There are many topics which provoke debate and even hostility, such as politics and religion. Wise barbers and taxi drivers avoid such topics for fear annoying or disturbing their customers. Better to stay with talk about the weather or sports, but even talk about football or baseball can be sensitive subjects. In my articles, I don’t avoid topics which can be minefields of controversy such as subjects of political thought, economic theory, religion and philosophy. Hopefully readers will find much to stimulate thought and even find some discussions to be educational.

The Puzzle of New Millennia, New Century, and New Decade:

Contrary to political and religious topics which are important and controversial, there are some which, although somewhat trivial, provoke much dispute. One of this was the question that some of us posed before the turn of the century: When does the new millennium really begin? As most of you recall, the world in general celebrated the start of the new millennium on January 1, 2000. But this bothered a few of us. We took on the role of spoil sports and pointed out that the world was premature in their celebration by one year. The new millennium did not start until January 1, 2001, we argued. People were annoyed, even irritated, by this reasonable dissent to the popular opinion. Personally I annoyed and irritated a few friends and colleagues by arguing that the transition to the new millennium was the transition from the year 2000 to 2001, not the transition from 1999 to year 2000, as most people thought. Along with other knit-pickers like myself, I argued that when we count decades (or ten of any countable items, e.g., coins in my pocket, beans in a bag, people in a room) we start with the first item and recite “one,” add 1 for each year, and finish with the tenth year (tenth item) and recite “ten.” In other words, the decade runs from 1 through 10, the century runs from 1 through 100, and the millennium runs from 1 through 1000. So, 10, 100, and 1000 are the numbers that indicate the end of the decade, century, and millennium respectively (not the numbers 9, 99, and 999 respectively). The logical conclusion is that years 11, 101, and 1001 mark the start of the succeeding decade, century, or millennium. From which it follows logically that year 2001, not year 2000, was the start of the new millennium ten years ago. The new millennium should have been celebrated on January 1, 2001!

But notwithstanding the logical validity of these arguments, the world celebrated the new millennium on January 1, 2000. My analytical, logical friends and I marveled at what appeared to us as a nearly universal confusion. Was this general confusion just a phenomenon connected with the excitement of the new millennium? Added to this great anticipation of the imminent new millennium was the general apprehension, even fear, about how our computer systems would handle the change of year designation from ‘1999’ to ‘2000’ (called the “Y2K” crisis). People feared that computer systems would crash and many vital functions disrupted. But, as most will recall, computer specialists prepared early for the transition and things went smoothly for most companies and government agencies. The world as we know it did not end on January 1, 2000! But getting back to the premature observance and celebration of the new millennium, let’s ask again: Was this error one that was unique to the transition to the new millennium?

After some reflection and brief study, I found that this general error and confusion was not limited to the transition to a new millennium at years 1999-2000-2001. It has occurred also with respect to transitions between centuries and decades. Newspaper and history book accounts from the year 1899 indicated that folks back then also celebrated the transition to the new century prematurely, on January 1, 1900. They did not wait for the correct start of the twentieth century, January 1, 1901. Were they simply too impatient? Likewise, most people think that this current year of 2010 marks the start of the second decade of the twenty-first century (as indicated by my informal, unscientific polling of relatives, friends and neighbors). But when we apply the logical arguments stated above, we see that the start of the second decade does not happen until the year 2011. Our high school graduating class of 1960 recently held a reunion. Most of my old classmates held that our graduating class was the first of the 1960 decade. But, again, simple logic shows that 1960 was the final year of the 1950 decade, not the start of the 1960s.

Grouping numbers with other numbers that look the same.

So what is happening here? Surely people are not that illogical. Are they simply too impatient and prematurely mark transitions from one decade, century, or millennium to the next, mistaking the transition from 1999 to 2000 for the correct transition from 2000 to 2001? My initial diagnosis relies on simple folk psychology: People tend to group numbers with sets of numbers that look the same, rather than consider what those numbers signify. For example, the year 1960 looks like the years 1961 – 1969. These are the sixties, after all! The year 2010 looks like the years 2011 – 2019, and therefore belongs with that group. And the year 1900 looks like the years 1901 through 1999, and belongs with that group. Finally, anyone can see that year 2000 resembles years 2001 through 2999, and therefore belongs with the new millennium! Most people are prone to this natural way of thinking; and are not too impressed by logical arguments that demonstrate that our natural inclination to group numbers by appearance results in error. Most of those friends and colleagues that I approached on this question of the correct transition time, either laughed at me or expressed annoyance that I spent any time on such a trivial topic. The transition to the new millennium happens when the world agrees that it happens, and we don’t have time for the technical dissent (on a triviality) of a few mathematicians and logicians!

Yes, they’re surely correct. This is surely not a crucial issue which can affect what happens in our world. People observe and celebrate transitions when they think it appropriate. “Let us move on to more interesting and important issues,” seems to be the prevailing attitude. I agree; there are more interesting and interesting issues to discuss. But before wrapping this one up, let me see whether we can find a significant philosophical point behind all this disputation about the start of the next decade, century or millennium. Hopefully the reader will exercise a little more forbearance and stay with me for a few more paragraphs.

