My review of a book about math

This links to my review of Mathematics is About the World by Robert E. Knapp.

Approaching Infinity

The title is the title of a book by Michael Huemer, which I read most of, but not all. It’s good in parts. It addresses infinity in several ways. He presents many paradoxes involving infinity and gives his solutions for some of them. He discusses Cantor’s diagonal arguments (see my previous post for one). He doesn’t seem to wholly agree with them, but he doesn’t argue against them either.

He does challenge Cantor concerning Galileo’s paradox. “The puzzle is that there is a compelling argument both that the two sets are equally numerous (the one-to-one correspondence argument), and that one set is larger than the other (for the squares are a proper subset of the natural numbers). Cantor embraces the first of these arguments and rejects the second. His only justification for this is fiat: he proposes to simply define the relations ‘equal to’, ‘greater than’, and ‘less than’ using the one-to-one correspondence criterion.

     Of course, one could not consistently brace both of the arguments Galileo mentions, since they entail contradictory results. But Cantor’s decision to embrace only the one-to-one correspondence argument is not the only alternative. One could embrace the proper-subset argument while rejecting the one-to-one correspondence argument. Or one could, like Galileo, reject both arguments and hold that ‘greater than’, ‘less than’, and ‘equal to’ relations do not apply to pairs of infinite sets. Cantor does not argue for his alternative over the others” (6.9.4).

Huemer did not say one could embrace the proper subset argument for comparing the integers and the rationals.

The weakest parts of the book in my opinion were about the empty set and geometric points.

Huemer misses the strongest reason for having the concept of an empty set. The two major operations on sets are union and intersection. The first forms a new set by combining the members of two or more sets. The second forms a new set by identifying the common members of two or more sets. Say we have set A = {1, 2, 3} and set B = {3, 4, 5}. The intersect of A and B is {3}. But what if B = {4, 5, 6}? Then A and B have no common members. In other words, the intersect of A and B is nothing, i.e. the empty set { }, often symbolized as {∅}. In other words, the empty set is needed to make defined operations complete. It is similar to a need for the number zero, for example 5 – 5 = 0.

Huemer is frustrated with geometric points. He says they are unimaginable, because they have size zero, yet they are supposed to be the building blocks of other geometrical objects (11.2).

I regard the building block perspective as from the wrong direction. Start with 3-dim space, e.g. a cube. Then disregard one dimension to get 2-dim space (a square plane), disregard another dimension to get 1-dim space (a line), and finally to 0-dim space (a point). Alternatively, each intersection of two lines of the square is a point.

Later in the book there is a section (14.6.3) about points as locations. He does not have any arguments against this view, but he doesn’t endorse it either and says he doesn’t understand it. 

In my opinion it is the best view. We often hear people say things like how to get from point A to point B. More concretely it might mean driving a route from one location, such as a town, to another location, another town. In the context of geometry, a point is a location in Cartesian or polar coordinates. For example, the point (2,3) in 2-dim Cartesian coordinates means 2 units to the right of the origin (0, 0) or y-axis and 3 units above the x-axis. Of course, Euclid’s Elements did not have this perspective, since Cartesian coordinates were invented much later. 

Infinity contra-Cantor #2

Georg Cantor also famously claimed that the cardinality of the set of rational numbers equals the cardinality of the set of integers because he could devise a 1-1 correspondence or mapping between them as shown here and here.

Relatedly, the continuum hypothesis is: There is no set whose cardinality is strictly between that of the integers and the real numbers (rationals and irrationals).

Accepting certain assumptions Paul Cohen showed that the continuum hypothesis is neither true nor false. I think it’s false, the quantity of rational numbers being greater than the integers, but less than the reals (rationals and irrationals combined). I base this on two  perspectives different from that of Cantor and Cohen. The first is the real number line. There is an unlimited count of rational numbers between 0 and 1, between 1 and 2, and so forth.

The second is part-whole logic. As shown in the above links, a 1-1 mapping can be made. However, note the first column in the array in either link, which is the integers. A different 1-1 mapping or function f(x) = x can also be made. Accordingly, the integers comprise a proper subset of the set of all rational numbers, implying the set of rational numbers is larger than the set of integers.

