text:
User:Ray Calvin Baker/HowToCount

 This is the HowToCount.txt file, created SAT 2011 JUL 09 04:44 PM, revised SAT 2011 JUL 09 04:44 PM, revised SAT 2011 JUL 15 07:43 PM, revised SUN 2011 NOV 27 06:09 PM.

FAQs:

Q. Do I know how to count? A. Actually, NO! NOBODY can count! However, I hope you will be interested to know what the problems are. Once upon a time, I thought I could count; but, now that I have considered some of the difficulties and complications of counting, I'm not so sure. I CAN count past a googol, and I think I know how to count past a googleplex, but I'm inclined to give up after that. It seems so repetitious! I definitely know that I will need some help to finish this article (Contributions from France and the UK on "numeration" are especially needed).

Q. How many kinds of counting are there? A. Good question! Most different ways depend a lot on WHY you are counting (or trying to). I'll try to list some of the various ways that I can think of.

(1) Tallying. I make a mark for each item I count. Then, there will be a one-to-one correspondence between items and marks. For each item, there is one mark. For each mark, there was one item. There is a funny thing about counting: whatever it is that I have done so far, eventually, I will need to do it again, "one more time".

(2) Say the names of the numbers. One, two, three, et cetera. One item, one number name. For each number name, there was one item. This brings up the problem of "numeration": how are the numbers to be named? I'll discuss this later. You'll see!

(3) Write down the digits of the number. 1, 2, 3, etc. Writing down something that I thought is usually better (for me!) than trying to remember something that I said. Writing down the digits of the numbers is the method I used to count to 2000, which is the basis for my discussion of small, medium and large numbers, below.

(4) Easy counting (my favorite). Arrange the things to be counted into patterns, then count the groups, and rows, and columns, and whatever you need to know about the pattern you have arranged. Usually, you will need only very small numbers to count columns and rows. I find calculation (multiply rows times columns, add in any "leftovers") is usually more accurate than trying to count every blessed item separately. (I am easily distracted, so I tend to "lose count".)

(5) Hard counting. This is counting items on a converor belt as they move past. Items move past, then go out of sight. Sometimes there are no items visible on the belt, but I need to remember "where I left off". Sometimes, items go past so fast that it is easy to "lose count". Often, there are noises and other distractions, too.

(6) I have scarecely begun! I just found a college-level course related to the subject of my essay.  Stanley, Richard. 18.S66 The Art of Counting, Spring 2003. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 15 May, 2012). License: Creative Commons BY-NC-SA

Ray Calvin Baker (talk) 14:51, 15 May 2012 (UTC)  A SIMPLE, MATHEMATICAL RECIPE FOR TALLYING

This material is based on information discussed in _The_Emperor's_New_Mind_, by Roger Penrose, and by many other sources, such as _The_Age_of_Spiritual_Machines_, by Raymond Kurzweil. Alan Turing was the first person to describe calculations in this way.

This is the description of a very simple counting machine. ,0"0"->0"0"R, 0"1"->1"1"R, 1"0"->0"1"STOP, 1"1"->1"1"R,

Each group of symbols between the commas represents (for instance): 0 the state the machine is in; "0" what the machine is reading from the tape; ->0 the state the machine goes into next,; "0" what the machine writes onto the tape; and R  which direction the machine moves.

We assume we start with an imfinite (or adequately long) tape containing the data: ....0001111000.... This is obviously a very simple representation of the number "four".

Let's attach our machine to the tape, and start it in state 0. ....00|0|1111000....     |state 0; reading "0"; starting calculation.

The machine description says, "stay in state 0, write "0", then move Right.

....000|1|111000....      |state 0; reading "1"

The machine description says, "change to state 1, write "1", then move Right.

....0001|1|11000....       |state 1; reading "1"

The machine description says, "stay in state 1, write "1", then move Right.

....00011|1|1000....        |state 1; reading "1"

The machine description says, "stay in state 1, write "1", then move Right.

....000111|1|000....         |state 1; reading "1"

The machine description says, "stay in state 1, write "1", then move Right.

....0001111|0|00....          |state 1; reading "0"

The machine description says, "change back to state 0, write "1", then (move Right) and STOP.

....00011111|0|00....           |state 0; reading "0"; STOPPED; calculation complete.

Our tally, which started at four, is now five. The machine has counted up by one.

Such a simple idea! Yet Alan Turing was able to demonstate that machines like this could calculate ANYTHING that can be calculated, if they had enough tape and enough time. (P. S. He also demonstrated that some things CANNOT be calculated!)

Furthermore, Alan was able to demonstrate that one special machine could process a suitably coded description of ANY other such machine, and the data to be given to that other machine, and produce exactly the same answer that the other machine would produce. This special machine is called, "The Universal Turing Machine". A coded description of this machine is given on pages 56 and 57 of _The_Emperor's_New_Mind_, or, if you prefer, in binary, on pages 71 through 73.

CONNECTIONS

Some of my personal favorites: The old classic movie, "Forbidden Planet" (not because it starred Robbie the Robot long before the silly "Lost in Space" TV series, but because the ancient, alien Krell civilization had left behind a science lab with lighted dials arranged by powers of ten (somewhat like an animated numeration table), to show power consumption. (The entire planet was a power supply for their "make almost anything" machine). Unfortunately, the Krell had forgotten how they became civilized, and even that they were civilized. They destroyed one another in jealous rages, fighting over the "make almost anything" machine.) Two connections: first, a favorite course at MIT is (according to things I have read), "How to Make Almost Anything", and second, MIT professor Marvin Minsky suggests that visual recognition, "this is a chair", and speech recognition, are very difficult to program in computers because most of us have forgotten how we learned these skills, and even that we learned them. Incidentally, the brand name on the player piano my mother's parents owned was, if I remember correctly, "Krell". Recommended listening: "Rachmaninoff Plays Rachmaninoff" -- a Russian composer (born 1873, performed about 1920, died 1943) plays his arrangement of "The Star Spangled Banner". Another favorite (toy) was a lever-operated mechanical counter, perhaps taken from a hay baling machine by my Uncle Leslie. This was a lot faster and more fun than waiting for the odometer on the family car to get up to a long row of 9's, then slowly turn over.

