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Book Description:

What is mathematics about? Is there a mathematical universe glimpsed by a mathematical intuition? Or is mathematics an arbitrary game of symbols, with no inherent meaning, that somehow finds application to life on earth? Robert Knapp holds, on the contrary, that mathematics is about the world. His book develops and applies its alternative viewpoint, first, to elementary geometry and the number system and, then, to more advanced topics, such as topology and group representations. Its theme is that mathematics, however abstract, arises from and is shaped by requirements of indirect measurement. Eratosthenes, in 200 BC, demonstrated the power of indirect measurement when he estimated the circumference of the earth by measuring a shadow at noon, in Alexandria, on the day of the summer solstice. Establishing geometric relationships, solving equations, finding approximations, and, generally, discovering quantitative relationships are tools of indirect measurement: They are the core of mathematics, the drivers of its development, and the heart of its power to enhance our lives.

About the Author:

Robert Knapp earned his Ph.D. in mathematics from Princeton University in 1972. He has published work on differential geometry and partial differential equations, and, after a year at the Insitute for Advanced Studies in Princeton, taught graduate and undergraduate mathematics at Purdue University. His study and appreciation of abstract mathematics began in high school and his conviction that mathematics, including abstract mathematics, is about the world began then, as well. Although he retired from the profession in the late 1970s, his study of the content, history and application of mathematics continues to this day. In recent years he has presented his unique perspectives on geometry and the number system in a series of lectures at Objectivist Summer Conferences organized by the Ayn Rand Institute. He has lived in the Philadelphia area for almost 30 years.

22 thoughts on “Just Published: Buy Now!

      • I’m expecting to have something I can review in about a week. The last attempt was dropping text because of various Greek letters (such as pi, epsilon, and delta) and mathematical symbolism.


    • I’m glad to see a Kindle incarnation, but would like to see the format improved before I buy and download the book. In the sample of the Kindle edition, paragraphs are often not indented–this is the case toward the end of the preface and then intermittently through chapter one. I can’t say about later chapters, having viewed only the sample. Kindle provides an app that enables publishers to preview how a book will appear in Kindle and Kindle software.


      • This appears to be an issue. I will investigate. Update; the kindle version was derived from a PDF, is faithful to the text, but has the noted formatting issues. I’m still looking into why that should be or what remedies might be possible.


  1. I have a very specific question related to an idea that I’m pursuing, and would appreciate your input.

    Where you state, “….he estimated the circumference of the earth by measuring a shadow at noon….”

    Does this not imply that our knowledge of an event exists in analog form? And by analog form, I do not mean a “metaphorical” analog form, but more along the lines of how an electrical engineer would describe an analog signal, i.e. a varying amplitude of energy, over time, recorded on a media. In this case, the projected shadow, on the ground, at a certain time.

    When we record a symphony, the mechanical energy of the sounds are converted to electromechanical energy by a microphone/transducer and stored on a rolling magnetic tape. When we hear a symphony, either live or recorded, the mechanical energy of the sounds are converted, by our ears, into bioelectric energy which are “recorded” on our neurons.

    What I would call an Analog Signal, you appear to be calling an Indirect Measurement.

    Is that a fair statement?


    • I don’t look at it that way. What’s important to me about the example of Eratosthenes is the use of a series of measurements, culminating in the one that I mention, to measure indirectly something he could never have measured directly. This series of measuremens includes day of the year, length of that day (summer solstice), distance of a particular town to the south, angle of the sun (overhead) in that town, length of a shadow (in effect) in Alexandria compared to the length of the vertical stick casting the shadow. Eratosthenes put this information together, in light of his understanding of Euclid’s geometry, to estimate the circumference of the earth. My thesis is that this is a dramatic example of the typical pattern of most measurement, the core reason we need mathematics, and a key to understanding its method.


  2. On p 336, paragraph 2, describing convergence of a series to X, you write, “for any set U containing an open ball around X there exists a number N such that every term after the Nth term is inside the set U.” I don’t follow this. What if the set U consists of {an open ball of (-1 to 1) plus -2 plus 2}, and the series consists of 2, -2, 2, -2 …? This never converges, yet every term is in U.


    • The key to this criterion is the “for *any* set U”. Whatever X might be, if the criterion fails for any open set around X, then the sequence does not converge to X. In this example, if U is, say, an interval of length 1 containing X, it cannot also contain both 2 and -2. So the criterion fails because there are values of both 2 and -2 no matter how far you go out in the sequence. So it does not converge to X. Since this argument holds for any X one might consider, the sequence does not converge.


  3. On page 336, paragraph 5, you define an open set as any set U with the property that every point in the set U is contained in a ball, centered at that point, that is also contained in U. On page 339, you describe a set of functions, g, as an open set. Are functions “points” in the sense of points contained in a ball?


