There is nothing intrinsically special about the names of the days of the week themselves. Instead of assigning each day a name like "Saturday" (Saturn's day), "Sunday" (the sun's day), "Monday" (the moon's day), and so on, we could just as easily have given each day of the week a different name like "Jovday" (Jupiter's day), "Venday" (Venus' day), etc. Or we might have even given each day a totally meaningless name like "Krepstin," "Shlugsty," "Dorkelding," or anything else we might have come up with. Insofar as that part of the naming process goes, we are free to do whatever we choose.

But while we may be free to assign and reassign absolute names to entities (such as days of the week), we are not completely at liberty to assign and reassign their interrelationships in a willy-nilly fashion. For example if we were to rename the days of the week, and if we were to define "Shlugsty" as being the day following "Krepstin," then "Dorkelding" (a completely separate day from "Shlugsty") could not also be defined as being the day following "Krepstin" or else we would have a conflict -- there would be an inconsistency in our system.

So there are some definite restrictions imposed on us whenever we attempt to define or redefine man-made "truths." We are not always completely at liberty to arbitrarily pick and choose the meanings and associations of all the words that we use. (Sorry about that, Mr. Dumpty!)

Getting back to our hypothetical question, if everybody agreed to call today "Saturday" would it really be Saturday? My answer is no, it would still be Wednesday. (We would all just be "pretending" that it's Saturday, either out of conformity or out of ignorance.)

Consistency is a very powerful concept -- it has almost a kind of mathematical rigor to it. In fact consistency is the "glue" which holds mathematics together. If an inconsistency were ever to be discovered in mathematics, the entire system of mathematics would collapse. (We'll discuss this further in Chapter Five when we talk about Fallacy & Paradox.) An inconsistent statement cannot be "declared" to be a "truth" any more than the value of 2 + 2 can be "declared" to be equal to 5, or that the value of pi can be "declared" to be exactly equal to 3.

We've all seen puzzle questions in which we are given a sequence of numbers, and we are asked to determine the next number in the sequence. For example, given the sequence:

11, 13, 15, 17, 19, 21, 23, 25, 27, __

what number would come next? Obviously the sequence represents the list of double digit odd numbers, so the correct answer would be 29.

In solving a puzzle of this form you must first try to detect a pattern of consistency between the items in the sequence. Once you determine a pattern it usually becomes a trivial matter to specify the next item in the sequence.

But now try to determine the number which comes next in the following sequence:

1, 2, 4, __

Now the correct answer is not so obvious because there is much less information to work with. In fact the lack of information in this problem results in the existence of many correct answers.

On the other hand, the sequence represents those integers which are not a multiple of three (1, 2, 4, 5, 7, 8, 10, ... ). So a correct answer in this case would be 5.

In fact it's easy to show (not only for this particular problem, but for all of these types of sequence puzzles in general) that any number can be the "correct" answer! First construct a polynomial whose roots are the numbers in the given sequence:

P (x) = (x - 1) (x - 2) (x - 4)

Then pick any number (we'll choose 7) and multiply the polynomial by (x - 7) to create a new polynomial:

Q (x) = (x - 7) P (x)

= (x - 1) (x - 2) (x - 4) (x - 7)

You can now justifiably argue that the "correct" answer (to the problem of trying to determine the next number in the sequence: 1, 2, 4, ... ) is therefore 7, because the numbers 1, 2, 4, and 7 all represent the roots of the polynomial, Q (x).

Correctness is therefore not something which is necessarily unique. There can be more than one correct solution to any problem. The only requirement for a solution to be correct is that it be consistent with the given data. However some correct solutions may represent better solutions than others.

In science, when two different hypotheses (one simple and one more complicated) can both be used equally well to explain an observed phenomenon, the simpler hypothesis is adopted as representing the correct explanation. This practice is commonly referred to as using Occam's Razor and is based on our empirical experience that nature (i.e., physical reality) actually does appear to have a fundamental simplicity to it.

Even though the concept of consistency is not based on a physical reality, we can nevertheless invoke the spirit of Occam's Razor to determine the relative correctness of two or more competing solutions to a problem. In this case, Occam's Razor will not be used to completely eliminate solutions by declaring them to be totally "incorrect." Instead it will merely declare some of the solutions to be less desirable (or "less correct") than others.

