Discussion threads, connections, and extensions

Day 1

We started our discussions for the semester by first writing down answers to the following questions:

· What is mathematics?

· What is culture?

· What is mathematics and culture?

· Is mathematics created or discovered?

In discussing what mathematics is, there appeared to be two types of definitions. Some say that mathematics is the study of patterns and form with an attempt to categorize and others say that mathematics is the science of number. This then led to a discussion concerning the nature of number. I think almost all agreed that the fundamental notion of counting would be present in any culture but more “higher mathematics” would be dependent upon what the group of people valued. The Incas are a great example of a group of people who had a very concrete notion of the concept of number.

(see http://www-groups.dcs.stand.ac.uk/~history/HistTopics/Inca_mathematics.html)

If one considers numbers as enumerative devices, then describing mathematics as the science of number might not work when one considers the synthetic approach to geometry that Euclid took. Euclid combined the work of years of Greek geometers into a single collection of documents called Euclid’s Elements (see http://aleph0.clarku.edu/~djoyce/java/elements/elements.html or http://en.wikipedia.org/wiki/Euclid's_Elements

As Stacy G. Langton explains in her review of the book “As Euclid: The Creation of Mathematics” by Benno Artmann,

Euclid's geometry, unlike ours, is not "arithmetized". When we think of line segments, or of regions of the plane, or of space, we naturally think of their lengths, areas, or volumes as numbers. We attach numbers, as it were, to the geometric objects. For Euclid, on the other hand, only the positive integers are "numbers", whereas line segments, rectangles, and so forth are themselves geometrical "magnitudes", independent of any numerical interpretation.

Suppose, then, that we have, say, two line segments. Euclid might have denoted them by "AB" and "CD". I cannot see that it would distort Euclid's thinking at all if we were to use instead "algebraic" symbols such as "x" and "y". But suppose we then wrote down an expression such as "xy" (or, for that matter, "AB.CD"). What kind of object would such an expression represent? We, today, would naturally think of x and y as numbers, so that their product, xy, is also a number. Not so for Euclid! He could, and frequently did, form the rectangle having AB and CD as sides, but this rectangle, a geometrical region or area, was not a number, and was a "magnitude" of a different kind from the segments AB and CD. (Thus, for Euclid, an expression such as "xy + x" could have no meaning.) We could, if we liked, denote the rectangle formed by AB and CD by "AB.CD", as long as we were clear what the symbolism stood for. However, many readers would have an impulse to read into this symbolism the modern idea of multiplication of {\it numbers}, and to think of the product as a number representing the area.

If one thinks of number as more than just an enumeration device and as a mental construct that encompasses quantity and is an object in its own right, then perhaps we could reconcile the two definitions with the following statement by Philip J. Davis and Reuben Hersh in their 1983 book “The Mathematical Experience”, “The study of mental objects with reproducible properties is called mathematics.” Some other cool quotes about mathematics can be found at: http://www.ewartshaw.co.uk/maths.html.

Most people in the class (75%) said that mathematics was discovered and cited examples such as the mathematics of gravity and different number names by various cultures.

About 20% thought that mathematics was created and the remainder thought it was a combination of both. It appeared, in class discussion, that most of the class thinks that mathematical objects and truths exist outside “our reality” and do not require acknowledgement by humans to exist. We then discover these truths and objects and name them. This would be a form of mathematical Platonism. To me, there appeared to be a mix of both discovering and creating that was being argued for in the entire conversation. The process of naming and finding the proof that such objects exist and behave a certain way would appear to be a form of discovery. This is a philosophical question that has not been resolved in the mathematical community either. I am not sure of the origins of the quote, but it goes something like mathematicians are Platonists during the week and formalists on the weekends. In fact, doing a google search “mathematicians as Platonists” will yield some seemingly contradictory statements.

For an interesting, sometimes combative, discussion thread that started with the post “Good Pure Mathematicians who are not Platonists?”, see

http://www.physicsforums.com/archive/t-81275_Good_Pure_Mathematicians_who_are_not_Platonists?.html

DAY 2

We continued our discussion about what is mathematics and the influence one’s culture and beliefs could have on the doing of mathematics. We also discussed the articles that were listed on the homework page. Some of the following observations were made:

A lot of mathematics seemed to stem from the religion and its needs (such as the keeping of a calendar) of the culture

A peoples’ religious belief system (or the one in power) appeared to influence what mathematics was done or explored

The articles seemed to be written from a biased (West is best) perspective

China entered the international mathematical community just recently, appearing to not want to get involved with the West. Someone mentioned that

China appears to have resisted the Western influence in many other areas as well.

