Some extra-curricular reading for you all

A comment (thank you Anonymous) on my previous post on counting included this link to an entry on the Britannica Blog: Why Math Geeks (Especially Immigrant Geeks) Rule. It's an interesting read.

I think I'm finished moving so I'll get back to all this stuff this weekend. :-)

I'm in moving hell

Will this never end?

Let's start with counting

In an overview of the history of mathematics, J. J. O'Connor and E. F. Robertson begin by saying that ”Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.” (MacTutor History of Mathematics, http://www-groups.dcs.st-and.ac.uk/~history/)

On the same note, I believe that the first brick in understanding mathematical cognition is counting. Fortunately (for me) there has been a great deal of research on this subject. There is a significant amount of evidence through studies done for at least the last 30 years (as far back as I looked) that show a recognition of numerosity and counting not only in people but also in a variety of animals – birds, rats, various monkeys/apes, etc. Granted the animals do not exhibit an ability to detect numerosity much above quantities of 3 – 5, but it’s there. This leads me to believe that a basic knowledge of counting is hard-wired into our brains. It’s an evolutionary trait handed down from our ancient ancestors.

There was a study reported on in SCIENCE in 1999 that sheds a little bit of light on how our brains are wired for counting. This is an illustration that appeared with the article.

DIAGRAM: Figuring out arithmetic. The principal brain regions involved in calculating exact and approximate mathematical problems. The left inferior frontal lobe is involved in verbally coded number facts that can be used in exact calculations. The intraparietal sulci of the left and right parietal lobes are implicated in estimations and approximate calculation, which are dependent on visuo-spatial representations of numbers. The intraparietal sulci are part of the circuit controlling finger movement and are likely to be crucial to finger counting, a near universal stage in learning arithmetic. (Butterworth, Brian. “A Head for Figures”. Science, 1999, 284(5416), p. 28-28-29)

subitize, v.
Psychol.
[f. L. subitus SUBITE a. + -IZE.]

intr. and trans. To apprehend immediately (the number contained in a small sample). Hence subitizing vbl. n.

1949 E. L. KAUFMAN et al. in Amer. Jrnl. Psychol. LXII. 520 A new term is needed for the discrimination of stimulus-numbers of 6 and below... The term proposed is subitize... We are indebted to Dr. Cornelia C. Coulter, the Department of Classical Languages and Literatures, Mount Holyoke College, for suggesting this term. Ibid., If no discontinuities had appeared in the results, no distinction between subitizing and estimating could have been drawn. 1971 Jrnl. Gen. Psychol. Jan. 121 The number of items in an array capable of being subitized. 1981 Nature 15 Oct. 569/2 Judgements of ‘small’ numerosities..are ordinarily attributed to subitizing.

Over the next couple of weeks, I'll review six relatively recent journal articles I've down loaded and also talk about some older studies that looked at counting in pre-verbal babies. The first article on tap is entitled "Analog Numerical Representations in Rhesus Monkeys: Evidence for Parallel Processing" - the study it reports on pretty much disputes the idea of subitizing.

Now aren't you just holding your collective breaths waiting for these?

Musical illusions

I thought this article from Science News/Math Trek was very interesting - Musical Illusions

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.

World's oldest adornments

This was on the Yahoo news page today. It's interesting and you'll see a little more about this in later posts from me.

RABAT (Reuters) - Perforated shells discovered in a limestone cave in eastern Morocco are the oldest adornments ever found and show humans used symbols in Africa 40,000 years before Europe, the kingdom's government said.
(Read more here.)

Origami fun for today

Trisecting an Angle with Origami

So the last time I sat down to write to you all, I said that I'd been at the library looking up information. And I pretty much tried to scare you away by saying that you'd see more about it. Well since then (it's been a tad busy at my house), I've decided to really scare you all.

