# The third millennium starts on January 1 of the year 2001

In this note I argue that the 3rd millennium starts on the the first of January of the year 2001, rather than on the first of January of the year 2000.

### When should the third millennium start if we were to design a calendar from scratch?

The purpose of a calendar is to provide a uniform way to denote points in time. The most natural way to model time is by means of the real numbers. All that is needed is an origin and a unit measure of time. Any point in time can then be described as r units past the chosen origin where r is a (positive or negative) real number. As an illustration, the point in time -2.7 is indicated on the timeline below.
```        -2.7                                         chosen..
|                     chosen               unit....
V                     origin               interval
---|--------|--------|--------|--------|--------|--------|----
-3       -2       -1        0        1        2        3
```
Most of humanity appears to agree on the year as a suitable measure of time for expressing events in the history of mankind. So let me from here on refer to the chosen unit interval as a year, even though the same observations could be made for any other choice of a unit interval. Once an origin is fixed, there is a natural notion of "the first year after the chosen origin", "the second year after the chosen origin", etc., as indicated below. Likewise, there is a "first year before the chosen origin", a "second year before the chosen origin", etc. This terminology is not a proposed definition, but a consequence of the meaning of the words "first", "second", etc. in the English language, or any other natural language for that matter. Just like Washington is the 1st president of the USA, rather than the 0th, the interval between 0 and 1 is the 1st year after the chosen origin.
```        third    second   first    first    second   third
year     year     year     year     year     year
before 0 before 0 before 0 after 0  after 0  after 0
---|--------|--------|--------|--------|--------|--------|----
-3       -2       -1        0        1        2        3
```
Likewise there is a notion of the first decade after the chosen origin, the first century, and the first millennium. A decade is a period of 10 years, so the first decade ends at the end of the 10th year. Likewise the first millennium ends at the end of the 1000th year, and the second millennium at the end of the 2000th year after the chosen origin. Most people understand that the beginning of the 3rd millennium is the beginning of the 21th century as well. Likewise, it is also the beginning of the 201th decade, and of the 2001th year.

Now what name should be give to the nth year? The most obvious choice is to name the nth year "the year n". This is consistent with the fact that the nth day of January is called "January n". As long as we are not willing to refer to the 9th day of January as "January 8", there is no ground for calling the 9th year "the year 8". Hence the 3rd millennium starts at the beginning of the year 2001.

This essentially concludes the argument. The remainder of this note is an elaboration intended for the people who are still not convinced.

### The necessity of an origin

One proposal might be to do without an origin, name the years after the integers, and abstain from talking about the nth year. However, without origin there can be no notion of a third millennium, and hence there can be no reason to let it begin in the year 2000. Whenever anyone maintains that the third millennium begins at a specific day, this automatically induces an origin, namely at the beginning of the first millennium. And once there is an origin, there is also a notion of the first year, the second year, etc. In fact, a year is an interval of time just like a millennium or a century. Thus if the third millennium and the 21st century are deemed meaningful concepts, then the notion of the nth year must be a meaningful concept as well.

### Naming the years

There is maybe no need to name the years, as one can simply refer the to the nth year as "the nth year". Under this point of view there is nothing more to say than that the 3rd millennium starts at the onset of the 2001th year.

Nevertheless, some people feel the need to give names to certain intervals of time that are exactly a year long. Suitable candidates for such names are real numbers, and 5 naming schemes come to mind: a period of 1 year, could be called

