The Wright Brothers #2

In 1896 Orville Wright got typhoid fever, and he was in bed for several weeks. During this time Wilbur began reading about German glider enthusiast Otto Lilienthal, who was recently killed in an accident. Wilbur read aloud to Orville.

A manufacturer of small steam engines and a mining engineer by training, Lilienthal started gliding as early as 1869, along with his younger brother. He took his lessons from the birds. He thought the "art of flight" was to be found in the arched wings of birds by which they could ride the wind. He had no use for balloons, since they had so little in common with birds. To fly one had to be "initimate" with the wind.

Over the years Lilienthal built many gliders. Most were monoplanes with an arched wing consisting of muslin stretched over a willow frame. As pilot he would wear padded knees, position himself below the wings, and begin by running downhill. He would swing his body and legs for balance and control.

He also had himself photographed in action, taking advantage of newly invented cameras. Photos of him gliding appeared the world over, more so in the United States. A long article in McClure's Magazine, with seven photos, reached the largest audience.

In 1894 Lilienthal crashed but survived. In 1896 he crashed and died. News of Lilienthal's death aroused a great interest in Wilbur Wright. He began reading intensely on the the flight of birds, including the book Animal Mechanism from the family bookshelf. He read Animal Locomotion; or Walking, Swimming, and Flying, with a Dissertation on Aeronautics. For most readers it was too daunting. For Wilbur the book was exactly what he needed. Wilbur drew upon and quoted the book for years. It opened his eyes and started him thinking in new ways. Orville recovered from his illness and read the same works.

The automobile appeared and gained popularity. For Wilbur it lacked appeal. His sight was upward.

The Wright Brothers #1

The Wright Brothers is a book by David McCullough. It's widely known that the Wright brothers had a bicycle shop, which they opened in 1893, before venturing into flying.

The following is an amusing historical note about bicycles.

"Bicycles had become the sensation of the time, a craze everywhere. (These were no longer the "high wheelers" of the 1870s and '80s, but the so-called "safety bicycles," with two wheels the same size.) The bicycle was proclaimed a boon to all mankind, a thing of beauty, good for the spirits, good for health and vitality, indeed one's whole outlook on life. Doctors enthusiastically approved. One Philadelphia physician, writing in The American Journal of Obstetrics and Diseases of Women and Children, concluded from his observations that "for physical exercise for both men and women, the bicycle is one of the greatest inventions of the nineteenth century."
     Voices were raised in protest. Bicycles were proclaimed morally hazardous. Until now children and youth were unable to stray very far from home on foot. Now, one magazine warned, fifteen minutes could put them miles away. Because of bicycles, it was said, young people were not spending the time they should with books, and more seriously that suburban and country tours on bicycles were "not infrequently accompanied by seductions."
     Such concerns had little effect. Everybody was riding bicycles, men, women, all ages and from all walks of life. Bicycle clubs sprouted on college campuses and in countless cities and towns, including Dayton" (p. 22).

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?

Information and Investment #4

Chapter VIII of Information and Investment is The Need for Adaptability. It includes a very extensive discussion of liquidity.

The author assumes an entrepreneur will adopt the investment plan which offers the highest mathematical expectation of income, irrespective of the degree of dispersion which the possible values of income may show. I know from personal experience in risk management this isn’t wholly true, but I don’t regard it as very wrong.

Ideally, an investment plan is designed to offer positive profits under all possible scenarios likely to develop. In practice, however, such perfectly adaptability is out of reach. So long as the investor holds his resources in the form of money, he remains free to choose to engage in a variety of activities, the range of which is limited by technical, legal, and financial restraints. Whenever he decides to commit his resources in a form other than money, the scope of future activities is to some extent curtailed. However, the committing of resources in that way is the only hope the entrepreneur has for a substantial return. The more capital-intensive is the process, the greater is the fixed cost, and the greater will be the per unit cost of output if the total volume of production has to adapted to a lower level of demand.

Adaptability is enhanced by the power to make net expenditures from a source of readily available purchasing power. This has two dimensions – amount and speed. This is the heart of liquidity. Of course, money is the most liquid asset. Further resources are trade credit granted by suppliers and credit from a bank or other lender. Urgent sales of assets typically mean getting less money in return than patiently waiting.

An actual liquidity position at a particular time may be accounted for not in terms of intentions but as the unplanned result of recent transactions – of unexpected variations in costs or receipts. Cash balances in this sense are ex post, not ex ante in terms of a ‘transactions motive.’

Information and Investment #3

Chapter VI of Information and Investment is The Assortment of Production, or in other words, product differentiation.

Usually economic models assume a fixed set of goods and services, for both consumers and producers. But it is important to deal explicitly with the qualitative composition, for the variety of production is quite great. If we wish to consider, as does an entrepreneur, what kinds and qualities of goods to produce.

Consumers buy goods because of the satisfaction they expect to receive. They also experiment when making purchases as part of an endeavor to find newer and better ways of meeting their desires. Businesses also experiment with product variations to find newer and better ways of meeting customer satisfaction. Most formal economic models, especially the perfect competition model, ignore this.

Imagination, rather than information in an ordinary sense, is what entrepreneurs require in order to discover new ways of combining resources to meet consumers’ desires. Undiscovered ways of production are somewhat like musical tunes awaiting discovery. Often the competitiveness of a market is associated with or defined in terms of the cross elasticity of demand for the products sold in it. A high degree of competitiveness in this sense is much greater in reality than in the so-called pure competition model, which recognizes only price.