There are two kinds of numbers I should talk about here. CARDINAL numbers are used for counting -- "how many?" -- one, two three, etc. ORDINAL are used to put things into order -- first, second, third, etc.

EXTENSIONS BEYOND "NATURAL NUMBERS"

In the several thousand years that people have been thinking about mathematics, several other kinds of numbers have come into use. Most of these are beyond what I intend to talk about in this article, but they have found some use in solving unusual problems. I will list some of these for completeness, so you won't feel cheated. Don't try to fit ordinal numbers against these extensions beyond natural numbers. It won't work. Most of these extensions to number theory were found while trying to use algebra to solve equations, so make a note to yourselves, "I want to learn algebra!"

NATURAL NUMBERS are what I am talking about in this article. Some people argue, "Is zero a natural number?". Leaving out "zero" avoids some unpleasant problems for a little while, but mathematicians and computer scientists are very happy to include zero. May I ask you a question, please? How many pink unicorns are dancing about the ceiling above your head right now? Please look now, and count carefully. However, if your count is other than "zero", take pictures! Ask your friends to verify your count! Call the news reporters! and the TV stations!

INTEGERS were invented for subtraction problems that don't quite work in the usual way. For example, "5 - 3 = 2", but "3 - 5 = ?". Using the newly invented negative numbers, "3 - 5 = -2". I mentioned algebra and equations above, so I'll show you the equations and their solutions here. If you notice a pattern, good for you! What you need to know: "you have solved the equation when you know what 'x' is." First equation: "5 - x = 3". First solution: "x = 2" You probably solved this problem.

Second equation: "5 + x = 3". Much trickier! NO natural number added to 5 makes a natural number smaller than 5! Second solution: "x = -2".

FRACTIONS were invented for division problems that don't quite work in the usual way. For example, "6 / 3 = 2" but "3 / 6 = ?". "3 / 6 = 1 / 2 (one half, a fraction)". NOTE" many computer programming languages use "*" as the sign for "multiply". First equation: "3 * x = 6 (This is 'three times what number makes 6?')". First solution: "x = 2". Maybe you solved this problem, maybe you didn't understand it yet.

Second equation: "6 * x = 3". Second solution: "x = 1/2".

UNSOLVABLE PROBLEM: Find a fraction, 'x', which when multiplied by itself yields 2. This problem introduces REAL NUMBERS and the concept of limits. You may (must) use your calculators here. 1 * 1 = 1 < 2 (FYI: for "< 2", you can say, "is less than 2"); 1.4 * 1.4 = 1.96 < 2; 1.41 * 1.41 = 1.9681 < 2; 1.4142 * 1.4142 = 1.99996164 < 2; 1.41421 * 1.41421 = 1.9999899241 < 2; 1.414213 * 1.414213 = 1.999998409369 < 2; 1.4142135 * 1.4142135 = 1.99999982358225 < 2. To see where all this is going, enter 2 into your calculator. Then, press the "square root" key. You should see something close to 1.4142135 in the display. You can get closer and closer to 2, maybe within the range where roundoff error inside the calculator makes it SEEM like you got 2 exactly, but there is NO fraction which, multiplied by itself, yields 2, EXACTLY.

Then problems were found with square roots. Another calculator exercise: enter 1. Press the "change sign" key. Your calculator should display "-1". Now press the "square root" key. My COMPUTER says, "Invalid input for function", and it's more sophisticated than most mere calculators. First equation: "x * x = 4". First solution: "x = 2".

Second equation: "x * x = -1". Second solution: most likely, "calculator ERROR". However, mathematicians were too resourceful to accept that answer forever, so eventually, most of them agreed to say that "i" is "the square root of -1". Newly invented solution, "x = i", an IMAGINARY NUMBER. Physicists and electrical engineers began finding that "i" and its friends, the COMPLEX NUMBERS, are very useful indeed. NOTE: electrical engineers usually use "i" for current or amperage, so THEY use "j" to represent "the square root of -1".

One final concept: VECTORS. Use GROUPS of numbers! This way, you can add apples, bananas, and cabbages and get lunch! Example: a VECTOR could be represented as (Apples, Bananas, Cabbages). Then if you have 2 apples (2, 0, 0), 3 bananas (0, 3, 0), and 7 cabbages (0, 0, 7), lunch = (2, 0, 0) + (0, 3, 0) + (0, 0, 7) = (2, 3, 7). (I'm sorry that you don't like cabbage so very much.)

You are welcome to extend things even further, if you think you can find a useful (or beautiful, or convenient) reason for doing so. Now, back to counting numbers.

A SECOND OPINION

Some of you readers may think that I'm just making up stuff as I go along. Not true! But perhaps you would like a differnt perspective -- a different point of view. Here are three brief quotations from a famous Computer Science author -- Douglas Hofstadter, in his book, _Metamagical_Themas_, Chapter 6, "On Number Numbness" (May, 1982).

First, he comments that our American system of numeration differs from that of many "countries" around the world.

In the United States, this last number [1,000,000,000,000]], with its twelve zeros, is called a "trillion"; in most other countries, it is called a "billion". People in those countries reserve ""trillion" for the truly enormous number 1,000,000,000,000,000,000 -- to us a "quintillion" -- though hardly anyone knows that term. (page 118)

I enjoy his very clear description of what I think of as "the central problem of counting".

If, perchance, you were to start dealing with numbers having millions or billions of digits [considering the storage capacity of a typical CD-ROM forces this upon you -- RCB], the numerals themselves (the colossal strings of digits) woulld cease to be visualizable, and your perceptual reality would be forced to take another leap upward in abstraction -- to the number that counts the digits in the number that counts the digits in the number that counts the objects concerned. Needless to say, such third-order perceptual reality is highly abstract. Moreover, it occurs very seldom, even in mathematics. Still, you can imagine going far beyond it. Fourth- and fifth-order perceptual realities would quickly yield, in our purely abstract imagination, to tenth-, hundredth-, and millionth-order percptual realities.

By this time, of course, we would have lost track of the exact number of levels we would have shifted, and we would be content with a mere estimate of that number (accurate within ten percent, of course). "Oh, I'd say about two million levels of perceptual shift were involved here, give or take a couple of hundred thousand" would be a typical comment for someone dealing with such unimaginably unimaginable quantities. (pages 124, 125)

Finally, here's what Douglas says about number names (I think he's being more silly than I am).

