MONT 104Q -- November 16,18,20 -- Proofs and Refutations Monday: The traditional view of mathematical proof (very much formed by the influence of Euclid's Elements) was: proofs are logical demonstrations of how, given certain starting assumptions, other statements must necessarily follow. For instance, if we accept the Postulates and Common Notions, then other theorems like I, 47 are necessarily true. In fact, for a long time, most mathematicians (using physical intuition) would have gone even farther -- they thought the Postulates were "obviously true" about the physical world, and hence did not require proof(!) So the proofs of statments like I, 47 accomplished over the course of Book I of the Elements were more: guarantees of absolute truth and certainty. Because of this, mathematics came to be seen as a uniquely successful product of human intellect. It was virtually the only scientific discipline where results could in principle, and in fact regularly were, established with complete certainty. Of course, this leaves a question: How do we find theorems/proofs? As we have seen through our work on the "power theorem," part of the answer is: by looking at examples, by intuiting general patterns, then trying to show those patterns hold in other cases too by reasoning in general. But is the result of any proof ever absolute certainty? Does mathematics really produce that sort of uniquely absolute knowledge? People's confidence that the answer was "yes" was first shaken in the early 1800's when C. F. Gauss (who didn't publish his work for fear that others would misunderstand it), and then J. Bolyai, Nikolai Lobachevsky, and others showed that Euclidean plane geometry is not the only possible 2-dimensional geometry. There are other mathematical systems where Postulate 5 of Euclid and the result of I, 31 (construction of a parallel to a given line through a given point not on the line) are not true. Moreover, this can happen in two ways; most interesting is the one called hyperbolic geometry. In hyperbolic geometry, there are infinitely many parallel lines to each line through a given point not on the line, including lines where the sum of the angles on one side of a transversal add to less than two right angles: This geometry is just as consistent as Euclidean geometry, because it is actually even possible to construct "models" of it within Euclidean geometry. Confidence was shaken again in a different way in the late 1800's and early 1900's in a way that we'll see at the start of the second semester. But that is a different story and we'll postpone talking about that in more detail. As a result of these and other comparable developments, many thinkers about mathematics, including Imre Lakatos (1922 -- 1974), the author of Proofs and Refutations, set about to try to develop different and more convincing accounts of what mathematics is and what it does. In particular, as we will see, Lakatos's view of what a proof is and what it is good for is quite different from the idea of a "guarantee of absolute certainty" that we expect on the basis of the example of Euclid. We want to explore these ideas, as presented in Proofs and Refutations. Sections 1 - 3: The setting is an (imagined) classroom; the book develops as a dialog between the teacher and the students -- with some pretty dense footnotes explaining how the stages of the discussion and the events that transpire are intended to correspond to stages in the actual development of understanding of a real mathematical problem. The subject is a famous relationship about polyhdedra first noticed by L. Euler (1707-1783 CE). By looking at the 5 regular or Platonic solids (also studied extensively in Book XIII of the Elements!), then at other examples: polyhedron V E F tetrahedron 4 6 4 cube 8 12 6 octahedron 6 12 8 dodecahedron 20 30 12 icosahedron 12 30 20 triangular prism 6 9 5 square pyramid 5 8 5 etc. Euler noticed a pattern V + F = E + 2, or V - E + F = 2. And this is the starting point for the Conjecture: For all polyhedra, V - E + F = 2. (Euler discussed this in a published article but said he was not able to find a general proof or reason why this should always hold.) The teacher starts out by proposing a proof (essentially following an article by the slightly later mathematician A. Cauchy (1798-1857 CE). A "though-experiment": [work out on board] Step 1: remove one face and deform the resulting figure to lie in a plane (have V-E+F = 1 now). Step 2: triangulate the resulting plane figure by subdividing each face into triangles (still V-E+F = 1 since each new edge divides an existing face into two new faces: E -> E + 1 and F -> F + 1) Step 3: Remove faces one by one until there's a single triangle left. For each use one of these "moves": Case (a) remove a single edge (but not the vertices at the ends) E -> E - 1 and F -> F - 1, so V - E + F = 1 is unchanged Case (b) remove a single vertex and two incident edges E -> E - 2, V -> V - 1, F -> F - 1, so V - E + F is unchanged Final configuration has E = V = 3, F = 1, so V - E + F = 1. Q: What happens next? A: The students question the individual steps of the proof and propose "local counterexamples" -- i.e. examples that seem to invalidate the method used in the proposed proof, but not the statement of the conjecture: V - E + F = 2. Examples: (1) page 10: Student Gamma says: what if we remove an interior face like a piece from a jigsaw puzzle? What does the teacher reply? (2) page 11: Gamma is not deterred -- proposes a sequence of operations shown in Figure 4, page 11. What is the issue there? Teacher: "To salvage this proof, we would need to show that there is always some sequence of operations as in Case (a) or (b) above that leaves V - E + F = 1 unchanged." Q: Is this like any classroom you have been in? How is it different? (Note: the teacher is not an absolute authority and doesn't present him/herself that way -- students consistently question and criticize. The whole group is clearly engaged in learning through discovery. The students are very motivated and engaged(!) No "assessments," standardized or otherwise!) So, it's fair to say this classroom is probably meant to represent the community of mathematicians working together to create knowledge more than an actual or potential educational institution. (Many teachers who read P+R can't help but wish their classrooms looked and felt more like this though!) Wednesday: Section 4. (a) - (c) At the start of this, student Alpha presents the first "global counterexample" Q. What is this? Why is it a "global counterexample?" Doesn't this mean we should just drop the question and go on to something else? Note the teacher's reaction: (p. 13 - 14) "Columbus did not reach India, but ... " Read this, discuss. Q: What is the main theme of this section? A: Importance of proper definitions above all -- the class started into this (as did the mathematicians who studied these things to begin with!) with only a rough, approximate idea of what a polyhedron should be. That needs to be nailed down before we can make any progress! In particular, need to add explicit conditions like: faces are "real" (not "star") polygons, faces meet along whole edges, only two faces meet along any one edge, edges meet only at vertices, etc. The students propose a series of definitions of what they think a polyhedron should be, but find "monsters" that violate the conjecture V - E + F = 2. Delta's "Method of monster-barring" -- Rough idea: adjust definition until we get a class of polyhedra for which the theorem is true. ("Monsters" or "exceptions"?) also see footnote 2, p. 43. Q: Is that so wrong? Discuss Beta's speech on p. 26. Leads to the introduction of class of convex polyhedra for which the originally proposed proof of the Conjecture does in fact work(!) [discuss convexity ] But there are several remaining questions! (1) There are clearly non-convex polyhedra for which the Conjecture is true. Should we try to characterize the cases where it does work? Or should we be satisfied with a statement that is true for that restricted set of polyhedra? (2) What is the correct generalization of the Conjecture to polyhedra with "cavities," "tunnels," etc. like the "picture frame" from Figure 9 on page 19? (3) What's so bad about the "monsters?" [If time, and if it seems propitious, maybe discuss development of analysis through 19th century, development of understanding of "wild behavior" of real functions in general. Note footnote 2, page 19.] Friday 4 (d) "monster-barring" vs. "monster-adjusting" [discuss] 4 (e) (attempted) proofs generate new or better conjectures, the process is really a cycle: conjectures proofs (see teacher's speech on page 37). "proof analysis" by "lemma incorporation" as a model for Book I of the Elements? Is that how Euclid developed his presentation? (No way to answer that, but it's somewhat plausible!)