Reviews - Books I Read - DRJG: MATHEMATICS IS ABOUT THE WORLD - HOW AYN RAND’S THEORY OF CONCEPTS UNLOCKS THE FALSE ALTERNATIVES BETWEEN PLATO’S MATHEMATICAL UNIVERSE AND HILBERT’S GAME OF SYMBOLS; by Robert E. Knapp.

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MATHEMATICS IS ABOUT THE WORLD

HOW AYN RAND’S THEORY OF CONCEPTS

UNLOCKS

THE FALSE ALTERNATIVES

BETWEEN

PLATO’S MATHEMATICAL UNIVERSE

AND

HILBERT’S GAME OF SYMBOLS

Robert E. Knapp

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MATHEMATICS IS ABOUT THE WORLD

By Robert E. Knapp

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This book isn't for mathematicians, unless one has no other preferences for time away from desks.

One begins to suspect, soon enough, that it's one of those clowning efforts to make believe that 'math' can be popularized with generic students in US who have been allowed to opt out and instead pick journalism or film-making as alternative in high school, as has been going on for decades, due, one suspects, to education authorities simply giving in to demands to make it easy.

But it's neither, in fact, and anyone wanting relaxation with this will be disappointed unless relaxation is from equivalent of comprehending proof of FLT, or equivalent. Similarly, if a high school student is seeking an easy way to learn by reading this, and isn't disappointed, he or she is half of the ShAmy from Big Bang Theory.

No matter how much anyone, everyone, pretends, there's no alternative to using mind, exercising brain, in doing Mathematics, any more than hard work is an option when following a career in US in, say, law; or putting in hours is a choice rather than a must when on a path to a career in medicine. Anyone attempting to climb a mountain cannot do so in a chair carried by others, normally.

" ... And my central message is this: You do not have to choose between mathematical abstractions and reality. Mathematical abstractions are a way of understanding the world, of deepening and enriching one’s perspective on the world. One understands the essence of a mathematical discipline when one grasps what it is trying to measure. Mathematics is about the world."

He's indulging here in philosophy and history in describing his journey into mathematics and more.

Knapp is the rare person who's familiar enough with mathematics to know the names unknown to rest of the world, but goes on to name them without any mention of the magic of beauty they weaved, enchanting forever.

One might question what hate agenda Knapp has in writing a whole book sprinkled with verbose abuse.

"In my view mathematicians, despite the absurdities of set theory, have continued, to this day, to do mathematics. But how is this possible? Assuming that I’m right, why has mathematics survived?"

Amazingly he's introduced at end as someone who did a doctorate from Princeton, 1973, and taught at Purdue, before retiring in 1970s, and working at Ayn Rand research institute!

Does that say more about the two universities than we know?

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CONTENTS

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Acknowledgements 7

Preface 9

PART 1: ELEMENTARY

1. Euclid’s Method 23

2. Measurement and the Geometry of Magnitudes 101

3. Geometric Area, Proportion, and the Parallel Postulate 177

4. Numbers as a System of Measurements 229

5. Geometry and Human Cognition 277

PART 2: ADVANCED

6. Set Theory and Hierarchy in Mathematics 293

7. Vector Spaces: A Study in Mathematical Abstraction 357

8. Abstract Groups and the Measurement of Symmetry 425

Index 489

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REVIEW

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Acknowledgements 7

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Author credits various people including Ayn Rand for a work of her's, Objectivist Epistemology, and Srikant Rangnekar.

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May 29, 2022 - May 29, 2022.

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Preface 9

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Author lists the books he read, from standard respected texts on Mathematics in US - Birkhoff and Maclane, Halmos, ... to Dieudonne, concluding every time that Mathematics was not an abstract world of its own but was about the world.

"At some point during my college freshman year, I realized that neither mathematicians nor philosophers of mathematics shared my perspective, offering only the alternatives of formalism (a game of symbol manipulation), Platonism (a separate world of mathematics), or, as a third, the Fregean view that mathematics is essentially a branch of logic. I could accept none of these choices.

"However, I had discovered Ayn Rand and during the following summer she began her epochal series of articles on Objectivist epistemology. It struck me, from her first essay linking mathematics and concept formation, that I had found my key to understanding mathematics. In regards to its specific application to mathematics, however, I also realized that I was on my own."

" ... Mathematical infinities that involve numbers, infinities that do not exist in the world, are, indeed, an indispensable part of mathematics. ... "

Lines, points and planes do exist in the world, and everyone comprehends their dimensions, as well as the fact of number of points on any segment of the line being infinite in number.