One place where we think logarithmically is number names. We in America have a new name every three zeros (up to a certain point), from "thousand", to "million", to "billion", to "trillion"....

In any case, we seem to run out of number names at about a trillion. To be sure, there are some official names for bigger numbers, but they are about as familiar as the names of extinct dinosaurs: "quadrillion", "octillion", "vigintillion", "brontosillion", "triceratillion", and so on. We are simply not familiar with them, since they died off a dinosillion years ago. (page 127)

NUMERATION

Q. You mentioned "numeration". What is that? A. Numeration is the art and practice of giving names to numbers. Very large dictionaries list a "numeration table". First problem with counting: the "There Are Always Lots More Where That Number Came From" theorem. For any natural number n greater than zero, there are always at least n^n (pronounced "n to the nth power") larger numbers. For n=1, 1^1 = 1, and 2 is 1 larger number. For n=2, 2^2 = 4, and 3, 4, 5, and 6 are the 4 larger numbers. For n-3, 3^3 = 3*3*3 = 27, and 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30, are the next 27 larger numbers.

But a dictionary can only hold some definite number of names for numbers, so eventually, we will run out of names for numbers. Sometimes, "words fail me".

Second problem with counting: How far do the names go? The numeration tables, as I remember them from an old dictionary, go something like these: TABLE ONE		TABLE TWO		TABLE THREE (the BIG, important table) Getting started		on to one hundred	name		number of (placeholder) zeroes	my index 0	zero		21	twenty-one	thousand	3	as in 1000		0 1	one		22	twenty-two	million		6	as in 1,000,000		1 "monoplane" 2	two			etc. billion		9	as in 1,000,000,000	2 "bicycle" 3	three		30	thirty		trillion	12				3 "tricycle" 4	four		31	thirty-one	quadrillion	15				4 "quadruped" 5	five			etc. quintillion	18				5 "quintuplets" 6	siz		40	forty		sextillion	21				6 "sextet" 7	seven			etc. septillion	24				7 "September" 8	eight		50	fifty		octillion	27				8 "octagon" 9	nine			etc. nonillion	30				9 "November" 10	ten		60	sixty		decillion	33				10 "December" 11	eleven			etc. undecillion	36				11 "uno, unit + decem = 10" 12	twelve		70	seventy		duodecillion	39				12 "duo + decem" 13	thirteen		etc. tridecillion	42				13 "tri + decem" 14	fourteen	80	eighty		quadrodecillion	45				14 "quad + decem" 15	fifteen			etc. quindecillion	48				15 "quin + decem" 16	sixteen		90	ninety		sexdecillion	51				16 "sex + decem" 17	seventeen		etc. septdecillion	54				17 "sept + decem" 18	eighteen	100	one hundred	octodecillion	57				18 "octo + decem" 19	nineteen	101	one hundred one	nondecillion	60				19 "non- + decem" 20	twenty			etc. vigintillion	63				20 "viginti"

centillion	303				100 "100 cents per $"

International students may recognize that I am using the "American" system of numeration. It is very restricted compared to the British and German system, but it's the only one I understand, until some British or German student can teach me THEIR system. The Americans and French were allies at the time of the American revolution, so it is somewhat natural that the American system differs from the British system. However, a footnote in a more recent dictionary suggests that the French may have alligned their system recently to be more like the German system. The US of A lags behind the rest of the world in NOT adopting the Metric system, and now, again, in retaining a restricted numeration system.

WARNING: I am trying to reconstruct this table from memory, so names and spelling may not be quite correct.

To TABLE THREE, I have appended the "my index" column. This includes the numbers 0 through 20, and 100, along with a word or phrase I use to help remember the number names. Note that the ordering of names of months have been altered by Roman emperors adding "July" (for Julius Caesar) and "August" (for Caesar Augustus). September was once the seventh month, November was once the ninth month, and December was once the tenth month. Is there evidence here of a conspiracy to make counting difficult? IMHO, probably.

I also added the gap between vigintillion and centillion. This is only my opinion, but I think the point of this table is to recognize that numbers go on "forever" even though, in effect, we only have names for numbers index 0 through index 20. Multiply "my index" times three, then add three. This gives the number of (placeholder) zeroes in the named number.

Q. What is your favorite number? A. After all this discussion, surely it should be "My Favorite Number", MFN = 999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999. 20 19  18  17  16  15  14  13  12  11  10   9   8   7   6   5   4   3   2   1   0 (I have taken the liberty of writing "my index" numbers under this, so I can say it is nine hundred ninety-nine vigintillion, nine hundred ninety-nine nondecillion, nine hundred ninety-nine octodecillion, nine hundred ninety-nine septdecillion, nine hundred ninety-nine sexdecillion, nine hundred ninety-nine quindecillion, nine hundred ninety-nine quadrodecillion, nine hundred ninety-nine nine tridecillion, hundred ninety-nine duodecillion, nine hundred ninety-nine undecillion, nine hundred ninety-nine decillion, nine hundred ninety-nine nonillion, nine hundred ninety-nine octillion, nine hundred ninety-nine septillion, nine hundred ninety-nine sextillion, nine hundred ninety-nine quintillion, nine hundred ninety-nine quadrillion, nine hundred ninety-nine trillion, nine hundred ninety-nine billion, nine hundred ninety-nine million, nine hundred ninety-nine thousand, nine hundred ninety-nine. Time yourself -- how long does it take you to say that 10 times? Prediction: counting gets slower as the numbers get larger.

A very serious problem is obvious -- I DON'T KNOW the name of the next number! It's not in the numeration tables! But I won't give up! Although I can't NAME it, I can DESCRIBE it. It is 10 to the sixty-sixth power (1 followed by 66 zeroes). (This is still quite a bit short of a googol.)

Also, you probably think that counting gets very boring after a while. I have to agree; there are so many numbers, that, indeed, counting can get boring! To liven things up, I invented small, medium and large numbers.

SMALL, MEDIUM, AND LARGE NUMBERS

Small numbers are numbers so small that an industrious second-grader can actually count to them. Decades ago, I was such a second-grader. Using all my recess periods during a week, I wrote down all the numbers from 1 to 2000. (This is type (3) counting, as described in the FAQ's section, above.) Thus, by my definition, 2000 is the largest small number.

