History of Mathematics
info on Reading Responses

Avoid... In favor of...
I liked/I didn't like ___
___ was interesting/amusing
My favorite was ___
or anything else that's essentially a movie review
___ was effective at illustrating ___
___ persuaded me that ___
___ conflicted with my previous intuition that ___
or anything else descriptive, using precise language

General writing tips

  • Make one main point or argument or observation and support it in a careful and relevant way.
  • Engage with the text: show that you have been thinking about it. Original or creative responses are great; summarizing the text does not help.
  • Try to write so that another classmate can follow your point and your outside references.
  • Avoid colloquial phrases (totally, amazing, pretty much), redundancy, sloppy grammar (run on sentences, comma splices, sentence fragments), and spelling errors.
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    "Best of": a sampling of check-plus responses

    Symbols and formulas unit: Lautze, Knapp

    Axiomatization unit: Palmer, Dragon

    Infinities unit: Wight, Ngai

    Algorithms unit: Keener, Puzak

    Intro unit: see below.

    Some examples of strong responses from the intro unit, with commentary

    Response: Julie Asperger

    I was most amused by the exchange on the bottom of page 12 where the teacher is arguing in favour of altering the instructions of the lemma to make the process of triangulation work and that despite, or actually because of the complexity and implausibly of the lemma it has the potential to become truth, and when questioned on what would happen if he couldn't do such a twist to save his complex proof teacher responds, "Good question- it will be put on the agenda for tomorrow." (12) This to me was an elegant summation of the pragmatism of the proofs and refutations process that hinges on bold assertions being knocked down and dissected until after much wading through the muck someone comes up with something coherent and simple enough to be dogmatically taught to the next generation. I think there is a great deal of wisdom in keeping an idea in this context relevant, however strange it may seem, until it can actually be disproved rather than discounting the notion because it doesn't appear that it will stand up to the next barrage of criticism. I find this so funny though because it seems to me that this concept that "'[p]lausible' or even 'trivially true' propositions are usually soon refuted: sophisticated, implausible conjectures... might hit on the truth," flies directly in the face of Occam's Razor, the Holy Grail (or at least Roc's egg) of science. Even though math and science are frequently thought of as being intimately acquainted, I find it fascinating and wildly counterintuitive that their methods of refinement appear to be such opposites, and yet both methods have so far been shown to be the most thorough and effective for their respective environments.--Julie A 15:28, 10 January 2007 (EST)

    Good: precise, vivid language; makes a strong point ("This to me was an elegant summation of the pragmatism of the proofs and refutations process that hinges on bold assertions being knocked down and dissected until after much wading through the muck someone comes up with something coherent and simple enough to be dogmatically taught to the next generation.").

    Could be improved: some instances of movie-review language ("I was most amused by...", "I find this so funny...") and run-on sentences.

    Response: Michael Steele

    Along the lines of what Ms. Wright wrote, I also found the evolution of the definition of polyhedron to be one of the more interesting aspects of this first chunk of reading. When I first opened Proofs and Refutations and reached the problem at hand, I had quaint images of cubes and other regular polyhedra dancing through my head. Then, when the various pupils began proposing their own concoctions of what exactly a polyhedron is, I shut the book and tried to construct my own definition, hoping to affirm or deny my idea within the next few pages. Well, more than 40 pages later, there's still no consensus! I find it quite interesting that, although the problem initially posed was of proving whether or not V-E+F=2 is valid or not, a parallel (maybe even more imperative!) problem has been defining on exactly what can we speak of V-E+F=2?

    This aspect of the debate ties quite firmly with another debate I had been following earlier this week, as to who deserved to win the National Football League Most Valuable Player award. Reading through the comments section in response to Michael David Smith's article that Peyton Manning deserved the award on FootballOutsiders, a debate analogous to that of this reading arose among the many individuals that commented. While the arguments first focused on whether or not Peyton (or Drew Brees, or someone else) deserved the award, very quickly the debate turned what exactly does the MVP award mean, IE: how to define what constitutes an MVP . Like in the Lakatos reading, definitions were posed (see comments #47 and #51) and pathological cases engineered (see #44 about a league of 31 Ryan Leaf's and one Joey Harrington), and this aspect of the debate, I feel, proved the most interesting.--Maxjsteele 18:55, 8 January 2007 (EST)

    Good: responds to classmate, well-crafted language, outside references that support the point in precise ways.

    Could be improved: repeated use of "X is interesting" construction.

    Response: Erin Rimando

    On page 13, Gamma says "A single counterexample refutes a conjecture as effectively as ten." Before I started reading this book, I would have agreed with this statement. On page 14, though, Alpha describes a counterexample as "a criticism of the proof." Alpha's view certainly is more useful than Gamma's view, for we can learn and improve our knowledge with the help of criticism. Alpha's view is what the students use to improve their proof. However, with Gamma's view, a counterexample delivers a decisively fatal blow from which a conjecture cannot recover.

    This battle between the students to discover who has the right idea about the "truth" of the teacher's conjecture reminds me of a lecture in a semantics class I once took. In this class, I saw that the "truth value" of a statement isn't necessarily restricted to either "true" or "false", as what is usually taught in classical logic (for example, if someone says, "Today is Wednesday," we recognize either "Yes, he's right," or "No, he's wrong."). The part of the lecture applicable to this discussion is that a statement or problem can have an answer besides "true" or "false": sometimes, all we can say is "it depends." For the students in Lakatos' classroom, "it" (the truth of their conjecture) depends on their definitions. --Erimando 01:02, 9 January 2007 (EST)

    Good: strong phrases ("decisively fatal blow"), good outside example, specific and well-chosen references from reading.

    Could be improved: Example could use one more illustrative detail: how about one particular instance of a statement with indeterminate truth value?