4/1/98
- Quiz
- We are discussing creativity as a property of language. We are using the finite state model of language. Even though we have only a finite inventory and a finite number of processes, recursion gives us an infinite number of sentences.
- But this is not really satisfying. Sentences with similar meanings can be arrived at by tracing very different paths through the finite states machine. In a sense, this model is therefore counterintuitive. It's a great engineering model, but a bad solution philosophically.
- Return to our starting point. We have been thinking of numbers as analogous with sentences. Maybe we should instead think of numbers as analogous to words.
Of course, numbers are made up of the digits 0-9 in any order, while words are made up of letters with various limitations. These are called co-occurrence restraints. One example is that no English word begins with the letters "DT."- If numbers are words, what is the mathematical equivalent of a sentence? We will call this equivalent a mathematical statement, and such a statement might consist of an equation or an inequality or any similar relationship between mathematical quantities.
- Words are formed from sounds, which have distinctive features that we have already analyzed. Numbers are formed from numerals, in some sense, but not in quite the same way. What possible analogues of distinctive features can we come up with?
4/3/98
- We can take a geometric approach to tuning. One possibility is to divide the scale into 6 equal parts
the whole tone or pentatonic scale. Another possibility, and the one which is used in Western music, is to divide the scale into 12 equal intervals
C D E F# G# A# c
which results in chromatic scale. We usually arrange it on paper the way that the keys on the piano are arranged:
C C# D D# E F F# G G# A A# B c
We make each step equal, and call each step the equal-tempered semi-tone s=. The twelve semi-tones have to add up to an octave, so s=12 = 2, or s= = 21/12, which is approximately 1.059463.
Black keys C# D# F# G# A# White keys C D E F G A B c
- Pitch discrimination: up to around 1000 Hz (1 KHz), the human ear can distinguish differences of approximately 3Hz. Note that this number is not relative. At 30 Hz (at the low end of human hearing), we can only hear differences of around 10%, which is approximately 2 semi-tones. At around 1000 Hz, we can hear differences of 0.3%, and at higher frequencies, we can hear differences of approximately 0.25%.
- The unit of measurement that is used is a cent. This is 1/100 of an equally tempered semi-tone. We write it as ¢. Our definition is that ¢100 = 21/12, or ¢ = 21/1200, which is approximately 1.00057779. At sufficiently high frequencies, the human ear can hear distinctions of approximately 5¢.
- Annoyingly, the geometric semi-tone s= is irrational. The proof is quite simple. Suppose that s= is some fraction a/b. Then we would know that (a/b)12 = 2, or (a/b)6 = sqrt(2). But we already showed that sqrt(2) is irrational, so it can't be a fraction.
- More about creativity and language: We have tried the analogy of number and word, and (mathematical) statement and sentence. We can also think of an analogy to computer languages. Consider the computer command
In computer terminology, the word PRINT <file> / COPIES=5
COPIES=5is the qualifier for the command. We can also draw an analogy to sentences, and think ofCOPIES=5as an adjective modifying the file name.
If instead, we wrote the command as
then PRINT/COPIES=5 <file>
COPIES=5would function as an adverb modifying the verb.- If we think of words as consisting of sounds, then we can similarly think of numbers as made up of digits. (Note: This analogy should not be extended too far.) Sounds have certain features. For example, [i] is a vowel sound which is syllabic, sonorant, not nasal, voiced, high, not low, not back, and not rounded. We can try to find a similar identifying features of numbers.
- Another commonality between mathematics and languages is that both are linear. That is, one word or sound always follows another, and cannot be simultaneous with another.
4/6/98
- Quizzes scheduled for Friday, April 17, and Monday, April 27.
- Octal or base eight: Just as we can write numbers in binary and decimal, we can also write them in base 8. Corresponding to a one's, ten's, hundred's place, we have one's, eight's, sixty-four's place, etc. For example, the number 4318 = 4*64 + 3*8 + 1*1 = 28110. We can convert from decimal to octal by repeated division by the number 8, just as we can convert from decimal to binary by repeated division by 2.
- We can convert rapidly from binary to octal (and vice versa) by group digits in triplets, and then converting each triplet from binary to an octal digit. For example, 100111012 = (010)(011)(101)2 = 2358. We can go the other direction as well: 3628 = (011)(110)(010)2 = 111100102.