Medium-sized numbers are numbers larger than small numbers, that have only a small number of digits. Thus, the largest medium-sized number consists of 2000 9's. Medium-sized numbers are numbers so small that you can actually write them down, in about a week of recess periods, if you care enough to do so.

Large numbers are numbers too large to be conveniently written down. Needless to say, most numbers are large numbers. (Proof of this statement is left to the industrious student.)

The numeration table from the big dictionary suggests a fourth class of numbers -- sporadically named numbers. One centillion is one of these. The dictionary also has entries for "googol" and "googolplex". A googol is written as 1 followed by 100 zeroes. Thus, a googol has only 101 digits, and is a medium-sized number. I could write it out, but I choose not to. Maybe later....

A googolplex, on the other hand, is a VERY large number! It is 1 followed by a googol of zeroes. No human could live long enough to count out the googol of zeroes, so a googolplex is much too large to ever write out. And it would require an enormous amount of paper and ink even to try that!

HOW TO COUNT PAST A GOOGOL

But can a person count PAST a googol? I think so. I can count PAST 100 like this: "99; 100; 101".

So, I can count PAST a googol like this:

first, I need to write down 100 9's, with commas in the right places to separate the digits into groups of three digits. This is the largest number with exactly 100 digits, the number before 1 googol. Next, I write the separating semicolon.

Now, write 1 followed 100 zeroes, putting commas in places as needed. This is the smallest number with 101 digits, one googol.

Write the separating semicolon.

Finally, write 1, followed by 99 zeroes, followed by 1. put in commas as needed. This is the number after one googol; it, too, has exactly 101 digits. Now, I could rush ahead, follow my recipe, and count past a googol. But, if I hold back, perhaps YOU could count past a googol before I show you how it's done (not the recipe, but the actual writing down of the digits and punctuation marks).

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) 999,999,999,999,999,999,999,999,999,999,999,999,999, 999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999, 999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999, 999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999; (3 * 20 = 60 digits in each complete row; 3 * 13 digits = 39 digits in the incomplete row. 60 + 39 = 99 digits.)

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000; (one googol, written out.)

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,001; (the number AFTER one googol.)

(The numbers in parentheses are guides, which I used to be as certain as possible that I did indeed write the numbers down correctly.)

DESCRIPTIONS INSTEAD OF NAMES

We have seen that the numeration table simply does not go far enough. Indeed, it cannot! We need an infinite number of names for an infinite number of numbers! But let's look again at the numeration table, in hopes of finding some regularities that may help us make useful DESCRIPTIONS of numbers, in place of the names we do not have. Here is a copy of TABLE THREE again, with some different comments and descriptions:

TABLE THREE (the BIG, important table) name		number of (placeholder) zeroes	a descriptive phrase which can replace the name thousand	3	as in 1000		"times 10 to the 3rd power plus" million		6	as in 1,000,000		"times 10 to the 6th power plus" billion		9	as in 1,000,000,000	"times 10 to the 9th power plus" trillion	12				"times 10 to the 12th power plus" quadrillion	15				"times 10 to the 15th power plus" quintillion	18				"times 10 to the 18th power plus" sextillion	21				"times 10 to the 21st power plus" septillion	24				"times 10 to the 24th power plus" octillion	27				"times 10 to the 27th power plus" nonillion	30				"times 10 to the 30th power plus" decillion	33				"times 10 to the 33rd power plus" undecillion	36				"times 10 to the 36th power plus" duodecillion	39				"times 10 to the 39th power plus" tridecillion	42				"times 10 to the 42nd power plus" quadrodecillion	45				"times 10 to the 45th power plus" quindecillion	48				"times 10 to the 48th power plus" sexdecillion	51				"times 10 to the 51st power plus" septdecillion	54				"times 10 to the 54th power plus" octodecillion	57				"times 10 to the 57th power plus" nondecillion	60				"times 10 to the 60th power plus" vigintillion	63				"times 10 to the 63rd power plus"

centillion	303				"times 10 to the 303rd power plus"

The "descriptive phrases" look fairly regular, except for "first" = "st", "second" = "nd", and "third" = "rd". Between "the" and "power", the numbers go up in increments of 3: 3, 6, 9, 12, etc. I think we can use this. But notice: we are using numbers in our "descriptive phrases". I predict problems ahead!

I propose to use the "descriptive phrases", because they obviously "keep going on and on", even after the names (e. g., "vigintillion") give out. Words have failed me! What can I say after "vigintillion"? There is no other consecutive listing in the large, unabridged dictionary.

Our next "landmark", a "sporadically named" number, is one centillion.

HOW TO COUNT PAST ONE CENTILLION

We have already seen how to count past one googol, so this should be easy!

