Misprints requested!

MWA may soon have a second printing.  If any of you have noticed misprints, I would be grateful if you could send them along to harris@math.columbia.edu, so they can be corrected.  A few of them have already been pointed out.  I’m sure there are many more.


5 thoughts on “Misprints requested!

  1. Anonymous

    This isn’t exactly a misprint, but in Chapter 3 you write

    “Mathematicians frequently write that they see a proof not as a goal in itself, but rather as a confirmation of intuition. Thus G. H. Hardy once claimed that “proofs are what [collaborator J. E.] Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils.” ”

    This quote is taken out of context. The full sentence in Hardy’s lecture is

    “If we were to push it [the analogy of a mathematician as an observer] to its extreme we should be led to a rather paradoxical conclusion; that there is, strictly, no such thing as mathematical proof; that we can, in the last analysis, do nothing but point; that proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the
    lecture, devices to stimulate the imagination of pupils.”

    He goes on to say “This is plainly not the whole truth, but there is a good deal in it.
    The image gives us a genuine approximation to the processes of mathematical pedagogy on the one hand and of mathematical discovery on the other;”

    A paragraph later he says “On the other hand it is not disputed that mathematics is full of proofs, of undeniable interest and importance, whose purpose is not in the least to secure conviction.”

    I think the way you quote this in your book is a misrepresentation.


    1. mathematicswithoutapologies Post author

      Ian Hacking calls this an “oft-quoted line” on p. 87 of his Why Is There Philosophy of Mathematics At All?. It appears in Lakatos’s Proofs and Refutations and in Morris Kline’s Mathematics, the Loss of Certainty, Claude Rosental’s Weaving Self-Evidence as well as in numerous articles and lectures by philosophers and sociologists. If the quotation in the book is a misrepresentation it is one with a long pedigree. But I think the context is not limited to the lines that immediately precede and follow the quotation. The third paragraph of Hardy’s section 13, in which he compares a mathematician to an observer of “a distant range of mountains,” already seems to me to justify the habitual use of this quotation. The question is whether it’s the first part or the second part of the sentence “This is plainly not the whole truth, but there is a good deal in it” that carries the meaning of the passage. It seems obvious to me that it’s the second part; otherwise Hardy’s analogy is pointless.

      So should the quotation always be accompanied by Hardy’s disclaimer? Clearly it should if the topic is Hardy’s philosophy; but that was not the topic in Chapter 3. Does the inclusion of this “oft-quoted passage” in Chapter 3 offer a misleading or incomplete picture of Hardy’s philosophy? Possibly it does, as it does whenever it has been quoted; in which case the quotation should be given a rest. However, the main purpose of Hardy’s article is to provide some satisfactory account of his “invincible feeling” that believing the statement of the Goldbach Conjecture amounts to believing something, and I think this accords well with the second part of the sentence about “the whole truth.”

      The context also includes the year 1929 in which Hardy’s paper was published, when Hilbert’s formalist program could still hope for success.


      1. Anonymous

        Thanks for your thoughtful reply, and for accepting my comment in spite of your objections to it. I didn’t realize you were following such established practice. Nonetheless, I disagree that, as you put it, the question is which part of “This is plainly not the whole truth, but there is a good deal in it” carries the meaning of the passage. Both parts carry the meaning. Hardy is arguing for a complex understanding of what “proof” is, and I think it is a misrepresentation to quote a fragment which removes all nuance.

        I was a bit startled by your list of examples of authors who quote the passage as you do. I looked up the quote in the four books you mentioned, and as far as I can tell they also remove all nuance, with the partial exception of Lakatos’ “Proofs and Refutations” (but I think even there the disclaimer is not strong enough). I see now that you are writing in a tradition which is unfamiliar to me, and I will try to accept it. I am finding your book very interesting, educational, and thought-provoking. I only looked up this particular quote because I am a fan of Hardy (I can’t wait to get to chapter 10!) and it seemed at odds with what I knew of him from “A Mathematician’s Apology” and from “Some famous problems in the theory of numbers” — perhaps I should have learned long ago to take quotes with more of a grain of salt.


  2. Pingback: On anonymous comments | Mathematics without Apologies, by Michael Harris

  3. Dieter Düsedau

    On page 115, line 10 should be corrected:
    3=4*0+3; 6=2*3; 7=4*1+3; 11=4*2+3; 12=4*3
    all equations should be separated by semicolons



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