- Remember that we are drawing analogies between mathematical statements and sentences, and numbers and words. We can try to extend the analogy to one between digits (or numerals) and sounds, and then we can search to see what might be analogous to distinctive features of sounds. We can, for example, break digits into even and odd, or look at other properties.
- Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, ... The nth Fibonacci number is written as Fn. So F1 = F2 = 1, F4 = 3, F6 = 8, and so on.
Two interesting properties: take a pair of neighboring Fibonacci numbers, and then take the surrounding pair (e.g., 8 and 13, with surrounding pair 5 and 21). The two products gotten by multiplying together each pair will differ by 1 (in this case, compare 8*13 with 5*21). And the square of a Fibonacci number will differ by 1 from the product of its two neighbors (e.g., compare 82 with 5*13).- The most interesting feature of the Fibonacci numbers is gotten by taking ratios of successive numbers: 2/1, 3/2, 5/3, 8/5, 13/8, etc. This sequence of fractions approaches 1.618, the Golden ratio known to ancient Greek mathematicians and artists.
4/8/98
- A bit about the history of various calendars. The solar year is approximately 365.25 days long, and the lunar month is (even more approximately) 29.5 days long. In order to get these two cycles synchronized, there are two possibilities. One is to lengthen the month, which is what happened in the Roman calendar. The other is to adopt some sort of periodic cycle of adding extra lunar months periodically to the year. Call these years with extra months "leap years."
If we take 12 years with 12 lunar months, and 7 years with 13 lunar months, the result is very close to 19 solar years. The Babylonians noticed this, and worked out a regular cycle of 7 leap years every 19 years.- The Jewish calendar (at the time of the birth of Jesus, at any rate) was a bit less systematic. The religious authorities would add leap years any time that the new year would appear to be starting "too soon," which was defined as before spring started (more specifically, before barley sprouted). After the Jewish diaspora, approximately 300 years after the birth of Jesus, the same Babylonian 19-year cycle was instituted, with leap years in years 3, 6, 8, 11, 14, 16, and 19 of the cycle. The intent of the rule was to make sure that the first month of the calendar (named Nisan, derived from the Babylonian) did not begin too early. In particular, Passover, celebrated at the fifteenth day of the first month (in other words, the first full moon of the new year) had to fall after the vernal equinox.
- Christianity faced the problem of celebrating Easter. Because Lent is observed for the forty days prior to Easter, the date of Easter needed to be determined well in advance. The rule that was adopted by Roman Catholics was that Easter would fall on the first Sunday after the first full moon after the equinox, which usually amounts to the first Sunday after Passover. Eastern Orthodox ritual began the Holy Week (i.e., Palm Sunday) on the first Sunday after Easter, which usually means that Eastern Orthodox Easter is a week after Roman Catholic Easter.
- C.F. Gauss derived the following formula for Easter, which works in nearly every year:
J = year (Jahre in German) M = 24 (Gregorian calendar, 1900-2099)
15 (Julian calendar)N = 5 (Gregorian calendar, 1900-2099)
6 (Julian calendar)a = J mod 19 b = J mod 4 c = J mod 7 d = (19*a + M) mod 30 e = (2*b + 4*c + 6*d + N) mod 7 P = 22 + d + e
Then P will be the date of Easter in March if P is less than 32. If P is greater than 31, then P-31 will be the date of Easter in April.- Here is the computation for the year 1902:
J = 1902 M = 24 N = 5 a = 2 b = 2 c = 5 d = 2 e = 6 P = 30
and indeed Easter in 1902 fell on March 30.- A bit more about our analogies, and in particular the difference between surface representations and deep representations. If we break our digits 0-9 down into components, we can think of them as follows:
0 1 2 3 4 5 6 7 8 9 1 - + - + - + - + - + 2 - - + + - - + + - - 4 - - - - + + + + - - 8 - - - - - - - - + +
If we think of a word as a set of linearly ordered sounds, and a number as a set of linearly ordered digits, this gives the form of each notion, but not really the content. We can define form as items and processes, but it is harder to discuss content in those terms.- We could instead define the words "dog," "truck," and "freedom" in terms of properties that distinguish each concept from the other two.
4/15/98
- Class was taught by Li Zhuqing.
- Chinese uses both tone and characters. It is important to distinguish characters from words, since a word might consist of two characters. Modern Chinese has around 48,000 characters.
