It seems almost inevitable that a paper advocating for the use of proof and computer analysis would include at least a brief overview of the controversy surrounding the Four Colour Theorem. Borwein (2004) writes that the proof and computer analysis can act harmoniously to support mathematical development. With a constructivist point of view, he argues that mathematics is a mainly human subject with some room for empirical testing and knowledge. Like science, Borwein argues that mathematics is an endeavour which, while it can be self-sufficient, can also be found to be false or incomplete. Computer analysis and computation can help ease the process of proofing by hand, which is supported by the examples in number theory which he provides for the reader.
Perhaps this speaks to my foundation in mathematics, but I have always found comfort in fields of mathematics in which calculations/programming supported mathematical understanding. I found that one of my most enriching courses (and one which seriously made me consider re-trying computer programming in the future) was a course which involved mathematical computation, Math 210. Although seen as an easy-way-out for avoiding the computer programming requirements put out by the math department, Math 210 was challenging because I had to know what to program. The programming helped me verify lemmas, create graphical arguments, and show different results visually - some of these I would not have understood as well as I do now, had I not had the opportunity to program them in two easier languages, Maple and Matlab.
What confused me with Borwein, however, is his claim that "extraordinary speed and enough space are prerequisite for rapid verification and for validation and falsification ('proofs and refutations'.) One cannot have an "aha" when the 'a' and 'ha' come minutes or hours apart" (p. 12). Borwein is likely aware of the misplaced nature of intuition in an empirical. This is to say, Borwein seems to encourage that mathematics become a "serious replicable enterprise" (p. 14). The creation of a scientific enterprise does not require the same kind of intuition as mathematics may, and truly, these two fields are separated except by applications of mathematics (engineering, computation, proofing in statistics, etc). The very nature of knowledge generation differs in mathematics from the positivist one of science, as the former requires personal interaction and interpretation of the problem. The final product in mathematics - in this case, a proof - often does not include evidence of the doer of mathematics, only their interpretation. It is, then, not the construction of the mathematics itself, but the choice of inclusion/omission of content and material that constructs mathematics rather than allows one to discover it.
Alex,
ReplyDeleteIt's interesting that you say you found comfort in computer based mathematics courses, as I myself have felt uncomfortable in those situations. Generally though, I feel uncomfortable in front of a computer or screen. I find comfort in sitting down with paper and a pen, but strained when I try to put the same effort in a digital environment. Part of me wishes I could have just hand written my entire thesis!
Mathematics is a beautiful human endeavour, a facet which is overlooked by many. Although computers do a significant amount of "work" within a proof, the bulk of the job is done creating the program which can do it! The four-colour theorem was lucky in that it boiled down to a finite number of maps. The problem with the twin prime conjecture is that we have infinitely many cases to go through. Even if we don't see twin primes for a very long time, that doesn't mean that we won't find a pair at some point.
I'm a little confused by your statement at the end about the inclusion/omission of content constructing mathematics. Certainly, if certain content is omitted, there could be a very large gap in the existing proof. Normally, reviewers would notice this and then the proof needs to be patched up in regards to that gap. In fact, this happened to a friend recently. He and a colleague were essentially ready to submit a paper they had been working on for the last six months on, but just now found a huge gap within their proof; a gap which they think they might not be able to alleviate.
I am very interested in the statement, "computer analysis and computation can help ease the process of proofing by hand." My paper that I am writing for this course is about how technology has influenced mathematics teaching and learning. One of the articles that I read discussed how very competent mathematics students can become less confident when using computers to solve problems. Only those who were already confident & motivated when it came to using computers, were able to maintain their mathematical confidence. Of course there will be exceptions, but it should be noted that computers do not always alleviate mathematical issues (much to my chagrin as I am the Chair of my school's Technology Committee). To allow technology to aid things like solving proofs, I believe that the user must first become comfortable with the basic functions of the technology before implementing it in a mathematical situation.
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