Notes on "Two, Three, Four and All That: The Sequel"

[dream_weaver]

Notes on: Two, Three, Four and All That: The Sequel” Lecture 1

Not a transcript. These notes paraphrase the speaker's points and are not accurate quotes unless in quote tags. {Curly brackets denote my comments}

Number, Infinity and Their Misadventures Within the 19^th and 20^th Century.

How are the various kinds of number are related and consistent to/with each other?

The calculation problem of continuous quantity was solved by the methods of calculus developed by Newton and Leibniz. How to understand and validate this has continued ever since.

Since math is widely regarded as the epitome of certainty and the model of reason, quantities with contradictory properties, and logical difficulties tend to undermine both.

“The root of all these logical quandaries in the foundations of math and in the set theory approach to number is, the failure to recognize that number is objective in Ayn Rand’s sense of the word.”

“Two, Three Four and All of That” focused on the ‘what’ and ‘how’ of the counting numbers.

This will deal with the methods by which we conceptualize numbers, and should be instructive in grasping continuous quantity and the real numbers, as well as sort through the problems associated with set theory.

Course outline.

• Equivalence of Sets (One-to-one correspondence)
  
• Postulational Method
  
• Actual Infinities
  
• Positive Development
  

Corvini quoting William Dunham: Journey for Genious

“Cantor: Two sets, M and N are equivalent if it is possible to put them by some law in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. Definition is a critical one since it in no way requires that sets M and N be finite. Instead, it applies equally well to sets containing infinitely many elements. And with this, Cantor moves into uncharted territory.”

1. Equivalence of Sets.

Identification of four differing types of sets:

• Pairing
  

Pairing is a one-to-one correspondence of two finite groups where the comparison takes place at the end.

Example 1.

Pairing two piles of pebbles.

You cannot tell by looking if the multiplicity of pebbles is the same by just looking.

Pairing the pebbles breaks the problem down into two manageable problems. It pairs a pebble from each pile, and it makes the pile smaller. Notice that the solution is in the piles. You repeat the process until you either run out of pebbles, or you can perceptually see that the remaining pebbles are the same. “Notice the importance of coming to the end.”

Example 2.

In auditorium with many rows of seats and full of people, you can count the number of seats and the number of people, or you could request that the people sit down. If there is one person for each seat, no empty seats or no standing people, then you have the same number of people as seats.

However, if it is dark in the back of the auditorium, and you cannot see the back of the room, the fact that you can see that several rows have no empty seats and no standing people does not tell you anything about the part of the auditorium you cannot see.

• Automatic
  

Two groups necessarily equal in number by virtue of how their members were formed. Can be open ended, but not necessary.

Example 3

“Going back to finite groups, can you ever know that two groups are equal in multiplicity, without coming to the end? The answer is yes; sometimes you can, if you rig the problem.”

Consider the auditorium again. What if, instead of comparing the number of seats to the number of individuals, we instead compare the number of bodies to the number of their heads? Does it matter if we can see the back of the room now?

• Functional relationships
  

Based on the identity and properties of the numbers, this comparison requires that both sets be ‘infinite’.

Example 4.

Consider the set of counting numbers to the set of even numbers.

However, for each finite group (1-100):(2-100) there are 100 counting numbers and 50 even numbers. We can identify a law that there are twice as many counting numbers as there are even numbers in each group.

• Positional relationships
  

Based solely in position in some kind of cleverly constructed list such as between the counting numbers and the rational numbers. Depends on the ‘infinity’ of the two sets.

Example 5.

Cantor found a clever way to use two sets of the counting numbers going in two directions. Starting at one corner and working diagonally, with some rules to eliminate duplicates and what not, by using one axis as the numerator and the other axis as the denominator, he concluded “that the rational numbers could be put into one-to-one correspondence with the counting numbers and that these two ‘infinite sets’ are this equivalent or equinumerous.”

“This may seem strange, if you look at the number line, because it is also true that there is an infinity of rational numbers between every two whole numbers.”