    • Yes and perhaps I should make this clearer in the text. When the focus is on convergence of sequences of functions, the functions, from the topological perspective are treated as points and an open set is a set of functions. Obviously, a function is still a function, so one still has to distinguish the points in its domain and range. The concept of an “open ball” generally does not apply to functions; it can only apply when a distance function has been defined. But, for functions, the notion of a filter, discussed in the text, plays essentially the same role as an open ball does in Rn.


  4. I just finished chapter 7 (which was a lot easier than chapter 6). It was exciting to see all the things that vector spaces can do.

    Here are the new typos I found:

    page 360, Para 1, line 7, “opposiste”

    page 371, Para 3, line 4, “vectors spaces” should be “vector spaces”

    page 375, Pare 4, line 5, “looked at measurements” should be “looked at as measurements”

    page 397, Para 4, line 2, “any two vectors u1 and u2 In U” should be “any two vectors u1 and u2 in U”

    And here are my new questions:

    page 397, In paragraph 1, is the duality you speak of between the vectors and the coordinate spaces?

    page 406, In the last paragraph, where you write “closure under addition of vectors and multiplication by numbers, is the defining property of a subspace” do you mean “of a vector space”?

    page 421, Is an inner product available whenever the components of the vectors are similar in kind (similar as discussed on page 416)?



    • Thank you.

      Regarding duality, yes you are correct. Duality is interesting, important, and occurs in many forms, It most standard use is in the concept of a dual space, which can be thought of as consisting of measures of the vectors in the original space. My first draft included a section on dual spaces, which I was persuaded to cut.

      Regarding subspace, I’ve changed the text for the next edition to make this clearer. I mean subspace of a vector space, in contrast to a subset of a vector space that is not closed under addition or multiplication and is, therefore not a subspace. A subspace is a subset that is, itself, a vector space.

      Regarding inner product I would agree. But inner products, more generally, are a way of capturing relative differences and can, for example, be utilized to define a topology on a vector space. Inner products are often useful as a calculational device and, in such contexts function as a concept of method even in cases with disparate axes.


  5. I finished the book a few days ago and just completed entering my glossary today. Here it is. Unfortunately, WordPress removed all the formatting.


    Mathematics The science of measurement 425
    Measure A measurement is the identification of a relationship to a unit, and expression of how the attribute or existent being measured differs from the unit in a particular respect. If the object of measurement is a magnitude, the measurement is a number. A concept identifies similarities among differences; a measurement identifies differences among similarities. 42, 306, 312
    Geometric Perspective The geometric perspective is a focus on the object of measurement. 104, 283
    Mathematical Perspective The mathematical perspective is a focus on the means of measurement
    Magnitude A magnitude is a continuous quantity that can be compared by ratio 105
    Multitude A multitude is a quantity of discrete units 105
    Number A number is a unitless ratio, a relationship between a quantity and the unit used to measure the quantity. A number is not a quantity; it is a type of measurement. 233
    Mathematical Domain A mathematical domain is a demarkation of instances of a valid, previously identified mathematical abstraction of mathematical possibilities of a particular kind, considered as an object of investigation. 306
    Set The concept “set” is a concept of method. A set is a specification within a delimited domain to isolate and consider a range of mathematical possibilities. To look at an isolated range of possibilities as a set is to take a geometric perspective on that range of possibilities. It is to look at the members of the set as, in some sense, objects of thought. 302
    Infinite Set When the distinguishable mathematical possibilities are infinite, the set is unlimited. 302
    Empty set An empty set is a relative concept pertaining to a specific domain. E.g. to have no apples is not the same as to have no oranges. 318
    Irrational Number “Irrational Number” is a concept of method, a device to keep track of distinguishable mathematical relationships through an indirect measurement without losing precision.
    Configuration Space A configuration space is a universe of potential constellations of measurement. 289
    System of measurement A system of measurement is a domain of measurement plus a means of combining particular measurements to obtain a new measurement (in the same domain?) I’m unsure of this, as I am inferring it from what Bob said, but he does not give a definition. 229
    Open set An open set is a set U with the property that every point in the set is contained in a ball, centered at that point, that is also contained in U.
    Standard of precision An open set, as it relates to the points it contains, is a standard of precision. Taken as a standard, invoking an open set U is to say that there is no material difference among its points. [They are an equivalence class.)
    Topology A topology is a specification of the open sets of a domain.
    Vector n-tuple. Vectors are not dimensionless ratios the way numbers are. They are not systems of measurements but systems of quantities, as an abstract perspective on something in the universe 374
    Vector Space n-tuples that have an addition property that functions the way addition is supposed to, can be multiplied by numbers, multiplication obeys the distributive law, addition of anything in the domain is in the domain, multiplication by a number is in the domain, there is a zero, and the negative of anything in the domain is in the domain. 370, 374