The number before one centillion is: nine hundred ninety-nine times ten to the 300th power plus nine hundred ninety-nine times ten to the 297th power plus nine hundred ninety-nine times ten to the 294th power plus nine hundred ninety-nine times ten to the 291st power plus nine hundred ninety-nine times ten to the 288th power plus nine hundred ninety-nine times ten to the 285th power plus nine hundred ninety-nine times ten to the 282nd power plus nine hundred ninety-nine times ten to the 279th power plus nine hundred ninety-nine times ten to the 276th power plus nine hundred ninety-nine times ten to the 273rd power plus nine hundred ninety-nine times ten to the 270th power plus nine hundred ninety-nine times ten to the 267th power plus nine hundred ninety-nine times ten to the 264th power plus nine hundred ninety-nine times ten to the 261st power plus nine hundred ninety-nine times ten to the 258th power plus nine hundred ninety-nine times ten to the 255th power plus nine hundred ninety-nine times ten to the 252nd power plus nine hundred ninety-nine times ten to the 249th power plus nine hundred ninety-nine times ten to the 246th power plus nine hundred ninety-nine times ten to the 243rd power plus nine hundred ninety-nine times ten to the 240th power plus nine hundred ninety-nine times ten to the 237th power plus nine hundred ninety-nine times ten to the 234th power plus nine hundred ninety-nine times ten to the 231st power plus nine hundred ninety-nine times ten to the 228th power plus nine hundred ninety-nine times ten to the 225th power plus nine hundred ninety-nine times ten to the 222nd power plus nine hundred ninety-nine times ten to the 219th power plus nine hundred ninety-nine times ten to the 216th power plus nine hundred ninety-nine times ten to the 213rd power plus nine hundred ninety-nine times ten to the 210th power plus nine hundred ninety-nine times ten to the 207th power plus nine hundred ninety-nine times ten to the 204th power plus nine hundred ninety-nine times ten to the 201st power plus nine hundred ninety-nine times ten to the 198th power plus nine hundred ninety-nine times ten to the 195th power plus nine hundred ninety-nine times ten to the 192nd power plus nine hundred ninety-nine times ten to the 189th power plus nine hundred ninety-nine times ten to the 186th power plus nine hundred ninety-nine times ten to the 183rd power plus nine hundred ninety-nine times ten to the 180th power plus nine hundred ninety-nine times ten to the 177th power plus nine hundred ninety-nine times ten to the 174th power plus nine hundred ninety-nine times ten to the 171st power plus nine hundred ninety-nine times ten to the 168th power plus nine hundred ninety-nine times ten to the 165th power plus nine hundred ninety-nine times ten to the 162nd power plus nine hundred ninety-nine times ten to the 159th power plus nine hundred ninety-nine times ten to the 156th power plus nine hundred ninety-nine times ten to the 153rd power plus nine hundred ninety-nine times ten to the 150th power plus nine hundred ninety-nine times ten to the 147th power plus nine hundred ninety-nine times ten to the 144th power plus nine hundred ninety-nine times ten to the 141st power plus nine hundred ninety-nine times ten to the 138th power plus nine hundred ninety-nine times ten to the 135th power plus nine hundred ninety-nine times ten to the 132nd power plus nine hundred ninety-nine times ten to the 129th power plus nine hundred ninety-nine times ten to the 126th power plus nine hundred ninety-nine times ten to the 123rd power plus nine hundred ninety-nine times ten to the 120th power plus nine hundred ninety-nine times ten to the 117th power plus nine hundred ninety-nine times ten to the 114th power plus nine hundred ninety-nine times ten to the 111st power plus nine hundred ninety-nine times ten to the 108th power plus nine hundred ninety-nine times ten to the 105th power plus nine hundred ninety-nine times ten to the 102nd power plus nine hundred ninety-nine times ten to the 99th power plus nine hundred ninety-nine times ten to the 96th power plus nine hundred ninety-nine times ten to the 93rd power plus nine hundred ninety-nine times ten to the 90th power plus nine hundred ninety-nine times ten to the 87th power plus nine hundred ninety-nine times ten to the 84th power plus nine hundred ninety-nine times ten to the 81st power plus nine hundred ninety-nine times ten to the 78th power plus nine hundred ninety-nine times ten to the 75th power plus nine hundred ninety-nine times ten to the 72nd power plus nine hundred ninety-nine times ten to the 69th power plus nine hundred ninety-nine times ten to the 66th power plus MFN.

One centillion is: one times ten to the 303rd power.

The number after one centillion is: one times ten to the 303rd power, plus one. Wasn't that fun!

I still predict serious problems ahead! Like, when we get into the neighborhood of "... times 10 to the 2001st power plus ...".

2001 is a medium-sized number. Eventually, medium-sized numbers get too large to write down conveniently! Is there a work-around? Yes, but you are not going to like it.

HOW TO COUNT PAST A MEDIUM-SIZED NUMBER

Let's try to count past 999 times 10 to the 1998th power plus 999 times 10 to the 1995th power plus 999 times 10 to the 1995th power plus 999 times 10 to the 1992nd power plus 999 times 10 to the 1989th power (I don't think I'm going to be able to finish writing this out! So I'll let you imagine the rest until we get to....) plus 999 times 10 to the 66th power plus MFN. (I'm glad MFN is conveniently available!)

The next number is: 1 times 10 to the (2 times 10 to the 3rd power plus 1)th power. (That's how to take care of 2001, the medium-sized number!)

The next number is: 1 times 10 to the (2 times 10 to the 3rd power plus 1)th power plus 1. (Gasp! we did (most of) it!)

Eventually, we will need to nest numbers to infinite depth, so I will just now admit, "I can't count!". Well, maybe if I keep trying, I can get past one googlplex; but, please! no more! Douglas Hofstadter expressed it so well in "A SECOND OPINION", above: "Oh, I'd say about two million levels of perceptual shift were involved here, give or take a couple of hundred thousand" would be a typical comment for someone dealing with such unimaginably unimaginable quantities.

HOW TO COUNT PAST ONE GOOGLEPLEX

NOTE: I do not want you to be disappointed here. Counting past a googolplex is simply not humanly possible, within the limits of the physical universe (as we currently think we understand it), and within the limits of human civilization (as we currently think we understand it). But I think we can try to reassure ourselves that we do understand (in principle) how we COULD count past a googolplex, were it not for the inconvenience of not having googols of years in which to complete the project, and not having googols of particles (atoms, electrons, quarks, whatever) upon which to write the digits.

I used a trick when I dealt with a googol -- I just showed how to count PAST a googol. That way, I only needed to account for one hundred digits, one hundred one digits, and one hundred one digits. Unfortunately, that trick will not suffice for any attempt to count past a googolplex. We will need to account for ALL of the digits -- (first) one googol of nines, (second) the digit one, followed by a googol of zeros, and (third) the digit one, followed by (one less that a googol) of zeros, followed by the digit one. Just to count PAST a googolplex, we would need to account for three googol plus two digits. It's simply not HUMANLY possible!

First, we need a googol of 9s.

Remember, earlier we only counted PAST a googol. We realized then that it would take much too long to count TO a googol. But now, to count past a googolplex, we will need to count out at least a googol of digits, and we will need to do that at least three times.

Second, we need 1 followed by a googol of 0s.

Finally, we need 1 followed by (one googol less one) zeros, followed by 1.