Returning once again to the question about whether or not today would actually be Saturday (if we all simply declared it to be so), Occam's Razor can guide us in determining that answer as well. If today would really be Saturday, then we would have to regard today as being an exception to our rule that Wednesday always follows Tuesday (since yesterday really was Tuesday). On the other hand if today would really be Wednesday (like it normally would have been), then there would not have to be any exceptions to the rule that Wednesday always follows Tuesday. Since consistent rules (ones without exceptions) are cleaner and simpler than inconsistent rules (ones with messy exceptions), Occam's Razor confirms that today would indeed still be Wednesday.

Consistency is a powerful tool. Besides allowing us to determine the correct day of the week, the principle of consistency can be used to determine the correctness (or incorrectness) of many other commonly accepted conventions currently used in everyday life.

If a person who is addicted to alcohol is referred to as being an alcoholic, then how should we refer to a person who is addicted to work? (I'll give the answer at the end of this chapter.)

July 20, 1969

or, using only numbers, as:

7/20/69

But is this the correct form in which to specify dates? Is the order in which the numbers are specified (the month, followed by the day, followed by the year) consistent with how we write numbers in general?

Let's look for example at a number like "three hundred fifty two." How would we express that number using numerical digits instead of words? The arbitrarily established convention that we've adopted is to first write down the most significant digit (the one which represents the largest portion of the number, in this case the 3, since it represents "hundreds"), followed to the right by the next most significant digit (in this case the 5, since it represents "tens"), followed to the right by the least significant digit (in this case the 2, since it represents "ones"). Therefore the number would look like this:

As an aside, the fact that we write the digits in this particular order is completely arbitrary. We could just have easily adopted some other convention, such as writing the same digits from right to left instead (i.e., 253). And, had we chosen to write all of our numbers that way, then that convention would have been just as "correct" as the "left-to-right" method that we actually use. (Of course 253 would then not indicate "two hundred fifty three.")

The left-to-right convention is also the way we express clock time. For example "four twenty seven and fifty one seconds" would be expressed by first writing the most significant portion (the hours), followed to the right by the next most significant portion (the minutes), followed to the right by the least significant portion (the seconds). Using colons to delimit each portion, the time would therefore be written as:

So then why do we express calendar time by writing the most significant portion (the year) in the right-most position? The correct (i.e., consistent) way to write a date is to first write the most significant portion (the year), followed to the right by the next significant portion (the month), followed to the right by the least significant portion (the day). Therefore the correct way to express the date of the first lunar landing is:

1969 July 20

or, using only numbers, as:

In fact, since clocks and calendars both indicate quantities of "time," the most consistent way of specifying time in general would be to combine those two separate concepts into one single concept. Any point in time could then be expressed in the single form:

year : month : day : hour : minute : second

where we would no longer require a separate delimiter (such as a "/") to be used only for the "date" portion.

Surely everyone knows how to count ... or at least they think that they do. Let's see if you know the correct way to count. I would like for you to count out loud the number of x's contained inside the parentheses below:

( x x x x x )

(No, I don't want you to just give the answer, "five." I want you to actually count each item.)

Did you say: "One ... two ... three ... four ... five"? Is that what you think of as being the correct way to count? Ok then, try counting (out loud again) the number of x's contained inside these parentheses:

(   )

No, I don't want you to just give me the answer: "There aren't any." I want you to actually count the items.

What's the matter? Having problems? (And I thought you said that you knew how to count!)

The correct way to count is not to start with one, but with zero. If you start counting with one, you are automatically making the unwarranted assumption that there actually is at least one item to be counted. (If you really want to make such groundless a priori assumptions about the number of items that you will be counting, then you might as well always start counting with some even higher number, like maybe three. Then you can simply skip over to the third item and begin your counting from there!)

To count correctly you need to first perform a very simple "initialization" step. Before you even look at the items to be counted, you merely say "zero." In mathematical terms this step can be thought of as counting the null set, an empty set that exists in every collection of items. You then look for the first item (if there is one) and continue counting from there.

