suggestive commentary (with apologies to Rucker, Pickover, Nagarjuna, and Hofstadter)

Okay, I'd written something and it got eaten. Let's see if I can recover it as best.




Here, have some quotes:

From wikipedia's article about class in set theory

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets). A class that is not a set is called a proper class.
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Moving onward to the article on intuitionistic type theory, the following:

"A category with families is a category C of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor T : C^op -> Fam(Set). Fam(Set) is the category in which the objects are pairs (A,B) of a set A and a set-valued function B, and the morphisms are functions f : A -> A' between the index sets to gether with, for all indices a : A a function g : B -> (B'.f)."

The important point is that there's a notion of context. I have a copy of Barendregt's opus about lambda calculus, in which he defines a context as a term with some holes in it. What's important to take from that is cohomology is mathematics (or at least algebraic topology's way of detecting holes in a space -- indeed an expression in lambda calculus is a kind of space, if I want to take the metaphor that far)

Moving on to the metaphysical theory of types:

"A type is a category of being. A human is a type of thing; a cloud is a type of thing (entity); and so on. A particular instance of a type is called a token of that thing; so Socrates was a token of a human being, but is not any longer since he is dead. Likewise, the capital A in this sentence is a token of the first letter of the Latin alphabet."


Going to the theory of types:
"

At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. Objects of a given type are built up from objects of the preceding type. Types in this sense are related to the metaphysical notion of 'type'. Bertrand Russell invented type theory in response to his discovery that naive set theory was afflicted with Russell's paradox. Type theory features prominently in Russell and Whitehead's Principia Mathematica.

• "...[T]he property of being a perfect square can be significantly predicated of both cardinals and ratios of cardinals; but the property of being odd (or even) is not defined for the ratios. We are thus unable to answer the question whether 2/3 is odd or even." ^[1] --Ernest Nagel"
  

The hierarchy of types here is a kind of kowtowing to the natural numbers. Are types countable? Could they lie on a 72 dimensional symplectic manifold?


From type -- model theory

"In model theory, a type is a set of formulae in first-order logic such that all the formulae could be consistent descriptions of a certain element (or several elements) of the model. A complete type is a maximal such set (that is, for every first-order question about the elements, the type answers either "yes" or "no")."

Okay, I'll see if I can reconstruct what I wrote before my browser ate my post.


Set theory is about sets, not about elements of sets. Category theory is about categories, not objects of categories. Group theory is about groups, not objects of groups. Type theory is a theory of types not of tokens of types. Projective geometry is about projective spaces, not about points and lines. Each one of these theories has a fundamental, atomic abstraction beyond which we'll consider inviolate. Each one of these is a particular way of understanding with respect to its fundamental abstraction, and based on that fundamental abstraction, it can produce results over some scope of mathematics. It's possible to relanguage and reframe the contexts of each so that discoveries in one are discoveries in the other, thus advancing the raft cooperative toward more understanding, making our local patch of math larger.

Along the edge is gnarly froth: the systems stick and have massive territories in each of them. In the edges of set theory -- Burali-Forti and Russel's paradoxes, sex cells for type theory are released, in the frothiest vapors of type theory come Turing, Church, Chaitin, and so forth. In the work on the lambda calculus and type theory sex cells for computer science and algorithms research. In the froth of set theory Cantor's continuum hypothesis remains. In number theory, Riemann's hypothesis remains in the froth.

One of the things I wanted to point out was that different languages or contextual framings in theory gave rise to different kinds of froth. The fundamental concern with something in some context is there. This blog entry no different. The interesting place is neither the raftspace we inhabit, our particular indranet, or the particular indranets/framings of certain branches of mathematics (it may be possible to topologize mathematics by converting the theorem trees into graph theoretical structures and then doing historical analysis of those structures given the tools of graph theory, which is a quirky kind of jumping out of the system).

These systems are like particular collectives of rafts, all intersecting. Each one has its fundamental abstraction. And those fundamental abstractions are oddly regarded as different things. There is an extend of sealing-off one branch from another.

Hofstadter notes, in GEB, that "no system can be its own metasystem", each one of the particular styles or abstraction languages which mathematics is performed cannot wrap around itself. The attempt here is not to provide some overarching language for mathematics, that's been attempted and proven pointless.

Now for the bold claim. Mathematics is still concerned about one object, the abstractions, transformations, evolutions, transcriptions and so forth of that object. Sequencing of copies of that object, etc. I think that Grothendieck is in the right direction when you hear clamoring that there's one prototypical point. Because if all the calculations and stuff you do that end up being numerically verified by sake of computations or integrations or so forth to particular numbers, whether they be real or complex or so forth, for as long as they are things which you can just hold in your hand in the sense that they're not apeiron, then, you're not really dealing with the froth. Rucker correctly interpreted that the concern is the gnarly froth. The answers to questions can be shapes, not numbers, they can be motions, not scribblings representing motions on paper. The kind of high-falutin' sensory escapades of math find their modus operandi in tiny scribblings of ink on paper. Questions like: "what is the area of the mandelbrot set?", and "what is the volume of this cone?" are exactly the kind of logos reductases I think are a kind of bad idea in the long run, no matter what kind of practical consequences they have, because they encourage the distilation of structure; they're object-centric kinds of things. Each one of the above mentioned disciplines attempts to impose/determine/detect/filter/distill structure from the objects in consideration, or rather, the object in consideration. This object is no-thing.

Mathematics is perhaps the best form of meditation available. Here's the point of departure analogy: mathematics offers a kind of portability which cannot be found in religion.