First, let's remember that one googol is 1 followed by 100 0s. This would be 10 followed by 99 0s. So 9 followed by 99 9s would be the number one less than a googol. Think of this as 99 followed by 33 groups of 3 9s. We need this for our exponent. Reasoning by analogy, one googolplex less one would be 9 followed by (approximately) one third of a googol of groups of three 9s. Good for you if you anticipate that we will be trying to count backwards from one googol before very long. I don't expect to be able to actually do that -- just show you how it would be done if we had enough time and patience.

The number before one googolplex is: nine times ten to the ([The number one less than a googol begins here] nine times ten to the 99th power plus nine hundred ninety-nine     times ten to the 96th power plus nine hundred ninety-nine      times ten to the 93rd power plus nine hundred ninety-nine      times ten to the 90th power plus nine hundred ninety-nine      times ten to the 87th power plus nine hundred ninety-nine      times ten to the 84th power plus nine hundred ninety-nine      times ten to the 81st power plus nine hundred ninety-nine      times ten to the 78th power plus nine hundred ninety-nine      times ten to the 75th power plus nine hundred ninety-nine      times ten to the 72nd power plus nine hundred ninety-nine      times ten to the 69th power plus nine hundred ninety-nine      times ten to the 66th power plus MFN) power plus 999 times ten to the ([The number three less than a googol begins here] nine times ten to the 99th power plus nine hundred ninety-nine      times ten to the 96th power plus nine hundred ninety-nine      times ten to the 93rd power plus nine hundred ninety-nine      times ten to the 90th power plus nine hundred ninety-nine      times ten to the 87th power plus nine hundred ninety-nine      times ten to the 84th power plus nine hundred ninety-nine      times ten to the 81st power plus nine hundred ninety-nine      times ten to the 78th power plus nine hundred ninety-nine      times ten to the 75th power plus nine hundred ninety-nine      times ten to the 72nd power plus nine hundred ninety-nine      times ten to the 69th power plus nine hundred ninety-nine      times ten to the 66th power plus [MFN won't work here -- we need three less than MFN.]                                       nine hundred ninety-nine times ten to the 63rd power plus nine hundred ninety-nine times ten to the 60th power plus nine hundred ninety-nine times ten to the 57th power plus nine hundred ninety-nine times ten to the 54th power plus nine hundred ninety-nine times ten to the 51st power plus nine hundred ninety-nine times ten to the 48th power plus nine hundred ninety-nine times ten to the 45th power plus nine hundred ninety-nine times ten to the 42nd power plus nine hundred ninety-nine times ten to the 39th power plus nine hundred ninety-nine times ten to the 36th power plus nine hundred ninety-nine times ten to the 33rd power plus nine hundred ninety-nine times ten to the 30th power plus nine hundred ninety-nine times ten to the 27th power plus nine hundred ninety-nine times ten to the 24th power plus nine hundred ninety-nine times ten to the 21st power plus nine hundred ninety-nine times ten to the 18th power plus nine hundred ninety-nine times ten to the 15th power plus nine hundred ninety-nine times ten to the 12th power plus nine hundred ninety-nine times ten to the 9th power plus nine hundred ninety-nine times ten to the 6th power plus nine hundred ninety-nine times ten to the 3rd power plus nine hundred ninety-seven) power plus 999 times ten to the ( [The number six less than a googol begins here] 9 times ten to the 99th power plus nine hundred ninety-nine times ten to the 96th power plus nine hundred ninety-nine times ten to the 93rd power plus nine hundred ninety-nine times ten to the 90th power plus nine hundred ninety-nine times ten to the 87th power plus nine hundred ninety-nine times ten to the 84th power plus nine hundred ninety-nine times ten to the 81st power plus nine hundred ninety-nine times ten to the 78th power plus nine hundred ninety-nine times ten to the 75th power plus nine hundred ninety-nine times ten to the 72nd power plus nine hundred ninety-nine times ten to the 69th power plus nine hundred ninety-nine times ten to the 66th power plus nine hundred ninety-nine times ten to the 63rd power plus nine hundred ninety-nine times ten to the 60th power plus nine hundred ninety-nine times ten to the 57th power plus nine hundred ninety-nine times ten to the 54th power plus nine hundred ninety-nine times ten to the 51st power plus nine hundred ninety-nine times ten to the 48th power plus nine hundred ninety-nine times ten to the 45th power plus nine hundred ninety-nine times ten to the 42nd power plus nine hundred ninety-nine times ten to the 39th power plus nine hundred ninety-nine times ten to the 36th power plus nine hundred ninety-nine times ten to the 33rd power plus nine hundred ninety-nine times ten to the 30th power plus nine hundred ninety-nine times ten to the 27th power plus nine hundred ninety-nine times ten to the 24th power plus nine hundred ninety-nine times ten to the 21st power plus nine hundred ninety-nine times ten to the 18th power plus nine hundred ninety-nine times ten to the 15th power plus nine hundred ninety-nine times ten to the 12th power plus nine hundred ninety-nine times ten to the 9th power plus nine hundred ninety-nine times ten to the 6th power plus nine hundred ninety-nine times ten to the 3rd power plus nine hundred ninety-four) power plus 999  times ten to the ( [The number nine less than a googol begins here] 9   times ten to the 99th power plus nine hundred ninety-nine times ten to the 96th power plus nine hundred ninety-nine times ten to the 93rd power plus nine hundred ninety-nine times ten to the 90th power plus nine hundred ninety-nine times ten to the 87th power plus nine hundred ninety-nine times ten to the 84th power plus nine hundred ninety-nine times ten to the 81st power plus nine hundred ninety-nine times ten to the 78th power plus nine hundred ninety-nine times ten to the 75th power plus nine hundred ninety-nine times ten to the 72nd power plus nine hundred ninety-nine times ten to the 69th power plus nine hundred ninety-nine times ten to the 66th power plus nine hundred ninety-nine times ten to the 63rd power plus nine hundred ninety-nine times ten to the 60th power plus nine hundred ninety-nine times ten to the 57th power plus nine hundred ninety-nine times ten to the 54th power plus nine hundred ninety-nine times ten to the 51st power plus nine hundred ninety-nine times ten to the 48th power plus nine hundred ninety-nine times ten to the 45th power plus nine hundred ninety-nine times ten to the 42nd power plus nine hundred ninety-nine times ten to the 39th power plus nine hundred ninety-nine times ten to the 36th power plus nine hundred ninety-nine times ten to the 33rd power plus nine hundred ninety-nine times ten to the 30th power plus nine hundred ninety-nine times ten to the 27th power plus nine hundred ninety-nine times ten to the 24th power plus nine hundred ninety-nine times ten to the 21st power plus nine hundred ninety-nine times ten to the 18th power plus nine hundred ninety-nine times ten to the 15th power plus nine hundred ninety-nine times ten to the 12th power plus nine hundred ninety-nine times ten to the 9th power plus nine hundred ninety-nine times ten to the 6th power plus nine hundred ninety-nine times ten to the 3rd power plus nine hundred ninety-one) power plus 999 times ten to the (