• A: after its starting point
• B: after its endpoint
• C: after its middlepoint
• D: after the one of its starting or endpoint closest to 0
• E: after the one of its starting or endpoint furthest from 0
If we take for example the year starting at 2.7 and ending at 3.7, as well as the year starting at -2.4 and ending at -1.4, the possible names for those two years, according the the naming schemes A-E, are:
• A: 2.7    -2.4
• B: 3.7    -1.4
• C: 3.2    -1.9
• D: 2.7    -1.4
• E: 3.7    -2.4
For each of the 5 naming schemes the years whose name is an integer are indicated on the timeline as follows:
```        third    second   first    first    second   third
year     year     year     year     year     year
before 0 before 0 before 0 after 0  after 0  after 0
---|--------|--------|--------|--------|--------|--------|----
-3       -2       -1        0        1        2        3

A:     |   -3   |   -2   |   -1   |    0   |    1   |    2   |
B:     |   -2   |   -1   |    0   |    1   |    2   |    3   |
D:     |   -2   |   -1   |   -0   |    0   |    1   |    2   |
E:     |   -3   |   -2   |   -1   |    1   |    2   |    3   |

|        |        |        |        |        |
C:    -3   |   -2   |   -1   |    0   |    1   |    2   |    3
|        |        |        |        |        |
```
What are the pro's and contra's for each of these naming schemes?
• Symmetry: The schemes C, D and E have the advantage that the years after the chosen origin are treated in the same way as the years before. The schemes A and B are asymmetric.
• Differentials: How much time expired between March 15 of the year -2 and March 15 of the year +2? The schemes A,B and C say 4 years, which is good. The scheme D says 5 years, and the scheme E says 3 years. Both answers may be considered confusing.
• Rounding: In which year falls the event 1233.6? Scheme C says in 1234. The algorithm is rounding to the nearest integer, which is a very plausible algorithm. Scheme D says 1233. Here the algorithm is leaving out the non-integer part, which is pretty plausible as well. The other 3 schemes also come with easy algorithms, but slightly less plausible than C and D.
• Ambiguity: Scheme D is the only one that allocates the same name to different years: there are two years 0. This can be resolved by calling one of them -0, but it remains annoying.
• Counting convenience: Scheme E is the only one in which the nth year is the year n, and likewise, the nth year before the chosen origin is the year -n. This is an extremely convenient property, as most people would dislike to distinguish between the nth year and the year n. It would be a continuous source of confusion. Scheme B has the same advantage for the positive years, although it fails to have the property that the nth year before the chosen origin is the year -n. This may be good enough for societies that have passed the chosen origin long ago anyway.
Evaluation: The fifth argument (counting convenience; n=nth) appears to be the most convincing. This would point to naming scheme E, with B as a second choice. The choice between E and B might be based on whether the arguments "symmetry" or "differentials" are considered more important. In either case the 3rd millennium would start with the year 2001. If we disregard the fifth argument, naming scheme C is the only one that is evaluated positively in the light of all other desiderata. According to this scheme, the 3rd millennium starts halfway the year 2000. Schemes A and D are the only ones that let the 3rd millennium start with the year 2000. According to the arguments listed above, naming scheme B is preferable to A, whereas E is preferable to D. Thus, if the notion that that the 3rd millennium starts with the year 2000 is to be supported at all, it should be with other arguments. I can imagine only two such arguments, and believe both of then should be rejected:
• Counting from 0: In mathematics and computer science there is a tradition of indexing the elements in a list of 4 items as 0, 1, 2, and 3. One advantage of this method is in the definition of the natural numbers: the number 3 is simply defined as the set {0,1,2}. Another advantage is that the technique scales well to infinite lists. None of these advantages has much relevance for the current problem. However, one might wish to argue that counting from 0 is a good habit in itself: the year n is then called after the number of years that happened before it.

My counterargument, is that anyone who takes this argument seriously, should be willing to apply it to all elements of the calendar. The 1st of January of 1998 should then be called January 0, or 0/0/1998, rather than 1/1/1998. Likewise, the 10th of January should have the name "9".

My feeling is that not many proponents of the year 2000 as start of the third millennium are willing to take this step. For it would be truly confusing. However, when the days of the month are to be counted from 1, the mere argument that years are to be counted from 0 does no longer apply.

• Millennianism: Another argument to select naming scheme A or D could be to ensure that the 3rd millennium starts in the year 2000. A more objective version of that desideratum is that any new millennium should start at the moment of change in the relevant digits of the year-numbering scheme. However, according to this criterion, scheme D is the only one that qualifies. For in scheme A the first millennium before the chosen origin starts on January 1 of the year -1000, which is the day after December 31 of the year -1001. Nevertheless, the change of the digits occurs only a year later, on January 1 of the year -999. Thus, for millennium reasons the case for scheme A is unconvincing. As most proponents of starting the 3rd millennium in 2000 would not want to endorse naming scheme D (because of the double occurrence of the year 0), the millennium argument fails to have the desired effect.
Conclusion: The 3rd millennium starts with the year 2000 only if one of the naming schemes A or D are chosen. However, there are better arguments for choosing one of the schemes E, B or C.