Therefore the correct way to count the first collection of x's would be:

"Zero ... one ... two ... three ... four ... five"

and the correct way to count the second "collection" would be:

"Zero"

By counting in this manner you can consistently count any number of items (including no items) without having to treat the case of "nothing to count" as a special case. (Remember that Occam's Razor abhors rules that require exceptions and special cases.)

Failure to acknowledge zero as the first step in counting has led to the establishment of many awkward conventions. For example the new millenium begins (or began, depending on when you read this) with the year 2001, and not with 2000 (as the general population seems to think). And the reason, of course, is because there was no year numbered "zero AD."

As another example of awkwardness, consider the popular question: "On a scale from one to ten how would you rate such and such"? Then, even though you absolutely hate such and such, you are still required to give it at least one point!

The letters AM and PM are abbreviations for the Latin expressions "Ante Meridiem" and "Post Meridiem," respectively. (Ante and post are Latin for "before" and "after," respectively. Meridiem is the Latin word for "meridian," the imaginary great circle on the celestial sphere, which passes through the point in the sky directly overhead, and through the point on the horizon directly south of the observer.) Every day the sun spends essentially half of its time to the east of the meridian (i.e., ante meridiem), and half of its time to the west (post meridiem). And the crossover point (the middle of the daylight portion of the day) defines local noon, the time when the sun reaches its highest point above the horizon.

Because the meridian is defined in terms of an observer, the exact time at which local noon occurs depends on where the particular observer is located. Therefore local noon seldom (if ever) occurs exactly at "clock-time" noon (i.e., twelve o'clock). So if one were to interpret the meaninings of AM and PM literally, then the exact time at which "AM becomes PM" would not be at 12 o'clock, but would instead occur at different times for observers at different locations.

Even adopting the concept of "standard meridians" (i.e., 24 fixed meridians, one for each of the time zones) would not solve the problem because the sun does not make meridian crossings (of any meridian) in exactly 24 hour intervals. Since the orbit of the earth around the sun is not a perfect circle, the times of the sun's meridian crossings slowly change from day to day. Over the course of a year, this daily drift (known as the "equation of time") can amount to as much as about plus or minus 18 minutes. (On globes of the earth the equation of time is sometimes represented graphically as an analemma, a skinny "figure-eight" shape that usually gets printed somewhere around the area of the Pacific Ocean.) Therefore, even measured relative to a standard meridian, the time at which "AM becomes PM" would still not be constant.

And to make matters even worse, six months out of the year are represented by Daylight Savings Time (when the sun actually crosses the meridian an hour later than our clocks would otherwise indicate). Should we therefore refer to the noon hour as being PM during the winter, but AM during the summer?

When specifying clock times it is clearly impractical to try and interpret the meanings of AM and PM in their literal senses. Instead we should interpret AM as simply being a reference to the first half of the day, and PM as being a reference to the second half of the day. Therefore specifying a time like 7:45 AM simply means that 7 hours and 45 minutes of time have elapsed since the start of the first half of the day (i.e., since midnight). Similarly specifying a time like 4:15 PM means that 4 hours and 15 minutes of time have elapsed since the start of the second half of the day (i.e., since noon).

We can now quite easily determine the answer to our question about whether 12:00 noon should be designated as AM or PM. (And our discussion pertains not only to the instant of noon, but to the entire noon hour.)

Let's first start with 2 o'clock in the afternoon. Should 2 o'clock in the afternoon be designated as AM or PM? Answer: PM. Why? Because 2:00 PM means that 2 hours of time have elapsed since the start of the second half of the day (i.e., since noon).

Now let's back up an hour, to 1 o'clock. Should 1 o'clock in the afternoon be designated as AM or PM? Answer: PM. Why? Because 1:00 PM means that 1 hour of time has elapsed since the start of the second half of the day (i.e., since noon).

Now let's back up one more hour, to noon. Should 12:00 noon be designated as AM or PM? Answer: AM. Why? Because 12:00 AM means that 12 hours of time have elapsed since the start of the first half of the day (i.e., since midnight). (To refer to noon as being 12:00 PM would be tantamount to making the clearly absurd claim that noon occurs 12 hours later than noon!) Therefore,

12:00 AM is actually Noon

and

12:00 PM is actually Midnight.