[We have a VERY, VERY, long way to go here. I hope you see the pattern]

) power plus nine hundred ninety-nine     times ten to the  6th power plus nine hundred ninety-nine      times ten to the  3rd power plus nine hundred ninety-nine; [the long-awaited end of the number before one googolplex]

one times ten to the one googol power;

one times ten to the one googol power plus one.

That's as much as I care to write down of counting past a googolplex! I give up. No more! I quit. I admit it, I can't count!

Oh, just one final reminder: There are at least (one googolplex to the (one googolplex) power) natural numbers larger than one googolplex.

The end.

title: User:Ray Calvin Baker/HowToCount


text:
User:Ray Calvin Baker/StarTrekvsStarWars

 This is the "StarTrekVsStarWars.txt" file, created WED 2011 AUG 10 04:45 PM, revised SUN 2011 NOV 27 06:24 PM.

Back in 1960, when I was a freshman in college, Tomlinson Hall at Case Institute of Technology was always crowded, standing room only, on Thursday nights at 7:00 PM. The original Star Trek series was playing to a crowd of engineering students.

Much as I appreciated Gene Roddenberry's science fiction creation, I have long felt that he failed to address properly two issues relevant to twentieth century mathematics. I offer these two "episodes" to correct these omissions. Of course, I am not the writer Gene was, so some Trekkies may take delight in "improving" upon my stilted dialog. But I do intend to lay out the basic story for these "missing" episodes.

1. Mister Spock Does the Logical Thing  RCB_LogicalThing  2. The Court-Martial of Captain Jean-Luc Picard  RCB_PicardsCourtMartial

I found some material in the sandbox that I thought was interesting. If I can retrieve some of this from the history files, it might provide a forum for Star Wars fans, too. Isn't creative American literature a reasonable course for University study?

3. Star Wars Fans and Star Trek Fans Continue the Story (or Stories)

The end.

 P. S. I found out why the links to the "missing" Star Trek episodes did not seem to work, and fixed the problem. I had attempted to use the links within a "pre-formatted section. Once I put in a few more HTML tags, I got the links to work.

Ray Calvin Baker 01:22, 2 December 2011 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
User:Ray Calvin Baker/LogicalThing

 This is the "LogicalThing.txt" file, created WED 2011 AUG 10 05:05 PM, revised WED 2011 AUG 10 05:05 PM.

Mister Spock Does the Logical Thing

As I begin this story, I do not know how it will end. I will emulate Mister Spock's logical decision, just as he did it.

Renegade is a science outpost established to study the nearby Von Neuman nebula, a region filled with a variety of unstable stars. This makes it both essential for study, and difficult to approach. Furhermore, as an outpost it has strategic value both for the Federation and for the Romulan Empire. It has not enough strategic value that either side wants to risk all-out war, but the Federation needs to keep sending supply ships, which the Romulans have been attempting to destroy. In fact, each of the last six Federation supply ships has been destroyed, and the scientists at Renegade are in a desparate situation. Hence the current mission of the USS Enterprise.

Science officer Spock has been studying the reports from all successful resupply missions, and the recovered message buoys from all but two of the failed missions. He concludes that there is only one safe way to approach the Renegade station (because of the proximity of unstable stars). He has established that five basic scenarios have been played out, as Federation ships attempt to resupply Renegade. 1. Federation ship approaches in "silent running" mode, finds Romulans waiting in ambush, fires torpedoes, and completes its resupply mission successfully. 2. Federation ship approaches in "silent running" mode, finds Romulans waiting with all weapons at full power, fires torpedoes, but is destroyed by Romulan phaser fire. 3. Federation ship approaches with all weapons at full power. The Romulans see them coming, and lay out a spread of torpedoes. One of these destroys the Federation ship. 4. Federation ship approaches with all weapons at full power, and finds Romulans waiting with all their weapons at full power. The Federation ship gets of the first shot, destroys the Romulan blockade, and resupplies the Renegade base. 5. Any other tactics lead to the destruction of the Federation ship.

"So, you mean to tell me that we have only two choices, "running silent", or "all weapons at full power"? And there's a 50-50 chance we will all be killed? And we'll have only a few seconds to fire if we find the Romulans first?"

"Yes, Captain. That is the situation as I see it."

"Well, Spock, I think the thing for me to do, is: give you full command of the ship so you will have every tactical advantage possible, then, trust your Vulcan logic to 'do the logical thing.'"

"Thank you for your confidence, Captain, But my notes do not indicate any pattern to the Romulan strategy. It seems completly random to me."

"Check your notes again. We're all counting on you to find something."

Spock returns to his science station and begins to review his notes.

"Hmm. This is interesting. The nearby nebula was named after John Von Neuman, who invented something called 'game theory' around the year 1940. Ah! here's something ... 'Theory of Games' ... 'zero-sum games' ... ... payoff matrices ... optimal strategies. Here's a strange strategy for a symmetric, zero-sum game. Yes! the payoff matrix for our situation looks like this:"

\  Federation choices:      | \ "silent     | weapons    | R \ running"    | powered    | O *++ M a|1. Romulan |3.Enterprise| U m| ship      | destroyed; | L b| destroyed; |           | A u| Resupply  | Resupply   | N s| mission   | mission    | h| sucessful | fails      | C +++ H p|2.Enterprise|4.Romulan  | O o| destroyed; | ship      | I w|           | destroyed; | C e| Resupply  | Resupply   | E r| mission   | mission    | S e| fails     | sucessful  | d|           |            | ===+++

"Attention all hands! Mister Spock has command of the Enterprise, until we're all killer, or until we dock at Renegade station in two hours. He says there is a 50% probability for either occurrence. He says he has found the optimum strategy in the archives. I am preparing a message buoy for launch in case the Enterprise is destroyed, so future Federation ships may learn from our experience. If you want to leave any messages for your next-of-kin, use your personal logs, and patch them through to Communications. Mister Spock says he is ready to make his tactical decision in fifteen minutes. I trust he will do the logical thing."

