The Unapologetic Mathematician

Mathematics for the interested outsider


I think I’m done with categories qua categories for now, and am ready to move on to another subject for a little while. But before I do, I’m going to take a little break and finally get down to this restructuring of the subjects listed on the right. It’ll also give me some time to catch up on some stuff in the real world that needs doing.

The RSS feed will probably be going nuts with updates to posts as I crawl through the archives and relabel things. My apologies in advance.

[UPDATE]: I’ve finished the group theory archives. Unfortunately, the “category” bar on the right seems to not indent nested categories in this theme. Sort of annoying…

[UPDATE]: There seems to be a bug with subcategories on WordPress today, and I’m not the only one having it. So I’ll have to hold off on refining the rings and categories topics until later. In the meantime, I’ve reworked the sidebars a bit, including a search field!


October 22, 2007 - Posted by | Uncategorized


  1. Good for you! Librarians have known for ages that knowledge is less useful if it ain’t ordered right.

    Comment by nbornak | October 22, 2007 | Reply

  2. Hi John Armstrong,

    1 – Thanks for the reference ‘The Principle of Relativity’, but to which are you referring?:
    a – by A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl (Paperback – 1923)
    b – by Albert Einstein and Frances A. Davis (Paperback – Jun 1, 1952)
    c – by Alfred North Whitehead (Paperback – April 1, 2007)

    I will add one to my reading list, and may eventually be able to read all three, but will not be able to get to any until after March 2009.

    2 – It was not my intent to duplicate Einstein but rather understand him in my own terms.
    I am not surprised that my effort looks “nothing like” the original work.
    By attempting to use applied mathematical techniques from engineering for my crude and perhaps naive analysis, I think that I have done what I meant to do.
    It appears Einstein allows for relative speeds or velocities of light to exceed c with a maximum at 2c but the maximum energy at c*sqrt(2), if c is obtainable.
    I acknowledge that I may not even be wrong.
    Certainly in my crude geometric representation, OA did not transform to be nearly equal to BC=B1C1 as desired.

    3 – Richard Bellman coined “dynamic programming” as a synonym for “optimization”.
    Wiki discussion of dynamic programming

    I have not read the original work of Richard Bellman, Dynamic Programming, Princeton University Press. Dover paperback edition 1957.
    When a Bellman colleague stated that his principle of dynamic programming was not rigorous, Bellman reportedly replied, “Of course not. It’s not even precise. A good principle should guide the intuition.”

    Archimedes was not precise when he attempted to determine the value of PI by comparing polygons to a circle.

    Comment by Doug | October 26, 2007 | Reply

  3. For a new topic: group field theory!

    Comment by angryphysicist | October 28, 2007 | Reply

  4. Nothing nearly so complicated. No, I’ll finally be talking about topology, which will let me define the real numbers at last, and thus on to analysis.

    Comment by John Armstrong | October 28, 2007 | Reply

  5. Doug, sorry your comment got caught in the spam filter.

    This is really in response to a comment you left somewhere else, where you did say that you were attempting “to understand how Einstein derived this equation.” And here you say that you were not attempting to duplicating his work. So which is it?

    Comment by John Armstrong | November 2, 2007 | Reply

  6. Hi John,

    I fail to see how I can be much clearer than:
    “It was not my intent to duplicate Einstein but rather understand him in my own terms”

    [clipped and sent back to the original forum]

    Comment by Doug | November 3, 2007 | Reply

  7. Again, this is really not the place. The proper place is on the forum where the discussion arose. When you posted there you said your intent was “to understand how Einstein derived this equation.”

    My response in that original forum was to this prompt. Now you come here and say that your intent is to come up with an alternate derivation. One of the following is true:

    (a) You honestly don’t understand the difference, which may well come down to your diffrent background. I don’t know what sort of standards medical schools push as regards logical rigor. But in this context, you’re trying to speak math, and your accent is horrible.

    (b) You know the difference and are attempting to pull a fast switch, changing your goals as your earlier ones are frustrated. This is incredibly dishonest, and so I hope this isn’t the case.

