Counting

I fell asleep last night thinking about counting and children and what we learn from counting.

It's time to come back

Last fall I joined two group son Ravelry - the Harry Potter Knit & Crochet House Cup and Fantasia. For the last three months, these groups have gotten me going on a small treasure trove of crochet projects. Several are math related.

My favorite is a mobius strip grown big and turned into a shawl/cowl (aka scowl). I've finished one and almost completed the second. The whole thing is supposed to represent something I love and is full of things to represent numbers (Fibonacci numbers, prime numbers, golden mean).

Here are a couple of pictures (the model is my gorgeous niece).





I have a second one almost completed in a beautiful silk/merion blend from Malabrigo. It's my sister's gift and she chose the yarn.

Is Mathematics language?

The answer is both “yes” and “no” depending on whether you are asking if the method of communicating mathematics is a language or if you are asking whether mathematics as language defines the cognitive processes necessary to perform mathematics.

Your real question should probably be “why are you asking this?” The reason why I feel that the question of mathematics as language needs to be answered has to do with theories published in 2000 by a mathematician (Devlin, K., The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip) and a linguist and cognitive psychologist (Lakoff, G. and Nunez, R. E., Where mathematics comes from: how the embodied mind brings mathematics into being). In both cases, language becomes the underlying explanation for mathematical cognition. Devlin posits that mathematics piggy-backed on language. Lakoff and Nunez use linguistics constructs to explain the cognitive processes underlying mathematics.

So let's decide if mathematics is a language using a general definition of language. Language is a complex system so no simple definition will cover all the nuances, but we can for the time simplify as much as possible. A definition of language includes producing speech, analyzing the speech we hear, the vocabulary and its symbolic references we use, grammar, and syntax. (Deacon, 1998, p.40) Deacon offers a generic definition for language as “a mode of communication based upon symbolic reference (the way words refer to things) and involving combinatorial rules that comprise a system for representing synthetic logical relationships among these symbols.” Deacon also states that within this definition mathematics “might qualify as having the core attributes of language.” (Deacon, 1998, p.41)

Let's see what the "core attributes" of language might be. You can say that language consists of basic sound units, phonemes that combine to form morphemes. These phonemes and morphemes when combined according to predefined rules become abstract symbols which we ascribe meaning to and understand. In other words, they become words. Language also allows various combinations of these symbols by following syntactic and pragmatic rules - sentences and paragraphs. (See Matlin, 2005, p.298) The result is communication that describes an object, event, or action, which need not be present or even exist. In other words, language symbolizes and creates meaning using a series of abstract symbols (letters or sounds) which have no particular connection to a concrete object other than those connections we agree exist.

Presented in these terms, the phonemes of mathematics are the basic digits, zero through nine. These digits are combinable following a few simple rules and form mathematics morphemes, numbers such as 123 or 3.1416 or even 4/5. The branch of mathematics in question defines the particular syntax used. For example, Algebra’s syntax determines how to write an equation and the order of operations used to solve the equation. Each digit, number, or equation symbolically refers to a quantity or describes a system, event, or form. The actual quantity need not be present nor does a quantity or an equation need to refer to a concrete object or collection of objects. Many of the systems, forms, and events described by mathematics are themselves abstract concepts such as equations which describe the multidimensional shape of the universe.

Do these similarities between language and mathematics provide a sufficient condition to call mathematics language? In some instances, mathematical and language processing take place in areas of the brain that are generally similar. Experiments done on bilingual individuals indicate that exact calculations occur in the left inferior frontal lobe of the brain. This area controls linguistic representations of exact numerical values. However, approximations of numbers occur in the left and right intraparietal sulci in areas associated with visuo-spatial tasks. (Butterworth, 1999; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999)

Given this, my answer to the question is “yes, there exists a well defined language of mathematics used to communicate mathematical knowledge.” However, I believe that it is a mistake to confuse the act of communicating mathematics with the cognitive processes taking place while doing mathematics so my answer to a mathematical language that defines the cognitive process is “no.”

References:

Butterworth, B. (May 7, 1999). A Head for Figures. Science, 284, 928-929.

Deacon, T. W. (1998). The Symbolic Species: The Co-evolution of Language and the Brain. New York: W.W. Norton and Company.

Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (May 7, 1999). Sources of Mathematical Thinking: Behavioral and Brain-Imaging Evidence. Science, 284, 970-974.

Devlin, K. (2000). The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. Basic Books.

Matlin, M. W. (2005). Cognition (Sixth). Hoboken, NJ: John Wiley & Sons, Inc.

Lakoff, George; Nunez, Rafael E. (2000). Where mathematics comes from: how the embodied mind brings mathematics into being. Basic Books.

Origami fun for today

Trisecting an Angle with Origami

Math and Genes

If you start reading about genetics, you start running into a lot of mathematics. Well, you don't usually know you have, but the math is there. Articles will appear that talk about the evolution of genes (or species) and refer to work that was done to try and determine when a gene mutated. The articles seldom (never?) actually show you the math, but here's one that does from MathTrek - A Grove of Evolutionary Trees.

Happy Birthday Leonard!

Leonard Euler was born on this day in 1707. He produced mathematics in virtually every area of math.

Here are some of my favorites.
Typing Euler into the search box at Mathworld produces 671 results.
Typing Euler into Google produces 12,300,000 results.

Riemann Hypothesis - Is it proved (or disproved)???

So this is the current "news" making the rounds on some of the math blogs - a draft paper proving that the Riemann Hypothesis is wrong.

A rehash for all of you who get bleeding brains when I go into math mode:

There is a function (the zeta function) that predicts the distribution of the prime numbers. In 1859 this pretty neat mathematician named Riemann presented a short paper in which he conjectured that the non-trivial roots of the zeta function would all have a real part = 1/2.

Lost you, didn't I? Okay, go back to high school algebra and those equations you had to solve. The ones that looked like x² + 3x - 4 = 0. You'd do some arithmetic and find out that you could rewrite the equation as (x + 4)(x - 1) = 0 and the solution was x = -4 or x = 1. Those were, more or less, the trivial roots.

Although it's more complicated (I'm guilty of major simplification in the previous paragraph), the trivial roots of the function in the Riemann Hypothesis are all negative numbers (-2, -4, -6, .....) and the non-trivial roots are all complex numbers. The hypothesis says that the non-trivial roots will all be 1/2 + yi (y is any real number and i is the square root of -1).

Fast forward to 2007 and nobody has proved or disproved the hypothesis. It's a really BIG deal for a whole lot of reasons. A draft paper has appeared on ArXiv claiming to disprove the hypothesis. Keep in mind that this is a draft and has not been peer-reviewed in any manner. Nevertheless, it's getting attention. Check the entries on Gooseania and Ars Mathematica.

Even though my interest in the zeta function has to do with that real part "y" of the non-trivial roots, I've downloaded the paper. I can tell you that 8 pages into it, I'm cross-eyed. LOL

And my friends think that I give them headaches

From Yahoo - Brainiacs Succeed in Mapping 248-Dimensional Object

For the more mathematically inclined among you, check out what The n-Category Cafe had to say today.

Now this is potentially very cool

From The n-Category Cafe, a link to what looks like a very cool paper. At least the first page caught my attention.

Mathematical knowledge: internal, social and cultural aspects

The author (Yu. I. Manin) describes the paper as follows: "I discuss some general aspects of the creation, interpretation, and reception of mathematics as a part of civilization and culture."