“Cantor was concerned to combat the Aristotelian view that there cannot be an actual infinity, mainly because Cantor believed that God was infinite.” – Michael Huemer, Approaching Infinity, p. 71.

Infinity contra-Cantor #1

In the late 19th century Georg Cantor chose the method of 1-1 correspondence to decide the different quantities of infinite sets. He called such quantities their “cardinality.” If the members of one set can be paired 1-1 with the members of another set, then they have the same “cardinality” (link). 

As the Wikipedia page shows, some famous mathematicians such as Kronecker and Poincaré objected to some of Cantor’s ideas. But most people who have an opinion about it accept Cantor’s arguments and consider that the cardinality of the positive integers equals that of the even positive integers, as explained on the linked page.

I was astounded when I became aware of that many years ago. Years later I realized Cantor's error. The assumed method of 1-1 correspondence implicitly rules out 2-1 correspondence, N-1 correspondence, and part-whole logic. 

A 2-1 correspondence or mapping can be formed from the natural numbers (integers) to the even numbers as follows: 1, 2 → 2;   3, 4 → 4;   5, 6 → 6;   7, 8 → 8;   9, 10 → 10; and so on. If this mapping did not include “and so on”, then the finite quantity of natural numbers would surely be regarded as being twice that of the even numbers. (See above link, Exercise 3). However, many abandon that idea when “and so on” is included. I do not. From a part-whole logic perspective, imagine having a container with all the integers in it, then removing the odd integers from the container. Wouldn’t the container then have half the quantity of numbers as before? Or partition the container in half vertically, putting the odd integers on one side of the partition and the even integers on the other side. Wouldn’t there be twice as many numbers in the whole container as there are on one side of the partition?

Aristotle's Wheel Paradox #3

The title includes #3 because I posted #1 and #2 in February (link). I am posting #3 because I edited the Wikipedia article about the topic (link).

I improved the first section. It had said the paradox was about two wheels. The section 'Wrong problem entirely ' on the Talk page says the paradox is about one wheel. I wholly agree. So my change to the first section says it is about two circles and one wheel (or a suitable substitute for the wheel).

I added the Analysis & Solution section to include my two solutions to the paradox. To the best of my knowledge the solutions are original. At least I didn't see or hear them anywhere else.

Since others can edit Wikipedia articles at any time, I cross my fingers that somebody won't impair it. I earlier added similar text to the Talk Page. I believe others can alter that too, but maybe they are less likely to do so.

P.S. I later added a third solution, labeled the first solution in the Wikipedia article.

The Great Math Mystery

Last night we watched The Great Math Mystery, a NOVA episode on PBS television. It was excellent and I recommend it. It can be watched on-line here at least temporarily. There is a full transcript, too. The mystery is:  Is math invented by humans, or is it the language of the universe? Reasons are given for both -- some math is invented and some is discovered.  I believe the best answer came near the end. Math concepts such as numbers are abstracted by humans, but then they and their relationships are found to apply beyond their origin and lead to further discoveries.

The topics include the Fibonacci sequence, the number pi, Galileo's mathematics of falling bodies, Maxwell's equations, Marconi's discovery of radio telegraphy, the quantitative intelligence of lemurs, and the difference between pure math and applied math. Regarding the last, pure math is exact and imaginative but becomes much more useful via approximating with short-cuts such as done by engineers.

Infinitesimal #4

A intellectual war involving math and politics also occurred between Thomas Hobbes and John Willis. Near the end of the English Civil War (1642-1651) Hobbes wrote Leviathan. In it Hobbes argued for a social contract and rule by an absolute sovereign. He wrote that civil war and a brutal state of nature ("the war of all against all") could only be avoided by strong, undivided government.

