1 = I
2 = II
3 = III
4 = IIII (in modern times, sometimes IV, but this is modern)
5 = V
6 = VI
7 = VII
8 = VIII
9 = VIIII (now sometimes IX)
10 = X
11 = XI
15 = XV
20 = XX
25 = XXV
Contents: Introduction * Accuracy and Precision * Ancient Mathematics * Assuming the Solution * Average: see under Mean, Median, and Mode * Binomials and the Binomial Distribution * Cladistics * Corollary * Definitions * Dimensional Analysis * [The Law of the] Excluded Middle * Exponential Growth * Game Theory * Curve Fitting, Least Squares, and Correlation * Mean, Median, and Mode * Necessary and Sufficient Conditions: see under Rigour * Probability * Arithmetic, Exponential, and Geometric Progressions * Rigour, Rigorous Methods * Sampling and Profiles * Significant Digits * Standard Deviation and Variance * Statistical and Absolute Processes * Tree Theory * Utility Theory: see under Game Theory *
Appendix: Assessments of Mathematical Treatments of Textual Criticism
Mathematics -- most particularly statistics -- is frequently used in text-critical treatises. Unfortunately, most textual critics have little or no training in advanced or formal mathematics. This series of short items tries to give examples of how mathematics can be correctly applied to textual criticism, with "real world" examples to show how and why things work.
What follows is not enough to teach, say, probability theory. It might, however, save some errors -- such as an error that seems to be increasingly common, that of observing individual manuscripts and assuming text-types have the same behavior (e.g. manuscripts tend to lose parts of the text due to haplographic error. Does it follow that text-types do so? It does not. We can observe this in a mathematical text, that of Euclid's Elements. Almost all of our manuscripts of this are in fact of Theon's recension, which on the evidence is fuller than the original. If manuscripts are never compared or revised, then yes, texts will always get shorter over time. But we know that they are revised, and there may be other processes at work. The ability to generalize must be proved; it cannot be assumed).
The appendix at the end assesses several examples of "mathematics" perpetrated on the text-critical world by scholars who, sadly, were permitted to publish without being reviewed by a competent mathematician (or even by a half-competent like me. It will tell you how bad the math is that I, who have only a bachelor's degree in math and haven't used most of that for fifteen years, can instantly see the extreme and obvious defects.)
One section -- that on Ancient Mathematics -- is separate: It is concerned not with mathematical technique but with the notation and abilities of ancient mathematicians. This can be important to textual criticism, because it reminds us of what errors they could make with numerals, and what calculations they could make.
"Accuracy" and "Precision" are terms which are often treated as synonymous. They are not.
Precision is a measure of how much information you are offering. Accuracy is a more complicated term, but if it is used at all, it is to measure of how close an approximation is to an ideal.
(I have to add a caution here: Richards Fields heads a standards-writing committee concerned with such terms, and he tells me they deprecate the use of "accuracy." Their feeling is that it blurs the boundary between the two measures above. Unfortunately, their preferred substitute is "bias" -- a term which has a precise mathematical meaning, referring to the difference between a sample and what you would get if you tested a whole population. But "bias" in common usage is usually taken to be deliberate distortion. I can only advise that you choose your terminology carefully. What I call "accuracy" here is in fact a measure of sample bias. But that probably isn't a term that it's wise to use in a TC context. I'll talk of "accuracy" below, to avoid the automatic reaction to the term "bias," but I mean "bias." In any case, the key is to understand the difference between a bunch of decimal places and having the right answer.)
To give an example, take the number we call "pi" -- the ratio of the circumference of a circle to its diameter. The actual value of p is known to be 3.141593....
Suppose someone writes that p is roughly equal to 3.14. This is an accurate number (the first three digits of p are indeed 3.14), but it is not overly precise. Suppose another person writes that the value of p is 3.32456789. This is a precise number -- it has eight decimal digits -- but it is very inaccurate (it's wrong by more than five per cent).
When taking a measurement (e.g. the rate of agreement between two manuscripts), one should be as accurate as possible and as precise as the data warrants.
As a good rule of thumb, you can add an additional significant digit each time you multiply your number of data points by ten. That is, if you have ten data points, you only have precision enough for one digit; if you have a hundred data points, your analysis may offer two digits.
Example: Suppose you compare manuscripts at eleven points of variation, and they agree in six of them. 6 divided by 11 is precisely 0.5454545..., or 54.5454...%. However, with only eleven data points, you are only allowed one significant digit. So the rate of agreement here, to one significant digit, is 50%.
Now let's say you took a slightly better sample of 110 data points, and the two manuscripts agree in sixty of them. Their percentage agreement is still 54.5454...%, but now you are allowed two significant digits, and so can write your results as 55% (54.5% rounds to 55%).
If you could increase your sample to 1100 data points, you could increase the precision of your results to three digits, and say that the agreement is 54.5%.
Chances are that no comparison of manuscripts will ever allow you more than three significant digits. When Goodspeed gave the Syrian element in the Newberry Gospels as 42.758962%, Frederick Wisse cleverly and accurately remarked, "The six decimals tell us, of course, more about Goodspeed than about the MS." (Frederick Wisse, The Profile Method for Classifying and Evaluating Manuscript Evidence, (Studies and Documents 44, 1982), page 23.)
Modern mathematics is essentially universal (or at least planet-wide): Every serious mathematician uses Arabic numerals, and the same basic notations such as + - * / ( ) ° > <∫. This was by no means true in ancient times; each nation had its own mathematics, which did not translate at all. (If you want to see what I mean, try reading a copy of The Sand Reckoner by Archimedes sometime.) Understanding these differences can sometimes have an effect on how we understand ancient texts.