How are they organized into a dictionary? One way is to count strokes of characters, or to divide characters into different parts. Some characters have left-right structure, and some have up-down structure; this helps to organize the dictionary.
Some characters are pictographs; mainly, these are pictures of concrete nouns. Some characters are gotten by combining forms (for example, adding a character that indicates a feminine noun to another character). Some characters are formed by adding an indicator to another character (for example, the symbol for wood can have an indicator added to turn it into the symbol for root). And some times, a character is simply borrowed to indicate a new word.- There are 214 radicals (parts of characters), and a dictionary might be organized in this way. This system is rare in modern China. One can also use features of the 4 corners of radicals, but this is even rarer.
- Chinese does not distinguish plural from singular nouns. Instead, "measure words" are used between the number and the noun. There is also a measure word for the notion "some."
Measure words exist in some English contexts (e.g., "two pieces of paper"), but their use in Chinese is mandatory.- Chinese has many "set phrases." These are four character combinations which are often taken from classical literature. We saw some examples of these which contain number words.
- Recall that Chinese is a tonal language. Mandarin has 4 tones. Some dialects have as many as 10 tones. Cantonese has 9 tones.
- Chinese also has many dialects. There are two mutually incomprehensible dialects in Fujian province, each with 7 tones.
- Linguists use a Middle Chinese dictionary (from the sixth century AD) to study phonology. It listed 8 tones, in 4 major groups which conveniently contain 2 tones apiece.
- Number the tonal values from 1 to 5, with 5 as high and 1 as low. That lets us write down the tones. Words which consist of more than one syllable can sometimes change the tone of the first syllable depending on the tone of the second syllable. For example, "ni hao" should have a falling-rising tone on each syllable, but the first syllable changes to just a rising tone.
The high point of this "tone sandhi" occurs in a Fujian dialect, in which nearly all 49 possible tone combinations (remember that in Fujian, there are 7 tones) change at least slightly. For example, the tone combination (242)(51) becomes (44)(51). It is always the first of the two tones that changes.- Singing in Chinese can be very hard to understand. The tones of the syllables are usually dropped in favor of the tones of the melody.
- Topics for Friday's quiz include calculating in octal (conversion, addition, subtraction, and multiplication), the irrationality of the twelfth root of 2, the principles behind the computation of the date of Easter, interval arithmetic involving equal, Pythagorean, and natural temperaments, and basic questions about Chinese numerals.
4/17/98
- Quiz and discussion of the homework for next class.
- The word for 100 in Indo-European is *kmtom (with accent marks over the k and o). Focus in the following on the initial consonant. In IE, it is a [c], which is a palatal voiceless plosive.
- Here are the words for 100 in various IE languages, grouped into two large groups:
The two large groups consist of those languages in which the initial sound is something like a [k] and those languages in which the initial sound is something like an [s]. This plosive vs. fricative split runs through the IE family of languages. Languages in the first category are called centum languages, and those in the second category are called satem languages.
Plosive Greek he-katon Latin centum [k]>[tj], palatalization Old Irish cét [k] Gothic hund [k]>[x]=<h>, Grimm's law Tocharian känt Fricative Sanskrit s'atám [ç], voiceless palatal fricative Avestan satem Lithuanian simtas [with accents] Old Church Slavonic approximately suto
The split shows up in many different words. The guess is that most likely the original sound in IE was a palatal plosive [c], like the initial sound of "kill," as distinguished from the [k] initial sound of "cool," which is velar.
There is a similar split for voiced sounds. The words for "winter" show this. Latin "hiems" has an initial h, which derived from [gh], while Old Church Slavonic uses the word "zima," and the [z] is a fricative.
- What about words for 8? The IE is *okto(u), with various accents which HTML can't manage. What happened in each of the above languages?
What about the dots under the "s" and "t" in Sanskrit? They indicate yet another place of articulation, retroflex:
Plosive Greek okto (with some accents) Latin octo Old Irish ocht [kt]>[xt], written <cht> Gothic ahtau <h>=[x]<[k] Tocharian B okt Fricative Sanskrit astau dots under s and t; see below Avestan asta, missing accents Lithuanian astuo(nì), missing accents Old Church Slavonic approximately osmi
Retroflex sounds are made by bending the tongue backwards.
labial dental retroflex palatal velar labiovelar p t t with a hook c k kw - One more point: the initial "o" in IE has become an "a" in Sanskrit because of vowel merger.