Hobbes also was a geometer of some repute. Similar to the Jesuits, he believed that the answer to uncertaity and chaos was absolute certainty and eternal order. They believed the key to both was Euclidean geometry. He set about trying to "square the circle" and solve two other long-standing geometry problems. "Square the circle", or "quadrature the circle," means construct a square with area exactly equal to the area of a given circle. (It can be done with great, but not perfect, precision.) Under the traditional restrictions of using only a compass and straight edge, this had been proven impossible. Hobbes tried anyway. Mathematician John Willis was well prepared to discredit any solution Hobbes proposed. Willis' political attitudes also reflected the chaotic years in England, but he believed in a state that would allow for a plurality of views and wide scope for dissent. Willis also sided with those who supported the use of infinitesimals.

Hobbes was also a sharp critic of the mathematical works of Willis. For Hobbes the infinitely small was an unwelcome intruder in mathematics. In contrast Willis considered practically all the features of the infinitely small to be clear advantages. His math was for investigating the world as it is. The world could be a little mysterious, unexplored, and ambiguous, but it invited new investigation and new discovery (p. 287).

Infinitesimal #3

For centuries before the Reformation the Roman Catholic Church had reigned supreme in western Europe. Empires rose and fell. There had been invasions and occupations, heresies and plagues, but the Church had survived and thrived. The Church oversaw the lives of Europeans and gave order, meaning and purpose to their existence, and ruled on everything from the date of Easter to the motion of Earth and much else (p. 24). The Reformation was a challenge to all that.

The Jesuits, formed in 1540, were not much interested in mathematics in their first few decades. The founder, St. Ignatius, had little interest in mathematics. But as they built their education system, they became committed to Euclidean geometry. "It was the core of their teaching and the foundation of their mathematical practice. ... [T]he whole point of studying and teaching mathematics was that it demonstrated how universal truth imposed itself upon the world -- rationally, hierarchically, and inescapably. Ideally, the Jesuits believed, the truths of religion would be imposed on the world just like geometrical theorems, leaving no room for avoidance or denial by Protestants or other heretics and leading to the ultimate triumph of the Church. For the Jesuits, mathematics must be studied according to the principles and procedures of Euclid, or it should not be studied at all. A mathematics that ran counter to these practices not only was useless to their purposes, but it would challenge their unconquerable faith that truth, handed down through the hierarchy of the universal catholic Church, would inevitably prevail" (p. 74). Infinitesimals are not part of Euclid's Elements.

"The Jesuits valued mathematics for the strict rational order it imposed on an unruly universe. Mathematics, particularly Euclidean, represented the triumph of mind over matter and reason over the untamed material world, and reflected the Jesuit ideal not only in mathematics but also in religious and even political matters" (p. 91).

"Euclidean geometry was the embodiment of order. Its demonstrations began with the universal self-evident assumptions, and then proceed step by logical step to describe fixed and necessary relations between geometrical objects: the sum of the angles in a triangle is always equal to two right angles ... These relations are absolute, and cannot be denied by any rational being" (p. 119). A group of five Jesuits, the Rectors, had a strong control over what was taught, and it issued prohibitions on the teaching and promotion of infinitesimals (p. 122).

Cavalieri's and Galileo's ideas ran counter to that. They held that lines were comprised of indivisible points, planes of indivisible lines, and solids of indivisible planes. Thus these geometrical objects  were little different from the material objects we see around us (p. 91). Instead of mathematical reason imposing order on the physical world, pure mathematical objects are created in the image of physical ones. They wanted to study the world and find the order within. They were willing to accept some ambiguity and even paradox as long as it led to a deeper understanding (p. 177).

In Discourses (or Two New Sciences) Galileo made use of Aristotle's wheel paradox (link) to arrive "at a radical and paradoxical conclusion: a continuous line is composed of an infinite number of indivisible points separated by an infinite number of minuscule empty spaces. This supported both his theory of the structure of matter and his view that material objects are held together by the vacuum that pervades them" (p. 89-91). How Galileo thought about the wheel paradox is described here.

Infinitesimal #2

What are infinitesimals? That term was coined around 1670. A related term is indivisibles.  The author of Infinitesimal says: "To understand why the struggle over indivisibles became so critical, we need to take a close look at the concept itself, which appears deceptively simple but is in fact deeply problematic. In its simplest form the doctrine states that every line is composed of a string of points, or "indivisibles," which are the line's building blocks, and which cannot themselves be divided" (p. 8-9).