The belief that the sun revolved around the earth kept some modern mathematics from being done. The belief otherwise was squashed by the Catholic church (the primary religious power in the West for a long time).

DAY 3

We postponed our discussion concerning the text because the books were not in for several people. Instead we began to talk about different numerals and numeral systems.. Several good questions were raised that some of the class members volunteered to explore and report back to the whole class.

DAY4

We had reports back:

Esther’s work

Phil’s research found the following sites:

Did the Chinese invent the first numeral system??

More on the Chinese Rod numeral system

Alex looked up some stuff on the abacus

Ashley found the most current way of writing Chinese numerals that had a cool converter

David found more on Mayan numerals

DAY5

We finished up our look at different number systems and then did some number crunching with different bases. We looked at the binary system (base 2) and did some conversions from base 10 to base 2. We also converted a number in our system to its Babylonian representation.

DAY 6 (9/19)

We discussed the reading excerpt from “Africa Counts”. Our discussion included “mystical” ways of thought, logical thinking, taboos on enumerating people or prized possession. It also raised many questions. For example, do “religious societies” stray away from logic. Some people argued that certain people in the government who are Judeo Christian appear to be anti-logical and that logic would seem to contradict religion. I brought up the point that the Judeo Christian God is said to be rational, hence logical- how does that fit in? What is logic? We also discussed the fact that industrialized societies seem to quantify everything. Many suggested that this was for efficiency and that in smaller communities, it is not as necessary.

We also looked at the Zulu peoples’ finger gestures for the numbers 1-10 and considered how they differed from our approach to holding up fingers for the corresponding quantities.

We then went on to look at the slow acceptance of negative number and this led to a tangential discussion of noun vs verb vs adjective. Can a number be all three? I was reminded of what a gerund! is.

Day 7

We looked at a game played by African children that involves placing rows of number down in a triangular shape and removing them one at a time on alternate sides. We also discussed why division by zero is not okay, remember, there are 2 cases to consider: 0 divided by 0 and N divided by 0 where N is nonzero.

Dr. Wang also showed us how to add 1+2+3+4+5+6+7+8+9+10 on the abacus. She also showed how to quickly add up the numbers 1 to 100:

1+2+3+4+….+97+98+99+100 = 1+100 + 2+99+ 3+98 +4+97 +………..+50+51 = fifty pairs of 101 = 50 x101=5050.

We also started to discuss infinity (but never quite got to the end-snicker, snicker). I posed the following scenario:

You have an urn. You have a timer set for 60 seconds. You have balls that each have a number on them.

At 60 seconds start the timer running.

After 30 seconds, so at 30 seconds—place balls 1-10 in the urn and take out ball #1. We will call this time 1.

After half of 30 seconds has elapsed, so at 15 seconds—place balls 11-20 in the urn and take out ball #2. We will call this time 2.

After half of 15 seconds has elapsed, so at 7.5 seconds—place balls 21-30 in the urn and take out ball #3. This is time 3.

At 3.75 seconds, place balls 31-40 in the urn and take out ball #4. This is time 4.

At 1.875 seconds, balls 41-50 go in, ball #5 comes out and this is time 5.

Continue on in this manner….

At seconds, balls 191-200 go in and ball #20 comes out and this is time 20.

So on and so on and so on…

You may have to work pretty fast (!) to do this, and might not be able to do so practically. This is where we will have to imagine an ideal world where we can move fast enough and we can continue the process on as described. Now, the timer does run out. There is a moment in time at which 60 seconds has elapsed. After this moment has passed, we ask:

How many balls are left in the urn?

How many balls were put in the urn?

How many times did we take out a ball?

Be careful with trying to reason by analogy from the finite to the infinite (they are not the same). This “jump” from the finite to the infinite is where your problems and the bridge between the practical and the theory occur. That is why an understanding of infinity is so hard to reach! In fact, we don’t quite reach it, we approach it.

I claim that there are no balls left in the urn. Why? If there were balls left, they would have numbers on them. Say ball x is left. Well, according to my process, ball x came out at time interval x.

Week #5 and Week #6

Discussed the Hersh book and infinity

Week #7

Monday: Watched the new CBS show Numbers

Wednesday Looked at some payoff matrices in game theory

Week #8

Finished discussing the Hersh book. Looked at set theory and the Axiom of Choice

Week #9

Started discussing the new book, Prime Obsession.

Week #10 and #11

Functions and Riemann

Some mathematical animals

A potpourri of math

Week #12, #13

Midterm project presentations

Week #14

Overview of complex numbers and some other fun math stuff

Week #15

RSA coding and encryption issues