Extension as the background of substance a formal approach

Extension as the background of substance a formal approach Bart lomiej Skowron Wroc law University The Pontifical University of John Paul II, Kraków bartlomiej.skowron@gmail.com February 11, 2013

Presentation plan 1. Substance

Presentation plan 1. Substance Substance

Presentation plan 1. Substance Substance Extension

Presentation plan 1. Substance Substance Extension Conclusions

The nature of corporeal substance is extension in length, breadth and depth; and any other property a body has presupposes extension as merely a special case of it. For example, we cant make sense of shape except in an extended thing, or of motion except in an extended space. (2) The nature of thinking substance is thought; and anything else that is true of a mind is merely a special case of that, a way of thinking. Descartes

What is the extension? Does it have any formal moments?

What is the extension? Does it have any formal moments? 1 being in a space

What is the extension? Does it have any formal moments? 1 being in a space 2 being impenetrable

What is the extension? Does it have any formal moments? 1 being in a space 2 being impenetrable 3 having a positive dimension

What is the extension? Does it have any formal moments? 1 being in a space 2 being impenetrable 3 having a positive dimension 4 being (everywhere) in between

What is the extension? Does it have any formal moments? 1 being in a space 2 being impenetrable 3 having a positive dimension 4 being (everywhere) in between 5 possibility to divide

How to analyze of these category?

How to analyze of these category? ontologically

How to analyze of these category? ontologically phenomenologically

How to analyze of these category? ontologically phenomenologically physically (material ontology?)

How to analyze of these category? ontologically phenomenologically physically (material ontology?) mathematically (formal ontology?)

How to analyze these category? ontologically phenomenologically physically (material ontology?) mathematically (formal ontology?)

What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension

What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension Geometry

What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension Geometry Topology

Topology is the study of those properties of geometric configurations which remain invariant when these configurations are subjected to one-to-one bicontinuous transformations, or homeomorphisms (...). We call such properties topological invariants. For example, the property of a circle to separate the plane into two regions is a topological invariant; if we transform the circle into an ellipse or into the perimeter of a triangle, this property is retained. On the other hand, the property of a curve to have a tangent line at every point is not a topological property; the circle has this property but the perimeter of a triangle does not, although it may be obtained from the circle by means of a homeomorphism. K. Kuratowski

Basic notions of topology Topology

Basic notions of topology Topology Open (closed) set

Basic notions of topology Topology Open (closed) set closure of A, interior of A

Basic notions of topology Topology Open (closed) set closure of A, interior of A Connectedness (all one piece)

Definition 1 A topological space (X, τ) is a set X together with a family τ of subsets of X, called open sets (or topology on X) and satisfying the following axioms: 1 and X are open. 2 Any union of open sets is open. 3 The intersection of any finite number of open sets is open. Definition 2 A complement of open set is called closed set. The closure of an entity is a kind of remedy deficiencies, additions towards the perfection of the object.

Definition 3 A topological space (X, τ) is said to be connected if there are no U, V τ (U V ) such that U V = and U V = X. Otherwise, X is called disconnected. For example [0, 1] and R n for n N are connected. Definition 4 A topological space (X, τ) is said to be compact if each of its open covers has a finite subcover. The real line R is not compact since the open cover ( n, n), n N of it has no finite subcover. The closed line segment [0, 1] is compact, but the open line segment (0, 1) is not.

Homeomorphism 1. Substance Definition 5 Homeomorphism: a one-to-one continuous bijection that has a continuous inverse

Homeomorphism 1. Substance Definition 5 Homeomorphism: a one-to-one continuous bijection that has a continuous inverse Topology is the study of invariants of homeomorphisms

Homeomorphism 1. Substance Definition 5 Homeomorphism: a one-to-one continuous bijection that has a continuous inverse Topology is the study of invariants of homeomorphisms Invariant under homeomorphism: dimension, being open, being closed, being dense, being separable, being connected, etc.

How to analyze extension in topology? reminder being in a space being impenetrable (impervious) (not admitting of passage through) having a positive dimension being (everywhere) in between possibility to divide

Being in space 1. Substance space shall be composed of an infinite number of points (without this it is difficult to talk about an extension)

Being in space 1. Substance space shall be composed of an infinite number of points (without this it is difficult to talk about an extension) Cantor Set

Being in space 1. Substance space shall be composed of an infinite number of points (without this it is difficult to talk about an extension) space shall be connected Cantor Set

but there are spaces connected with holes

but there are spaces connected with holes Topologist s sine curve

but there are spaces connected with holes Topologist s sine curve simply connected space (?) (A space is simply connected if every loop in the space is contractible).

way of thinking: dispose of pathologies

Impenetrability 1. Substance Possible lines of analysis:

Impenetrability 1. Substance Possible lines of analysis: the most difficult from the perspective of formal techniques (qualitative)

Impenetrability 1. Substance Possible lines of analysis: the most difficult from the perspective of formal techniques (qualitative) but let s try

Impenetrability 1. Substance Possible lines of analysis: the most difficult from the perspective of formal techniques (qualitative) but let s try impossibility of being at the same time on the same place (in a given space) by two objects extended in the same way

homogeneity (?)

homogeneity (?) Definition 6 A topological space X is called homogeneous if for all x, y X there is a homeomorphism f : X X such that f(x) = y.