4/22/98
- One last point about the etymology of words for 8: Latin "octo" becoming Italian "otto" is an example of regressive contiguous total assimilation.
- Another interesting thing about the number 8: the square of any odd number is one more than a multiple of 8. Symbolically, if u is an odd number, then u2=8n+1. Here's a bit of a chart of u and n values:
Notice that the numbers in the "n" column are actually triangular numbers.
u n 1 0 3 1 5 3 7 6 9 10
We can prove this without much trouble. If u is an odd number, then it is one more than an even number, so we can write u=2k+1. Then some algebra gives u2=4k2+4k+1 = 4(k2+k)+1. So we see right away that u2 is one more than a multiple of 4.
But we can do a bit better. Write k2+k as k(k+1), and then we can see that either k is even or else k+1 is even. Either way around, k2+k is even. So 4(k2+k)+1 is 4(even number)+1 = 8*number + 1.
Also, we can see that we can write u2 as 8(k(k+1)/2)+1, which is 8*(triangular number)+1.
- 8 is a lucky number in Chinese culture.
- "Jesus" in Greek is written "iesous" (transliterated), which adds to 888 in the Greek system. Jesus was circumcised on the eighth day after his birth (as are all Jewish boys). Italian baptismal fonts often have eight sides for this reason.
- Roman numerals:
It is natural to wonder about the origins of those letters. C is the first letter of "centum," the Latin word for 100, and M is the first letter of "mille," the Latin word for 1000, so one might think that this is an example of the acrophonic principle: using the first letter of a word to stand for the word. But in fact, the use of these two letters is an accident, and has nothing to do with the related Latin words!
I 1 V 5 X 10 L 50 C 100 D 500 M 1000
- The use of I for 1 is a symbol of a single finger, and the V is most likely a symbol of an open palm. X is probably two V's put together.
- Numbers 1000 and higher were often written with a line (vinculum) above the number (which HTML won't let me do), so L with a line over it would stand for 50,000. I with a box mostly around it would symbolize 100,000.
(I) was used to stand for 1,000 up until the 1500's (though M was used as far back as the second century as well). The origin of M was probably from "mille pasuum," `1000 paces,' meaning 1 Roman mile. The letter M is probably a script form of (I). In this light, D is probably derived from I), half of (I).
Also, ((I)) was used for 10,000, and I)) was used for 5,000. Finally, (((I))) was used for 100,000, and I))) was used for 50,000.
What is the origin of (I)? It is a corruption of the Greek letter phi.- There are three other Greek letters that don't correspond to letters in the Roman alphabet: ks, kh, and ps. To complicate matters, the Greek alphabet split into Eastern and Western European styles. The chart is
The letter X, not used for other purposes, was used for 10, and the letter V with a vertical line was used for 50. Eventually, it was stylized into an L.
xi [ks] chi [kh] psi [ps] East three horizontal lines X V with vertical line through it West X V with vertical line ps (actually, pi-sigma)
Finally, the Greek letter theta, also not used in the Roman alphabet, turned into the letter C for 100.- Another feature of Roman numerals is subtractive notation. We represent 352 as CCCLXXII. But if a smaller numeral is followed by a larger one, the smaller is subtracted from the larger. Hence, IX means 9, IV means 4, and there are even some times when Romans wrote IXX for 8.
This is also a feature of Latin names for some numbers in the teens. The names for numbers from 11-17 are additive, but 18 is "duodeviginti" (2 from 20) and 19 is "undeviginti" (1 from 20). Note that in Roman numerals, though, 19 is written as XIX and not IXX.- Hexadecimal is the name for base 16, even though it is a combination of Greek and Latin roots and not a historically accurate word. We need symbols for digits from 10 to 15, and we use letters from A to F, so that
A16=1010 Convert a number from decimal to hexadecimal by repeated division by 16 (akin to binary and octal conversions).
B16=1110
C16=1210
D16=1310
E16=1410
F16=1510
- It is simple to convert a number from binary to hexadecimal and vice versa. Given a binary number, group the bits in fours, and convert each group of 4 bits to a single hexadecimal character. For example, 1101011012 = (0001)(1010)(1101)2 = 1AD16, because 00012 = 116, 10102 = 1010 = A16, and 11012 = 1310 = D16.
The reverse conversion is just as simple. To turn A416 into binary, turn each "digit" into 4 bits, and get A416 = (1010)(0100)2 = 101001002.