    Either way, I’ll say again (three times now) that this is not the appropriate place. If you want to have this discussion, go back to where you started it.

    Comment by John Armstrong | November 3, 2007 | Reply

  8. Hi John,

    I hope to say this once only. I will repeat as many times as you request as long as you understand that I must stop as of 11 November 2007.

    I can understand why a student of yours writes “… I’m withdrawing anyway” on exam of yours without an effort to flatter you.

    I heartily agree with you that I fail to understand your criticism that I did not plagiarize Einstein.

    I will not have a discussion with you about a topic on another person’s blog, even if the topic originated on another’s blog. That is a waste of that bloggers’ time and storage space.

    I have as much respect for your mathematical knowledge as I have for William A. Dembski, PhD math U-Chicago.

    I am perfectly happy and very satisfied with you critique of my mathematical ability.


    Comment by Doug | November 5, 2007 | Reply

  9. Let me see if I’ve got this straight:

    You won’t continue the discussion you started on another weblog because it would waste that host’s time, but I’m supposed to continue it here.

    You respond to the fact that I won’t indulge your whims by attacking my teaching style, with absolutely no information about the situation other than one offhand comment I quoted, such as the student’s prior performance or what help of mine he or she had sought.

    You further try to compare me somehow with a noted ID supporter in a backhanded attempt at slander.

    And you still refuse to accept that my response to your original comment in another forum was purely about your statement that you were attempting “to understand how Einstein derived this equation”, which you now implicitly deny while offering no retraction of your original comments.

    I suppose next you’ll threaten to hold your breath?

    Comment by John Armstrong | November 5, 2007 | Reply

  10. Hi John,

    Well I am still breathing.

    At least you got it partly correct.

    On another blog about examples, I made my comment. You did not respond to this as an example, but criticized my manner of deriving E=mc^2 in my effort to gain personal insight into Einstein’s derivation. This aspect of the discussion therefore belongs on your blog.
    I agree 100% per cent that my derivation looks nothing like that of Einstein.
    If it looked like that of Einstein then one would either be parroting or plagiarizing his work.

    On another blog, you made an effort to correct my understanding of grayscale imaging, the Monster and M-theory. I tried to move such a discussion to your blog, but you ended up wasting everyone’s time on that blog by demonstrating that you had no knowledge of:
    a-grayscale imaging, you really should read the literature before commenting, literature previously provided, “8 bit grayscale, 8 bit indexed, 24 bit RGB color”
    b-I had to provide you to literature on Borcherd’s 1989 Monstrous Moonshine proof for which he received the 1998 Fields medal.
    c-I was remiss in not referring to literature on the 11-dimensions of Witten’s M-theory.

    Read Brian Greene, The Elegant Universe, chapters 12-13. This was also a PBS NOVA special.
    There is also a M-theory synopsis on wiki,
    “It seems plausible, then, that there is some quantum theory — which Witten dubbed M-theory — in eleven-dimensions which gives rise at low energies to eleven-dimensional supergravity, and is related to ten-dimensional string theory by dimensional reduction.”

    On that blog you stated:
    a-“Then a “picture” of would be a collection of 248 points, each a different shade of grey.” and “We have a 248-dimensional manifold that may need a 500-dimensional space to be embedded smoothly in.”
    b-“M-theory has nothing. M-theory hasn’t been defined, much less rigorously laid out.”
    c-”“Monstrous Moonshine” is a conjecture. Are you thinking of the “Moonshine Module”, defined by Frankel et. al.? In that case, it was not originally related to string theory, so invoking strings here is completely spurious. The moonshine module is just a representation of a very large, but finite (not continuous) group. is related, but very obliquely.”
    d-”Look, I’m far from some sort of mathematical elitist that wants to keep these ideas the province of some sanctified priesthood, but the hallmark of the field is specifically-defined terms used in very specific ways. You can’t just take whatever words sound familiar and throw them together.“

    For d, please do a word analysis.
    I agree that you are no “mathematical elitist“.
    I still doubt that I will ever have the ability to fully understand your points.
    I see no need to retract any comments that I have ever made to you, but instead reiterate all of them.

