Heschl's Gyrus

This report from Science Daily describes the discovery of a correlation between the size of “Heschl’s Gyrus” in the brain and the ability to learn foreign languages. It also mentions an intriguing study that found that musical training started at an early age contributed to more successful spoken foreign language learning.

I have been pushing the idea that learning abstract math requires (among other things) learning a foreign language. So, I wonder, does the size of Heschl’s Gyrus correlated with ability at higher math? Also, the study about musical training suggests a reason for the commonly noted tendency for mathematicians to be musical, but I suppose that idea is pretty longshottish.

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Shape note singers use illegal vowels

According to the rules of English phonetics, certain vowel sounds cannot occur in an open syllable – a syllable that does not end in a consonant. Among these are the vowels we used to call the “short” vowels in school – the vowel sounds in pat, pet, pit, pot, put and putt. Other vowel sounds, such as those in made, need and mode, can occur in open syllables, witness may, me and no. I can’t give references for this claim since my linguistics books are in Ohio and I am in Wisconsin.

It sounds to me that traditional Sacred Harp singers in the south use the sound of the vowel in “pet”, or perhaps “pat”, to pronounce the indefinite article “a” and the vowel in the definite article “the”. This is intriguing. Could they be preserving the old sound of the vowel in those words before it became schwa? There is precedent for preserving sounds in singing that have been lost in speaking, for example in French: “Frère Jacques” is four syllables in the song and two in speech.

I wish I could give a link to a recording that shows this phenomenon with the articles. I listened to some of the videos of southern singings on YouTube but the sound quality is too bad to be evidence. I am posting this on the shape note mailing list asking whether other singers agree with my observations and whether anyone knows convincing examples of recordings with this pronunciation.

Schwa is written “ə” in linguistics. It is the sound of many unaccented vowels in English, for example the first vowel in “about”. By the way, many American southerners, including me, have a second schwa-like sound in many words that in midwestern speech are pronounced with the usual schwa. It is a very short sound like the “i” in “pit”. The only minimal pair I can think of for my speech is “carrot / caret” (I thought about it for a whole ten minutes). The spelling is only partly correlated with the sound – I use the i-colored schwa in “senate” and “minute” (the 60th of an hour) but the regular schwa in “venal” and “sinus”.

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Thinking without words

Several times in my life I have been infuriated by people contradicting something I said that I knew was true. (“You can’t cross the border between Georgia and North Carolina. They don’t border each other”. I have only done it about fifteen times.)

One of the most annoying are the people who tell me I can’t think without words. This seems to be the opinion mostly of logicians and computer scientists (but I think only a minority of them). When I am concentrating on math or on a physical repair job I USUALLY think without words. And in many other situations as well. The result is that when someone asks me what I am doing I am literally at a loss for words. I have to deconcentrate and come up with a verbal explanation of the nonverbal thinking I was doing. Which makes me look as if I don’t “know” what I am doing.

When I need to memorize the sequence 6785 (part of our car’s license number) I visualize the numbers 5678 with the five leapfrogging over the other numbers to end up on the right. I don’t say the numbers, I picture them. This has enabled me to write down the license number on the motel application without having to drop my bags and dash out the door to look at the car, which is usually parked the wrong way for me to see the back end.

When I stare at a chain of gears to see which way one of them goes when I turn another one, I visualize the turning of each intermediate one, one at a time. I don’t say or think “clockwise, counterclockwise” and so on, I see them turning and I feel kinetically the top of one going clockwise moving to the right – I sort of feel MY top (shoulders and arms) moving to the right.

When I see a pullback diagram I feel the upper left corner being pushed down and to the right so as to be the last corner of all the squares with the same bottom and right edge. I don’t think the words “pullback square” unless I am in the process of trying to formulate a claim about it.

I learned that I do this from reading Zen and the Art of Motorcycle Maintenance. That really wasn’t the main point of that book but it is what I remember most vividly from reading it.

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Harry Potter's English

Last night I saw the latest Harry Potter movie, Harry Potter and the Order of the Phoenix. As with the previous one (Goblet of Fire), which I watched on DVD, I had a lot of trouble understanding much of the dialog. Because of that previous experience I paid more attention to my problem this time and discovered that it was primarily the young people that I had trouble understanding. I don’t remember having this problem with the first four films.

Relevant background: I am a retired professor and have known and spoken with British academics for 40 years, and have spent several months living in Britain as well (Oxford and London). I am also hard of hearing.

Evidently, British young people, even educated ones, speak quite differently from their parents and grandparents. This is not just my experience: linguists have noticed it, for example the phonetician John Wells here (for 29 August). That site provides a clip of a young member of the aristocracy speaking. I can’t understand her either.

I had no problem understanding the actors who played the adults in the HP movies; I have had a lot of experience with that sort of accent (Scottish as well as southern English).

The American audience in the movie theater (in rural northern Wisconsin) had no problem with the kids. They laughed several times at verbal interactions between them that I didn’t understand.

I had less trouble understanding Daniel Radcliffe and Emma Watson in the interviews here. Is it possible that in the movie they deliberately spoke like British teenagers and during their interview they used their usual speaking-to-adults dialect?

I expect to watch these movies a second time on a DVD player with the subtitles turned on!

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The languages of mathematics

Conjecture: Mathematical English (ME) and the symbolic language of math (SL) are two distinct languages, not dialects of the same language.

