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Posts posted by merjet

  1. 45 minutes ago, StrictlyLogical said:

    EDIT:  Just reread a previous post of yours.  Note, a 2 dimensional matrix can be used to solve N unknowns in N equations.  A matrix with 3 or more dimensions is quite a different kind of thing. 

    In one sense the dimension of a matrix is always 2 -- it has rows and it has columns. The usual meaning of a matrix's dimension is the number of rows and the number of columns -- e.g. 2x2, 3x3, 4x4, ... mxn (link). My saying "matrices with more than 3 dimensions" was less than exact. My intent was a 4x4, 5x5, etc. which do not have spatial counterparts. I don't know what you mean by "matrix with 3 or more dimensions."

  2. 21 hours ago, merjet said:

    Any idea about content, e.g. more of this and less of that?

    I finished reading Knapp’s book, Mathematics is About the World.

    I rate it 5 stars, but with some room for improvement.

    Knapp barely mentions arithmetic and counting. More about arithmetic would strengthen his thesis that mathematics is about the world. The positive integers used for counting (and zero) form the foundation for the real numbers. Understanding addition and subtraction of fractions call upon the important concepts of unit and transformation, which he does use extensively for different topics – measuring and vector spaces.

    As an aside, as I have already indicated, mathematics is also about the way we think about the world. Mathematicians “extrapolate” concepts beyond perceptual reality. Examples are complex numbers and matrices with more than 3 dimensions.

  3. 26 minutes ago, StrictlyLogical said:

    Perhaps he has a slightly different audience in mind?  More for a layperson who is philosophically inclined, rather than a technical or scientific person who is mathematically inclined?

    Okay. Any idea about content, e.g. more of this and less of that?

  4. 1 hour ago, William O said:

    It looks like Harry Binswanger has a new book on philosophy of mathematics in the works. That should be interesting.

    Regarding Robert Knapp's book, the author acknowledges Binswanger's help, and Binswanger wrote a 5-star review of it for Amazon. So I'm curious if you have any clues about how a book by Binswanger would differ from Knapp's?

  5. 1 hour ago, StrictlyLogical said:

    Yes induction works nicely.

    and yup, it isn't necessary in my view...

    I'd like to know the reasons why it would be necessary.. in someone else's view.

    I can't give think of any good reason why somebody else would think it necessary for 1+a = a+1.  On the other hand, there are other P(n) that could be proven by mathematical induction where the truth of P(n) for all n is not so intuitively obvious. Problems 3-7 here are examples. In the case of 1+a = a+1, using mathematical induction is akin to computing the area of a circle using integral calculus instead of using the simple formula pi*r^2.

  6. 34 minutes ago, StrictlyLogical said:

    Anyone want to explain, in view of what I’ve stated above, how induction would be necessary here?  And IF necessary how it would be possible?

    It isn’t necessary -- at least in your view -- but it is possible.

    Task: Prove (1 + a) = (a + 1) is true for all natural numbers.

    Method: mathematical induction

    Base case: a = 1. (1 + a) = (a + 1) is obviously true.

    Inductive step:

    Show that if P(k) holds, then also P(k + 1) holds.

    (1 + k) = (k + 1)

    (1 + k) + 1 = (k +1) + 1

    (1 + (k + 1)) = ((k +1) + 1)


    From the linked page: "Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. ... Proofs by mathematical induction are, in fact, examples of deductive reasoning." 

    In other words, mathematical induction relies on a chain of deductions.

  7. Hilbert per Stephen: “Thus algebra already goes considerably beyond contentual number theory. Even the formula (1 + a) = (a+ 1), for example, in which is a genuine number-theoretic variable, in algebra no longer merely imparts information about something contentual but is a certain formal object, a provable formula, which in itself means nothing and whose proof cannot be based on content but requires appeal to the induction axiom."

    Why not and why? Let a = 4. The formula tells me that I can (1) start with 1 dime and add 4 dimes, or (2) start with 4 dimes and add 1 dime. Either way, the result is 5 dimes.  Also, if a equals some other integer > 1, then I can (1) start with 1 dime and add a dimes, or (2) start with a dimes and add 1 dime. Either way, the result is the same count of dimes.

  8. On 7/21/2019 at 11:51 AM, GrandMinnow said:

    But I strenuously recommend that it is folly to read a volume such as this without first learning the basics of symbolic logic and then at least introductory mathematical logic.

    I have no plan to do so. The above also indicates how far this thread has strayed. The title is Math and Reality. Mr. Knapp’s book’s title, sans subtitle, is Mathematics Is About the World. I agree it is very much about the world, but think it’s a little more than that. More concretely, Knapp’s thesis is that arithmetic and geometry, especially analytic geometry, pertain to the world. He defines mathematics as the science of measurement. (Analytic geometry and calculus enable indirect measurement.) I think mathematics is a little broader than that, but measurement is a big part. Functions and vector spaces also pertain to the world. His book is not about symbolic logic, mathematical logic, predicate calculus, or finitary vs. infinitary.