    The attempt at insult was not backhanded, it clearly states an example of how much respect that I have for you.
    Your offhand quote is apparent in the nature of your blogging style.

    I want to thank you for sharing all your that you have to teach.
    I find it sad that I have absolutely nothing more to learn from you.

    Remember that the last day that I will post to your blog is 11 NOV 2007.
    I probably won’t read any more drivel on your blog until 8 NOV 2007.

    Have you considered using “uniformed” rather than “unapologetic”?

    Well I am still breathing.

    Comment by Doug | November 6, 2007 | Reply

  11. (a) You’re restating your points and mine completely out of context, again. You originally brought up using “color” as another dimension to use in drawing “pictures” of E_8. I pointed out that yes, one could use n greyscale pixels as coordinates on an n-dimensional cube, or one could use n RGB pixels as coordinates on a 3n-dimensional cube, but the resulting picture wouldn’t necessarily have anything to do with a projection of E_8 (or of the root lattice, which you kept confusing for the Lie group itself).

    The “248” I mentioned had nothing to do with the number of bits involved in the representation, but instead refers to the number of dimensions of the underlying manifold of E_8.

    (b) There is much disagreement, and I’m far from alone in my position. M-theory as it stands is not a theory. It is a collection of conjectures pointing towards the possible existence of a theory. Invoking it to explain anything at this point, especially outside of the very specialized area of study in which it arose, is pure crackpottery.

    (c) As I understand it, Borcherds proved the original version of a conjecture which has been generalized since its formulation, and I’ve heard the generalizations referred to under the same name. As an analogy, the Goldbach conjecture has been generalized to the Hardy-Littlewood conjectures, which (sensibly) have a different name than the original special case. Someone proving the Goldbach conjecture need not prove all the rest of the generalization.

    However, this is only tangentially related to my area of study, and my familiarity with the current terminology is gained from passing mentions in conversations with people who pay more attention to it. If I have been using the term “monstrous moonshine” to refer to a wider problem than is specifically warranted, I offer a correction of my earlier terminology.

    Your arguments consist wholly of term-dropping substituted for actual understanding of the concepts you claim to be interested in, name-calling, selective quotation, and other well-known cheap rhetorical tricks. You’re sloppy, faddish, and petulant when those facts are pointed out.

    Comment by John Armstrong | November 6, 2007 | Reply

  12. Hi John,

    Well points a,b,c are relatively well argued, even though I so not necessarily agree.

    I will skip a since we are too far apart in our different perspectives.

    For b, I think that I understand what you are trying to say.
    Since Witten formulated what is called M-theory, I really think it has to have 11-D as he formulated it.
    I think that I see a means of modifying [by insight, not rigor] his M-theory to only 8-D [6-complex-D + 1-coupling-D + 1-time-D] which has a form that seems to be relatively consistent with Borcherd’s Monster of 24-complex-D + 1-string-D + 1-time-D.
    This may explain why Witten recently embraced the Monster.
    I could be wrong.
    Cumrun Vafa formulated F-theory of 12-D with a second time-D.

    For c, I only know that many people are interested in the Monster and examining it in various ways.
    I once posted to Borcherd’s blog asking how the Monster may be related to game theory.
    He responded that he was not an expert in game theory.
    This very much surprised me since he was a student of John Horton Conway [‘On Numbers and Games‘] and wrote some chapters of ‘Sphere Packings, Lattices and Groups‘.

    The last paragraph is drivel.
    I have engaged in drivel with you, although I maintain that you started it.
    I do not like drivel since there is no meaningful exchange.
    I do not mind apologizing for using drivel, but will reluctantly resort to such in unnecessary confrontation.

    My last post will be tomorrow 9 NOV.
    I have decided to watch American college and professional football over Veteran’s Day weekend.

    Thanks for the information exchange in a,b,c but not for the words of the last paragraph.

    Comment by Doug | November 9, 2007 | Reply

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