Numbers as Labels Versus Numbers as Counting Numbers:

A curious fact about our use of numbers is that we use them in two very different ways, often without noticing this difference. A number can function as a label or name for something; and a number can function as a member of series of numbers. Examples of the use of numbers as labels or names are easy to find: the street number ’15’ may be used to identify a particular city street; the number ‘2502’ may identify a specific property; or the number “18” a particular floor in a high-rise building. In each case, the fact that the street may not really be the fifteenth street in the city (there are only fourteen streets), or that there aren’t 2501 properties lined up prior to property 2502, or that the high-rise building omits the thirteenth floor, going from floor 12 to floor 14, does not affect the identification of 15th street, or address 2502, or the identification of floor 18 on our high-rise. Here the numbers function as labels or names; they are simply identifiers. They could just as well be names using only alphabetical characters. The number ’15’ for our street functions the same as the name “Elm” in identifying a specific Avenue. The number ‘2501’ functions the same as a name — e.g. the James place – would work to identify a specific property. This use of numbers (as labels or identifiers only) can apply to our telephone or cell phone numbers, to our social security numbers and other numbers that are assigned to us at various stages of our lives. In earlier years of the telephone, telephone identifiers included words or letters along with numbers. The numbers were not counting numbers.

With the second use of numbers, the function of the number is not limited to identifying the item at issue. If I tell you correctly that the new Chevrolet parked outside my house is the eighth car I have owned, the number ‘8’ could identify the car as car number 8. But the number ‘8’ also tells you that if you count the number of cars I have owned starting with my first as ‘1’, the Chevrolet would be number ‘8’; in other words ‘8’ is part of a series of numbers (counting numbers) representing a series of cars that I have owned. Along the same line, if I tell that I was the fourth child born to my parents, I not simply labeling myself as the “fourth child,” although this label would be accurate enough. I would also be telling that, counting each child from the first to me, assigning a number to each, you would arrive at the number ‘4’. Child number 4 indicates ‘4’ as a usable label, but more importantly, it indicates my birth as occurring fourth in line. In short, ‘4’ functions as a member of a counting series of numbers. Likewise, if I tell you correctly that this year marked my 41st wedding anniversary, I’m not simply naming or labeling this year as marriage number ’41’; I’m telling you that 41 years have passed since my wife and I were wed. Count them, one by one, starting with year 1969, and you get ’41’. The same is true for the use of numbers to indicate our age. This year I am 50 identifies this year as year ’50’ for me; but it also tells you that if you count the years from my birth starting with ‘1’ and adding ‘1’ for each year until the current year, you will derive my age. The number ’50’ is part of a series of counting numbers, 1 through 50.

Now let us apply this distinction between numbers as labels and as counting numbers to the question of new millennia, centuries, and decades. The number 2010 is not just a label for the current year; it also functions as a counting number (count the number of years since the start of the decade at year 2001, adding ‘1’ for each year and the total is ’10’ and you arrive at number ‘2010.’ Hence, 2010 marks the end of the first decade. The number 1900 is not just a label assigned to a specific year. It represents a number in a counting series; in principle, were you to count years from 1 through 1900, adding ‘1’ for each year, after 1900 additions you would arrive at year 1900. But 1900 divided by 100 yields 19, with 0 remainder. So ‘1900’ marked the end of 19 centuries, with the year 1901 marking the start of the 20th century. Likewise, the year 2000 was not just a label for that year ten years ago. It also indicated that 2000 years had passed since our conventional start of the Julian calendar at year ‘1’. Now when we divide 2000 by 100 we get the answer ’20’ with 0 remainder; so 2000 marked the end of 20 centuries, year 2001 as the start of the new twenty-first century. The case for correct identification of the new millennium is easier. Divide 2000 by 1000 and you get ‘2’ with 0 remainder. So the year ‘2000’ marked the end of the second millennium, with year ‘2001’ marking the start of the new millennium. Hence, when we take into account the use of the numbers ‘2000’ and ‘2001’ as counting numbers, and not simply labels, it follows that January 1, 2001 was the start of the new millennium, and the general observance of the new millennium on January 1, 2000 was an error. Observing the new millennium at the start of year 2000 betrayed a general failure to distinguish between two distinct functions of numbers. This is moderately offensive to anyone who likes to see things done rationally and cleanly.

Does this matter to anyone besides people like me who dwell on these oddities? Does it ever matter whether we understand when a number used as a label and when as a counting number? Probably not, but one can imagine situations in which it would matter. Suppose that a person on floor fifteen of a high rise needs emergency attention and the paramedics are sent to floor fourteen instead, because someone did not notice the difference between ’15’ used as name or label and its use as part of counting series, a straight count of floors from 1 to 15. Or imagine a stranded soldier who is told that the third platoon will rescue him, but counts only two platoons passing his position and fails to signal his presence and is not rescued. The second platoon passing by was really the ‘third’ platoon. Does this ever happen? I imagine that it does happen and has happened in the past.

At any rate, those readers with an analytical, philosophical bent of mind should find it interesting to consider the difference between number used as a label and as a counting-number.