Michaela Fleming and Victoria Pierce -- What Do Mathematicians Think About Mathematics?

  As I said when you practiced your talk, you have a lot of good and thought-provoking
  material here.  But I don't think you have really thought about how a lot of it
  connects or considered the questions that are raised here in a deep enough way.  
  For example, if mathematics is so potentially addictive and dangerous as a study, 
  should we be encouraging young people to study it for the reasons you give in your
  opening quotation?  Or is it potentially the case that people who are susceptible to 
  addictions are especially drawn to the subject because of the beauty, order,
  and refuge from the real world that they see in it?  I'm not saying you should
  have tried to answer those questions, because I don't think there are any nice,
  clean answers to them.  But you essentially describe both sides of mathematics
  without trying to reconcile them or address the obvious questions about how
  they are related.

  The paper as a whole is not organized very well -- see comments 7 and 8
  below.  In a way, I think some of this might be due to the fact that you 
  constructed the Prezi for the talk before you wrote the paper.  I don't think
  Prezi forces you to think about the linear ordering of topics very well, so
  it looks to me as though you did not have a clear outline in mind when you 
  started to write.   

  There are also quite a few small writing/proofreading issues.  Look at
  the very first sentence closely, for instance(!)

  Your presentation was good, but it felt a bit "flat" on the day -- almost as though
  you had lost interest.  I hope that was not the case.

  Specific Comments

  1)  page 2: If you think about the quotation from G.H. Hardy that you used here:
      "Real mathematics must be justified as art if it can be justified at all"
      that ends up sounding quite *different* from the "Imagining the Mathematician"
      quotation you started with.  Did you stop to compare those points of view
      and think how they are related?  I think the way to reconcile them is that
      Hardy is talking almost exclusively about *pure mathematics,* i.e. mathematics
      studied only for its own sake, not for applications to real-world questions.
      He says elsewhere in *A Mathematician's Apology* that "there is no permanent
      place for ugly mathematics" (or something similar, I'm quoting from memory 
      here!)  and he lumps almost all aspects of real-world, applied mathematics
      in with "the ugly."  In other words, while Hardy's testimony is important
      for what it reveals about the way his mind worked, it's important to realize
      that he was also a rather extreme case(!)

  2)  page 2:  "One of the most fascinating parts about the study was that mathematicians 
      were able to separate meaningful versus meaningless statements given to them"

      > Why is that so fascinating?  Anyone who studies mathematics has to learn
      how to interpret symbolic expressions and extract meaning when it is there.
      That's a big part of the job, so to speak.

  3)  page 3: "Frenkel tries to prove that math is indeed important and beautiful 
      just like art" 

      > See comment 1 above for this too. The initial quotation you cited is
      saying that mathematics is not just beautiful like art, it's also useful 
      in a practical sense.  I don't think you are appreciating how different some 
      of these statements are and seeing that you ought to think about how they might 
      be reconciled with one another!

  4)  page 4:  "Some mathematicians view mathematics as harmful."  I have the 
      same comment here that I did in the run-through of your talk.  You aren't
      preparing this transition very well.  Everything that comes before is 
      mainly positive statements about mathematics.  Even the quotation from 
      Paul Halmos about the "high" that can come from proving a new result and
      the way that mathematics can potentially be addictive is not really preparing
      for this change in tone.  

  5)  page 7:  "[my work] represents all by itself the greatest work on the foundations 
      of mathematics ever done in the whole history of mathematics and undoubtedly 
      one of the greatest achievements in the whole history of science."  

      > Doesn't this sound rather "unhinged" to you?  Does anyone who has a 
      *healthy* degree of self-respect need to make such wild-sounding claims about
      his or her own work?  The odd thing is that Grothendieck is probably right
      about this, to some extent(!)  But this statement is so far "over the top"
      compared to what most scientists or mathematicians would say about their
      own work that it really calls his mental balance into question.  

  6)  page 8:  "Doing mathematics keep[s] mathematicians sane (before they go 
      insane from all the stress)."

      > Very funny!  (Did you mean it that way?) 

  7)  The paragraphs at the bottom of page 9 and the top of page 10 are 
      essentially repeating points you made before.  I think they could
      have been folded into the earlier discussions for a clearer overall
      organization.  

  8)  page 12:  This material really goes with the discussion of Frenkel's
      story about how he got "turned on" to mathematics by that one special
      teacher.
  
  9)  page 13: "Landau is stating that Hardy and Littlewood collaborated so much 
      that people considered them to be one mathematician."

      > He was well-aware of the fact that they were two separate people.  He's
      just making a joke here.

 
  Final Project Grade Computation 
  
   Bibliography:   10/10  

   Paper:          50/60

   Presentation:   28/30

   Total:          88/100