There is evidence that the earliest peoples had only two "numbers" --
one and two, which we might think of as "odd" and "even" --
though most primitive peoples could count to at least four: "one, two,
two-and-one, two-and-two, many." This theory is supported not just by
the primitive peoples who still used such systems into the twentieth century
but even, implicitly, by the structure of language. Greek is one of many
Indo-European languages with singular, dual, and plural numbers (though of
course the dual was nearly dead by New Testament times). Certain Oceanic
languages actually have five number cases: Singular, dual, triple,
quadruple, and plural. In what follows, observe how many number systems
use dots or lines for the numbers 1-4, then some other symbol for 5. Indeed,
we still do this today in hashmark tallies: count one, two three, four, then
strike through the lot for 5: I II III IIII IIII.
But while such curiosities still survive in out-of-the-way environments, or for quick tallies, every society we are interested in had evolved much stronger counting methods. We see evidence of a money economy as early as Genesis 23 (Abraham's purchase of the burial cave), and such an economy requires a proper counting system. Indeed, even Indo-European seems to have had proper counting numbers, something like oino, dwo, treyes, kwetores, penkwe, seks, septm, okta, newn, dekm, most of which surely sound familiar. In Sanskrit, probably the closest attested language to proto-Indo-European, this becomes eka, dvau, trayas, catvaras, panca, sat, sapta, astau, nava, dasa, and we also have a system for higher numbers -- e.g. eleven is one-ten, eka-dasa; twelve is dva-dasa, and so forth; there are also words for 20, 30, 40, 50, 60, 70, 80, 90, 100, and compounds for 200, etc. (100 is satam, so 200 is dvisata, 300 trisata, etc.) Since there is also a name for 1000 (sahasra), Sanskrit actually has provisions for numbers up to a million (e.g. 200,000 is dvi-sata-sahasra). This may be post-Indo-European (since the larger numbers don't resemble Greek or German names for the same numbers), but clearly counting is very old.
You've probably encountered Roman Numerals at some time:
1 = I
2 = II
3 = III
4 = IIII (in modern times, sometimes IV, but this is modern)
5 = V
6 = VI
7 = VII
8 = VIII
9 = VIIII (now sometimes IX)
10 = X
11 = XI
15 = XV
20 = XX
25 = XXV
etc. This is one of those primitive counting systems, with a change from one form to another at 5. Like so many things Roman (e.g. their calendar), this is incredibly and absurdly complex. This may help to explain why Roman numerals went through so much evolution over the years; the first three symbols (I, V, and X) seem to have been in use from the very beginning, but the higher symbols took centuries to standardize -- they were by no means entirely fixed in the New Testament period. The table at right shows some of the phases of the evolution of the numbers. Some, not all.
In the graphic showing the variant forms, the evolution seems to have been fairly straightforward in the case of the smaller symbols -- that is, if you see \|/ instead of L for 50, you can be pretty certain that the document is old. The same is not true for the symbols for 1000; the evolution toward form like a Greek f, in Georges Ifrah's view, was fairly direct, but from there we see all sorts of variant forms emerging -- and others have proposed other histories. I didn't even try to trace the evolution of the various forms. The table in Ifrah shows a tree with three major and half a dozen minor branches, and even so appears to omit some forms. The variant symbols for 1000 in particular were still in widespread use in the first century C. E.; we still find the form ⊂|⊃ in use in the ruins of Pompeii, e.g., and there are even printed books which use this notation. The use of the symbol M for 1000 has not, to my knowledge, been traced back before the first century B.C.E. It has also been theorized that, contrary to Ifrah's proposed evolutionary scheme, the notation D for 500 is in fact derived from the ⊂|⊃ notation for 1000 -- as 500 is half of 1000, so D=|⊃ is half of ⊂|⊃. The ⊂|⊃ notation also lent itself to expansion; one might write ⊂⊂|⊃⊃ for 10000, e.g., and hence ⊂⊂⊂|⊃⊃⊃ for 100000. Which in turn implies |⊃⊃ for 5000, etc.
What's more, there were often various ways to represent a number. An obvious example is the number 9, which can be written as VIIII or as IX. For higher numbers, though, it gets worse. In school, they probably taught you to write 19 as XIX. But in manuscripts it could also be written IXX (and similarly 29 could be IXXX), or as XVIIII. The results aren't really ambiguous, but they certainly aren't helpful!
Fortunately, while paleographers and critics of
Latin texts sometimes have to deal with this, we don't have to worry
too much about the actual calculations it represents. Roman
mathematics didn't really even exist; they left no texts at all
on theoretical math, and very few on applied math, and those very
low-grade. (Their best work was by Boethius, long after New Testament
times, and even it was nothing more than a rehash of works like
Euclid's with all the rigour and general rules left out. The poverty
of useful material is shown by the nature of the works in the successor
states. There is, for example, only one pre-Conquest English work with
any mathematical content: Byrhtferth's Enchiridion. Apart from
a little bit of geometry given in an astronomical context, its most
advanced element is a multiplication table expressed as a sort of
mnemonic poem.)
No one whose native language was not Latin would ever
use Roman mathematics if an alternative were available;
the system had literally no redeeming
qualities. In any case, as New Testament scholars, we are
interested mostly in Greek mathematics, though we should glance
at Babylonian and Egyptian and Hebrew maths also. (We'll ignore,
e.g., Chinese mathematics, since it can hardly have influenced
the Bible in any way. Greek math was obviously relevant to the
New Testament, and Hebrew math -- which in turn was influenced
by Egyptian and Babylonian -- may have influenced the thinking
of the NT authors.) The above is mostly by way of preface: It indicates
something about how numbers and numeric notations evolved.