"Yet, amazingly, the circumference of the earth was measured (within about 16%) by Eratosthenes in 200 BC, without leaving Alexandria. He did it by measuring the angle the sun’s rays made at noon on the day of the summer solstice. He made just this one measurement, but relied, for his calculation on two others: first, the distance of a particular town to his south and, second, the known fact that, in that town, at that time, and on that day of the summer solstice, the sun was directly overhead, a circumstance manifested by the known observation that the one could see, at the stroke of noon, the reflection of the sun at the bottom of a very deep well."

"Progress in mathematics consists in finding the connections, in finding the geometric and mathematical relationships that make indirect measurement possible. Every geometric theorem, every algebraic or differential equation expresses a relationship that can provide a bridge to an indirect measurement.

"But an equation is also something one needs to solve. Whenever certain variables of an equation are regarded as known, while others are regarded as unknown, the challenge is to “solve” the equation, to discover the corresponding values of the unknown variables. And this challenge, the search for solutions and for general methods of finding solutions, has driven progress in mathematics. Every mathematical abstraction and every new mathematical relationship provides one more step on this complex journey. In general, every part of mathematics, from the most elementary to the most abstract, began with a problem in indirect measurement and can be better understood in relation to measurement. Indirect measurement is the heart of mathematics, its reason for being, and the source of its power to enrich our lives."

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May 29, 2022 - May 29, 2022.

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PART 1: ELEMENTARY

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1. Euclid’s Method 23

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Author indulges more in philosophy than in mathematics or an introduction thereof for students.

" ... are Euclid’s pictures merely imperfect, suggestive renderings of ideal geometric figures? And do such geometric figures constitute a world of their own consisting of ideal points that have no extension and of ideal lines that have no width? The common fallacy that mathematics pertains specifically to a mathematical universe; that mathematics applies to the world, but is not about the world begins with the first page of Euclid’s monumental work."

" ... This competing view that mathematics consists of deductions from arbitrary axioms, goes back to David Hilbert in the late 19th and early 20th century. ... "

Author goes into history of mathematics too, from Greeks on to Hilbert.

"Ultimately the classical Greek answer to the Pythagorean dilemma of incommensurate ratios (in modern terms, irrational numbers), as well as the Greek approach to limits (the “method of exhaustion”), was provided by Eudoxus,19 born after Plato but before the birth of Aristotle. Although none of Eudoxus’s work survives in written form, his work was essential to the theory of proportion later presented in Euclid’s books V and VI ... It is a commonplace today that Eudoxus’s theory of ratio anticipates Dedekind’s approach to irrational numbers, developed in the 19thcentury.22 And Eudoxus’s method of exhaustion, exploited by Euclid and, later, by Archimedes23 captures the key insight embodied in the modern theory of limits, precisely defined for the first time by Cauchy in the early 19th century."

" ... Archimedes was the first to finally square the circle, and he went on from there to determine, with modern rigor, the volume and surface area of the sphere and the volume of a cone. Applying Eudoxus’s method of exhaustion, Archimedes’s methods in this regard anticipate the integral of Newton’s calculus.24"

"In Ayn Rand’s terms when one measures triangles, color is an omitted measurement. A triangular object must have some color, but it may have any. The particular color doesn’t matter; it does not affect one’s study of shape. In the same way, when one measures triangles, any microscopic or irrelevant imperfections of the triangle are omitted measurements. If these imperfections were relevant, we couldn’t count them as triangles. But if something doesn’t matter, one doesn’t measure it. One omits it from one’s analysis.31"

"The new conceptions of mathematics that arose during the nineteenth and early twentieth centuries included David Hilbert’s view of mathematics as the study of formal systems,46 Frege’s and Bertrand Russell’s attempt to reduce mathematics to symbolic logic,47 and the development of set theory as a purported foundation of mathematics.48 This latter set theoretic approach has many fathers, but a key milestone on that path was Georg Cantor’s conception of an actual, completed infinity.49 ...

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May 29, 2022 - May 29, 2022.

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2. Measurement and the Geometry of Magnitudes 101

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He takes up championing here for Ayn Rand and her views on Mathematics.

" ... It is Ayn Rand’s theory, in my view, that enables one to avoid the false alternative between the Platonic view of a separate mathematical universe and the more modern view that mathematical concepts are arbitrary conventions requiring no existential referent."