    Vectors are not systems of measurements. They are the object or attribute that is being measured.
    Coordinate space n-tuples that are systems of measurement
    Isomorphic Having the same structure, the same relationships among the elements 373
    Base (of a vector space) a set of vectors that can be combined arithmetically to form any vector in the space. A vector is a quantity and a basis is a set of quantities. 386
    Matrix An array that captures a set of instructions, a set of calculations on a column vector 389
    Linear transformation A mapping L from one vector space V to another vector space W exactly when, for any two vectors v1 and v2 in V and any two numbers a1 and a2, the following relationship holds: 393
    L(a1v1 + a2v2) = a1L(v1) + a2L(v2)
    Homogeneous Equation A matrix equation of the form Av = 0. (Av = w is an inhomogeneous equation.) 405
    Kernel The set of solutions to the homogeneous equation Av = 0 405
    Equivalence Relation Tw vectors are regarded as equivalent, in the context of a linear transformation A, if and only if their difference is in the kernel K of the linear transformation A 409
    Quotient space If V is any vector space and K is a subspace of V, then vectors are considered equivalent, with respect to K, if their difference lies in K. The vector space that ignores distinctions among vectors occupying the same equivalence class, is called the quotient space of equivalence classes with respect to K. In standard notation, the vector space is written V/K. 410
    Bilinear form An expression is a bilinear form if, for all x, y, z, and c, the following formulas hold: 421
    = +
    = +
    = c
    = c
    Symmetric Bilinear form A bilinear expression is symmetric if = 420
    Inner Product The inner product of two vectors, x and y, written = 1/2 ( |x + y|2 – |x|2 – |y|2) where |X|2 = (x12 + x22) 414, 421
    Symmetry An aspect of something that is identical from one particular perspective yet different from a second, equally valid, perspective. 427
    Group A system of measurements of symmetry. All groups have three important properties: 427, 439
    1. Associative multiplication: A(BC) = (AB)C
    2. An identity element: AE = EA = A
    3. Every element has an inverse: AA-1 = E
    Transformation Group A group of transformations (rotations along various axes?) of a geometric object 437
    Subgroup A group that is a subset of a larger group 443
    Quotient group A quotient qroup G/N, of a larger group G and a subgroup N, is a new group in which one or more elements of the group N are now in an equivalence class, and each equivalence class becomes an element in the new group. 444


  6. I just received the book yesterday and have only nibbled a bit. I did, however, try to find “complex number” in the idex; it wasn’t there. This is surprising because these have drawn a good deal of mystery mongering. My high school teacher just told us that i = the square root of -1, showed usage in quadratic equations, and left the matter there. I even took a course in the theory of functions of a complex variable in college, but the mystery remained. G. Chrystal starts his treatment about the same way as my high school teacher: defining i=sqr root of -1 and saying that it’s needed to maintain the generality of algebraical operations (Vol I, p 132) The usefulness of complex numbers in Electrical Engineerin leaves no doubt about a tie to reality, but the above defininition left a nasty taste anyway. (I see that E.W. Hobson has geometric constructions of complex operations in his trig book.)

    However, I recently found a savior on the matter in a little book titled “College Algebra” by A. Adrian Albert. He simply defined complex numbers as points in a plane. He then defines operations on these ordered pairs of reals. i turns out to be (0,1) and i^2 is -1 when (0,1)X(0,1) is performed. Mystery basically gone.

    Nevertheless, I see a large gap between introducing complex number the way Chrystal does and the way Albert does. So, I’m sure any comments on this issue in your book would have been valuable.


    I find the quality of this paperback binding to be quite good.

    Have you considered setting up a twitter account? You can engage your readers that way, and they may even spread your tweets around to their “followers.” This can help spread your ideas (and Rands’s) and publicise your book.


    • There are things I want to say about complex numbers and I expect to devote a chapter to in my next book. But, briefly:
      • I agree with Ayn Rand’s characterization of them as concepts of method (pp 304-306 of Introduction to Objectivist Epistemology (paperback).
      • Historically they arose as a means of finding (real!) roots of third degree polynomial equations. But mathematicians were only happy with them when the geometric interpretation that you mention was discovered. There is an excellent history of imaginary numbers (An Imaginary Tale) by retired Professor of Electrical Engineering Paul J. Nahin.
      • I do have some comments on page 381 that I forgot to index.


      • Thanks for the reply.

        To be accurate, it wasn’t the geometrical interpretation of complex numbers that helped me: it was seeing them as derived mathematical entities with defined operations such that i^2 = (-1,0), a complex number and i=(0,1) another complex number. That left Albert’s strange looking operations as something yet to be defended, but Hobson’s geometric constructions left me with the grasp that they were at least sane.

        I read Rand’s comments in ITOE.
        I found nothing about complex numbers on page 381 of your book.
        The book by Nahin looks fascinating and is in my amazon shopping cart for my next order in July.


  7. Sorry for the typo: 481. Mathematicians wanted a quantity that complex numbers applied to. They saw the geometric interpretation as answering that requirement.


    • No, there isn’t. Nor is it in the kindle version.

      But I’m considering posting something on point set topology. The interesting questions are: What does it mean? How does it relate to epsilon-delta? Why does it work? I.e., what is the underlying principle? I would expect my observations to overlap the new material in Chapter 6.


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