The Greek system of numerals, as used in New Testament and early Byzantine times, was at least more compact than the Roman, though it (like all ancient systems) lacked the zero and so was not really suitable for advanced computation. The 24 letters of the alphabet were all used as numerals, as were three obsolete letters, bringing the total to 27. This allowed the representation of all numbers less than 1000 using a maximum of three symbols, as shown at right:
Thus 155, for instance, would be written as rne; 23 would be kg, etc.
Numbers over 1000 could also be expressed, simply by application of a divider. So the number 875875 would become woe,woe'. Note that this allowed the Greeks to express numbers actually larger than the largest "named" number in the language, the myriad (ten thousand). (Some deny this; they say the system only allowed four digits, up to 9999. This may have been initially true, but both Archimedes and Apollonius were eventually forced to extend the system -- in different and conflicting ways. In practice, it probably didn't come up very often.)
Of course, this was a relatively recent invention. The Mycenaean Greeks
in the Linear B tablets had used a completely different system:
| for digits, - for tens, o for hundreds,
for thousands.
So, for instance, the number we would now express as 2185 would
have been expressed in Pylos as
o====|||||. But
this system, like all things about Linear B, seems to have been
completely forgotten by classical times.
A second system, known as the "Herodian," or "Attic," was
still remembered in New Testament times, though rarely if ever used. It
was similar to Roman numerals in that it used symbols for particular numbers
repeatedly -- in this system, we had
I = 1
D = 10
H = 100
X = 1000
M = 10000
(the letters being derived from the first words of the names of the numbers).
However, like Roman numerals, the Greeks added a trick to simplify, or at least compress, the symbols. To the above five symbols, they added P for five -- but it could be five of anything -- five ones, five tens, five hundreds, with a subscripted figure showing which it was. In addition, in practice, the number was often written as G rather than P to allow numbers to be fitted under it. So, e.g., 28,641 would be written as
M M GX X X X GH H D D D D I
In that context, it's perhaps worth noting that the Greek verb for "to count" is pempw, related to pente, five. The use of a system such as this was almost built into the language. But its sheer inconvenience obviously helped assure the later success of the Ionian system, which -- to the best of my knowledge -- is the one followed in all New Testament manuscripts which employ numerals at all.
And it should be remembered that these numerals were very widely used. Pick up a New Testament today and look up the "Number of the Beast" in Rev. 13:18 and you will probably find the number spelled out (so, e.g., in Merk, NA27, and even Westcott and Hort; Bover and Hodges and Farstad uses the numerals). It's not so in the manuscripts; most of them use the numerals (and numerals are even more likely to appear in the margins, e.g. for the Eusebian tables). This should be kept in mind when assessing variant readings. Since, e.g., s and o can be confused in most scripts, one should be alert to scribes confusing these numerals even when they would be unlikely to confuse the names of the numbers they represent. O. A. W. Dilke in Greek and Roman Maps (a book devoted as much to measurement as to actual maps) notes, for instance, that "the numbers preserved in their manuscripts tend to be very corrupt" (p. 43). Numbers aren't words; they are easily corrupted -- and, because they have little redundancy, if a scribe makes a copying error, a later scribe probably can't correct it. It's just possible that this might account for the variant 70/72 in Luke 10:1, for instance, though it would take an unusual hand to produce a confusion in that case.
There is at least one variant, in fact, where a confusion involving numerals is nearly a certainty -- Acts 27:37. Simply reading the UBS text here, which spells out the numbers, is flatly deceptive. One should look at the numerals. The common text here is
In B, however, supported by the Sahidic Coptic (which of course uses its own number system), we have
Which would become
This latter reading is widely rejected. I personally think it deserves respect. The difference, of course, is only a single omega, added or deleted. But I think dropping it, which produces a smoother reading, is more likely. Also, while much ink has been spilled justifying the possibility of a ship with 276 people aboard (citing Josephus, e.g., to the effect that the ship that took him to Rome had 600 people in it -- a statement hard to credit given the size of all known Roman-era wrecks), possible is not likely.
We should note some other implications of this system -- particularly for gematria (the finding of mathematical equivalents to a text). Observe that there are three numerals -- those for 6, 90, and 900 -- which will never be used in a text (unless one counts a terminal sigma, and that doesn't really count). Other letters, while they do occur, are quite rare (see the section on Cryptography for details), meaning that the numbers corresponding to them are also rare. The distribution of available numbers means that any numeric sum is possible, at least if one allows impossible spellings, but some will be much less likely than others. This might be something someone should study, though there is no evidence that it actually affects anything.
Of course, Greek mathematics was not confined simply to arithmetic. Indeed, Greek mathematics must be credited with first injecting the concept of rigour into mathematics -- for all intents and purposes, turning arithmetic into math. This is most obvious in geometry, where they formalized the concept of the proof.
According to Greek legend, which can no longer be verified, it was the famous Thales of Miletus who gave some of the first proofs, showing such things as the fact that a circle is bisected by a diameter (i. e. there is a line -- in fact, an infinite number of lines -- passing through the center of the circle which divides the area of a circle into equal halves), that the base angles of an isoceles triangle (the ones next to the side which is not the same length as the other two) are equal, that the vertical angles between two intersecting lines (that is, either of the two angles not next to each other) are equal, and that two triangles are congruent if they have two equal angles and one equal side. We have no direct evidence of the proof by Thales -- everything we have of his work is at about third hand -- but he was certainly held up as an example by later mathematicians.
The progress in the field was summed up by Euclid (fourth/third century), whose Elements of geometry remains fairly definitive for plane geometry even today.