I have asserted this in several places (Handbook, abstractmath.org) but I am not a linguist and it could be that linguists would disagree with this conjecture, or that the study of a mathematical corpus would reveal that another theoretical take on the situation would be more appropriate.

Some relevant points are listed below. I intend to expand on them in later posts.

1) Is ME a dialect of English or a register of English? Or does it have some other relationship to English?

2) ME appears to have several dialects or registers. One register is that used for what mathematicians call “formal proofs”. These are not formal in the sense of first order predicate logic, but their language is constrained, with the intent of making it easier to see the logical structure of the argument. Another register is that of “intuitive [or informal] explanations”. This is more like standard English.

3) The SL is clearly not a spoken language. It is a two-dimensional written language using symbols from English and other languages and some symbols native only to math. People do try to speak formulas aloud occasionally but this is well known to be difficult and can be done successfully only for fairly simple expressions.

4) There are other non-spoken languages such as ASL for example. I don’t know whether there are other non-spoken languages that are written. I don’t think dead languages count.

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Mass nouns in math writing

Mass nouns seem to be rare in math writing. I have done a little poking around the math journals in JStor and have several observations to make. All are tentative observations based on a small amount of evidence (and thinking).

Space” as a mass noun

“Space” as a mass noun was common before WWII but is rare now. A search for “in space” (in quotes to make it the phrase that is searched for) gives mostly references to outer space and to very old papers, mostly before 1930. Conjecture: The disappearance of mass nouns in math writing is a consequence of the rise of structural thinking in math.

One recent paper where “space” occurs seemingly as a mass noun, in the title no less, is: F. W. Lawvere, Categories of Space and Quantity, in J. Echeverria et al. eds. The Space of Mathematics: Philosophical, Epistemological and Historical Explorations, DeGruyter, Berlin (1992), 14-30. However, the word “space” appears as a mass noun only once in the body of the paper (according to my hasty scan) and many times as a count noun. Anyway I am not sure it is being used as a count noun in the title. It is paired with “quantity”, which is surely an abstract noun, not a count noun.

Areas of math as mass nouns

Areas of math are commonly used as mass nouns, for example, “Using calculus, we see that the function has one maximum”, or “the result follows by straightforward algebra”. The language of math contains several sublanguages with different uses (symbolic language, rigorous language, “rich” language) and one of them is the metalanguage used for talking about doing math, as those examples surely are.

Mass nouns and plurals

In the paper La Palme Reyes M., Macnamara J. and G. E. Reyes (1999). Count nouns, mass nouns and their transformations: a unified category-theoretic semantics, in Language, logic and concepts: Bradford Book, MIT Press, Cambridge, Ma, 1999, pp 427-452, the authors say that plural nouns are mass nouns, in fact they are the free mass nouns corresponding to count nouns under the adjunction developed in that paper. (The Wikipedia article on mass nouns doesn’t seem to regard plurals of count nouns as mass nouns.) Now plurals are mass nouns with atoms (like “furniture” rather than like “water”). Of course, plurals occur all over the place in math writing. Conjecture: In rigorous math prose the only mass nouns that occur are plurals, or at least are mass nouns with atoms.

I am suspicious of the way Reyes, Macnamara and Reyes smush together mass nouns with atoms (furniture) and mass nouns without atoms (water). (“Atom” means in the lattice of parts. “Some of my furniture” can include a bed and two tables, but not the leg of a table. “Water” is treated in language as if it were infinitely divisible. Of course it really does have atoms in the physical sense.)

These two kinds of mass nouns behave differently in many ways. The most important is that plural nouns can refer to either distributive plurals or collective plurals. (“All groups have identities” is distributive, “the voters were in favor of the proposition” is collective.) I doubt that these different kinds of mass nouns constitute a natural grammatical class.

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Comma Rule Found Dysfunctional (reposted)

One morning recently, I looked out the window and said, “The snow that fell last night has already melted.” When I spoke, I paused after “night”:

The snow that fell last night (pause) has already melted.

Most declarative sentences have a two-part structure: a noun phrase (“the snow that fell last night”) which is the subject, and a predicate (“has already melted”). Each of these two parts have further structure, but since the top-level structure consists of these two parts, it is natural to pause between them when the subject is complicated. This helps the listener to parse the sentence in real time.

However, when we write such sentences we must not do this:

The snow that fell last night, has already melted.

There is a rule about English writing that forbids putting a comma between the two main parts of a sentence. This rule is dysfunctional! It makes very complicated sentences hard to read:

The book with the pictures of the baby sits usually in the pink bedroom dresser.

“Baby sits” is a familiar phrase but it is confusing here because the two words are in different main parts of the sentence. You may have to back track while reading to make sense of it. Another famous example is

The horse raced past the barn fell.

Since the Powers That Be insist we mustn’t separate the two main constituents of a sentence by putting a comma between them, perhaps in our modern world of computers we could use color.

The snow that fell last night has already melted.
The book with the pictures of the baby sits usually in the pink bedroom dresser.
The horse raced past the barn fell.

Charles Wells
April, 2004

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Start again

In gyre&gimble I will comment on ideas that occur to me, centering around math and language but not limited to that. This is a revival of a blog I ran a few years ago and I expect to repost here a few of the postings from the former incarnation.

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math, language and other things that may show up in the wabe