A Critique of the Diagonal Method

But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite understanding...

Galileo Galilei

The diagonal method is said to do an amazing thing. It is said to prove that, while the rational numbers are "denumerable" the real numbers are not. This dichotomy between the rational and reals is considered the basis of arithmetic, calculus and all higher mathematics.

I find myself on the unpopular side of the issue. From my view, transfinite theory is essentially a dichotomy couched between two paradoxes. Those admiring the Aristotelian tradition of logic dislike the theory as it builds the foundations of mathematics on paradox. This section of Descriptive Mathematics explores the paradoxes of transfinite theory and suggests that we would be better served by stopping the modern habit of treating paradoxes as a foundational issue and treat them as an interesting aside.

The site has gone through several iterations. The last iteration, A Tale of Two Paradoxes, is the most readable of the essays. However, I've left the first versions of the essays online. The Tale of two Paradoxes emphasizes that the diagonal method is simply a convoluted form of the liar's paradox. Transfinite theory begins with Bolzano's interpretation of Galileo's paradox to assert that the rational numbers are denumerable, the concludes with the liars paradox to show that the reals are not.

Transfinite theory is extremely seductive in that it gives mathematicians a feeling that they can derive the fundamental nature of the universe from contemplating paradoxes. However, paradoxes let you prove anything you desire. We can encapsulate paradoxes in volumes of symbolic logic. We can mask paradoxes as fundamental definitions. However, paradoxes are paradoxes.

I have also left the previous essays online, and two brief essays on the infinite hotel and repeating nines. I suggest strating with A Tale of Two Paradoxes.

Site History

The first Iteration of this site was a simple parody of the diagonal method. The goal was to show a few of the problems I had with the theory and to inject enough cynicism into the presentation of the diagonal method to allow students room for critical thinking.

As I was getting a good amount of traffic on the site, I started thinking about creating a more serious refutation of theory. I admit, after reading David Foster Wallaces book Everything and More (which presents the party line on transfinite theory) my blood was boiling a bit.

In the second iteration of the theory, I decided to move from simple parody of the diagonal method to a more serious effort of refutation. The iron curtain fell. The transfinite wall might fall as well. This second iteration starts by exploring the role that Hegelian other dialectical methods played in forming transfinite theory. Conversely, it looked at the role transfinite theory played in reinforce the new paradox based dialectics that served as the foundation of the regimes that devastated the Soviet Union and Germany. People who enjoy writing incendiary literature about intellectuals run amock could pen a great article about the evolution of transfinite theory.

The corner stone of the refutation was an essay about Galileo's paradox. I wanted to show that the diagonal method "disproved" Bolzano's interpretation of Galileo's paradox. Cantor thought that the diagonal method created a dichotomy between the rationals and reals. I wanted to show an interpretation that showed that the diagonal method shows that we cannot place infinite sets in a 1-1 correspondence with a subset of itself.

The problem with this second approach is that I was resorting the to same tricks as the "dialecticians" I hated. Even worse, I was trying to refute the paradoxes of the infinite with more paradoxes.

Refuting paradoxes with paradoxes cannot be done. The problem with paradoxes is that they let you prove anything you desire. Of course, different groups might have different interpretations of the fundamental paradoxes. But everyone holding the myriad of views we can generate from paradoxes will see their arguments as invincible.

Half way through this second iteration. I threw my keyboard down and said NO MORE! I haven't even a clue about what is in the second iteration. I simply stopped mid sentence.

It then finally dawned on me. I don't think very many people realize that the diagonal method is simply a convoluted version of the liar's paradox. I thought about how math, literature and politics has been dominated by paradox for the last two centuries. I decided to steal a title from Dickens and give one more go at an article on transfinite theory that emphasized that the diagonal method was nothing but a twist of paradoxes.

The point of the diagonal method is to show the difference between discrete and continuous mathematics. So in the article I wanted to emphasize that the differences between these subjects flow from the definitions of the subjects.