Euclid also produces the (surprisingly easy) proof that the number of primes is infinite -- giving, incidentally, a nice example of a proof by contradiction, a method developed by the Greeks: Suppose there is a largest prime (call it p). So take all the primes: 2, 3, 5, 7, ... p. Multiply all of these together and add one. This number, since it is one more than a multiple of all the primes, cannot be divisible by any of them. It is therefore either prime itself or a multiple of a prime larger than p. So p cannot be the largest prime, which is a contradiction.
A similar proof shows that the square root of 2 is irrational -- that is, it cannot be expressed as the ratio of any two whole numbers. The trick is to express the square root of two as a ratio and reduce the ratio p/q to simplest form, so that p and q have no common factors. So, since p/q is the square root of two, then (p/q)2 = 2. So p2=2q2. Since 2q2 is even, it follows that p2 is even. Which in turn means that p is even. So p2 must be divisible by 4. So 2q2 must be divisible by 4, so q2 must be divisible by 2. And, since we know the square root of 2 is not a whole number, that means that q must be divisible by 2. Which means that p and q have a common factor of 2. This contradiction proves that there is no p/q which represents the square root of two.
This is one of those crucial discoveries. The Egyptians, as we shall see, barely understood fractions. The Babylonians did understand them, but had no theory of fractions. They could not step from the rational numbers (fractions) to the irrational numbers (endless non-repeating decimals). The Greeks, with results such as the above, not only invented mathematical logic -- crucial to much that followed, including statistical analysis such as many textual critics used -- but also, in effect, the whole system of real numbers.
The fact that the square root of two was irrational had been known as early as the time of Pythagoras, but the Pythagoreans hated the fact and tried to hide it. Euclid put it squarely in the open. (Pythagoras, who lived in the sixth century, of course, did a better service to math in introducing the Pythagorean Theorem. This was not solely his discovery -- several other peoples had right triangle rules -- but Pythagoras deserves credit for proving it analytically.)
Relatively little of Euclid's work was actually original; he derived most of it from earlier mathematicians, and often the exact source is uncertain (Boyer, in going over the history of this period, seems to spend about a quarter of his space discussing how particular discoveries are attributed to one person but perhaps ought to be credited to someone else; I've made no attempt to reproduce all these cautions and credits). That does not negate the importance of his work. Euclid gathered it, and organized it, and so allowed all that work to be maintained. But, in another sense, he did more. The earlier work had been haphazard. Euclid turned it into a system. This is crucial -- equivalent, say, to the change which took place in biology when species were classified into genuses and families and such. Before Euclid, mathematics, like biology before Linnaeus, was essentially descriptive. But Euclid made it a unity. To do so, he set forth ten postulates, and took everything from there.
Let's emphasize that. Euclid set forth ten postulates (properly, five axioms and five postulates, but this is a difference that makes no difference). Euclid, and those on whom he relied, set forth what they knew, and defined their rules. This is the fundamental basis to all later mathematics -- and is something textual critics still haven't figured out! (Quick -- define the Alexandrian text!)
Euclid in fact hadn't figured everything out; he made some assumptions he didn't realize he was making. Also, since his time, it has proved possible to dispense with certain of his postulates, so geometry has been generalized. But, in the realm where his postulates (stated and unstated) apply, Euclid remains entirely correct. The Elements is still occasionally used in math classes today. And the whole idea of postulates and working from them is essential in mathematics. I can't say it often enough: this was the single most important discovery in the history of math, because it defines rigour. Euclid's system, even though the individual results preceded him, made most future maths possible.
The sufficiency of Euclid's work is shown by the extent to which it eliminated all that came before. There is only one Greek mathematical work which survives from the period before Euclid, and it is at once small and very specialized -- and survived because it was included in a sort of anthology of later works. It's not a surprise, of course, that individual works have perished (much of the work of Archimedes, e.g., has vanished, and much of what has survived is known only from a single tenth-century palimpsest, which obviously is both hard to interpret and far removed from the original). But all of it? Clearly Euclid was considered sufficient.
And for a very long time. The first printed edition of Euclid came out in 1482, and it is estimated that over a thousand editions have been published since; it has been claimed that it is the most-published book of all time other than the Bible.
Not that the Greeks stopped working once Euclid published his work.
Apollonius, who did most of the key work on conic sections, came later,
as did Eratosthenes, perhaps best remembered now for accurately measuring
the circumference of the earth but also noteworthy for inventing the
"sieve" that bears his name for finding prime numbers.
And, their greatest mathematician was no more than a baby when Euclid
wrote the Elements. Archimedes -- surely the greatest scientific
and mathematical genius prior to Newton, and possibly even Newton's
equal had he had the data and tools available to the latter -- was
scientist, engineer, the inventor of mathematical physics, and
a genius mathematician. In the latter area, several
of his accomplishments stand out. One is his work on large numbers in
The Sand Reckoner, in which he set out to determine the maximum
number of sand grains the universe might possibly hold. To do this, he
had to invent what amounted to exponential notation. He also, in so
doing, produced the notion of an infinitely extensible number system.
The notion of infinity was known to the Greeks, but had been the subject
of rather unfruitful debate. Archimedes gave them many of the tools they
needed to address some of the problems -- though few later scholars made
use of the advances.
Archimedes also managed, in one step, to create one of the tools that would turn into the calculus (though he didn't know it) and to calculate an extremely accurate value for p, the ratio of the circumference of a circle to its diameter. The Greeks were unable to find an exact way to calculate the value -- they did not know that p is irrational; this was not known with certainty until Lambert proved it in 1761. The only way the Greeks could prove a number irrational was by finding the equivalent of an algebraic equation to which it was a solution. They couldn't find such, for the good and simple reason that there is no such equation. This point -- that p is what we now call a transcendental number -- was finally proved by Ferdinand Lindemann in 1882.
