Ever since being introduced to mereology by my friend and philosophical co-presenter/mentor, Stephen P. King, (you can hear one of those introductions in our recent radio chat), I have thought in the back of my mind that there is either something promising or distracting about it. Is it a piece to the puzzle, or a piece that doesn’t belong? This itself is something of a mereological question.
Underlap Uxy =df ∃z(Pxz ∧ Pyz)
Overlap Oxy =df ∃z(Pzx ∧ Pzy)
Proper Parthood PPxy =df Pxy ∧ ¬ Pyx
Equality EQxy =df Pxy ∧ Pyx
Proper Parthood PPxy =df Pxy ∧ ¬ Pyx
Proper Extension PExy =df ¬Pxy ∧ Pyx
Transitivity(Pxy ∧ Pyz) → Pxz
Antisymmetry (Pxy ∧ Pyx) → x=y
Mortality and mereology seem like an unlikely pairing at first glance. Mortality is such a powerfully real and ubiquitous influence on living beings and this philosophical study of parthood relations is so abstract and obscure, but what is it death except a cessation of wholeness? A bullet hole will perforate and fragment, a heart attack will stop the circulatory support for the brain, old age will chip away at all of the systems until one part fails to prop up the whole. Death, decay, and disintegration are closely related.
With Stephen’s knowledge of mathematics and philosophy, we have long been trying to put our finger on the precise nature of the subject-object dualism. While my mind favors word pairings rooted in direct experience, like literal-figurative, sensorimotive-electromagnetic, perceptual-relativity, and significance-entropy, his intellectual territory covers more formal models of analytical mathematical truth, the Stone duality, The Pontryagin duality, Bisimulation, Non-wellfounded Set Theory, etc.
Stone Duality relates algebra to geometry:
Putting Stone’s programme in categorical language, let A be some category of “algebras” and S one of “spaces”, the exact nature of which we leave open. Then by a Stone duality we mean an adjunction
in which TX is the algebra (maybe of open subspaces) associated with a geometrical object X, and PA is the space of primes of an algebra A.
It is a revolutionary theory of topological spaces and continuous functions that treats them directly, just as traditional geometry was about lines and circles, without smashing the continuum into dust. ASD provides a natural language for real analysis that describes the solution-space of an equation continuously in its parameters, even across singularities. Since it is presented syntactically, in a way that generalises ordinary algebraic notation, it is inherently computable. It was inpired by Marshall Stone’s study of the categorical duality between topology and algebra, taking his slogan “always topologize” seriously by topologising the topology. It also exploits the analogy between continuous and computable functions, on which Dana Scott built the theory of denotational semantics of programming languages.
Pontryagin duality (file under ‘how can anyone understand this):
Non-wellfounded sets and Bisimulation:
From the trusty SEP:
“The term non-wellfounded set refers to sets which contain themselves as members, and more generally which are part of an infinite sequence of sets each term of which is an element of the preceding set.
…The topic of bisimulation is one of the earliest goals in a treatment of non-wellfounded sets.
Let (G,→) be a graph. A relation R on G is a bisimulation if the following holds: whenever xRy,
- If x → x′, then there is some y → y′ such that x′ Ry′.
- If y → y′, then there is some x → x′ such that x′ Ry′.
These are sometimes called by the suggestive names zig and zag.
The Stone duality in particular has come up early and often as a natural fit for what I see as the ACME-OMMM continuum (is it too pretentious to call it the אΩc?). Stephen has suggested that my conception of the אΩc maps to the Stone duality; that the relation between subjecthood and objecthood is equivalent to the equivalence between topological spaces and logical algebras.
I agree that topological spaces are a good match for the ‘Occidental/West’ OMMM side, but I’m not sure that logical algebras could define the subjective picture completely because logic does not persuade us in our poetic modes. Algebra might offer something, given the etymology of the word:
Algebra (from Arabic al-jebr* meaning “reunion of broken parts”)
This is important because it fits with my idea of qualia as subtractive gestalts. By subtractive I mean that our experience of making sense is of recovering gestalt wholeness through eliding or subtracting out the gaps. In this sense, qualia can be described as the ur-algebra, from which all algebra follows.
In the Stone duality too, algebra is distinguished from geometry especially because of its ability to represent a continuous process rather than a static grouping of vectors. This continuous nature of algebraic process fits with my understanding that ‘time’ is not a natural primitive but an artificial derivative of experienced qualities like sequence, symmetry, and repetition.
Repetition requires at minimum that something can identify that something seems:
- identifiable as different from everything or anything
- identical in some way, even though it is instantiated separately
- identical to an instantiation that is remembered as a previous instantiation
Here I am trying to get under the floorboards of the Church-Turing and question what mathematics takes for granted: Pattern, pattern recognition, sense-making. Symbol grounding and realism will come later as a consequence of multiple sense channels.
The word algebra then has two important pieces to the ACME side - the idea of subtractive gestalts and continuous process. So far so good. The problem I have is with the mathematical constraint. Logic, binary or otherwise, is the life’s blood of all of mathematics and while life is filled with logics, I am convinced that we cannot get to feelings and participation from number operations alone.
With the poetic heights and psychotic depths of far ACME (‘Oriental/East’) phenomenology, it must be recognized that there is a reason that logic seems to fail us. Logic gives us ideal truth values of 1 or 0, but not experiential-particpatory values of good or bad. Our sense of ‘awesome’ or ‘horrifying’ is not simply doubleplus true/untrue. I think it is more accurate to say that our qualities of experience cast shadows that can be quantified, but it is not possible to reconstruct the experience from only those shadows.
If someone had to recreate our universe from scratch using only our descriptions of it, there is no way they could wind up with anything like what we experience using only a logical, rational framework. We need something beyond forms and scripted processes, beyond parts and wholes, something like Trans-Rational Algebras and Immereology.