Some mathematicians, especially Cavalieri, held that a line is composed of an "infinite" number of points, a plane is composed of an "infinite" number of lines laid parallel and a solid is composed of an "infinite" number of planes laid parallel. Of course, for any polygon, sets of parallel lines can be drawn in different directions, e.g. vertical or horizontal or diagonal, which gives a different number of lines.

Infinitesimal mentions that the math symbol used for infinity, ∞, was invented by John Wallis (who has a prominent role in the book). Wikipedia confirms this. Wallis used 1/∞ for an infinitesimal.

One conundrum is how many indivisibles (or points) there are in a given line and how small they are. Assume a given line has a huge, yet finite, number, we can call a gazillion. That suggests each indivisible is 1/gazillion in size. Problems start arising when considering other lines of different length. A "deep" problem -- discovered by the Greeks -- is that some lines can not be subdivided into an integer number of uniform-sized units, no matter how small they are. For example, the square root of two length units (e.g. inches) cannot be converted to an integer number of anything, because the square root of two is an irrational number. Ditto for pi, the circumference of a circle with a diameter of 1 length unit, and Euler's number e, the base base of natural logarithms.

A related topic is the continuum. The usual meaning of continuous is “unbroken” or “uninterrupted.” Thus a continuous entity—a continuum—has no “gaps.” Aristotle addressed it in book 6 of his Physics. He concluded that the concept of infinitesimals was erroneous, and that continuous magnitudes can be divided ad infinitum (p. 10). The Jesuit Benito Pereia proposed the thesis that a line is composed of separate points and presented all the arguments in its favor by others. He then demolished them one by one, and concluded, like Aristotle, that the continuum is infinitely divisible, and not composed of  indivisibles (p. 121). The Jesuits thought Cavalieri's were untenable.

The Epilogue of Infinitesimal describes how the use of "infinitesimals" later grew to have a prominent part of mathematics. I believe it would have been improved if it explained the concept of a limit as used in math, such as in calculus. Said concept was developed in the 19th century. It involves, but is not identical to, "infinitesimals" as described in the book. The similarity is stronger in integral calculus.

A much longer history of infinitesimals and related topics is here. The historical period the book is mainly about is a very small part of it.

Infinitesimal #1

I read the book Infinitesimal: How A Dangerous Mathematical Theory Shaped The Modern World. In mathematics, infinitesimals are things so small that there is no way to measure them. I thought the history in Infinitesimal – both political-religious and mathematics from about 1500 to 1675 – was very interesting. I believe the author makes the tie between them stronger than what they actually were, but there were parallel ideas - parties opposing one another in two very different realms.

The Society of Jesus, more commonly called the Jesuits, has a prominent role. Before Martin Luther initiated the Protestant Reformation, the Catholic Church was the dominant power in society. Kings and their lower ranking brethren depended on approval by the Catholic clergy. The anti-Reformists believed that the Reformation would bring about disorder and war. The Jesuits became the leading defenders of Catholicism. In large part their success was due to their building of educational institutions.

A leading Jesuit, Christopher Clavius, was almost single-handedly responsible for the adoption of a rigorous mathematics curriculum – Euclidean based -- in an age where mathematics was often ridiculed by philosophers and religious authorities. While Clavius clearly opposed the heliocentric model of Copernicus, it was mainly other Jesuits who opposed infinitesimals.

A leading proponent of infinitesimals was mathematician Bonaventura Cavalieri. He was a Jesuat, which is different from a Jesuit.

Except as noted below, the author summarizes the book's thesis very well as follows.

"Why did the best minds of the early modern world fight so fiercely over the infinitely small? The reason was that much more was at stake than an obscure mathematical concept. The fight was over the face of the modern world. Two camps confronted each other over the infinitesmal. On the one side were ranged the forces of heirarchy and order – Jesuits, Hobbesians, French royal courtiers, and High Chuch Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesmals. On the other side were comparative "liberalizers" such as Galileo, [John] Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesmals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come" (p. 8).

Most of the history presented in the book happened before Isaac Newton published his revolutionary Principia  in 1687, so Cavalieri instead of "the Newtonians" arguably fits better.