Archimedes didn't know that p is irrational, but he did know he didn't know how to calculate it. He had no choice but to seek an approximation. He did this by the beautifully straightforward method of inscribed and circumscribed polygons. The diagram at right shows how this works: The circumference of the circle is clearly greater than the circumference of the square inscribed inside it, and less than the square circumscribed around it. If we assume the circle has a radius of 1 (i.e. a diameter of 2), then the perimeter of the inner square can be shown to be 4 times the square root of two, or about 5.66. The perimeter of the outer square (whose sides are the same length as the diameter of the circle) is 8. Thus the circumference of the circle, which is equal to 2p, is somewhere between 5.66 and 8. (And, in fact, 2p is about 6.283, so Archimedes is right). But now notice the second figure, in which an octagon has been inscribed and circumscribed around the circle. It is obvious that the inner octagon is closer to the circle than the inner square, so its perimeter will be closer to the circumference of the circle while still remaining less. And the outer octagon is closer to the circle while still remaining greater.
If we repeat this procedure, inscribing and circumscribing polygons with more and more faces, we come closer and closer to "trapping" the value of p. Archimedes, despite having only the weak Greek mathematical notation at his disposal, managed to trap the value of p as somewhere between 223/71 (3.14085) and 220/70 (3.14386). The first of these values is about .024% low of the actual value of p; the latter is about .04% high; the median of the two is accurate to within .008%. That is an error too small to be detected by any measurement device known in Archimedes's time; there aren't many outside an advanced science lab that could detect it today.
Which is nice enough. But there is also a principle there. Archimedes couldn't demonstrate it, because he hadn't the numbering system to do it -- but his principle was to add more and more sides to the inscribing and circumscribing polygons. Suppose he had taken infinitely many sides? In that case, the inscribing and circumscribing polygons would have merged with the circle, and he would have had the exact value of p. This is the principle of the limit, and it is the basis on which the calculus is defined. It is sometimes said that Archimedes would have invented the calculus had he had Arabic numerals. This statement is too strong. But he might well have created a tool which could have led to the calculus.
An interesting aspect of Greek mathematics was their search for solutions even to problems with no possible use. A famous example is the attempt to "square the circle" -- starting from a circle, to construct a square with the same area using nothing but straight edge and compass. This problem goes back all the way to Anaxagoras, who died in 428 B.C.E. The Greeks never found an answer to that one -- it is in fact impossible using the tools they allowed themselves -- but the key point is that they were trying for general and theoretical rather than practical and specific solutions. That's the key to a true mathematics.
In summary, Greek mathematics was astoundingly flexible, capable of handling nearly any engineering problem found in the ancient world. The lack of Arabic numbers made it difficult to use that knowledge (odd as it sounds, it was easier back then to do a proof than to simply add up two numbers in the one million range). But the basis was there.
To be sure, there was a dark -- or at least a goofy -- side to Greek mathematics. Plato actually thought mathematics more meaningful than data -- in the Republic, 7.530B-C -- he more or less said that, where astronomical observations and mathematics disagreed, too bad for the facts. Play that game long enough, and you'll start distorting the math as well as the facts....
The goofiness is perhaps best illustrated by some of the uses to which mathematics was put. The Pythagoreans were famous for their silliness (e.g. their refusal to eat beans), but many of their nutty ideas were quasi-mathematical. An example of this is their belief that 10 was a very good and fortunate number because it was equal to 1+2+3+4. Different Greek schools had different numerological beliefs, and even good mathematicians could fall into the trap; Ptolemy, whose Almagest was a summary of much of the best of Greek math, also produced the Tetrabiblos of mystical claptrap. The good news is, relatively few of the nonsense works have survived, and as best I can tell, none of the various superstitions influenced the NT writers. The Babylonians also did this sort of thing -- they in fact kept it all secret, concealing some of their knowledge with cryptography, and we at least hear of this sort of mystic knowledge in the New Testament, with Matthew's mention of (Babylonian) Magi -- but all he seems to have cared was that they had secret knowledge, not what that knowledge was.
At least the Greek had the sense to separate rigourous from silly, which many other people did not. Maybe they were just frustrated with the difficulty of achieving results. The above description repeatedly laments the lack of Arabic numbers -- i.e. with positional notation and a zero. This isn't just a matter of notational difficulty; without a zero, you can't have the integers, nor negative numbers, let alone the real and complex numbers that let you solve all algebraic equations. Arabic numbers are the mathematical equivalent of an alphabet, only even more essential. The advantage they offer is shown by an example we gave above: The determination of p by means of inscribed and circumscribed polygons. Archimedes could manage only about three decimal places even though he was a genius. François Viète (1540-1603) and Ludolph van Ceulen (1540-1610) were not geniuses, but they managed to calculate p to ten and 35 decimal places, respectively, using the method of Archimedes -- and they could do it because they had Arabic numbers.
The other major defect of Greek mathematics was that the geometry was not analytic. They could draw squares, for instance -- but they couldn't graph them; they didn't have cartesian coordinates or anything like that. Indeed, without a zero, they couldn't draw graphs; there was no way to have a number line or a meeting point of two axes. This may sound trivial -- but modern geometry is almost all analytic; it's much easier to derive results using non-geometric tools. It has been argued that the real reason Greek mathematics stalled in the Roman era was not lack of brains but lack of scope: There wasn't much else you could do just with pure geometric tools.
The lack of a zero (and hence of a number line) wasn't just a problem for the Greeks. We must always remember a key fact about early mathematics: there was no universal notation; every people had to re-invent the whole discipline. Hence, e.g., though Archimedes calculated the value of p to better than three decimal places, we find 1 Kings 7:23, in its description of the bronze sea, rounding off the dimensions to the ratio 30:10. (Of course, the sea was built and the account written before Archimedes. More to the point, both measurements could be accurate to the single significant digit they represent without it implying a wrong value for p -- if, e.g., the diametre were 9.7 cubits, the circumference would be just under 30.5 cubits. It's also worth noting that the Hebrews at this time were probably influenced by Egyptian mathematics -- and the Egyptians did not have any notion of number theory, and so, except in problems involving whole numbers or simple fractions, could not distinguish between exact and approximate answers.)
