Coincidentally, #42

Is this obscene?

Current (global?) status on Quora for those topics (Nathan Coppedge). See: https://www.quora.com/profile/Nathan-Coppedge/highlights to view all rankings.

View my papers at: http://www.southernct.academia.edu/NathanCoppedge

Important Writing on Coherence July 2016

Perhaps the clearest I have ever written about the incoherence problem.

https://www.quora.com/Does-a-link-exist-between-the-theorem-of-incompleteness-and-the-circularity-of-definitions-in-every-language/answer/Nathan-Coppedge

Systems Expressed in Modulo 2

Historical Theories (dialectic structure)

1. History is a material progression.
2. Critical theory implies a dialectic of history.

Math (linear structure)

1. + / - operators define space.
2. The result is meaningful, because it expresses numeric values in two dimensions.
3. Two-dimensionality allows mapping of more complex equations.
4. Because math is more useful than before, math must be useful.

Coherent Categorical Deduction (logical structure)

1. Opposites are exclusive of all but neutrals.
2. The only pure neutral is zero.
3. All words have opposites, however absurd.
4. A word can be found for anything that has a concept.
5. Opposites must be opposed diagonally, to have the longest possible distance.
6. Either comparisons cancel out (opposite vs. opposite), or direct comparisons must take place between non-opposites within the set.
7. In quadra, therefore, there are only two combinations (with A arbitrarily selected as primary, and secondary choices having no express set order): AB-CD and AD-CB, since opposites must remain in opposite positions, but secondary opposites can exchange places.
8. Since no other combinations are possible in term of the expressed categories without producing either opposition or a different, equally exclusive, and non-contradicting category set (since only opposites contradict, and opposites are always expressed within the existing set no matter how many sets there are), the two combinations can be deemed coherent for quadra.

This writing was a product of the question What is Theoretical Justification? on Quora: https://www.quora.com/What-is-theoretical-justification/answer/Nathan-Coppedge

PHILOSOPHICAL THINGS I HAVE LEARNED FROM MATH

Applicationism: so far as it can't prevent pain, philosophy is basically a reasoning enterprise.

Typological Wholeness: whole numbers are fairly sacred.

Process = Progress: if you want to add more process, it has to mean something.

Whole-Part Relations: subsets matter for the set.

Binding Formulas: oftentimes, logic requires specific formulations.

Maximal Incompleteness: if you want to be complete, you have to be good.

Coincidental Genius: what makes something good is that it is good, not that you think it's good.

Tropism: different rules hold under different conditions.

Standardism: accept or reject the rules, but formalism first and informalism second.

Populating the Data: structuring a system may require populations of lesser concepts.

Degrees of Abstraction: modes are clearly less than systems, but more than variables.

Conquer the Problem of Identity: avoid arbitrariness. Use acceptable categories.

Be Rigorous: make sure a system is a system, and not an arbitrary system.

Use Readable Language: more than being simple, being legible.

HOW TO DO MATHEMATICS WITHOUT DOING MATHEMATICS

Predicate calculus can be pretty limiting.

Take everything you know (maybe infinity), and think again!

Forms of Deduction Based On An Introduction to Higher Order Logic

(J. Lambek and P.J. Scott. Introduction to Higher Order Categorical Logic. Cambridge: Cambridge U, 1986.)

An introductory manual dating from 1986 suggests A can be derived from B in certain cases in which a bounded coordinate system is used.

The text suggests that one of the key importances of the method is to derive the significance of pi in terms of phi (typical mathematical explanations).

This points towards an at least three-part method of deduction based on mathematics, expressly:

1. Relative identity to pi.
2. Phi translation of pi (formal equivalence).
and,
3. Modular equivalences between phi and pi.

However, irrational numbers are not what we need with universal knowledge, so I'm afraid these people must be secretly stuck on a 'Kantian boat'!

I argue that the relation to irrational numbers is arbitrary except when 'pi' is translated to mean 'coherence', since irrational numbers lead to a problem with trans-finite boundaries. Thus, the number-relation has no definite significance except modularly.

In terms of coherency theory, the 'three deductions' mentioned above may be expressed as:

1. Relative absoluteness (qualified and quantifiable nominalism).
2. Formal equivalence again.
and,
3. Philosophically-systematic variations of geometry.

There is no indication that math grants these deductions to be exclusive.

In fact, doubt could even be thrown on whether deductions of such a simple form are deductions in the first place.

Therefore, it seems justified to limit the importance ascribed to this earlier text in establishing methodological coherence, even while at the same time the authors do appear to grasp some of the fundamental workings of a coherent system.

In the context of this work, it appears that the philosophical importance of the methods has been ignored in favor of a mathematical explanation.

But it is significant that the authors held that B could be derived from A in such a way where B ^ 2 = A. This equation is functional for square Bounded Cartesian Coordinates in which it is assumed that A refers to  the total number of categories, and B refers to the number of deductions.

However, it is unclear whether this is what the authors intended, for although the context relates to closed cartesian coordinates, the context is also most directly related to graph theory and Hilbert Spaces.

The philosophy application could obviously look simpler, but it also looks to me to be more relevant to higher order categorical logic.

Categorical Introduction to Set Theory

Let {} represent a set.

{} is boundless or represents a definition.

By Wittgenstein, definitions are atomical facts. Conceptually, a definition can have any volume, and any shape a definition takes is a formality which seeks justification.

The argument that {} represents more than a point in space is untenable unless a formality is adopted.

{However, a point in space may contain any form of representable information}

Now consider three variables, which are NOT necessarily mathematical:

'{}, info, formality'

[{}] is the form of [formality]

[info] is the contents of the [{}]

[{}] is the observable [formality]

[{}] is the formal definition

There are infinite points in a non-typological line.

If [{}] is finite, it does not take up space, unless that space is typological.

To be an observable formality, the [{}] must be infinite or typological, or both.

However, [info] necessarily has a definition, a necessary bound.

[info] without definition would be an unbounded unbounded [{}], an empty [{}].

[info] must be bounded, while the [{}] must be unbounded.

Essentially, the [{}] must formalize the [info], or the result is an empty [{}].

The obvious answer is that the formality of a [{}] consists of typological [info].

At this point, the missing variable is mathematics.

I commented at M-Phi (Mathematical Philosophy Blog)

http://m-phi.blogspot.com/2013/08/jacquettes-argument-for-inconsistency.html

http://m-phi.blogspot.com/2013/08/isomorphism-groupoid-in-gr.html