Reminds of a colleague's puzzled disdain about a paper published by someone claiming a result that was an obvious consequence of a well known one, and therefore well understood by most mathematicians in the field.

"In contrast, hardness is a continuous quantity, but it isn’t a magnitude, at least as it was measured in the past.5 Traditionally, one measured hardness of minerals and gem stones along a comparative scale called Moh's Scale of Hardness. One said that a diamond had a hardness of 10 while quartz had a hardness of 7 because a diamond can scratch a quartz crystal but a quartz crystal cannot scratch a diamond. Between quartz and diamond in hardness are Topaz and Corundum. The differences were ordinal. There was a way to determine relative hardness, but we could not say, on such a basis, that a diamond is twice as hard as a quartz crystal.

"Lengths, areas, weight, acceleration (in a particular direction), force (in regards to the strength of the force), density, and water pressure are all magnitudes. The pitch of a sound is also a magnitude because one can relate pitch to the frequency of a vibration. But we do not perceive it that way: when one vibration is twice the frequency of another we perceive the difference as a musical interval, specifically as an octave. Finding frequency as a unit of measure was a scientific discovery."

"The Axiom of Archimedes

"In Aristotle’s Physics, as part of his argument against the existence of actual infinite magnitudes, one finds:

"“…for every finite magnitude is exhausted by means of any determinate quantity however small.”14

"Today, Aristotle’s statement is known as the Axiom of Archimedes,15 after the greatest mathematician of antiquity, considered one of the very greatest of all time. Keep in mind, though, that Aristotle preceded both Euclid and Archimedes, though he didn’t precede Eudoxus. Whatever else may be true, Archimedes did not originate the axiom of Archimedes and Aristotle grasped its import.

"To paraphrase, given any two magnitudes of the same kind, one can obtain a magnitude that exceeds the larger by taking a sufficiently high multiple of the smaller magnitude. If A is the smaller magnitude and B is the larger, there is some whole number N such that N times A exceeds B."

"Descartes’ enterprise includes two basic steps. The first is the introduction of coordinates, the ancestor of the real number line and also the ancestor of the modern form in which we utilize Cartesian coordinates. The second step is the use of equations, such as y = x2 to specify geometric shapes algebraically. ... "

" ... Greeks did not have the benefit of our perspective. Qua numbers, they had no concept of irrational numbers, no systematic vocabulary to specify them. And, for them, irrational numbers always arose in geometric contexts, as the incommensurate relationship of two magnitudes, such as the relationship of a diagonal to the side of a square. Geometric ratios, in Euclid, were specified, always, by a pair of magnitudes, usually lengths."

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May 29, 2022 - May 30, 2022.

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3. Geometric Area, Proportion, and the Parallel Postulate 177

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Here, the book improves.

"In regards to non-Euclidean geometry, the twentieth century added a final twist with Einstein’s development of the General Theory of Relativity. General Relativity is built on the geometry of light rays, a geometry in which light rays, propagated across space are treated as straight lines. And the geometry of these straight lines, according to Einstein’s theory, is not Euclidean.5"

"On an astronomic scale, light rays (or electromagnetic waves) are an essential part of all geometric measurement. The path of a light ray through space is the straightest path known to man. As such, a light ray is used to establish a line of sight, a specification of the direction from earth to other objects in the universe.

"But the path of a light ray is not always straight. For example, light rays bend, are refracted, when passing from one medium to another. More critically, light rays bend when passing through gravitational and other fields. To understand the meaning of a measurement by light rays requires understanding the effects of such factors. For example, in the case of refraction, this means understanding the angle of refraction.

"The effect of gravitation is more subtle. We know that the direction of a light ray, as it approaches earth from a distant star, will depend, for example, upon the position of the sun in relation to the direction of the distant star. Our best current understanding of just how the position of the sun will influence this angle is provided by the General Theory of Relativity. So the relevant context in measuring the direction of a distant star involves, first of all, the fact that one is measuring the direction of the distant star as the light ray flies. Secondly, the context includes the position of the sun and of any other massive bodies between us and the distant star. Thirdly, it includes our understanding of how various massive bodies influence our measurements by means of light rays. Taken together, our determination of the direction of incoming light rays from the distant star is a specification, as such, of the direction of the distant star, the only kind of specification that is physically possible today. And if we had some alternative way of measuring its direction, that alternative would be just that, an alternative way of measuring its direction that would involve a different set of contextual factors."