    His book presents an alternative view of mathematics that is very different from formalism, logicism, Platonism, and others. Regarding the philosophy of mathematics schools of thought surveyed here, his is most similar to Aristotelian realism or empiricism. My view is much like Knapp’s.

  9. 11 minutes ago, Boydstun said:


    I don't know why there is no spot for putting a like on that post, so I'll just state that I appreciate it, as well as other posts of this contributor.

    Try signing in. I don't see the heart icon when not signed in, but do when I am.

  10. 14 hours ago, GrandMinnow said:


    "'The symbols may represent intuitively meaningful percepts or concepts, but they are not to be so interpreted in pure mathematics.' That raises the possibility that the symbols are not always meaningless, but only that they should be so regarded at times."

    That is well put.


    * But at least one primary source is:

    Grundlagen der Mathematik I - Foundations of Mathematics I - Part A


    For showing that Hilbert did not take mathematics to be merely a meaningless symbol game, such a quote is QED, don't you agree?


    * From section 3.3 here:


    "Hilbert makes [the distinction] between the finitary part of mathematics and the non-finitary rest. The finitary part Hilbert calls 'contentual,' i.e., its propositions and proofs have content. The infinitary part, on the other hand, is not meaningful from a finitary point of view."


    Thank you.

    The quote is not clear or extensive enough for me to accept as proof Hilbert regarded no part of mathematics to be merely a meaningless symbol game.’ You even quoted Zach: “The infinitary part, on the other hand, is not meaningful from a finitary point of view."

    How did you get an English translation of Grundlagen der Mathematik? According to this page, some of Volume 1 has been translated to English. Anyway, it’s difficult for me to cite Hilbert himself about meaningful/meaningless without a full English translation available.

    Returning to the SEP entry Formalism in the Philosophy of Mathematics again: "The Hilbertian position differs because it depends on a distinction within mathematical language between a finitary sector, whose sentences express contentful propositions, and an ideal, or infinitary sector. Where exactly Hilbert drew the distinction, or where it should be drawn, is a matter of debate. Crucially, though, Hilbert adopted an instrumentalistic attitude towards the ideal sector. The formulae of this language are, or are treated as if they are, uninterpreted, having the syntactic form of sentences to which we can apply formal rules of transformation and inference but no semantics."

    "No semantics" means no meaning.

  11. 13 hours ago, GrandMinnow said:

    One fine point, just to be clear:

    You mentioned "David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning."

    I said that it is not fair to ascribe that view to Hilbert without citing it in his writings. And, so far, we not been given such a citation. 


    As I mentioned, Hilbert took finitary mathematics to be contentual and reliable beyond reasonable dispute. So I don't know in what sense he would regard finitary mathematics as most reliable when viewed as divorced from content, or even in what sense he would regard finitary mathematics as having more reliability when viewed as divorced from content. But those would be less implausible than saying he took finitary mathematics as reliable only when divorced from content. 

    But as to infinitary mathematics, I probably wouldn't quibble with saying that Hilbert took it to be reliable only in terms of formal symbol rules. Indeed, it would be fair to say that, more or less, formalists don't accept that infinite mathematical objects (such as infinite sets, infinite sequences, et. al) can be taken as reliable concepts other than as informal notions as extrapolations from formal systems. 

    But even this does not imply that Hilbert didn't recognize that infinitary mathematics is useful for the sciences. Hilbert, like just about any mathematician, was steeped in infinitary mathematics and would recognize that, say, infinite sequences for calculus are used for framing the mathematics for the physical sciences. 


    Fair enough. I did write, "David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning" twice. The second time was a direct quote from Kline. The first was copied from something I wrote several years ago, and was likely influenced by Kline.

    I once also wrote elsewhere following the above quote: "The symbols may represent intuitively meaningful percepts or concepts, but they are not to be so interpreted in pure mathematics." That raises the possibility that the symbols are not always meaningless, but only that they should be so regarded at times. 

    I can't remember ever reading anything by Hilbert himself. There is a risk in that, but relying on secondary sources is very hard to avoid due to limitations of time and interest.

    Page 48 here is Philip Kitcher regarding Hilbert's formalism. Kitcher regards Hilbert as an apriorist. I think that Hilbert's epistemology has some bearing on Hilbert's view of meaningful/meaningless. Do you agree with that? This referenced by Kitcher might be an interesting read. I didn't find the full article anywhere.

    Maybe we can more agree on our views of Hilbert's view of meaningful/meaningless if you will cite Hilbert himself or at least a secondary source you judge to be better than, or as good as, Kline and Kitcher. Maybe you have relied on Hilbert himself or a secondary source with the distinction you make between finitary and infinitary mathematics. I won't quibble with what you have said about that.

  12. 1 hour ago, GrandMinnow said:

    Formalism has variants.

    Of course, Formalism designates a group of people who don't all hold identical views. Hilbert was a major figure of the school. I suggest a more charitable reading of Kline and Knapp.  "Formalism -- a major figure being Hilbert -- holds that ....."

    Note that in both quotes I gave from Knapp's book, Knapp does not even use Hilbert's name. 

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