Extension as the background of substance a formal approach
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1 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
2 Presentation plan 1. Substance
3 Presentation plan 1. Substance Substance
4 Presentation plan 1. Substance Substance Extension
5 Presentation plan 1. Substance Substance Extension Conclusions
6 Presentation plan 1. Substance Substance Extension Conclusions
7 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
8 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. Res extensa is extened, res cogitans is not. Descartes
9 What is the extension? Does it have any formal moments?
10 What is the extension? Does it have any formal moments? 1 being in a space
11 What is the extension? Does it have any formal moments? 1 being in a space 2 being impenetrable
12 What is the extension? Does it have any formal moments? 1 being in a space 2 being impenetrable 3 having a positive dimension
13 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
14 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
15 How to analyze of these category?
16 How to analyze of these category? ontologically
17 How to analyze of these category? ontologically phenomenologically
18 How to analyze of these category? ontologically phenomenologically physically (material ontology?)
19 How to analyze of these category? ontologically phenomenologically physically (material ontology?) mathematically (formal ontology?)
20 How to analyze these category? ontologically phenomenologically physically (material ontology?) mathematically (formal ontology?)
21 What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension
22 What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension
23 What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension Geometry
24 What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension Geometry
25 What branch of mathematics should be taken into account? Space, being in beetwen, being everywhere (dense?), dimension Geometry Topology
26 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
27 Basic notions of topology Topology
28 Basic notions of topology Topology Open (closed) set
29 Basic notions of topology Topology Open (closed) set
30 Basic notions of topology Topology Open (closed) set closure of A, interior of A
31 Basic notions of topology Topology Open (closed) set closure of A, interior of A
32 Basic notions of topology Topology Open (closed) set closure of A, interior of A Connectedness (all one piece)
33 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.
34 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.
35 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.
36 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.
37 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. Connectedness and compactness are the invariants under homeomorpisms.
38 Homeomorphism 1. Substance Definition 5 Homeomorphism: a one-to-one continuous bijection that has a continuous inverse
39 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
40 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.
41 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.
42 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
43 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)
44 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
45 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
46 but there are spaces connected with holes
47 but there are spaces connected with holes Topologist s sine curve
48 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).
49
50 way of thinking: dispose of pathologies
51 Impenetrability 1. Substance Possible lines of analysis:
52 Impenetrability 1. Substance Possible lines of analysis: the most difficult from the perspective of formal techniques (qualitative)
53 Impenetrability 1. Substance Possible lines of analysis: the most difficult from the perspective of formal techniques (qualitative) but let s try
54 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
55 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 have a boundary or a bound or a limit
56 homogeneity (?)
57 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.
58 Positive dimension 1. Substance
59 Positive dimension 1. Substance There are three (in general non-equivalent) theories of dimension.
60 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.
61 Dimension can be infinite
62 Dimension can be infinite the dimension of possible worlds in Hilbert Cube combination topo-ontology is infinite
63 being (everywhere) in between being (everywhere) in between = being dense
64 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.
65 Possibility to divide 1. Substance Imposition on the space stronger and stronger separation axioms T 0, T 1, T 2, T 3, T 4
66
67
68 noncontradictory combinations of these conditions
69 noncontradictory combinations of these conditions
70 noncontradictory combinations of these conditions different types of extension
71 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
72 Type of extension of res cogitans in the space thoughts
73 Type of extension of res cogitans in the space thoughts
74 Type of extension of res cogitans in the space thoughts
75 Topology of res cogitans. The general idea of the topology of acts (X, τ) topological space
76 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.
77 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)
78 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) neighbourhood system = topology
79 Change of the res cogitans Change of the res cogitans = homeomorphic moving to yourself
80 Change of the res cogitans Change of the res cogitans = homeomorphic moving to yourself Being a hole is invariant under homeomorphism
81 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...
82 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
83 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.
84 Question to Jeff: What do you think about this plasma of intentionality?
85 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.
86 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
87 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 but unexpectedly
88 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 but unexpectedly these types are identical in terms of topology. That is they are homeomorphic.
89 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 but unexpectedly these types are identical in terms of topology. That is they are homeomorphic. The property of being extended is not the property which differentiates things and thoughts.
90 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
91 Open problems 1. Substance Is the type of extension invariant under homeomorphism?
92 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?
93 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?
94 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? Is it possible to examine the impenetrability in the topology?
95 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
96 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)
97 Formal Philosophy
98 Formal Philosophy Formal philosophy philosophy that uses a formal language
99 Formal Philosophy Formal philosophy philosophy that uses a formal logic
100 Formal Philosophy Formal philosophy = philosophy that uses a mathematical structures to philosophize
101 Formal Philosophy Formal philosophy = philosophy that uses a mathematical structures to philosophize
102 Formal Philosophy Formal philosophy = philosophy that uses a mathematical structures to philosophize philosophizing by structures,
103 Formal Philosophy Formal philosophy = philosophy that uses a mathematical structures to philosophize philosophizing by structures, rotation of the structures
104 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.
105 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.
106 Thank you!
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