(No, 12:01 in the afternoon is not PM either. It doesn't become PM until one o'clock.)

"B-b-but ..." you protest, "That's just the opposite of the way that everybody does it. We've defined noon as being 12:00 PM, and midnight as being 12:00 AM."

"I'm sorry, but they were defined incorrectly," is my reply. And it's not just my opinion that they're incorrect. They are incorrect. Referring to noon as 12:00 PM (and to midnight as 12:00 AM) is inconsistent. And inconsistencies cannot be ''declared" to be truths. And even if everybody agrees to call noon 12:00 PM, that still won't make it so. (Recall our prior discussion about everybody agreeing to call today Saturday when it's really only Wednesday.)

Still, there remains a nagging compulsion in the back of your mind. You're thinking: "If it's even as little as a fraction of a second into the afternoon, then it's just somehow got to be PM. After all, PM refers to the afternoon, doesn't it? So why can't we legitimately use PM?"

The answer is: We can! (But if we do, then we can no longer refer to the time as being 12 o'clock.) Let me explain:

As we have already indicated, whenever we express a time in terms of PM hours, we are specifying (or counting) the number of hours (and minutes and seconds) that have elapsed since noon. So if we want to express noon itself as a PM time then all we have to do is count (out loud, if you please!) the number of hours between noon and itself.

Yes, "ZERO!"

Therefore:

Noon can be expressed as Zero o'clock PM

And of course, by identical reasoning:

Midnight can be expressed as Zero o'clock AM

What? You say that your clock doesn't have a zero on it? Well, that's easy enough to fix. Just scrape off the twelve (which never should have been put there in the first place) and paste on a zero instead! After all, who in their right mind would ever want to suggest that we should begin counting things (such as the hours of a new day) by starting with -- twelve!

(When I presented this information on one of my radio shows several years ago, a listener came down to the station the next day and presented me with just such a clock on which he had carefully replaced the twelve with a zero! That clock is still mounted on the wall in the lobby at KSCO.)

Now that we've opened the door to the concept that clock hours do not have to be restricted to the range 1 through 12, let's carry that idea a little further.

When we count things (be they apples, marbles, or whatever), there is no restriction on us that says we must stop counting when we reach twelve. The same holds true for hours. (In fact we often do use expressions such as "48 hours" in our everyday speech.) Therefore we can count past 12 AM and ligitimately refer to the suceeding times as being 13 AM, 14 AM, etc. (This should come as no surprise to anybody who is familiar with "military" time.)

But there's no reason why we can't also extend this concept to the PM hours as well. In this case, 13 PM, 14 PM, etc. are equivalent to 1 AM, 2 AM, etc. of the following day. Therefore:

There is also no need to limit ourselves to only positive numbers. For example we can refer to 11 AM as being -1 PM, 10 AM as being -2 PM, etc. These times may look funny and they may be awkward to use. But it is not incorrect to specify them, if we so choose.

When viewing time in this unrestricted way, the true significance of "AM" and "PM" becomes readily apparent. It is no longer meaningful to simply declare that a given point in time, such as sunset, occurs "in the PM hours." Because:

The terms "AM" and "PM" are simply reference indicators that allow you to specify which "zero point in time" you have chosen to use.

As we have already mentioned, calendar dates are just another expression of time. However, instead of having only two reference points (AM or PM) to choose from, we now have twelve (January, February, etc.). Therefore, just as in the case of clock time, we can express calendar time in an unrestricted fashion. (Dates need not be restricted to the "actual" number of days in the month.)

No discussion about consistency and inconsistency would be complete without at least mentioning the English language with all of its inconsistencies and exceptions to rules. To delve into all of its inconsistencies, however, would make this chapter longer than the entire rest of the book! But the messy state of the language can probably be best summed up and illustrated by the old joke that the word "fish" could have been spelled as "g-h-o-t-i" (gh as in the word "laugh," o as in the word "women," and ti as in the word "motion")!

A person addicted to work would properly (i.e., consistently) be referred to as being a workic (not a workaholic). The term "workaholic" would properly refer to a person who is addicted to "workahol" (whatever that is!)