Next I'm going to argue that the 3rd millennium is still more than a year away (speaking on December 2 of the 1999th year A.D.), even if naming scheme A or D is chosen.

### When does the third millennium start?

Both the length of the unit interval (the 3651/4-day year) and the starting point of the years (January 1) have not changed significantly since 45 B.C., when Julius Caesar's calendar reform took effect. The numbering of the years used today stems from the Roman abbot Dionysius Exiguus in 527 A.D. He made an estimate of the year in which Jesus Christ was born, and called that year the first "Year of Our Lord Jesus Christ" ("Anno Domino Nostri Jesu Christi") - in short "the first year A.D." - thereby implicitly choosing the origin of his calendar to be almost a year before he believed that Christ was born (which was on December 25 of the first year A.D.). Starting from there, the year in which he did this became "the 527th year of our Lord Jesus Christ". This way of naming years gradually became accepted all over the western world. Although it turned out that Christ was in fact born 4 to 6 years earlier than Dionysius has estimated, people stuck to the counting of Dionysius, reasoning that the choice of the origin is not very relevant, as long as most people agree on what origin to use.

The year before "the first year of Our Lord Jesus Christ" became referred to as "the first year before Christ", or "the first year B.C.", which fits perfectly with the choice of the beginning of the first year A.D. as the origin of the calendar.

As for naming years, many people choose to refer to the first year A.D. as "the year 1 A.D.". Likewise, the first year B.C. was called "the year 1 B.C.". This amounts to a naming scheme such as E above.

After the invention of negative numbers, the astronomer Jacques Cassini proposed to name years after integers, instead of using the annotations A.D. and B.C. Motivated by argument 2 above ("differentials") he argued that if any given year was named n, the previous year should be named n-1. Thus as soon as he would give a name to any given year, the names of all other years would be determined. Cassini did this in the 1740th year A.D. Naturally, he was faced with the choice of naming this year 1740 or 1739. He decided to call it 1740. As a consequence, the first year A.D. was called 1, the first year B.C. was called 0 and the second year B.C. was called -1. This is naming scheme B, as explained above. Cassini argued that as negative numbers were a new invention anyway, not many people would automatically identify the year 3 B.C. with -3; a difference of 1 between the B.C,'s and the negative integers would therefore be much less confusing than a difference of 1 between the A.D.'s and the positive integers. Today, astronomers all over the world follow Cassini's convention, whereas historians tend to go by the B.C. notation. This never leads to confusion, as both parties are aware that -n translates to n+1 B.C.

Cassini's proposal did not shift the origin of the time axis by one year. By deciding that the nth year past the chosen origin should be called "n", he chose naming scheme B. Therefore the 3rd millennium according to Cassini starts at exactly the same time as the 3rd millennium according to Dionysius.

Proponents of the idea that the third millennium should start at the beginning of year 2000 can justify their proposal only by arguing for naming schemes A or D. (In fact, most of these people appear to be in favor of naming scheme A, as there is very little support for the idea that there should be two years 0.) Thus, such proponents will have to admit that there is a fundamental difference between the year n and the nth year. In fact, they should argue that the year n is in fact the n+1th year.

Would it be the case that we are confronted with the fact that the current year is called 1999, without there being evidence that it is also the 1999th year, said proponents could postulate the year 1999 to be the 2000th year, thereby laying the origin of the time axis at the beginning of the year 0. This would make the 3rd millennium start at the end of the current year (1999).

However, this position is historically incorrect. The current year is, according to a centuries-old tradition, known as the 1999th year. It is only a derived notion that it is also called the year 1999, a notion due to people that believe the nth year should be called the year n. As the proponents of starting the 3rd millennium in the year 2000 must reject that notion, the only consistent thing for them to do is to take issue with Cassini's naming convention, and insist that the current year should be called 1998 instead of 1999, given that it is the 1999th year. By insisting on such a convention the 3rd millennium will indeed start on January 1, 2000. However that moment will then be still more than a year away; it is the same moment that most people call January 1, 2001.

 Rob van Glabbeek December 2, 1999 rvg@cs.stanford.edu