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7-Way Venn Diagram

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yet another awesome but barely supported google product…. (plzzzz fix google reader!!!)

Super nice video of the Hopf Fibration.

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Make your own fractals online with Fractal Lab….

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So far, we’ve introduced sets as the foundation of mathematics and we’ve shown how to construct the natural numbers as power sets of the empty set. I don’t think we can go much further without introducing functions. The use of functions is ubiquitous in mathematics. When most people think about functions, they think of functions of real numbers and picture graphs like this:

However, the idea of a function is foundational and far precedes the construction of the real numbers in formal mathematics.

In order to precisely define functions, we need to talk a little about products and relations between sets. The product of two sets X and Y is the set consisting of ordered pairs of elements from X and Y:

X [pmath size=12]*[/pmath] Y = {(x,y) such that x [pmath size=12]in[/pmath] X and y [pmath size=12]in[/pmath] Y}

For example, {1,2} [pmath]*[/pmath] {a,b,c} = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)}. A relation R between two sets X,Y is just a subset of X [pmath]*[/pmath] Y:

[pmath]R subset X * Y[/pmath]

A relation is called left total if for every x [pmath] in [/pmath] X there exists y [pmath] in [/pmath] Y such that (x,y) [pmath] in [/pmath] R. A relation is called functional if for all [pmath] x in X[/pmath], [pmath] y,z in Y [/pmath] then y=z whenever [pmath](x,z) in R[/pmath] and [pmath](x,y) in R[/pmath]. A function is a relation that is both functional and left total. Ie. A function is a relation that relates a unique y [pmath] in [/pmath] Y to each x [pmath]in[/pmath] X. We generally denote a function “f” from a source set X to a target set Y by:

[pmath]f: X right Y[/pmath]

We often use graphs to visualize functions. For example, the following image represents a function “f” from the set {0, 1, 2, 3, … , 8} to itself:

In coming posts we’ll use functions to describe the cardinality of sets, as well as the addition and multiplication of natural numbers. Together, sets and functions are the basic tools of most mathematics.

Prev: Math II: Natural Numbers

So, now that we developed an idea of sets, we should start classifying them. We’ll start by constructing the natural numbers.

= {0, 1, 2, 3, … }

At this point we only have sets to work with, so we define the empty set as our zero.

0 = ∅ = {}

We take the power set of the empty set and define this set as one.

1 = P(∅) = {∅}

We continue taking power sets to construct the natural numbers

2 = P(P(∅)) = {{∅},∅}

3 = P(P(P(∅))) = {{{∅},∅},{∅},∅}

4 = P(P(P(P(∅))))  …

This gives us a class of sets with a successor operation P which together satisfy the Peano axioms, the essential algebraic properties of the natural numbers. Ie. With this construction mathematicians are able to derive all the known properties of the natural numbers (eg. distributivity).

Even though I’ve ignored some technical details in this construction, this still seems like a lot of work to make the most basic numbers. However, this is an important construction in the foundations of mathematics. Sets are the raw material of math and the natural numbers have significantly more structure. For example, the naturals are well-ordered with unique predecessors. The naturals also admit the binary operations of addition and multiplication. We can’t talk about this structure in a vacuum and that’s why we build these numbers using power sets of the empty set.

We’ll continue from here to talk about the integers and the rational numbers…

Prev: Math I: Sets

Next: Math III: Functions

IMHO mathematics is the study of mathematical elements using the rules of logic. So, the foundational idea of mathematics is a collection of elements known as a set. There are a variety of set theories. For example, naive set theory is perhaps the most intuitive, while ZFC set theory provides the most common axiomatic foundation for math.

The basic idea is that a set is a collection of elements. If a is an element of Z, then we write “a [pmath] in [/pmath] Z”. If the set Z contains the elements a, b, c, then we write this as

Z = {a,b,c}

The number of elements in a set is called the cardinality. The empty set, ∅ = {} , contains no elements and thus has zero cardinality. Next there are finite sets with finite cardinality. Then there are infinite sets of various infinite cardinalities. Georg Cantor hypothesized the continuum hypothesis about possible infinite cardinalities. However, Kurt Godel showed that the continuum hypothesis can neither be proved or disproved using ZFC set theory (Ie. the continuum hypothesis is independent of ZFC).

You can make new sets from existing set using various set operations. For example, if you have one set, you can make subsets or power sets. If you have multiple sets, you can make new sets by using intersections, unions, complements or products. These operations have countless applications. For example, we use power sets to construct the natural numbers and more generally to construct higher cardinalities. We use products of sets to define relations. In particular, the function relation has been used throughout mathematics and the sciences.

From here, I’ll probably write some posts on functions, numbers, infinity, algebra and eventually topology and category theory. I’ll salute Kurt Godel, Georg Cantor and a bunch of other math dudes. Lots of people hated on Cantor for creating set theory and transfinites, but he definitely kept it real throughout.

If you want to solidify your understanding of sets, try these Exercises in Set Theory.

Next: Math II: Natural Numbers

I was trained as a topologist. Since 1904, the Poincare conjecture has been the greatest unsolved problem in topology. In 2002-2003 a Russian mathematician, Grigory Perelman, essentially solved the Poincare conjecture. For his research he was awarded the Field’s medal, which is basically the Nobel prize of mathematics. He was also the first person to solve a Millennium Prize Problem and so he was awarded $1,000,000 from the Clay Math Institute. Grigory then TURNED DOWN both the Field’s medal and the $1,000,000. Grigory hasn’t been too stoked about talking to the media but he basically gave three reasons:

  1. Mathematical research is collaborative. Grigory couldn’t have solved the Poincare conjecture without the work of hundreds of fellow mathematicians. In particular, Grigory’s research relied heavily on the work of Richard Hamilton whom Grigory thought should be a co-recipient of any prizes. “To put it short, the main reason is my disagreement with the organized mathematical community,” Perelman, 43, told Interfax. “I don’t like their decisions, I consider them unjust.”
  2. Grigory did it for the math, not the fame or the money. Topology is an obscure and difficult subject. Grigory was offered fancy jobs at prestigious universities, but he was happier doing research in his hometown of St. Petersburg. The dude just wanted to chill at home with his moms and think about math. “I’m not interested in money or fame,” he is quoted to have said at the time. “I don’t want to be on display like an animal in a zoo. I’m not a hero of mathematics. I’m not even that successful; that is why I don’t want to have everybody looking at me.”
  3. Grigory was pissed at the biters. In 2006 Cao and Zhu published their own “proof” of the Poincare conjecture and declared it “the crowning achievement of the Hamilton-Perelman theory of Ricci flow.” A few months later they published various errata and retracted that “crowning achievement” statement. Grigory tried to keep his cool but made a variety of statements like, “I can’t say I’m outraged. Other people do worse. Of course, there are many mathematicians who are more or less honest. But almost all of them are conformists. They are more or less honest, but they tolerate those who are not honest.”

After all this drama, Grigory has reportedly QUIT professional mathematics. He’s now just another unemployed topologist living with mom. Since this is the last year of my contract with UConn,  I’ll probably be following in Grigory’s footsteps next year – minus the prizes and all…. As a stupidier and less successful topologist, I basically agree with Grigory on all his beefs. But I think he should have given the $1,000,000 to charity. There’s alot of hungry babies out there. Anyway, overall I salute you, Grigory! Keepin it real. 1luv.

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