" ... That understanding, however, involves solving, at least approximately, the Einstein Field Equations, a nonlinear set of equations for which very few exact solutions are known. ... "

"First, there is a special circumstance in which the effect of the sun can be pretty much eliminated. If the sun, at one instant is on one side of the star and, at the next, is on the other side, one can thereby quantify the influence of the sun and, by taking the average of the two positions, eliminate it. Indeed, this circumstance, exploiting a total eclipse of the sun, provided important validation of Einstein’s Field Equations. However, the gravitational field of the sun affects the observed angles of all incoming light rays, even rays that don’t pass so near the sun, and the Field Equations are needed generally to quantify the effect of the sun’s mass on all incoming light rays.

"Secondly, specifying the direction of the light ray and of the factors that are known to influence the result of one’s measurement does , in fact, specify the direction of the distant star, so long as one maintains the context of that measurement. However, without the use of the Einstein Field equations to quantify the effect of the sun on incoming light rays, one has no way to integrate that knowledge. One can specify that direction today, but one would have no way of relating the measurements one makes today to the measurements one will make tomorrow when the sun is in a different position. So one’s understanding of the direction would be limited to a very special context, namely the particular position of the sun at the time of the measurement. It is Einstein’s Field Equations that provide the critical link; that integrates one’s measurements and makes them meaningful, taken as a whole.

"With all that said, one’s measurements of the directions of the stars and their distances, based upon the light reaching us from those stars, offers a coordinate system applicable to the universe at large. Spatial relationships in the universe are measurable and the base of their measurement is Euclidean geometry.

"These light-coordinates do not, however, provide a Euclidean coordinate system. For example, one cannot simply rely on the Pythagorean Theorem, together with one’s measurements of the direction and distance of various stars, to compute the distance between any two of those stars from each other. Not in isolation from the laws of physics! One needs relativistic corrections to make these determinations.

"So the question presents itself: Is there some formula, some recipe for appropriately adjusting our measurements of distances and direction? Is there a universal way to convert our measurements of stellar position to the measurements that a Euclidean coordinate system would provide? Is there a way to compute the “real” unadulterated “measurements” of direction and distance?

"Now, obviously, such a recipe would be a sort of relativistic correction; a way of identifying the impact of massive bodies on the paths of light rays. Such a correction would be similar to the way one adjusts for refraction of light on earth. Having made these corrections, if these were possible, one would be in a position to apply the Pythagorean Theorem (in its three dimensional version) to find the distances between any two stars in the galaxy. And one could apply standard trigonometry to find the angles in any triangle formed by any three stars in the galaxy.

"But there is a final requirement for such a formula: One’s recipe would need to provide consistent answers regardless of which planet or star one took as one’s vantage point. It would be illegitimate to say, for example, that a determination of distances and angles based on data acquired on Sirius would, by the same recipe, give different answers from a determination based on data acquired on earth. The distance between two stars cannot depend upon which Euclidean coordinates one uses."

"Attempts to prove the Parallel postulate from the other postulates has a long history. That history ended in the nineteenth century with several independent discoveries that such a proof is not possible.11

"There is a kind of surface known as a hyperbolic surface. Roughly speaking, from an external perspective, it looks like a saddle at every point. But, on a sufficiently small scale, it looks flat. The first four Euclidean Postulates all hold on a hyperbolic surface, without qualification, and they mean essentially the same thing that they do in Euclidean geometry. In particular, they have the same measurement implications."

" ... Just as great circles serve as geodesics on the earth; just as the lines that we draw on a concrete slab on the earth do not curve or bend and look straight; a hyperbolic surface has geodesics. Any two points on a hyperbolic surface can be connected by a unique geodesic. Any geodesic can be extended, as needed, in either direction. A circle of prescribed radius can be drawn at any point on the hyperbolic surface. And all right angles are equal.

"Yet Postulate 5 fails on this surface. It is entirely possible on a hyperbolic surface for geodesics that are approaching each other at one point to ultimately veer off without ever intersecting in either direction. So, as a matter of deductive logic and of measurement, Postulate 5 must be independent of the others. So far as Euclid’s postulates and the interpretation of those postulates are concerned, a hyperbolic surface differs from a flat plane only in that Postulate 5 is valid on the plane and invalid on a hyperbolic surface.12"

"The earth is a curved surface, yet on a small scale it looks as though it were flat. And, to add to the confusion, we have established a coordinate system on the Earth, a non-Euclidean coordinate system, consisting of longitudinal lines running North and South and latitudinal lines running East and West. At any point, except for the two poles, the latitudinal East-West lines intersect the longitudinal North-South lines at right angles."

