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merjet last won the day on July 25

merjet had the most liked content!

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About merjet

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    United States
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    I have several articles published in Objectivity and the Journal of Ayn Rand Studies.
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    University of Illinois
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    actuary (retired)

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  1. Crystal Fire #1 Crystal Fire #2 Crystal Fire #3
  2. Democratic demagogues about health insurance
  3. I don't know, but crab apples are legit. 😊
  4. I put a review of Mathematics is About the World on Amazon. I will consider suggestions for improving it.
  5. 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."
  6. Spheres of Justice #13 Spheres of Justice #14
  7. 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.
  8. Okay. Any idea about content, e.g. more of this and less of that?
  9. 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?
  10. Decades-Old Computer Science Conjecture Solved in Two Pages This is not directly related to previous posts, but it shows a connection between very abstract mathematics and concrete reality. So it fits the title of the thread.
  11. 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.
  12. 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) QED. 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.
  13. Reply to MisterSwig. Something to consider: https://math.stackexchange.com/questions/1058596/in-plain-language-whats-the-difference-between-two-things-that-are-equivalent
  14. I didn't say or imply you might try to derail it.
  15. The nature of mathematical induction assures that it won't be derailed by infinite cardinals or non-commutative ordinals (link). 😲
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