For Tuesday 15th January.
Euclid’s Elements .
From Book I, read Definitions, Postulates and Common Notions, Propositions 1-3.
Using the notions of congruence, betweenness and point, and the definition of a region as a collection of points, define the following notions.
The notion of a straight line (which could be infinite, finite, closed or open).
The notion of a straight line that is infinite in both directions.
xy is at least as long as zw.
In proposition 1 of the elements, Euclid constructs a pair of congruent circles each of whose center lies on the edge of the other. He then shows, from his basic postulates, that the the centers of these two circles, along with one of the points it which the circles intersects is an equilateral triangle (a triangle with equal sides). How does Euclid know that there is a point at which the circles intersect? Do you think this can be proven from his postulates?
Using the notions of congruence, betweenness and point, and the definition of a region as a collection of points, define the following notions.
Sphere.
A plane.
A circle.
Three points, xyz, forming a right angle (with y at the angle).
Parallel lines.
Let us say that three balls in ordinary Euclidean space are “right angled” iff their centres are at right angles. Does this notion make sense in a (non-Euclidean) pointless geometry, where there are no points, and thus no centres? Justify your answer: you may freely use any notions we defined in class from ball and parthood.
In Euclidean geometry, do you think it is possible to define the relation of $x$ being 1 meter from $y$ using congruence and betweenness? Explain your answer either way.
Can you define the primitive of Kantian geometry — $w$ is on the left of the plane defined by $xyz$ — using only congruence and betweenness. Justify your answer.
Watch youtube video on Pythagoras’ theorem
Chapter 1 of Maudlin.
The papers by Kant and van Cleve in the dropbox folder.
Describe your own example of a pair of incongruent counterparts. (Formally, a counterpart is something that is related by a isometry: a mapping preserving congruence and betweenness. Congruent entities can be superimposed, so incongruent counterparts are two individuals that are related by an isometry but can’t be superimposed.)
Could a hand in otherwise empty space be left or right in virtue of its relation to a Euclidean space? Explain your answer using the tests for definability we have studied in previous classes.
The Leibniz Clark correspondence (in dropbox). Leibniz’s second letter to Clark section 1, Clark’s second reply, section 1. Leibniz’s third letter to Clark section 1-section 7, paying special attention to section 5 and section 6.
Maudlin Chapter 2.
Is a ball in empty Euclidean space a homomorph, or an enantomorph?
What about a hand in empty Euclidean space?
What about a hand in Moebius space (the 3d analogue of a Moebius strip)?
What about a hand in four dimensional space?
Are there any enantomorphs in four dimensional space?
Are there are enantomorphs in Moebius space?
The primitives of Newton’s physics include: (i) congruence and betweenness relations between points of space (corresponding to distance), (ii) congruence and betweeness relations between times (corresponding to duration), (iv) property of being a material particle (iv) A three place location relation “particle p is located at x at time t”.
Using the Newtonian primitives define the following notions:
a. A particle being at absolute rest.
b. A particle moving at a constant velocity .
c. A two particles always moving at the same velocity (i.e. the same speed in the same direction).
d. A particle moving in circular motion at a constant speed.
An example relationist theory: particles are composed of a succession of instantaneous particle events. In addition to particle events that are mathematical objects like real numbers; but no space or time.
The primitives of the relationist theory include: (i) a distance relation mapping two simultaneous particle events to a real number, the distance between them. (ii) a temporal relation mapping two particle events to the time that elapses between their occurance (a positive number if the second event occurs after the first, and negative other wise).
Assume the relationist theory. For the following notion, either define the notion or show that it can’t be defined by finding a mapping that preserves the relationist primitives but not the notion in question.
a. Can you define the notion of a particle moving at the same velocity as another particle?
b. Can you define a notion of absolute rest?
c. Can you define the notion of a particle moving in uniform circular motion?
Chapter 2, sections on Newton’s bucket.
Explain why the relationist, using the primitives of spatial congruence and betweeness between simultaneous events, and temporal congruence on events, cannot distinguish the two versions of the yolked globes.
Choose one extra primitive to add to the relationist theory, and explain how, using that primitive, you can explain the difference. (Try to come up with examples different from class.)
Given your answer to question 2, answer the following two questions. Is the new primitive acceptable given a relationist view according to which there are only material objects (and no regions of space or time)? Does you new primitive allow you to introduce the notion of absolute rest.