" ... To go west or east, on a large scale, is to maintain a constant distance from the poles."

" ... Earth’s surface does not admit a grid of straight lines intersecting at right angles. The reason is that, unlike the Euclidean plane, the Earth is not flat and the lines on its surface do not satisfy the Parallel Postulate. On such a surface, one has a choice. One can walk in a great circle. Alternatively, one can maintain a constant distance from a great circle such as the equator. But one cannot, simultaneously, do both."

"The Eudoxus/Euclid theory of proportion is one of the towering achievements of Greek geometry. It is the base of trigonometry, of our ability to measure astronomic and microscopic distances and other geometric relationships. It provides the mathematical foundation for astronomy, navigation, geographic mapping, and all of the physical sciences. We appeal to it whenever we make an architectural drawing, a scale model or scale drawing of any kind. And the entire edifice rests on Eudoxus’s theory of ratio and Euclid’s theory of area, resting, in turn, on his theory of parallel lines.

"Euclid’s development of area and of geometric proportion is clear and beautiful in its elegance. But, as I noted in Chapter 1, Euclid had a tendency to focus on the means of measurement at the expense of the object of measurement, focusing, for example on lines and circles without mentioning the directions and distances that they measure. This tendency is particularly noticeable in Euclid’s treatments of ratio and area. Thus Euclid expounds Eudoxus’s theory of ratio without ever telling us what a ratio is. For example, are ratios numbers? Or are they only sort of like numbers? Or are they something else entirely?"

Knapp discusses Euclid’s treatment of area, ratio and more, and gives the beautiful geometric illustration that instantly makes this obtuse statement crystal clear -

"Euclid begins with Proposition 35, which reads: “Parallelograms which are on the same base and in the same parallels are equal to each other.”20"

There's another beautiful illustration, alternative to proof given by Euclid, of Pythagoras theorem, Knapp concluding

"As a final comment, it is well known that the validity of the Pythagorean Theorem depends upon the Parallel Postulate and the properties of parallel lines. Thus, it is appropriate that it comes out of Euclid’s treatment of area which, itself, depends upon the Parallel Postulate."

"For Euclid, finding a square equal to a given area is measuring the area. Indeed, when the Greeks spoke of “squaring the circle,” they were asking for a way to construct a square that would have the same area as a circle. The problem, as posed, as being executed by straight edge and compass, cannot be solved. It was left for Archimedes to do the next best thing: He proved that the area of a circle was equal to that of a right triangle with one leg equal to the circumference of the circle and the other equal to its radius.29

"In this, Archimedes looked both forwards and backwards. He looked backwards in the way that he specified his answer. He looked forwards in using a limiting process (Eudoxus’s method of exhaustion) to measure something that straight edge and compass could not. Archimedes could specify the dimensions of the triangle; but he could not construct it.

"Euclid’s strengths were his focus on the geometry and his systematic, if only implicit, use of abstract measurement to reach his conclusions. His essential weaknesses, beyond the Platonic elements of his work, were two. The first, introducing fundamental quantities without even naming them, I have already mentioned. The other was his steadfast avoidance of units. In order to provide a formula, e.g., a formula for area, one must multiply numbers. But even before that, one must have assigned numbers to lengths and, continuing the example, to assign a number to a length, one must first choose units."

"The theory of geometric proportion is at least as old as Pythagoras. But the Pythagorean theory of geometric proportion had collapsed, ironically by virtue of two path-breaking discoveries by the Pythagoreans themselves. One of these was the celebrated Pythagorean Theorem relating the sides of a right triangle to its hypotenuse. And the other was the finding that√2 is irrational, that the diagonal of a square is incommensurate with its sides."

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May 30, 2022 - May 30, 2022.

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4. Numbers as a System of Measurements 229

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"Notice that the scope of the concept of number (i.e., of the natural numbers), once grasped, is independent of one’s context. The scope of the concept does not change when one discovers a need for particular numbers that one hasn’t previously encountered.5 Typically, one does not even notice that the new number is new, for that number is already subsumed within the number system, as such. One does not name each number individually. Rather, the decimal number notation provides a system for designating any number that one will ever encounter."

Knapp spends an inordinate number of pages justifying why one would want irrational numbers, perhaps in response to an idiotic question by a student who merely wished to avoid learning without a low grade.