Positive dimension 1. Substance

Positive dimension 1. Substance There are three (in general non-equivalent) theories of dimension.

Positive dimension 1. Substance There are three (in general non-equivalent) theories of dimension. Intuitions Roughly speaking these theories are based on two basic ideas. First, the points have dimension of 0, the curves have dimension of 1, a dimension of surfaces is 2, dimension of space is 3, etc. Second, objects of dimension n are bounded by objects of dimension n 1, i.e. points are bounded by the empty set, the curves are bounded by the points, the surfaces are bounded by the curves, the spaces are bounded by the surfaces, etc.

Dimension can be infinite

Dimension can be infinite the dimension of possible worlds in Hilbert Cube combination topo-ontology is infinite

being (everywhere) in between being (everywhere) in between = being dense

being (everywhere) in between being (everywhere) in between = being dense Definition 7 B X is called dense (in X) if every point x X either belongs to B or is a limit point of B or equivalently cl(b) = X.

Possibility to divide 1. Substance Imposition on the space stronger and stronger separation axioms T 0, T 1, T 2, T 3, T 4

noncontradictory combinations of these conditions

noncontradictory combinations of these conditions different types of extension

noncontradictory combinations of these conditions different types of extension a special type of extension: R n connected, arc-connected, simply connected, T 0 T 4 separation axioms are satisfied, n-dimensional space

Type of extension of res cogitans in the space thoughts

Topology of res cogitans. The general idea of the topology of acts (X, τ) topological space

Topology of res cogitans. The general idea of the topology of acts (X, τ) topological space Elements of X = acts Definition 2.1 (Neighbourhood) If p X, then a neighbourhood of p is a subset V X, which includes an open set U containing p. Neighbourhood of an act p = all acts that may occur with p (eg. at the same time)

Change of the res cogitans Change of the res cogitans = homeomorphic moving to yourself

Change of the res cogitans Change of the res cogitans = homeomorphic moving to yourself Being a hole is invariant under homeomorphism

Change of the res cogitans Change of the res cogitans = homeomorphic moving to yourself Being a hole is invariant under homeomorphism Subject may increase, decrease, widen, shrink...

Change of the res cogitans Change of the res cogitans = homeomorphic moving to yourself Being a hole is invariant under homeomorphism Subject may increase, decrease, widen, shrink... Identity = homeomorphicness

Consciousness 1. Substance Jan Szewczyk: Universally infinite (infinitely spreadable) field of fields, finite as well as infinite, establishing the certain closed in itself and internally integral unity of existence, not being the material part of nature, neither connected to it causally nor structurally, centered in its subjective source. This field spouts or radiates from Self in all directions (therefore undoubtedly three dimensionally) and constitutes itself in unceasingly and multi dimensionally undulating medium. It makes the certain environment or plasma of intentionality, having its own temporality, and therefore, necessarily, it own spatiality or sort of phenomenological quasi corporeality.

Question to Jeff: What do you think about this plasma of intentionality?

Is the consciousness radically non stretchable? Consciousness stretches out towards the world, hitting the objects to whom it intends it approaches them meets them in some way In this way we attribute some kind of extent to the consciousness.

Thought experiment 1. Substance Simplifying the matter suppose that the extensionality of res extensa is identical to the extensionality of the R 3. Simplifying the matter suppose that the extensionality of res cogitans is identical to the extensionality of the open three-dimensional sphere with appropriate dents and bulges, but without holes through and through. The res extensa has the type of bounded extension and the type of res cogitans is unbounded

Claims and conclusions 1 Substance is extended 2 We should use the topology in the study of extension 3 Extension can be modeled as a special case of a topological space 4 There are many forms of extension ( substances) 5 The moment of being extended is not the property which differentiates things and thoughts. 6 Extension of things extension of thoughts

Open problems 1. Substance Is the type of extension invariant under homeomorphism?

Open problems 1. Substance Is the type of extension invariant under homeomorphism? So, among others: Is a being bounded an essential moment of extension or not?

Open problems 1. Substance Is the type of extension invariant under homeomorphism? So, among others: Is a being bounded an essential moment of extension or not? What is the role play of manifolds in these studies?

M. Rosiak described extension as a possibility of coexistence of a plurality of objects, the multitude of differences between places. A possibility of coexistence of a plurality of objects topology

M. Rosiak described extension as a possibility of coexistence of a plurality of objects, the multitude of differences between places. A possibility of coexistence of a plurality of objects topology (formal ontology)

Formal Philosophy

Formal Philosophy Formal philosophy philosophy that uses a formal language

Formal Philosophy Formal philosophy philosophy that uses a formal logic

Formal Philosophy Formal philosophy = philosophy that uses a mathematical structures to philosophize

Formal Philosophy Formal philosophy = philosophy that uses a mathematical structures to philosophize philosophizing by structures,

Formal Philosophy Formal philosophy = philosophy that uses a mathematical structures to philosophize philosophizing by structures, rotation of the structures

What kind of structure should be used in Formal Philosophy? Complex structures because the philosophical issues are complex Examples: Algebraic fields, rings, posets, lattices, differentiable manifolds, Hilbert spaces, closure spaces, model Monster, categories (category theory), etc.

What for a formal philosophy? We need a formal philosophy, we need a structures, because the philosophy is just so hard to practice without it.

Thank you!