“What Cantors definition does is to integrate all these disparate situations under a single concept of equivalence. By that device, he contrives to extend the applicability of concepts of number, ideas of comparing multiplicity, or in some sense comparing sizes of sets; he contrives to extend that to the infinite.”

“But, is it valid?”

Deferring her answer to lecture 3 she asks:

“Is it ok to apply one to one correspondence to infinite sets, if so, what makes it ok, and if it’s not ok, what is wrong with it?”

The real payoff {what gives his method credibility} is in the treatment of continuous quantities.

Consider two line segments of different lengths. The two line segments can be proportioned similarly to each other yielding the same amount of points on both. The points on one line can be brought into a one to one correspondence with the points on the other.

This can also be done to equate the number of points on a line with the number of points in a square or in a cube. Cantor, in a letter to Dedekind wrote, “I see it, but I don’t believe it.” with regard to this matter.

Combining this idea with continuous quantities as point sets, he shows that the set of rational numbers is countable, as well as the algebraic numbers including the square roots, cube roots and so on, arguing that it cannot be done to the real numbers which he consider transfinite numbers (real numbers). From this he deduced the existence and the non-countability of the transcendental numbers.

His conclusions were met with opposition from the mathematical establishment of his time.

Galileo, 300 years before Cantor, uses one to one correspondence of counting numbers and square numbers to conclude that the concepts of greater and less - the comparison of quantities - just do not apply in the context of infinite sequences. His view was characteristic of the dominant opinion of Cantor’s day.

This opinion is obviously not shared today. Some view the validation of calculus as reliant on the theory of infinite sets. Today the main debate is on how to handle the problems to which Cantor’s approach leads. It is debated today because people view it as being crucial to the foundations of calculus.

Corvini wants to know how this view has come to be viewed as so foundational.

• Corvini’s review of negative and positive integers.
  

Review of last years course referencing negative numbers as ‘a unit of opposition’. Viewing opposites as members that would cancel one another out such as: a dollar of asset and a dollar of debt, or an increase of speed and a decrease of speed, or powers and roots; where the effects of one are cancelled out by the effects of the other.

This extends the Greek notion of number as the relation of a group to one of its members taken as a unit. Now we have the idea of number as the relation of a group to a unit that is not in the group, a unit in opposition to the group, a unit that cancels one of the members of the group.

The negative and positive integers represent a compound relationship made up of two simpler relations. One of them is the multiplicity relation, while the other is an either/or type relation. It asks what the relationship is of the unit we counted in the multiplicity and how does it relate to the base or reference unit where it is either

same as

or

opposite to

.

{Corvini covers this review quite extensively in her earlier presentation “

Two, Three, Four and All That

”}

Edited February 7, 2013 by dream_weaver

[ruveyn1]

Newton and Leibniz developed calculus based on the notion of infinitesimals. And infinitesimal is a "number" or "quantity" that is smaller than any real number but is not zero. Bishop Berkley had a field day deriding that notion. He referred to infinitesimals as "the ghosts of departed quantities" The Bishop was a rather good mathematician himself and he wrote a book which sliced and diced the concept of the infinitesimal. There is only one problem. The notion of the infinitesimal gave the right answers even though the concept was not well found by either Newton or Leibniz. In the early and middle 19 th century (around 1840) Cauchy remedied the situation by providing a rigorous definition of limit. With limit one can define continuity of a function and can also define the derivative of a function (its slope at a point) This turns out to be operationally identical to Leibniz dy/dx, the "quotient" of two infinitesimals. Cauchy finally provided a mathematically sound basis for calculus.

Now here is the strange thing. Abraham Robinson dealt with the puzzle --- if infinitesimals are nonsense, why did they give the right answers? In 1960 Abraham Robinson finally gave a kosher definition for infinitesimals in his book "Non-Standard Analysis". This involved extending the real number system using a trick of set theory and formal logic to include infinitesimals. Every genuine real number is surrounded (so to speak) by a swarm of extended real numbers that are infinitesimally close. If you have the chops to deal with the mathematical logic and set theory by all means read the book. It is a hard read, but it is mathematically sound and finally put a foundation under Newton and Leibniz approach.

ruveyn1