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.