Still, Hebrew mathematics was quite primitive. There really wasn't much there apart from the use of the letters to represent numbers. I sometimes wonder if the numerical detail found in the so-called "P" source of the Pentateuch doesn't somehow derive from the compilers' pride in the fact that they could actually count that high!
Much of what the Hebrews did know may well have been derived from the Babylonians, who had probably the best mathematics other than the Greek; indeed, in areas other than geometry, the Babylonians were probably stronger. And they started earlier; we find advanced mathematical texts as early as 1600 B.C.E., with some of the basics going all the way back to the Sumerians, who seem to have been largely responsible for the complex 10-and-60 notation used in Babylon. How much of this survived to the time of the Chaldeans and the Babylonian Captivity is an open question; Ifrah says the Babylonians converted their mathematics to a simpler form around 1500 B.C.E., but Neugebauer, generally the more authoritative source, states that their old forms were still in use as late as Seleucid times. Trying to combine the data leads me to guess the Chaldeans had a simpler form, but that the older, better maths were retained in some out-of-the-way places.
It is often stated that the Babylonians used Base 60. This statement is somewhat deceptive. The Babylonians used a mixed base, partly 10 and partly 60. The chart below, showing the cuneiform symbols they used for various numbers, may make this clearer.
This mixed system is important, because base 60 is too large to be a comfortable base -- a multiplication table, for instance, has 3600 entries, compared to 100 entries in Base 10. The mixed notation allowed for relatively simple addition and multiplication tables -- but also for simple representation of fractions.
For very large numbers, they had still another system -- a partial positional notation, based on using a space to separate digits. So, for instance, if they wrote | || ||| (note the spaces between the wedges), that would mean one times 60 squared (i.e. 3600) plus two times 60 plus three, or 3723. This style is equivalent to our 123 = one times ten squared plus two times ten plus three. The problem with this notation (here we go again) is that it had no zero; if they wrote IIII II, say, there was no way to tell if this meant 14402 (4x602+0x60+2) or 242 (4x60+2). And there was no way, in this notation, to represent 14520 (4x602+2x60+0). (The Babylonians did eventually -- perhaps in or shortly before Seleucid times -- invent a placeholder to separate the two parts, though it wasn't a true zero; they didn't have a number to represent what you got when you subtracted, e.g., nine minus nine.)
On the other hand, it did allow representation of fractions, at least as long as they had no zero elements: Instead of using positive powers of 60 (602=3600, 601=60, etc.), they could use negative powers -- 60-1=1/60, 60-2=1/3600, etc. So they could represent, say, 1/15 (=4/60) as ||||, or 1/40 (=1/60 + 30/3600) as I <<<, making them the only ancient people with a true fractional notation.
Thus it will be seen that the Babylonians actually used Base 10 -- but generally did calculations in Base 60.
There is a good reason for the use of Base 60, the reason being that 60 has so many factors: It's divisible by 2, 3, 4, 5, 6, 12, 15, 20, and 30. This means that all fractions involving these denominators are easily expressed (important, in a system where decimals were impossible due to the lack of a zero and even fractions didn't have a proper means of notation). This let the Babylonians set up fairly easy-to-use computation tables. This proved to be so much more useful for calculating angles and fractions that even the Greeks took to expressing ratios and angles in Base 60, and we retain a residue of it today (think degrees/minutes/seconds). The Babylonians, by using Base 60, were able to express almost every important fraction simply, making division simple; multiplication by fractions was also simplified. This fact also helped them discover the concept (though they wouldn't have understood the term) of repeating decimals; they had tables calculating these, too.
Base 60 also has an advantage related to human physiology. We can count up to five at a glance; to assess numbers six or greater required counting. So, given the nature of the cuneiform numbers expressing 60 or 70 by the same method as 50 (six or seven pairs of brackets as opposed to five) would have required more careful reading of the results. As it was, numbers could be read quickly and accurately. A minor point, but still an advantage.
Nor were the Babylonians limited to calculating fractions. The Babylonians calculated the square root of two to be roughly 1.414213, an error of about one part in one million! (As a rough approximation, they used 85/60, or 1.417, still remarkably good.) All of this was part of their approach to what we would call algebra, seeking the solution to various types of equations. Many of the surviving mathematics tablets are what my elementary school called "story problems" -- a problem described, and then solved in such a way as to permit general solutions to problems of the type.
There were theoretical complications, to be sure. Apart from the problem that they sometimes didn't distinguish between exact and approximate solutions, their use of units would drive a modern scientist at least half mad -- there is, for instance, a case of a Babylonian tablet adding a "length" to an "area." It has been proposed that "length" and "width" came to be the Babylonian term for variables, as we would use x, y, and z. This is possible -- but the result still permits confusion and imprecision.
We should incidentally look at the mathematics of ancient Mari, since it is believed that many of the customs followed by Abraham came from that city. Mari appears to have used a modification of the Babylonian system that was purely 10-based: It used a system exactly identical to the Babylonian for numbers 1-59 -- i.e. vertical wedges for the numbers 1-9, and chevrons ( < ) for the tens. So <<II, e.g., would represent 22, just as in Babylonian.
The first divergence came at 60. The Babylonians adopted a different symbol here, but in Mari they just went on with what they were doing, using six chevrons for 60, seven for seventy, etc. (This frankly must have been highly painful for scribes -- not just because it took 18 strokes, e.g. to express the number 90, but because 80 and 90 are almost indistinguishable).
