i wonder if space couldn't be thought of here as a kind of container (in the sense that a set is a container -- which of course it isn't, but you see perhaps what i mean). that's all i've got, i'm afraid. a quick flip to the index of _logics of worlds_ reveals only names, so no quick help there... Jeffery Fisher
One line that struck a chord with me was Morris Kline's point that somewhere in the mid-19th century the emphasis shifted from "physical explanation" to "mathematical description". ravi
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I do see what you mean, but you have to go deeper, or more technical.
My preferrence is for ideas on the concepts of space, rather than number. I just don't think all that well with numbers. So then concepts like being for me make more sense if I conceive it as an abstraction like space. So here is how I approach the idea that mathematics is ontology.
While we are discussing some relation possibly a dualism between number and being, there is another aspect involved, and that is the dualism between number and space. Let's say there is a valid dualism or equivalence between number and being, and another dualism between number and space. Then I think it follows there is a dualism between being and space. I think this follows some set theory theorem on the equality of sets and is deeply related to the problem of Existance axioms. In algebra its called the associative law in algebra..
Nevermind that for now.
You can construct this number or set dualism to space in mathematics several different ways and I am not at all sure which one is technically the correct or the most fundamental form for meta-mathematics. The usual math course method starts with order pairs <a,b> and uses the Cartisian Product. For every ordered pair <x,y> there exists a point in 2-d Euclidean space, i.e. the Cartisan plane.
One of the more fundamental but relatively easy ways to grasp the dualist nature of number and space is called incidence geometry. You can map the numbers {1, 2, 3} to points on the plane via the Axioms of Incidence:
1) For every point P and for every point Q there exists a unique line L that passes through P and Q
2) For every line L there exists at least two distinct points incident with L.
3) There exists three distinct points with the property that no line is incident with all three of them.
http://math.uncc.edu//~droyster/math3181/notes/hyprgeom/node28.html
This is going to take a drawing and some mediation. You are going to draw each of the above axioms. First, put a pencil point on the paper and make it pretty big. Second, put another point, not incident with first one on the paper. Now draw a line between between the points. Put a third point on the paper not incident with the line. Now connect this last point with the end points of the line. See you have a triangle. Whoopy fucking doo.
Now the mediation. Notice that there is a hierarchy of order here. First, there are points, then there are lines, then there is a plane, which is the triangle (and not the paper). This drawing represents a complete incidence geometry since it satisfies the first three axioms. You can now label the points of the triangle (1, 2, 3). However, there is something missing. We need an equivalent to (0).
There is another dualism working underground here. It is the equivalence between this drawing or representation of an incidence geometry and the axioms on sets. What is missing is the Null element, which must be with every ordered set. So we can add the null, and call it the `center' of the triangle. Put a little dot in the center not as big as the points so you can pretend the dot is null, and not the same as a point. We could have started by simply imaging where the null would go, and put our first point somewhere else.
But now we have a problem with our isomorphism or mapping one to one, onto, between numbers and a minimal geometry of incidence. We count (cardinality) four elements total, (1,2,3)->(0,1,2,3) if we add zero or null.
We now switch to naive sets and perform a mapping between between natural numbers and ordered sets. We start with the map of zero to null, null <-> zero then follow:
1 = 0+ ( = {0}), 2 = 1+ ( = (0,1}), 3 = 2+ ( = {0,1,2}),
and so on.(Halmos, Naive Set Theory 44p) I should mention that in the theory of incidence and mapping it to sets we can write the following correspondence using parenthesis:
a,b,c -> three distinct points, i.e points
(a,b), (b,c), (a,c) -> lines
(a,b,c) -> plane
(Basic Concepts of Geometry, Prenowitz w, Jordan M, 141p) The above triangle with its labels is called a model of incidence geometry, or M1. The set of all (x,y) is also a model, called M19. To see the depth of this version of geometry go here and scroll down to the table were each interpretation corresponds to euclidean, hyperbolic, and or elliptic geometries.
There is a very difficult to understand, consequence to the postulate of null. Null has a mirror or dual image which is infinity. This dualistic dependency is captured in the:
Axiom of Infinity. There exists a set containing 0 and containing the successor of each of its elements. (Halmos)
This is straight out of Hegel, and captured in his method that starts with Nothing and then per force generates Being out of Nothing. In his Science of Logic he has a whole several sections that consitute a catagory, that follows his doctrine of Being beginning with the category called Quality. The mathematical or `scientific' category is called Quantity. The category begins with Pure Quantity and develops Continuous and Discrete Magnitudes, or the mathematics of infinities and finitudes. I would love to find out if Dedekind, Cantor, maybe the later Frege had read this work or other ever read Hegel..
Here is a reference to that work and its connect to infinities:
http://broodsphilosophy.wordpress.com/2006/06/28/hegel-and-infinite-series/
It really was a dis-service to effectively ban Hegel from US academies. My copy of Science of Logic was written in 1812-16 in German. It was first translated into English in 1929. My copy dates from 1966, which I bought used at Moe's about fifteen years ago and never got further than the opening sections on Nothing, Being and Becoming. I immediately noticed the potential mapping of the elementary theory of sets natural numbers and incidence geometry to this teriary logic.
To see the connection with current cosomological theory of course we begin with nothing, then bang, we get being-space or matter-spacetime which is dynamic or directional time, in the concept of becoming.
CG