Open Society and Its Enemies, Karl Popper puts forward an interesting thesis: Popper suggests that Plato was a reactionary, and that Plato was not quite that enthralled with messy open discourse and democracy (which includes input from the merchant class and the rabble along with the philosophers). By taking on the position of interpreter for Socrates, Plato was able to run under the guise of a great "liberal" thinker while injecting subtle twists that lead to the preservation of the upper class.

Although Plato carved his niche by being an interpreter of Socrates, Plato is most noted for his theory of forms. The theory of forms holds that there is an ideal form of things behind the illusion. The philosopher discovers the hidden forms ruling the universe with intellectual investigations. Many mathematicians consider themselves as Platonic. Mathematicians play with abstract models. Although such models might only exist in our minds, it is apparent that they have some sort of form outside our minds. We can talk about Klein Bottles even if we cannot construct them.

The theory gets messy when applied to government. People holding the theory of forms for government are often opposed to democracies. Democracy is a free for all where the people toss their government up in the air every four years or so to see where the dimpled chads fall. Platonic thinkers would prefer a government derived from a small group of insiders. Popper speculates that Plato believed that man was in the process of degeneration. Plato's Republic describes a way to get back to the ideal government where a philosopher king rises from the military ranks and dictates his will to the under classes.

Popper essentially holds that Plato laid the philosophical foundation for many of the brutal dictatorships that have dominated western history.

One interesting side note: Apparently, the American founding fathers did not read Plato. They were far more interested in characters like the Cato, Cicero, Aristotle, etc.. Meanwhile, the French and the various proletariat revolutions that followed were based on Platonic ideals via Rousseau, Kant, Hegel, Marx and others.

It is interesting that this theory of forms which makes sense when applied to mathematical models, tends to lead to wide scale oppression when applied to political organization.


In my one sided history, I hold Aristotle as a hero. Aristotle had many faults, he was the source of many misconceptions about the nature of physics, etc.. Even worse, he was quite Platonic in his attempts to derive things from first principles. However, Aristotle did one thing that forgives all his faults. Aristotle helped lay the foundations for what we now call logic.

There were many advantages to Aristotle's logic. Logic demands clearly stated premises that are intuitive (by intuitive I mean easy to understand) and refutable. Ideally, premises should be logically independent of each other. The final great advantage of Aristotle's logic is that it was language based. The main feature of Aristotelian logic, the syllogism, fits in well with standard sentence structure and human thought process.

While Zeno's method of finding absurdity in other people's conclusions made him intellectually invincible, Aristotle's method of a language based logic and efforts to uncover simple, but refutable, premises opened the door to a golden age of communication.

Of course, to make the works palatable, Aristotle needed to the address the paradoxes and reductio ad absurdum arguments that mired philosophical discourse. He realized, like Goedel a few millennia later, that it was impossible to have a complete logical system. He also realized that paradoxes exist.

Aristotle's solution to the problem of infinity was quite ingenious. He separated arguments about infinity into potential and actual. Aristotle allows the use of potential infinity, but rejects the use of actual infinities. A potential infinity is simply an unbounded argument. It goes on forever without completion. The limit in Calculus is the best example of a potential infinity. The potential infinity leaves the actual completion of an infinite series as a mystery. A good example is the sequence .99999.... This sequence pretty much behaves as if it were 1. For practical purposes we can treat it as one, but Aristotle's potential/actual split would leave open the absolutist statement that .99999... = 1.

Contrary to what many transinfite theorists contend, I do not believe that Aristotle was trying to ban infinity. I think he was trying to find a way to salvage the use of infinity in arguments without getting mired in the absurd. I hold as proof of this supposition that fact that Aristotle's predecessors (Archimedes, Eudoxes and Euclid) all made use of the potential infinity. By avoiding the great philosophical debates that encumber the completed infinity, Aristotle's actual/potential split opened the door to better mathematics.


The greatest gem from Ancient Greek mathematics is Euclid's axiomatic system for geometry. Euclidean geometry is one of the greatest examples of logical thought. From a small number of logically independent axioms, Euclid was able to create phenomenally diverse and logically consistent universe.

Euclid got around the problems of the infinite simply referring to points and lines...lines just happen to be continuous. Rather than jumping into reams and reams and reams of arguments trying to justify the existence of a line, Euclid simply accepts that we can form a rather solid intuitional notion of a line.

