(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).
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.
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.
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