For numbers in the hundreds, they would go back to the symbol used for ones, using positions to show which was which -- e.g. 212 would be ||<||. (Interestingly, they used the true Babylonian notation for international and "scientific" documents.) But they did not use this to develop a true positional notation (and they had no zero); rather, they had a complicated symbol for 1000 (four parallel horizontal wedges, a vertical to their right, and another horizontal to the right of that), which they used as a separator -- much as we would use the , in the number 1,000 -- and express the number of thousands with the same old unit for ones.
This system did not, however, leave any descendants that we know of; after Mari was destroyed, the other peoples in the area went back to the standard Babylonian/Akkadian notation.
The results of Babylonian math are quite sophisticated; it is most unfortunate that the theoretical work could not have been combined with the Greek concept of rigour. The combination might have advanced mathematics by hundreds of years. It is a curious irony that Babylonian mathematics was immensely sophisticated but completely pointless; like the Egyptians and the Hebrews, they had no theory of numbers, and so while they could solve problems of particular types with ease, they could not generalize to larger classes of problems. Which may not sound like a major drawback, but realize what this means: If the parameters of a problem changed, even slightly, the Babylonians had no way to know if their old techniques would accurately solve it or not.
None of this matters now, since we have decimals and Arabic numerals. Little matters even to Biblical scholars, even though, as noted, Hebrew math probably derives from Babylonian (since the majority of Babylonian tablets come from the era when the Hebrew ancestors were still under Mesopotamian influence, and they could have been re-exposed during the Babylonian Captivity, since Babylonian math survived until the Seleudid era) or perhaps Egyptian; there is little math in the Old Testament, and what there is has been "translated" into Hebrew forms. Nonetheless the pseudo-base of 60 has genuine historical importance: The 60:1 ratio of talent: mina: shekel is almost certainly based on the Babylonian use of Base 60.
Much of Egyptian mathematics resembles the Babylonian in that it seeks directly for the solution, rather than creating rigourous methods, though the level of sophistication is much less.
A typical example of Egyptian dogmatism in mathematics is their insistence that fractions could only have unitary numerators -- that is, that 1/2, 1/3, 1/4, 1/5 were genuine fractions, but that a fraction such a 3/5 was impossible. If the solution to a problem, therefore, happened to be 3/5, they would have to find some alternate formulation -- 1/2 + 1/10, perhaps, or 1/5 + 1/5 + 1/5, or even 1/3 + 1/4 + 1/60. Thus a fraction had no unique expression in Egyptian mathematics -- making rigour impossible; in some cases, it wasn't even possible to tell if two people had come up with the same answer to a problem!
Similarly, they had a fairly accurate way of calculating the area of a circle (in modern terms, 256/81, or about 3.16) -- but they didn't define this in terms of a number p (their actual formula was (8d/9)2, where d is the diameter), and apparently did not realize that this number had any other uses such as calculating the circumference of the circle.
Egyptian notation was of the basic count-the-symbols type we've seen, e.g., in Roman and Mycenean numbers. In heiroglyphic, the units were shown with the so-very-usual straight line |. Tens we a symbol like an upside-down U -- ∩. So 43, for instance, would be ∩∩∩∩III. For hundreds, they used a spiral; a (lotus?) flower and stem stood for thousands. An image of a bent finger stood for ten thousands. A tadpole-like creature represented hundred thousands. A kneeling man with arms upraised counted millions -- and those high numbers were used, usually in boasts of booty captured. They also had four symbols for fractions: special symbols for 1/2, 2/3, and 3/4, plus the generic reciprocal symbol, a horizontal oval we would read a "one over" So some typical fractions would be
| 1/4: ⊂⊃ II II | 1/6: ⊂⊃ III III | 1/16: ⊂⊃ ∩II IIII |
It will be seen that there is no way to express, say, 2/5 in this system; it would be either 1/5+1/5 or, since the Egyptians don't seem to have liked repeating fractions either, something like 1/3+1/15. (Except that they seem to have preferred to put the smaller fraction first, so this would be written 1/15+1/3.)
The Egyptians actually had a separate fractional notation for volume measure, fractions-of-a-heqat. I don't think this comes up anywhere we would care about, so I'm going to skip trying to explain it. Nonetheless, it was a common problem in ancient math -- the inability to realize that numbers were numbers. It often was not realized that, say, three drachma were the same as three sheep were the same as three logs of oil. Various ancient systems had different number-names, or at least three different symbols, for all these numbers -- as if we wrote "3 sheep" but "III drachma." We have vestiges of this today -- consider currency, where instead of saying, e.g., "3 $," we write "$3" -- a significant notational difference.
We also still have some hints of the ancient problems with fractions, especially in English units (those on the metric system have largely solved this): Instead of measuring a one and a half pound loaf of bread as weighing "1.5 pounds," it will be listed as consisting of "1 pound 8 ounces." A quarter of a gallon of milk is not ".25 gallon"; it's "1 quart." (This is why scientists use the metric system!) This was even more common in ancient times, when fractions were so difficult: Instead of measuring everything in shekels, say, we have shekel/mina/talent, and homer/ephah, and so forth.
Even people who use civilized units of measurement often preserve the ancient fractional messes in their currency. The British have rationalized pounds and pence and guineas -- but they still have pounds and shillings and pence. Americans use dollars and cents, with the truly peculiar notation that dollars are expressed (as noted above) "$1.00," while cents are "100 ¢"; the whole should ideally be rationalized. Germans have marks and pfennig. And so forth. Similarly, we have a completely non-decimal time system; 1800 seconds are 30 minutes or 1/2 hour or 1/48 day. Oof!
