Tuesday, October 09, 2007
Thursday, July 05, 2007
I think I'm finished moving so I'll get back to all this stuff this weekend. :-)
Friday, June 15, 2007
Monday, June 11, 2007
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)
[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?
Saturday, June 09, 2007
Wednesday, June 06, 2007
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.”
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.
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.)
Monday, June 04, 2007
Sunday, June 03, 2007
(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).
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.
Sunday, May 20, 2007
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.
Monday, May 14, 2007
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?
Monday, May 07, 2007
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.
Thursday, April 19, 2007
Sunday, April 15, 2007
Typing Euler into the search box at Mathworld produces 671 results.
Typing Euler into Google produces 12,300,000 results.
Wednesday, April 11, 2007
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
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 firstname.lastname@example.org . Please use Research Scientist as the email subject
Tuesday, April 10, 2007
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.
Sunday, April 08, 2007
Thursday, April 05, 2007
Tuesday, April 03, 2007
So I had the interview - lasted about 1/2 an hour. Since I had the phone on speaker, the erstwhile housemate heard the whole thing. I felt like I'd done badly, but he thought it went well. I guess it went good enough because a face-to-face interview will be scheduled this week. *crossing fingers*
Now if I could just figure out why the two people I've spoken with are both trying to point me away from the analyst position and towards a Six Sigma one.
Monday, April 02, 2007
I discovered that nature was constructed in a wonderful way, and our task is to find out the mathematical structure of the nature itself. It is a kind of faith that has helped me through my whole life. - Albert Einstein
So on to today's interview.
Thursday, March 29, 2007
The first is tomorrow morning for a company I'd really like to work for and have been trying to get a job with for some time. The job is moderately interesting, but I don't know what the pay range is. Somehow asking over the telephone today didn't feel right.
The second interview is on Monday. It's actually a second interview of sorts since the initial contact turned into a telephone interview. I was doing some mental scrambling on that one. LOL. But I must have done fairly well since they got back to me two days later to schedule the "second" interview. At any rate, it's a really interesting job, the company initiated the contact with me which I think is a major plus, and the pay range is very nice.
So we'll see what happens. Given my current disastrous financial situation, I'm inclined to go for money.
Wednesday, March 28, 2007
Monday, March 26, 2007
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
Wednesday, March 21, 2007
The joy didn't last long. My firewall had an upgrade. I downloaded the file and ran it. Something went totally caca so with pounding head, I got to try and restore the 'puter and get the expletive deleted upgrade installed. It's done now, but the expletive deleted Norton System Works isn't recognizing the upgrade and thinks that I don't have a firewall. *banging head*
If my car was running, I'd run away from home.
Tuesday, March 20, 2007
I'm pretty sure now that I am totally unemployable so the cat and I are making a list of couches we can sleep on starting April 1st. Please let me know what dates are available at your house.
Monday, March 19, 2007
So why, you might ask, am I asking if this is a sign? Well when I sat down to search I put my crochet project and the very nice bamboo crochet hook from my sister on my lap. At some point I heard the hook fall on the floor. Okay, so all I have to do is pick it up. Right? Wrong. Apparently it fell into a mini-black hole because I can't find it.
Is this a sign? Will my resume fall into the HR equivalent of a black hole?
Sunday, March 18, 2007
Before the cake, my sister and BIL took me to Wally World so I could get some yarn and buttons I need to finish the two hooded sweaters for the Pixie's boys. When they're done this week, I'll post some pictures.
Saturday, March 17, 2007
I read, write, and speak English. That's it. Just English. (I had 6 years of French, but that was more than 30 years ago. How much do you think I remember?) A number of years ago on my first day at a new job, my new boss asked me what languages I knew. As I listed off half a dozen computer languages, he looked more and more confused. So I asked him what languages he knew. His answer - English, French, Italian, Spanish, Russian.
I've also noticed when looking at the Stat Counter data that my blog is accessed by people from other countries and I bet that they can read English and probably at least one or two languages other than their native language.
How sad is it that the average American barely speaks English? (This rambling post is a really good example of that.)
Warming Up to Criticality: Quantum change, one bubble at a time Physicists can now observe matter as it gradually turns into a Bose-Einstein condensate--the exotic state of matter that displays quantum behavior at macroscopic scales. http://www.sciencenews.org/articles/20070317/fob3.asp
I don't have the background to understand anything that is more in depth than this article, but isn't it cool? I wonder what the mathematics that describes this system looks like?
Friday, March 16, 2007
Two links to investigate.
The first from my blogless friend Sue. She calls it "Feeding the Dark Side". I call it a really neat combination of math, crochet, and art. So, go here and check it out.
The second from blog mama, Teresa is called Romantic Math. The section starts with this quotation - "Beauty is the first test: there is no permanent place in this world for ugly mathematics." (G. H. Hardy) If nothing else it's an interesting use of functions. *grin*
Thursday, March 15, 2007
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."
Saturday, March 10, 2007
Monday, March 05, 2007
Sunday, March 04, 2007
Harvey tells me that I have to explain my absence. He recommends making up something dramatic and suggests that jungle adventures are always good.
Now I was trying to spare you all the gory details of my long absence, but I suspect that Harvey will not let the matter rest. So I will begin the tale.
Many moons ago in a land that existed outside of the bounds of normal time, a land of mathematical magic, beautiful princesses, and really really hot princes, an old woman crept through the trees in search of that one perfect Fibonacci flower. Or perhaps a snow flake born out of the chaos of a fractal storm. But alas, her quest was leading to the dark places where the spiders spun their webs to capture human dreams for food.
Lost within the darkness, the old woman sat down to reflect on her life. Perhaps within its tunnels she would find a way out of the spiders lair and back into the light.
(Should I really go on? LOL)
Wednesday, February 28, 2007
With all this job stuff I don't get to do the really fun things like go over to the MIT site and check out their Open Courseware. There are some really fun math courses there.
Tuesday, February 27, 2007
My sister and a very good friend (she sent me the coolest row counter!) just started blogs about their knitting and needlework adventures. So I've added links to them. Go encourage them to write more. *grin*
So I need to work. Any good ideas?
For what it's worth, when the temp agency called to say the assignment was ending the guy kept on saying how happy the client had been and what an excellent job I'd done. But really how hard is it to accurately query databases and enter the information into another database?