Else it's obvious they were no more invented than was zero. They exist, as do all mathematical entities and structures.

But having spent that inordinate amount of space and with nothing but seemingly philosophical empty verbosity, next he jumps to say that

"There is another, very general, method of identifying irrational numbers and finding rational approximations to any required degree of precision. ... "

And thus jumps to introduce convergent series. Having introduced the concept by handwaving, Knapp goes spazieren again into philosophy and history.

"But is there, in fact, something for it to converge to? What is the ontological status of the limit of the sequence? How, if at all, does it relate to the world? This was the mathematical issue that confronted mathematicians, such as Dedekind and Cantor during the second half of the nineteenth century. ... "

And proceeds to define Cauchy sequence, before jumping back, perhaps to reassure the balking pupil.

"One never encounters irrational numbers when one makes concrete numerical measurements. ... "

Isn't diagonal of a square the first irrational number anyone would come across?

Then he invokes Ayn Rand to introduce imaginary numbers!

Having gone back and forth, he jumps to serious discussions of mathematics of slightly closer past.

"The proposals of Weierstrass, Dedekind, Cantor, and Heine were (and are) all regarded as satisfactory proposals. Dedekind’s proposal became the most standard construction and the clumsier construction of Weierstrass is no longer taught today. But, from a formal perspective, they are all regarded as equivalent.

"For example, the proposals of Dedekind and Cantor are formally equivalent in the following sense: One can put Dedekind’s real numbers into one-to-one correspondence with Cantor’s real numbers, a correspondence that preserves all arithmetical and ordering relationships."

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May 30, 2022 - May 30, 2022.

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5. Geometry and Human Cognition 277

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"Prior to Descartes, the Western world used Euclid’s Elements as its mathematical framework to study question in physics. This can be seen in Two New Sciences11 by Galileo and even Newton’s Principia,12 written after the time of Descartes. But ultimately, the tables were turned (so to speak) and geometric questions began to be formulated and studied algebraically."

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May 30, 2022 - May 30, 2022.

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PART 2: ADVANCED

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May 30, 2022 - May 30, 2022.

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7. Vector Spaces: A Study in Mathematical Abstraction 357

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Knapp goes to linear algebra to assert that, despite everything done in mathematics since 18th century, mathematics remains what he calls 'science of measurement ', known to Greeks.

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May 30, 2022 - May 30, 2022.

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8. Abstract Groups and the Measurement of Symmetry 425

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Knapp continues hate discourse.

"Abstract algebra, algebraic topology, and non-Euclidean geometry helped bury, over a century ago, the science of quantity, as a characterization of mathematics.1 Among the earliest, and among the most important, of such disciplines, and part of abstract algebra, is the theory of abstract groups.

"There was never a need for this burial ... "

Then he proceeds to discuss symmetry and permutation groups before proceeding to cube roots of one.

"Mathematics, as Ayn Rand put it, is the science of measurement. ... "

Oh, that's where he got that! But Rand wasn't a mathematician by any definition of mathematics, any more than she was an architect. Her pronouncements on the latter, including descriptions, are lifted from Frank Lloyd Wright. In or about mathematics she obviously couldn't lift from anyone recent. They did mathematics - not stage shows pretending to discourse thereon.

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May 30, 2022 - May 30, 2022.

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Mathematics is About the World: How

Ayn Rand's Theory of Concepts Unlocks the

False Alternatives Between

Plato's Mathematical Universe and

Hilbert's Game of Symbols

by Robert Knapp (Author)

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May 29, 2022 - May , 2022.

Purchased May 28, 2022.

Format: Kindle Edition

Kindle Edition

Language:‎ English

ASIN:- B00QJLBNU0

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MATHEMATICS IS ABOUT THE WORLD

HOW AYN RAND’S THEORY OF CONCEPTS

UNLOCKS

THE FALSE ALTERNATIVES

BETWEEN

PLATO’S MATHEMATICAL UNIVERSE

AND

HILBERT’S GAME OF SYMBOLS

Robert E. Knapp

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https://www.goodreads.com/review/show/4751059171

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Reviews - Books I Read - DRJG

Monday, May 30, 2022

MATHEMATICS IS ABOUT THE WORLD - HOW AYN RAND’S THEORY OF CONCEPTS UNLOCKS THE FALSE ALTERNATIVES BETWEEN PLATO’S MATHEMATICAL UNIVERSE AND HILBERT’S GAME OF SYMBOLS; by Robert E. Knapp.

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