Mister Spock jots down a note: "HT silent running; TH full power". Then he begins talking to himself, "Heads, heads, Inconclusive. Tails, tails, Inconclusive. Tails, heads, full power." He turns on the intercomm, "Attention all hands. This is Mister Spock taking command of the Enterprise, per Captain Kirk's orders. Full power to shields. Full power to all phasor banks. Prepare to fire all phasors as soon as I verify targeting. Course laid in. Steady as she goes."

Spock's targeting of the Romulan vessel is precise. He verifies targeting within two seconds. The Romulan's shields hold for a few seconds, but give way before the Romulan's can return fire. Seconds later, the Romulan ship explodes in a devastating matter-antimatter core breach. The Enterprise shields hold firm, and the inertial dampers buffer the incoming shock wave.

"This is Mister Spock commanding USS Enterprise. Requesting permission to dock."

"Permission granted!" (Wildly enthusiastic cheers in the background.) Boy, are we glad to see you!"

Two hours and ten seconds after he made is original ETA, Spock brings the Enterprise in for a smooth docking. Within minutes, supplies are being unloaded into the Rengade station's storage areas.

"Mister Spock, did I see you tossing a coin while you were working there at the Science station?"

"Yes, Captain, you did. You had asked me to do 'the logical thing,' so I did."

Mister Spock had been concerned that the coin he tossed might be biased, so he took a sensible precaution. This requires some simple calculations to explain. Let's assume a biased coin comes up "heads" 60% of the time. "Heads, heads, probability .6 * .6 = .36, inconclusive." "Tails, tails, probability .4 * .4 = .16, inconclusive." "Heads, tails, probability .6 * .4 = .24, proceed with 'silent running'", but this is not what happened. "Tails, heads, probability .4 * .6 = .24, proceed with all weapons at full power" Each of these last two cases has exactly the same probability, .24, so this eliminates the effect of a biased coin, leaving only the desired 50-50 result.

Spock and the crew of the Enterprise were lucky. But Spock did "the logical thing" under the circumstances. The end.

title: User:Ray Calvin Baker/LogicalThing


text:
User:Ray Calvin Baker/PicardsCourtMartial

 This is the "PicardsCourtMartial.txt" file, created WED 2011 AUG 10 11:26 PM, revised SUN 2011 NOV 27 06:49 PM.

The Court-Martial of Captain Jean-Luc Picard

In one of the Star Trek: Generations episodes, Captain Picard made a statement consistent with the following text, taken from a twentieth century computer science textbook.

Statement presented in evidence by the defense:

3.1.1 History of Computability

The idea of having an algorithm, or recipe for performing some task, has existed for thousands of years. For many years, people also held the following belief: if any problem could be precisely stated, then with enough effort a solution could eventually be found (or else a proof that no solution exists could eventually be provided). In other words, it was believed that there was no problem so intrinsically difficult that, in principle, it could never be solved.

One of the great supporters of this belief was the famous mathematician David Hilbert (1862 - 1943). He once said, "Every definite mathematical problem must necessarily be suseptible of an exact settlement either in the form of an actual answer to the question asked, or by the proof of the impossiblity of its solution and therewith the necessary failure of all attempts ... one of the things that attracts us most when we apply ourselves to a mathematical problem is precisely that within us we always hear the call: here is the problem, search for the solution; you can find it by pure thought, for in mathematics there is nothing which cannot be known."

Hilbert's aim was to devise a formal mathematical system in which all problems can be precisely formulated in terms of statements which are either true or false. His idea was to find an algorith which, given any statement in the formal system, would determine whether or not that statement was true. If Hilbert had achieved this objective, then any problem which was well-defined could have been solved simply by executing the algorithm. Deciding the truth of a given statement in the formal system was called the Entscheidungsproblem by Hilbert, who considered it to be a fundamental open problem in mathematics.

(Source: _Computer_Science_A_Modern_Introduction_, by Les Goldschlager and Andrew Lister, Prentice Hall, 1982, ISBN 9-13-165704-8, page 73)

As a student of mathematical logic, I was shocked by Captain Picard's statement! I was also shocked that his senior offices seemed to take no notice of the statement. How would this scene have played out, had the senior officers regarded Picard's statement with the alarm that a twentieth-century student felt? Could it be that Gene Roddenberry (or his associates) had missed one of the most important developments in the history of mathematics? I believe Picard's careless (and completely false!) statement should have been perceived as sufficient reason for his officers to relieve him from duty immediately!

Would the following have been brought forth by a prosecutor at Picard's court-martial hearing?

Statement presented in evidence by the prosecution:

Unfortunately for Hilbert's objective, the 1930s brought a wave of research which showed that the Entscheidungsproblem is not computable. That is, no algorithm of the type for which Hilbert longed, exists. A cynic might say that mathematicians could heave a sigh of relief, for if such an algorithm did exist, they would be out of a job just as soon as the algorithm was found! In fact, however, mathematicians were stunned by the remarkable discovery.

The first news of the discovery came in 1931 when Kurt Godel published his now famous incompleteness theorem. Among other things, this showed that there is no algorithm whose input can be any statement about the integers and whose output tells whether or not that statement is true. Closely following on Godel's heels, further mathematicians such as Alonzo Church, Stephen Kleene, Emil Post, Alan Turing and many others, found more problems which had no algorithmic solution. Perhaps the most remarkable feature of those early results on problems which cannot be solved by computers is that they were obtained in the 1930s, before the earliest [electronic] computers had been built!

(Source: as above, pages 73, 74)

How do you think this episode should have continued? Should Picard have been reprimanded for conduct unbecoming a Star Fleet Captain? Should his officers have been commended for relieving the Captain of duty, an unpleasant and distasteful task, but correct and in accord with essential Star Fleet Regulations?

I don't know, because this episode was never written. But I think it should have been, because of the importance of the issues in modern mathematics.

The end.


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