Assignments and Further Reading

• General guidelines
• How to cite sources and avoid plagiarism
• How to cite web sources – check with us before using any that are not on the 105 web site
• University guidelines on academic dishonesty

• Answer two of the following:

1. Consider the following argument: "At the beginning and end of a change the thing that changes must be both different -- so that change has occured -- and the same -- so that some one thing has changed; thus at the beginning and end it is the same and it is different, which is impossible, so nothing can change." How does Aristotle's distinction between the 'is's of identity and predication solve this paradox?
2. Explain Grünbaum's 'staccato' run supertask: include a graph showing (for the first few stages) the runner's position against time, and be sure to explain what is 'super' about the task.
   
3. Suppose that space and time are discrete so that the shortest time interval is T seconds long and the shortest distance is X metres. (a) Our space and time are probably like this – why don't we notice? (b) Explain with examples how to define the speed of an arrow moving in a discrete space and time. (c) Explain how two runners can pass each other without ever being exactly abreast, in a discrete space and time.
   

• Further readings:
  
• Zeno's Paradoxes by Nick Huggett on the Stanford Encyclopedia of Philosophy.
• Space From Zeno to Einstein by Nick Huggett, on reserve in the Daley Library.
  

• Answer all of the following (giving equal weight to each):
1. Describe the shape of space according to Aristotle. Explain Archytas' argument that such a shape is impossible. Give one response to Archytas that shows how space could have an edge.
2. Assume (falsely, I'd say, but ignore me for now) that intelligent life can only exist if space has three dimensions. Then an 'anthropic' (meaning to do with humans) explanation of why there are three dimensions is because there is intelligent life in the universe, namely us. Is this a reasonable explanation? Justify your answer.
3. In the familiar Euclidean geometry through any point there is only one line that is parallel to a second given line; explain, using an illustration, why this is not the case in a space with Lobachevskian geometry (i.e.,  the geometry of a saddle). Describe an experiment that could detect which parallel postulate holds. Suppose that the experiment reveals the existence of many parallels. One might conclude that space is non-Euclidean, but Poincaré offered an alternative explanation; explain, using an illustration, that explanation.
   

• Further readings:
  
• Flatland By Edwin A. Abbott.
  

• Answer all of the following (giving equal weight to each):
1. What are the different ways that motion should be understood (a) if space is matter (as Descartes said), (b) if matter is a state of space (as Newton considered), (c) if space and matter are distinct substances (as Newton said) and (d) if space is a relational construct as Leibniz said?
2. Describe Newton's bucket experiment. Why does the fact that the water can only ever rise to one height at any time pose a problem for a relational account of space? How might one fix the problem?
   
3. Left and right footed shoes are incongruent counterparts. According to the fitting account of handedness, what difference between them do we refer to when we call 'left' or 'right'?
   

• Answer all of the following (giving equal weight to each):
1. Explain what problem McTaggart poses for the idea that the same property of 'presentness' is possessed by each moment in the block, when it is the present. How does relativizing the property of 'presentness' to each moment solve the problem?
2. Describe a time travel paradox (different from the one in the course pack), explaining just what is paradoxical about it.
3. It's logically impossible for such a paradox to arise, so assuming that someone (or something) does time travel as in your story, there must be some step at which something that happens appears possible but is actually impossible. Identify such a step, and explain how it is possible in the everyday ('coarse grained') sense, but physically impossible (i.e., in the fine grained sense).
   

• Answer all of the following (giving equal weight to each):
1. Explain why a pair of non-relativistic bullets that reach their targets simultaneously in one frame also reach their targets simultaneously relative to others. Explain why a pair of relativistic light particles ('photons') that reach their targets simultaneously in one frame do not reach their targets simultaneously relative to others. (This explanation will need diagrams, but not necessesarily equations.)
   
2. Draw your own version of the figure in lecture 24, and use it to show that an object is shorter relative to a frame (e.g., that of A) when it moves relative to that frame than when it is at rest relative to that frame.
   

• There is no Assignment 6! Instead, rewrite any one of your previous assignments to receive a better grade on that piece of work (assuming your rewrite is better than the original). That is, we will calculate your final grade based on assignments 1-5, and you have the chance to improve your grade on one of those pieces of work. It's probably a good idea not to the first assignment, since that is worth a smaller portion of the grade than the others.