4/24/98
- The explanation of the "interrupted game" problem: Remember that we are considering a game of chance in which either player has an equal chance of winning any given round. The first player to win 3 of the 5 rounds wins the game and $64. We can agree that if the game is interrupted with the scores of 0-0, 1-1, or 2-2, then the money should be split 32-32.
What if the game is interrupted with the score 2-1? Consider what happens if we play instead one further round. There is a 50% chance that the score will become 2-2, and a 50% chance that the score will become 3-1. In the first case, we agreed that the money should be split 32-32, and in the second case, obviously the money is split 64-0. Since each of these cases is equally likely, we should average them, and then the money should be split 48-16 if the game is interrupted with a 2-1 score.
What happens if the game is interrupted with the score 2-0? Consider again what happens if we play one further round. There is a 50% chance that the score will become 2-1, and a 50% chance that the score will become 3-0. In the first case, the money should then be split 48-16, and in the second case, the money should be split 64-0. Therefore, if the game is interrupted with the score 2-0, we should determine how to split the money by averaging these two amounts, giving a split of 56-8.
What happens if the game is interrupted with the score 1-0? What would happen if we played one further round? There is a 50% chance that the score would become 1-1, and a 50% chance that the score would become 2-0. In the first case, the money would be split 32-32, and in the second case, the money would be split 56-8. The average of these is 44-20, and that is how the money should be split if the game is interrupted with the score 1-0.
- Monday's quiz will contain 3 questions. The first will ask for an evaluation and explication of some section taken from Dr. Matrix, chapters 12, 18, and 22; the choice of subject matter is yours, and you may refer to the book during the quiz. The second question will ask you for a presentation of some subject from class, and the third will ask for a proof of some fact given in class. You may not refer to your notes for either of these.
- The contents of a computer can be divided into instructions and data, with instructions=commands=processes, and data=parameters=items. But in fact, instructions are also data; this was the great insight that made computer programming possible.
- Within a computer, ASCII (the American Standard Code for Information Interchange) is the code used to turn characters into numbers. Standard ASCII consists of 7 bits, representing numbers from 0 to 127 (016 to 7F16 in hexacdecimal). Today, most implementations use 8 bits, and the remaining bit can be either ignored, or used for parity checking, or used to extend the code to contain 256 characters (written in decimal from 0 to 255, and in hexadecimal from 016 to FF16. The following discussion is just of 7-bit ASCII.
The characters included are both control characters (instructions to the communication device to behave in a certain way, such as ringing a warning bell or executing a carriage return) and printable characters (letters a-z and A-Z, digits 0-9, blank space, and other symbols on a keyboard). The printable characters are coded from 32-126 (2016-7E16), and the non-printing control characters are coded from 0-31 (016-1F16) and 127 (7F16), which is the DELETE symbol.
4/27/98
- Quiz.
- Magic squares of order 4 (and some other even orders) can be constructed as follows: start by putting the numbers from 1 to 16 in order in a square:
One method involves drawing the two diagonals, and swap the numbers on each diagonal with their peripheral complement. That produces the following square:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1
In the other method, again draw the diagonals, but this time, take each number not on either diagonal, and subtract it from 17 (in general, subtract it from one more than the largest number in the square). That produces the following:
1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16
- We have not discussed the numerology of numbers from the Bible. There are two particularly famous New Testament examples.
- In John 21:1-14, particularly 9-11, there is a reference to the number 153. The Book of John normally speaks of numbers in round terms, so there is much written about the significance of this number. One appealing observation is that
153 = 1 + 2 + 3 + ... + 17 = 13 + 33 + 53.
- Surely the most famous number in the Bible is 666, "the Number of the Beast," from Revelations 13:18. One possible explanation is that it is a reference to Nero. In the Hebrew/Aramaic alphabet currently used at that time, Nero would be referred to as QSR (Caesar) NRWN (Neron). The letters were assigned numbers as follows:
and these numbers add to 666.
Q S R N R W N 100 60 200 50 200 6 50
Another interpretation uses the Greek name of Jesus, IHCOVC, which gets these numerical values:
which adds to 888. Since 7 is agreed to be a lucky number (in Western culture, anyway), this suggests that Jesus's name is 7+1 (in triplicate), whereas the Number of the Beast is 7-1 (in triplicate).
I H C O V C 10 8 200 70 400 200