One of the most intriguing aspects of the geometry was the fifth posulate. This posulate stated that given line A and a point B not on A, that there was only one line running through B parallel to A.

Many people thought that the fifth postulate must some how be derivable from the first four. However, the mere fact that Euclid named the fifth postulate as a postulate seems to indicate that Euclid understand that the fifth postulate was logically independent of the first four postulates.


certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof.

Archimedes, Weber State

Archimedes was killed when the Romans conquered Athens. His last request was simply to be left alone with his circles. Archimedes had discovered the volume of many interesting geometrical shapes. He wrote about his methodology in a work called The Method. Unfortunately, The Method was lost for several centuries. The PBS seres Nova has an interesting production on the rediscovery, translation and presevation of this great document.

Around 1000 AD, a scribe created a hand written copy of the document. circa 1200, the document was scraped, then turned into a prayer book (a scraped and reused document is called a palimpsest). The palimsest was identified as a work of Archimedes by Johan Ludvig Heiberg. He took pictures which he transported to England. The book disappeared again in World War I.

Further adventures include a con man wanting to make a fortune from the book creating forgeries over the priceless work of Archimedes and selling the forgeries to a French collector. The owners of the palimpsest sell the document at auction in 1991. Researches have been working to read the lost words of Archimedes since.

In this work, Galileo describes the approach to his discoveries that includes physical experiments and recursive methods (potential infinities) to discover the volumes of geometrical shapes. The work seems to indicate that the Greeks had not banished use of infinity. They simply tried to minimize the times they referred to the infinite.

The Aftermath

The Ancient Greek culture waned. In many ways, the Roman culture with its Ceasars and colloseums was the personification of the Platonic ideal. In Rome, the leaders rose from the military ranks. The lower classes and slaves were clearly in their place. The rulers placated the rabble with gladiator fights, etc.. Of course, Rome also produced Catos, Cicero and others who argued for the open society. Rome had its own unique characteristics and style of rule. It is difficult to label it as either Aristotelian or Platonic.

Black deaths, wars, mini ice ages and other blights dotted time. The empire declined, the Catholic Church rose. The Catholic church was more interested in spiritual and political matters. Science and math merged with superstition. Wallace's book Everything and More seems to imply that the dark ages were a direct result of Aristotelian thought. To me, the Ceasars, and the "One Universal Catholic Church" seems closer to the Platonic ideal than to the Aristotelian ideal. Timewise, Euclid and Archimedes better qualify for the title of heir to Aristotle than Attila the Hun. Regardless, the period between Archimedes and the Renaissance generally holds little interest for modern mathematicians. The greatest contributions during this period seems to come from the Arabia with the invention of Algebra and Arabic counting systems.

In the 13th century two names appear: Thomas Aquinas (1225-1274) and Roger Bacon (1214-1294). Thomas Aquinas was instrumental in reviving interest in Aristotle and logic within the Catholic Church. As a extremely devote follower of the Catholic Church, he added his spiritual twist to debates about infinity. Doctor Mirabilis (Roger Bacon) was dismayed with the dogmaticism of Oxford. Giving up on the ability to derive good chemistry from first principles, Bacon realized the importance of experimentation and dialog in Aristotle's works. Rather than debating in an intellectual vacuum, Bacon developed the foundations for modern science.

This revived interest in the works of the ancients took a century to permeate society. Promising discoveries in the wake of Bacon's scientific method and a deepening appreciation for logic by the followers of Aquinas had built to a critical mass in the 1400s. When Brunelleschi redisovered the principles of linear perspective and Gutenberg invented movable type, the world was already a powder keg looking for a match.

The Renaissance

I like to pretend that the Renaissance began when Filippo di ser Brunelleschi drew his famed perpective picture of the Baptistry of Saint Giovinni. Of course, the Renaissance was not simply the creation of a single man, a group of men or ideology. It was a cultural reawakening to the wonders of discovery.

The academic community was reading the works of the ancients and were learning to master the new numbering systems and algebras from the Arabs. Painters and scientists were looking to make their own mark. It was a time of great discovery made possible in part by the works of Bacon and Aquinas.

Although I would love to talk about linear perspective, for the sake of brevity, our history needs to move next to a paradox discovered by Galileo.

Refutation ~ Critique of the Diagonal Method ~ Links Alive