Biology S1002E (Summer 2002)
Notes for Week #1
Roger B. Blumberg

Wednesday, May 29th

• An introduction to the course, Theory and Practice of Science, and to this summer's version of it. We incorporate more mathematics in this course than in most biology courses, so that original scientific papers, rather than textbooks or popular science books, can be used as the basis for study.

• Distinguishing mathematics from "number crunching" and from the practice of experimental/empirical science. Although all sciences (and perhaps all academic disciplines) can be described as a search for patterns and explanations for these patterns, in mathematics the method of "solving" problems and analyzing patterns is rather unique. Examples:
  1. We consider a single-digit sequence of integers, generated by a known rule (i.e. 2,3,6,1,8,6,8,4,8,....), and notice that no matter how many digits we compute the digits 9, 7, 5, and 0 do not appear. We use a basic truth about odd and even numbers to prove that in fact these digits will never appear in the sequence no matter how long we compute the sequence.

  2. We consider the famous "7 Bridges of Konigsburg" problem, and what begins as a proof that there is no "once and only once" solution becomes a generalized theory of graphs and when we can trace them in such a way that each edge is drawn once and only once.

     (For more on these examples, analytic solutions, and other other examples we might have used, see the Week #1 Notes for "Foundations of Mathematics")

• In both of our examples, we can see a method that characterizes the search for an analytic solution to a problem, and the way mathematical solutions are distinguished in their power and elegance from those of natural and social sciences. In each case the method might be characterized as follows:
  • Identify (and experiment with) a phenomenon or problem.
  • Identify a pattern or structural feature that seems relevent to being able to solve the problem and/or understand the behavior of the phenomenon.
  • Represent or re-represent the problem in a way that retains these relevent features and discards the others.
  • Prove that the pattern is real/true and not an accident of misperception or limited calculations.
  • Generalize the proof in order to understand problems and phenomena similar to but well beyond the original problem.

• For Next Time: Finish the analysis of all 2-D graphs so that you can say whether any particular graph has a "once and only once" solution *without* putting pen to paper.

Thursday, May 30th

• We begin with a sequence of integers generated according to yesterday's rule, but with different digits at the start:

  3,4,1,2,4,2,8,8,1,6,.....

  Can we say whether the same pattern(s) we saw and proved were real yesterday will be true of this sequence as well?
• Similarly, we consider a large 2-D graph we have never seen before, and ask whether we *know* if there is a path through the graph that uses each line once and only once (without trying to make a path ourselves).

• We finish the analysis of the three categories of graphs, explaing why a graph with more than two vertices with odd degree cannot have a "once and only once" solution, and "prove" that a graph must have an even number of vertices with odd degree.

• The focus of today's class is on what constitutes a mathematical proof. Like an argument made in a piece of expository writing, a proof can be described as a relationship between one or more premises, postulates, or hypotheses, and one or more conclusions. Unlike the arguments we encounter in most expository prose, however, mathematical proofs are explicit about their premises and are structured so that it is clear that the conclusions are necessarily true when the premises are true.

• The clearest examples of the deductive nature of mathematical proof are those "direct" proofs we used in class to: 1) derive a formula for the sum of consecutive integers (i.e. Gauss' sum); and 2) prove that in any graph the number of vertices with odd degree must be even. (Details of both proofs can be found in the Notes for Week #2 of the "Foundations of Mathematics" course.)

• Another example of mathematical proof that is similarly deductive is the method of "indirect" proof. Here we try to prove a statement by assuming it's contrary is true, and then showing that this assumption is self-contradictory. Because mathematical statements are either true or false, by showing that this contrary assumption must be false, we prove that the original statement was true. The example we used was a proof of the statement: "The square-root of 2 is not a rational number."

• Finally, we learn a method of proof which is arguably the most elegant in all of mathematics: proof by mathematical induction. Although it is called "induction", it actually has the force of the deductive forms of proof we've considered so far. Some use an analogy to the "Domino Principle" to describe how mathematical induction works, but the elegance of the method comes from the fact that it involves only two steps:
  I. Prove the statement true for a particular first case.

  II. Prove that, if the statement is true for some general case (e.g. n), then it will also be true for the "next" case (e.g. n+1).

  Taken together, these proofs allow you to conclude that a given statement is true for all cases (e.g. all n).

  We'll do two simple examples, one a proof that all integers greater than 5 can be represented as multiples of 3s and 4s (this disguised as a postage problem), and the other an inductive proof of one of the formulas we derived earlier using direct proof.

• For Next Time:: Have a look at the Notes about proof at the Foundations of Mathematics site, and check the syllabus on Monday for an outline of next week's classes.