Just about three and a half years ago I started looking for information on mathematical cognition. The search started with a book recommended to me by a high school principle, lead to a research project and rather long (one hour) presentation, and then became something of an obsession. The biggest problem I’ve been having is that the bulk of the research being done on mathematical cognition looks at arithmetic but not mathematics (geometry, algebra, and so forth). I have a couple of other problems right now. The first is that the more I’ve looked at one of the few theories of mathematical cognition that addresses “higher” mathematics, the more I think they’ve got it wrong. The other problem is my lack of access to people who study the subject which means that I can come up with all kinds of ideas, but have no way to get any feedback.

(And if I actually ever arrived at a coherent theory, I’d have no way of testing it. But I guess that’s another story.)

The lack of feedback leads directly to all of you. I decided to pretty much write papers and put them here for your comments, questions and ability to find the holes in my logic. There are a number of areas to looks at – the evolution and development of the brain and of cognition in general, the historical development of mathematics, studies of brain “use” while doing mathematics (or arithmetic), studies of mathematical learning disabilities, and possibly the effect of changes in society.

I’ll post the first of these “papers” in the next few days. Just to get it out of the way, I’ll talk about my ideas of whether or not mathematics is a language and whether or not mathematical cognition can be understood that way. (Aren’t you just way too excited?)

In the meantime, here are the definitions for arithmetic and mathematics from The Oxford English Dictionary, Second Edition (1989).

arithmetic, n.1
1. The science of numbers; the art of computation by figures.

2. Arithmetical knowledge, computation, reckoning.

3. A treatise on computation.

mathematics, n.
1. Originally: (a collective term for) geometry, arithmetic, and certain physical sciences involving geometrical reasoning, such as astronomy and optics; spec. the disciplines of the quadrivium collectively. In later use: the science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis; mathematical operations or calculations. Colloq. abbreviated maths, (N. Amer.) math.

When the modern subject is studied as an abstract deductive science in its own right, it is often referred to more fully as pure mathematics (see PURE a. 2d); when applied to the modelling of physical objects and processes (e.g. in astronomy, various branches of physics, engineering, etc.) and random processes (in probability), and to the handling of data, its full name is applied mathematics (see APPLIED a.), or (in early use) mixed mathematics (see MIXED a.2 5).

2. The mathematical considerations or principles relating to a specified phenomenon, process, etc. With of.

What I did at work today

On Sundays I'm in the library student computer lab from 1:00 to 8:30. I like the regular student computer lab better - mostly because I'm sitting at a desk with space to spread out my stuff. The library is somewhat less "user friendly." But the librarian got my laptop connected to the college Internet, so that was pretty cool. I spent a couple of hours scanning journal articles about genetics and evolution of the brain. Obviously there weren't a lot of students needing assistance. LOL

What I really started looking at (but got a tad side tracked) was information on the calculations used to determine when in the past a mutation occurred in a particular gene. I did read some interesting stuff in relation to that.

You can expect to see more about this since this is the only forum I have.

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.

What are you doing on your summer vacation?

I'm moving, looking for a permanent job, and working!!!!!!!!!!!! I start a summer stint (30 hours/week for 12 weeks) in the college's student computer lab. Because of the nature of the job, there's a lot of down time so I will be making a concerted effort to read and understand H. M. Edwards' book Riemann's Zeta Function. I'm really looking forward to this.

I'm not, however, looking forward to getting everything sorted out and moved. I hope that getting my internet connection moved goes easily.

So what marvelous plans do you all have?

Still looking for a job

Sometimes I wonder if I shouldn't just give it up and apply for well fare.

I was contacted by a company and went through two telephone interviews - one with HR, one with the relevant manager. Then they brought me in for four face-to-face interviews - HR again and three managers. But did I get the job? Nooooooooooooo. According to the email I received (almost a week later than they told me I would hear and after contacting them), I was an excellent candidate, but they hired some one with some unique skills (translation - younger than me).

The supervisor for the college computer lab contacted me to see if I wanted to work there for the summer (no students available for the work). So that's what I'll be doing for 12 weeks. Not a lot of money, but the basic bills starting in June will be covered.

The worst part of all this is that my lack of income means I'm losing my house. I'm trying to see if I can stave it off long enough to try and sell the house myself, but it's not looking good. See my first sentence.

