I fear this portion of the web will appear a bit reactionary. In my opinion, the goal of mathematics is simply to describe abstract spaces and logical models. Math is not a secret cypher from which a seer can derive the hidden conflicts of nature and truth. This essay is harsher than what should appear in math texts. Unfortunately, the diagonal method helped bring oppositional logic and rhetoric into the foundation of mathematics and into forefront of thought in German, Russian and American society.
The diagonal method was one of many similar foundational theories based on oppositional logic that appeared in the 1800s. The goal of this essay is to present the method within the context of its day. As with Marx, Cantor had some valuable insights on the nature of irrational numbers and continuous spaces. The problem with both theories lie with the process of elevating a single dichotomy to the forefront of the subject.
This particular essay does present any strong mathematical arguments regarding the theory, it is simply a look at the pathology that had affected a large number of subjects including mathematics.
Transfinite Theory was one of a number of foundational theories produced in the 19th century based on oppositional logics, the dialectics from Zeno and other sources outside the classical Aristotelian tradition. The 1800s intellectual environment in Europe was dominated by efforts to cast aside the Enlightenment's naive belief in a fixed, rational reality, and to build new methods of thoughts on the edifaces of the works of Rousseau and Kant. Chief among these foundational theories were: historicism propounded by Hegel, communism developed by Marx, psycho-analysis espoused by Freud.
The grand new philosophical systems were not simply creations of the left. People of all political bents were finding and building their political beliefs on a variety of oppositional terms. A good example of this is Ayn Rand (of the 20th) built a libertarian philosophy that incorporated the conflict between the individual and society into the foundations of objectivism–The reverse of Marx.
The goal of oppositional logic is to explain subjects through the opposition of terms. Cold only has meaning in relation to heat. Good only has meaning in relation to evil. There are two sides of the force—dark and light. Each of these programs begin with a series of profound observation about the nature of a subject. The dialectician then creates a dichotomy to explain the deep inner workings of the system. The dichotomy would often have the form of a thesis, antithesis and catharsis. For example, Freud gave us an id, an ego and super ego. Kant had something going on with a priori, a posteriori and synthetic. Cantor's dichotomy is between the rational and reals. He gives us the denumerable, non-denumerable and transfinite. Hegel and Marx were a little more complex. Both involved a world spirit resolving several thesis/antithesis splits in monumental bloody clashes, then moving on to the next conflict. Marx's main program gave us the bourgeoisie, proletariate and avant guard. In common terms, the capitalist, the worker and revolutionary.
The oppositional programs often gained a devote following by providing adherents with magificient powers. Hegel promised this his "scientific" method would allow leaders to predict and control the forces of history. Marx promised his followers that they would lead the oppressed to a workers' paradise. Freud promised the ability to see others' hidden motives, and Cantor promised the ability to see infinity.
The adherents to the different philosophies generally formed tight groups that actively defended their philosophies against all attacks and had clever ways of getting their viewpoints pushed into the center of their subjects. The word elitism jumps to my tongue. Regardless of motivation, the effect has been to create a world filled with isolated, often brutal camps vying for power.
As with Zenos dialectics in ancient Greece, modern dialectics had a strange fascination with paradoxes. The reflexive paradox will exist in all systems that are sufficiently complex as to allow for self reference. For example, what does a democracy do if a majority voted to disband the democracy. The ability of a democracy to collapse is often given as proof of its inferiority to other stable systems. However, this same paradox occurs when a totalitarian leader dictates that there should be an election.
The study of infinity brings in its own mix of paradoxes when we looked at the Galileo's Paradox). Other paradoxes occur when we try to speak of all inclusive entities such as the set of all sets, the world spirit or the will of the people.
Rhetoricians love paradoxes. When you have a good set of paradoxes at your disposal, you can prove anything you desire.
Traditional logical systems tried to push the study of paradoxes to the fringes. When you pull the paradoxes into foundations (or meta level) then you create a situation where the prize goes to those with the most forceful will, and not to those with sound arguments or to those with sound judgement. When we pushed the paradoxes into the foundation of logic and math, by default we end up handing the world to rule by madmen.
The methodology of many modern dialecticians is to start with desirable outcomes, then to look back and find a set of opposition terms and paradoxes that will support the desired outcome. By starting with desired conclusions and working backward, it is easy to create a seductive system. In Marx's case, the desired outcome was a system where workers received a greater share of the produce of their labor. Cantor desired a logical foundation that would let him explain the nature of continuity.
When it comes to the actual rhetorical battles, systems that start with the desired outcomes often gain an upper hand over those system starting with premises and deriving theorems through logic. For example, if you start with the premise that people should be free to make their own decisions and live with the consequences of their actions, then you will arrive at a society where there is an uneven distribution of wealth and a large number of people suffering the consequences of their actions. A person who starts with the desired outcome of butterflies of happiness fluttering through open fields, will win out over those promoting freedom.
