assignments
Regular Writing Assignments should be no more than 1000 words. Cite sources as appropriate. The form of the regular Math Assignments will vary.
due Thu Jan 29
Writing Assignment 1
- What does mathematics mean to you? What is mathematics and what does doing math entail? Including at least one example of your experience doing math is a good idea.
- Given the snow-day awkwardness, we're asking that you upload this to your wiki page (see the wiki link on the left) by Tuesday, read the other submissions and comment on at least two before Thursday.
due Thu Feb 12
Math Assignment 1
The intention is not that you do /all/ of these questions. Have a go at the ones that seem most interesting to you and will stretch you (but not too much).
- Prove that \( \sqrt{3} \) is irrational.
- Take a mathematical concept that you know well (it does not need to be very advanced at all, but it can be) and describe it from both sides (or multiple sides) of one of the divisions we've discussed (Poincare's intuitionism vs. logicism, platonism vs formalism (vs predicativism vs etc.),...).
- Show that there are five Platonic solids (regular convex polyhedra---i.e. same number of edges per face and same number of edges per vertex).
- Hint: There are at least five, as proved by role-playing dice. To show that there's at most 5, denote the number of edges for each face by \(x\) and the number of edges that meet at each vertex by \(y\). Use these to relate \(F\), the number of faces in the polyhedron, to \(E\), the number of edges in the polyhedron and also to relate \(V\), the number of vertices in the polyhedron, to \(E\). Substitute into \(V-E+F=2\) to get an equation with just \(x\), \(y\), and \(E\). What possibilities are there for \(x\) and \(y\) that make the equation true?
- Beyond where we read to, Lakatos's class went on to consider the "crested cube" which satisfied the definitions given so far but did not have \(V - E + F =2\). Here's one:
Calculate \(V - E + F\). What do you think about it (is it a polyhedron? a monster?) and what do you think the class thought about it?
- We talked about Russell's paradox in class. A similar one (with similarly devastating consequences for Frege/Russell's original hopes for building math out of logic) is Berry's paradox. Consider the phrase "The smallest positive integer not definable in under eleven words." Why is this a paradox? What might you do about it?
- Conjecture from class: Given a polyhedron with regular sides, we can take the dual twice and recover (a smaller version of) the shape we started from. Prove, refute, refine, etc.
Recall that in the Poincare reading he developed the idea of proof by mathematical induction. The idea is that given a statement \(P(n)\) that depends on \(n\) you show two separate things:
- \(P(1)\) is true,
- if \(P(k)\) is true for an arbitrary \(k\) then \(P(k+1)\) is also true.
And from these we conclude that \(P(n)\)is true for all positive numbers \(n\). The next two problems are about using this method.
- Show by induction that \( 1 + 2 + 3 + \cdots + n = n(n+1)/2 \). (There is also a more "moral" proof---can you find it?)
- Find the error (I presume you believe there is one) in the following proof by induction of the statement "All horses are the same colour".
- Let \(P(n)\) be the statement that every set of \(n\) horses is monochromatic (i.e. are all the same colour).
- \(P(1)\) is that a single horse is the same colour as itself. This is true.
- Suppose that \(P(k)\) is true for some number \(k\). That is, all sets of \(k\) horses are monochromatic. Now take an arbitrary set of \(k+1\) horses. Make them stand in line. The first \(k\) horses are monochromatic as there are \(k\) of them and we have supposed that all such sets are monochromatic. Similarly the last \(k\) horses are monochromatic. So the first is the same colour as the second, which is the same colour as the third and so on all the way to the \((k+1)\)th.
- By induction, we have that a set of \(n\) horses is monochromatic for any number \(n\). In particular, it is true for \(N\), the number of horses that exist in the world right now.
- Three logicians walk into a bar. The bar tender asks "would you all like a beer?" The first logician says "I don't know". The second logician says "I don't know". The third logician says "Yes. Three beers please".
- Explain why this joke (is it a joke? I'm not sure) makes sense.
due Fri Mar 6
Writing Assignment 2
Write a brief response to the idea of Biography as a tool of the history of science/mathematics. What benefit do we gain in understanding both of contemporary science as well as of history by writing and researching biographies? What do you think of some of the pitfalls described especially by Greene: the hero's quest format, the choice of only great scientists, etc.? If there was something else in the reading that really struck you, feel free to respond to that also.
due Thu Apr 2
Math Assignment 2
due Mon Apr 13
Social Issues Paper
due Sat May 9
Final Paper
