Axes of Formal Thought

In Search of a Comprehensive Taxonomy

There isn’t one, single way of formulating a taxonomy of formal methods, but there are ways that are more or less helpful for how a given individual approaches formal problems. For years I’ve been casting about for such a taxonomy, and I have worked out different ways of doing this.

At one time I wrote a series of posts on the Banach-Tarski paradox (which is a paradox in the sense of being counter-intuitive, but not a paradox in the sense of being contradictory; cf. the list of resources, below, under “Wordpress Posts on the Banach-Tarski Paradox”), and in working through these ideas I came up with the following classification of formal thought:

This was obviously built around the motivation of understanding the Banach-Tarski paradox, which involves decomposing a sphere into a finite number of parts, and then reassembling the parts into two spheres. I realized that this was interestingly complementary to what is going on with fractals, in which there is an infinite iteration of a finite operation. Thus the Banach-Tarski paradox and fractals are kitty-corner from each other in the above table, with the other two permutations of these possibilities filled in by primitive recursive arithmetic (PRA), with its finite operations finitely iterated, and the possibility of infinitely iterating infinite operations, which would be infinite fractals.

While this is interesting, and it was the attempt at a taxonomy that pointed out to me the possibility of infinite fractals (I don’t know if anyone else has suggested this possibility), as noted above this taxonomy is highly derivative from my thoughts on the Banach-Tarski Paradox, and I wanted something less bound by a particular problem. Searching for other approaches, I also formulated the following table:

I honestly don’t remember what I was working on when I drew up the above table, though it obviously embodies my ongoing interest in trying to formulate big picture concepts, with the lower right permutation being the widest possible permutation of formal thought, and the other spots in the table indicating other approaches that are more limited in some respect than a formal overview (cf. The Overview Effect in Formal Thought, also linked below in “Studies in Formalism”).

While this table has some interesting features (classical mathematics appears as “method without any unifying conception,” in the upper right of the permutations), this, too, was too limited and unsatisfying, so I eventually formulated a table that lays out as its two axes the ideas that have long held the greatest fascination for me: formal/informal and constructive/non-constructive:

Here the permutations don’t quite sound as interesting as the above two attempts at a taxonomy of formal thought, but further elaboration allowed me to employ these axes in a comprehensive way that allows us to lay out a variety of formal approaches in one table. This more detailed exposition of the immediately above table is the table at the top of this post. Here the x axis is constructivity, from the constructive to the non-constructive, while the y axis is formality, from the formal to the informal.

With this taxonomy I am able to put Platonism, Neo-platonism, mysticism, computer science, dialethism, insolubilia, the ineffable, intuitionism, classical mathematics, and much more all in one table, and showing the various relations of these disciplines to each other. I could, and maybe someday I will, further expand this table in order to place more instances of distinctive approaches to formal thought within the same matrix.

In the same way that this table didn’t sound immediately as interesting as my other attempts at taxonomy, it also didn’t suggest anything new to me, as the Banach-Tarski derived table suggested infinite fractals to me, but it does seem to furnish a more comprehensive and flexible taxonomy, and its further elaboration and exposition may offer insights to matters now hidden merely on account of our deficient comprehension of the totality of formality.

Tumblr posts on Constructivism:

• P or Not-P
• What is the Relationship between Constructive and Non-Constructive Mathematics?
• A Pop Culture Exposition of Constructivism
• Intuitively Clear Slippery Concepts
• Kantian Non-Constructivism
• Constructivism without Constructivism
• The Vacuous Identity Principle
• Permutations of Infinitistic Methods
• Methodological Differences
• Constructivist Watersheds
• Constructive Moments within Non-Constructive Thought
• The Principle of Vacuous Pragmatism
• Precise Generality
• Addendum on Precise Generality
• The Two Philosophies of Mathematics
• From Gödel to Historiography
• Gödel between Constructivism and Non-Constructivism
• The Natural History of Constructivism

Wordpress posts on constructivism:

• Cosmology: Constructive and Non-Constructive
• Saying, Showing, Constructing
• Arthur C. Clarke’s tertium non datur
• A Non-Constructive World

Wordpress posts on the Banach-Tarski Paradox:

• A Question for Philosophically Inclined Mathematicians
• Fractals and the Banach-Tarski Paradox
• A visceral feeling for epsilon zero
• A Note on Fractals and Banach-Tarski Extraction

Studies in Formalism:

• The Ethos of Formal Thought
• Epistemic Hubris
• Parsimonious Formulations
• Foucault’s Formalism
• Cartesian Formalism
• Doing Justice to Our Intuitions: A 10 Step Method
• The Church-Turing Thesis and the Asymmetry of Intuition
• Unpacking an Einstein Aphorism
• The Overview Effect in Formal Thought
• Einstein on Geometrical intuition

Aristotle, the founder of logic, as depicted in the Nuremberg Chronicles.