This combination of logical paradoxes, opposition of terms and loaded outcomes has a unique ability to make a person feel intellectually invunerable.
Unfortunately, creating a defenseable wall around your personal belief system does not necessary create the desired outcome of peace and happiness. The invincible system of argumentation leads to a totalitarian mind set. Often the practicianers of the totalitarian mind set find themselves victims of their invincible methodologies when the theorems spin out of control. We end up burying ourselves in a hole hoarding our useless treasures.
Regardless, The foundational systems built on oppositional logic are often extremely seductive. The theories include profound insights. The program generally has extremely admirable goals. The followers have wonderful rewards for defending the theory. Unfortunately, the program tends to lead to closed societies. By elevating single conflict into the core of existence, we create an imbalanced system. Such one dimensional theory built upon the opposition of terms usually falls far short of a system that takes the harder route of identifying attributes and premises built into a system.
For this reason, I consider Cantor's theory build on the opposition of the denumerable and non-denumerable to have less substance than one that would exist by building on a study of arbitrarily large sets, and dedicated to finding the richness of integers, rationals, algebraic and real numbers on their own. Denying the dichotomy does not invalidate Cantor's astutes insights. Denying the dichotomy gives the world a rich theory that appreciates the unique characteristics of integers, rationals and reals. It also gives us the ability to better understand the nature discrete and continuous sets.
Unfortunately, when a oppositional theory is accepted as a foundation for a subject, it takes a long time to rebalance the subject. There is hope. The Berlin wall did fall, and there has been a wide acceptance that Communism, Naziism and Fascism (all three systems with foundations in oppositional logic) were a failure. The scientific community has gradually been debunking Freud. Mathematics might someday reject the denumberable/non-denumerable dichotomy.
In The Open Society and It's Enemies, volumn II, Karl Popper does a great job of debunking the so called "scientfic foundations" of Hegel's historicism.
Unfortunately, this process takes an extremely long time. It is even harder in the cases of Marx and Cantor, as both of these intellectuals made substantial real contributions to their subjects.
It is a difficult disease to cure. Cantor has given us some great insights into the nature of space and the irrational numbers. It is simply that in his struggle to explain his ideas, he took the short cut of using oppositional logic.
I promised some reactionary literature. So here goes.
I suspect that the very reason that you are reading this article at the moment is because you realize that the ideas you develop in your life are important and that you are trying to create a mental map of the world that will help lead you and the people you care about to a better future. This is why we study and read books, because there apparently is a connection between learning and prosperity.
Although intellectuals pretend that thought is benign, history shows quite clearly that there are some systems of thought which lead to abundance and others that lead to deprivation.
The body counts show clearly that the great foundational theories that issued from 19th and 20th century Germany and Russia led to inhumanities on a scale never seen before.
If you accept that different foundational systems have different outcomes, then the challenge for the intellectual community is to promote a foundational methodology that is conducive to the welfare of man, and to reject those that do not. I contend that oppositional logic and oppositional rhetoric lead to negative results.
The problem is not with the methods employed in the research. The problems I see occurs in the implementation. The problems occur when ideas come out of the academic community, and start being integrated with other ideas and movements. This is especially true with foundational issues like language and logic.
Achieving prosperous ends for society is a very complex and difficult subject. For example, I believe that academic freedom should be one of the primary tenets of a progressive system. In accepting academic freedom, I are confronted with the possibility that some scholars would reject the premise of academic freedom (i.e., feel the need to impose their ideas on others). In a free society, how do we accomodate those who wish to deny freedom to others? The reflexive paradox rears its ugly head.
Fortunately, the academic freedom paradox is easy to resolve. There is a clear distinction between an individual and society. An individual holding totalitarian beliefs is pretty much a non issue. We can draw the line at imposing one's will on others. I can believe that I am Napolean reborn, but when I get off my Elba to raise my army, I cross the line.
Discussing foundational issues is still tricky. In order to communicate, we need a common foundation of language and logic. Addressing foundational issues requires more deliberation and a participatory process. What you see happening in Cantor's day is a large number of people working at this foundational level. We see people trying to impose their view of the world at this foundational (meta) level.
A good example of this is study and classification of species and races: It is well known that animals evolve. When a population of animals or plants is held in isolation for a long period of time, the population often evolves into a distinct species.
Different groups of people developed in isolation from each other. It is good science to look at these different populations and ask to what extent speciation occured. Such questions are good science. Problems occur when people take the information from such studies of race and try to form a race based social policy. People claiming to be followers of Darwin have actually advocated removing entire groups for the gene pool!!!! This is not the fault Darwin, it is the fault of the process of perverting scientific ideas.