This is for Rave!

Back in October Rave asked me in the comments to this post what an equation was. I had no idea. But tonight an answer was provided. So Rave, go read the comments and you will have your answer. *grin*

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.

A different DNA

VW Bug had this on her blog. I just couldn't resist and I have to say, this is so me. *grin*

Read my VisualDNA™ Get your own VisualDNA™

Wouldn't this be a GREAT job?

Still on the job hunt and still checking the various job boards. This morning's search hasn't been particularly productive, but HotJobs has the following listing. This is a job that I would absolutely love to have, but there is no way in the world I can even pretend to be qualified. Although it's pretty scary that I know what the ad is talking about. LOL

So here's the ad. Doesn't this sound like way too much fun?

Computational Research Scientist (Biomedical)
Company Name: Delta Search Labs, Inc.
Job Category: Pharmaceutical/Biotech; Technology
Location: Cambridge, MA
Position Type: Full-Time, Employee
Salary: Unspecified
Experience: 2-5 Years Experience
Desired Education Level: PhD
Date Posted: April 10, 2007

Solutions Labs (www.solutionslabs.com ) a Cambridge MA based R&D company, is recruiting one or two candidates with the following qualifications. Ph.D. or M.S. level researcher (M.S. absolutely must have several years relevant experience) with degree in Computer Science, Chemistry, Biophysics, Physics, Molecular Biology or Chemical Engineering wanted for bioinformatics, chemoinformatics, and chemometrics algorithm development. Will also consider exceptional candidates from Applied Mathematics, Statistics or related Engineering fields. Documented accomplishments in computational research and strong analytical, statistical, and numerical analysis skills required. Experience in spectroscopic data analysis (NMR is especially desirable) and mass spectrometry data analysis is prefered. We are seeking individuals whose primary background is computational and mathematical but who have experience analyzing experimental data and understand problems of preprocessing chemical data from various analytic methods. Desired background includes some of the following: pattern recognition, machine learning, multivariate statistics, numerical analysis, etc. Sufficient knowledge of biochemistry and molecular biology to interpret results of data analysis within a systems biology context (e.g. metabolic pathway analysis) would be a big plus. Experience with analysis and interpretation of Affymetrix microarray studies is also a big plus. Initial projects will be for both small molecule spectroscopic data pattern recognition (metabolomics) and DNA microarray analysis but might eventually expand to include mass spec proteomics. Position will involve both algorithm development and extensive programming. Programming may be done in Matlab and other scripting languages (e.g. Python). Position could involve collaboration with leading academic research groups. Ideal candidate will be creative with the desire and ability to help identify and aggressively pursue new opportunities in emerging fields. Strong written and verbal communication skills are a must. Prefer either US citizens, US permanent residents, or NAFTA TN applicants (but not an absolute requirement for exceptional candidates). Please send a cover letter and resume (pdf or doc file) to jobs@solutionslabs.com . Please use Research Scientist as the email subject

Okay...

I have heat and hot water.

The headache is gone. (Probably because the erstwhile housemate brought me cigarettes. I was out. Have I mentioned that I don't have a working car so if I run out of something, it's just too bad so sad.)

The snow is still on the way.

Some days it doesn't pay to get out of bed

I'm out of heating oil. No heat. No hot water.

Oh, and I have a headache.

And did I mention that there is a snow storm on the way?

We did the Easter dinner

and I am STUFFED!

It actually was a lot of fun. Friday was the erstwhile housemate's birthday so we celebrated his birthday today. (There's a joke in there, but ... ) So besides the traditional Easter ham we had a fabulous birthday cake that my sister made with help from my youngest nephew (9 years old) on the marzipan decorations. Youngest nephew put the candles on the cake just before I took this picture.

There was also a funny birthday card with a hunky man on the front. *grin*

BTW - the cake was a carrot cake like you've never had. Sister added raisins, nuts, and pineapple. Yummy.

Sweater

This is a sweater for me. It took me forever to get around to weaving in the ends. LOL
Front and back.