Individual people exploring ideas is not a problem. The problems occur when the theories become a mass social movement. It is important that this process of disseminating ideas in society is open and allows for deliberation.
The dangerous ideas are those aimed at by passing the process of deliberation. Rhetoreticians that try to redefine subjects at both the meta and practical level generally have the goal of short cutting the process of deliberation, and often result in oppressive systems of thought. It is important to note that the oppositional rhetorics of Hegel and Marx were designed to inspire mass social action by simultaneous changes in both the process and substance of discourse.
Cantor is a very interesting study in this regard. As a mathematician he was deeply interesting in foundational issues, and was caught up in the millenial old debate about the nature of continuity and infinity. Yes, he wished to make his mark on the world. That is not a problem. For that matter, I admire Cantor for his independent thought and for facing the hardships of getting his ideas published and accepted. Cantor's biography is a great example of mathematical inquiry at its finest.
From a biographical perspective, the most interesting thing about Cantor was that he was locked in a battle of wills with a pointy haired boss [Dilbert Reference]. This boss seemed to be holding to Schopenhauer's belief that the will creates reality. Kronecker's intent was to completely ban discussion of irrational numbers and infinity from mathematics.
Cantor's work on trigonometric series led him to some extremely profound observations about the nature of irrational numbers and continuity. The difficulty Cantor encountered was trying to find a way to explain the difference between discrete and continuous sets. This problem was magnified by the fact that any explanation he gave would be automatically dismissed by his boss and former mentor.
Many biographers conjecture that Cantor suffered from from manic depression. This means Cantor's highs were higher and his lows lower. Magnify all the feelings you have on a typical day by ten. When Cantor was manic it is possible that he felt he was being inspired by God. There are interesting points in Cantor's biography. For example, he felt that the Aleph's had been specifically preserved by God for use as the first transfinite number.
When Cantor stumbled on the diagonal method, it would seem natural that this was the tool that would explain to the world the difference between the rationals and reals; Hence, Cantor developed the dichotomy between denumerable and non-denumerable sets.
All of this is well and good. This is the way mathematics and science advances. People develop a theory. The theory goes out to the scientific community for review. When the scientific community finds a problem, the theory returns to the lab for further refinement.
Unforunately, the turn of the 20th academic community was rather dysfunctional. Cantor's theory was swept into the great social movements of the day without the proper deliberation needed for a foundational issue.
Although Cantor was not a fan of Hegel, the young Hegelians took to Cantor's work with a vengeance. Cantor's life confirmed their world viewed. The Hegelians were anxious to cast aside the constrainst of traditional logic. The fact that Cantor's work included a dichotomy made the theory extremely attractive. Even better, it was easy to cast the oppressive boss as the standard bearer for traditional logic.
I have almost no objection with Georg Cantor. The problems lie with the great wave of activity that caused a theory with a rather blatant flaw to be accepted as a new foundation of logic and mathematics.
Transfinite Theory became one more brick in the great dialectical wall built by the Hegelians. Having a dichotomy at the foundation of the number system further legitimized the oppositional rhetoric used in the nationalism that swept Europe and that used in the Bolshevic revolution. It is this oppositional dialectic that was the foundation of both Communism and Nazism.
I hope you can see how the appearance of a person claiming to be able to see infinity had an enormous effect on racist doctrines in Germany. There was perhaps even thought that the ability to see a completed infinity was something unique to the Germany psyche. Casting asside traditional logic made Germany and Russia more susceptible to totalitarian governments.
It was not the improved understanding of the irrational numbers that caught the world's attention. It was the promise of being able to see infinity and the dichotomy that caught the world's imagination.
The strangest twist in the history of transfinite theory occurred when Bertrand Russell pointed out that Cantor's work on classes (the set of all sets...) was subject to the reflexive paradox.
Now, in traditional science, when a problem like this is discovered, the theorists go back into the lab for further investigation. The single most interesting thing about transfinite theory was that the discovery of a paradox at the basis of the theory did not result in the great academic deliberation. Instead it propelled the theory in the stratosphere.
All the people fascinated with paradoxes came out of the woodwork to add their humorous twist to the liar's paradox.
As for those intellectuals with a totalitarian mindset, the fact that there was now both a false dichotomy and a paradox at the foundations of mathematics and logic was a God send.
As Zeno realized in ancient Greece, once you have a paradox at the meta level, you become invincible.
The tragedies that occurred in Russia and Germany were not about liberal v. conservative. It was people totalitarian mindset that employed the oppositional dialectics that caused the great miseries of the 20th century.
Next ~ Refutation ~ Critique of the Diagonal Method ~ A Fountain of Bargans