We of course are used to these funny cases. But it should alsways be kept in mind that the ancients used this sort of system for everything -- and had even less skill than we in converting.
But let's get back to Egyptian math....
The hieratic/demotic had a more compact, though more complicated, system than hieroglyphic. I'm not going to try to explain this, just show the various symbols as listed in figure 14.23 (p. 176) of Ifrah. This is roughly the notation used in the Rhind Papyrus, though screen resolution makes it hard to display the strokes clearly.
This, incidentally, does much to indicate the difficulty of ancient
notations. The Egyptians, in fact, do not seem even to have had a concept
of general "multiplication"; their method -- which is ironically
similar to a modern computer -- was the double-and-add. For example, to
multiply 23 by 11 (which we could either do by direct multiplication or
by noting that 11=10+1, so
23x11 = 23x(10+1)=23x10 + 23x1 =230+23=253), they
would go through the following steps:
23x1 = 23
23x2 = 46
23x4 = 92
23x8 = 184
and 11=8+2+1
so 23x11 = (23x8) + (23x2) + (23+1) = 184 + 46 + 23 = 253
This works, but oy. A problem I could do by inspection takes six major steps, with all the chances for error that implies.
The same, incidentally, is true in particular of Roman numerals.
This is thought to be the major reason various peoples invented the
abacus: Even addition was very difficult in their systems, so they
didn't do actual addition; they counted out the numbers on the abacus
and then translated them back into their notation.
That description certain seems to fit the Hebrews. Hebrew mathematics frankly makes one wonder why God didn't do something to educate these people. Their mathematics seems to have been even more primitive than the Romans'; there is nothing original, nothing creative, nothing even particularly efficient. It's almost frightening to think of a Hebrew designing Solomon's Temple, for instance, armed with all the (lack of) background on stresses and supports that a people who still lived mostly in tents had at their disposal. (One has to suspect that the actual temple construction was managed by either a Phoenician or an Egyptian.)
The one thing that the Hebrews could call their own was their numbering system (and even that probably came from the Phoenicians along with the alphabet). They managed to produce a system with most of the handicaps, and few of the advantages, of both the alphabetic systems such as the Greek and the cumulative systems such as the Roman. As with the Greeks, they used letters of the alphabet for numbers -- which meant that numbers could be confused with words, so they often prefixed ' or a dot over the number to indicate that it was a numeral. But, of course, the Hebrew alphabet had only 22 letters -- and, unlike the Greeks, they did not invoke other letters to supply the lack (except that a few texts use the terminal forms of the letters with two shapes, but this is reportedly rare). So, for numbers in the high hundreds, they ended up duplicating letters -- e.g. since one tau meant 400, two tau meant 800. Thus, although the basic principle was alphabetic, you still had to count letters to an extent.
The basic set of Hebrew numbers is shown at right.
An interesting and uncertain question is whether this notation preceded,
supplanted, or existed alongside Aramaic numerals.
The Aramaeans seem to have used a basic additive system. The numbers
from one to nine were simple tally marks, usually grouped in threes --
e.g. 5 would be || ||| (read from right to left, of course);
9 would be ||| ||| |||. For 10 they used a curious pothook,
perhaps the remains of a horizontal bar, something like a ∼ or ∩
or 
. They also had
a symbol for 20, apparently based on two of these things stuck together;
the result often looked rather like an Old English yogh
(
) or perhaps
≈. Thus the number 54 would be written
| ||| ∼.
There is archaeological evidence for both forms. Coins of Alexander Jannaeus (first century B.C.E.) use alphabetic numbers. But we find Aramaic numbers among the Dead Sea Scrolls. This raises at least a possibility that the number form one used depended upon one's politics. The Jews at Elephantine (early Persian period) appear to have used Aramaic numbers -- but they of course were exiles, and living in a period before Jews as a whole had adopted Aramaic. On the whole, the evidence probably favors the theory that Aramaic numbering preceded Hebrew, but we cannot be dogmatic. In any case, Hebrew numbers were in use by New Testament times; we note, in fact, that coins of the first Jewish Revolt -- which are of course very nearly contemporary with the New Testament books -- use the Hebrew numerals.
There is perhaps one other point we should make about mathematics,
and that is the timing of the introduction of Arabic numerals. An early
manuscript of course cannot contain such numbers; if it has numerals
(in the Eusebian apparatus, say), they will be Greek (or Roman, or
something). A late minuscule, however, can contain Arabic numbers --
and, indeed, many have pages numbered in this way.
Arabic numerals underwent much change over the years. The graphic at right barely sketches the evolution. The first three samples are based on actual manuscripts (in the first case, based on scans of the actual manuscript; the others are composite).
The first line is from the Codex Vigilanus, generally regarded as the earliest use of Arabic numerals in the west (though it uses only the digits 1-9, not the zero). It was written, not surprisingly, in Spain, which was under Islamic influence. The codex (Escurial, Ms. lat. d.1.2) was copied in 976 C. E. by a monk named Vigila at the Abelda monastery. The next several samples are (based on the table of letterforms in Ifrah) typical of the next few centuries. Following this, I show the evolution of forms described in E. Maunde Thompson, An Introduction to Greek and Latin Paleography, p. 92. Thompson notes that Hindu/Arabic numerals were used mostly in mathematical works until the thirteenth century, becoming universal in the fourteenth century. Singer, p. 175, describes a more complicated path: Initially they were used primarily in connection with the calendar. The adoption of Arabic numerals for mathematics apparently can be credited to one Leonardo of Pisa, who had done business in North Africa and seen the value of the system. He'll perhaps sound more familiar if we note that he was usually called "Fibonacci," the "Son of Bonaccio" -- now famous for his series (0, 1, 1, 2, 3, 5, 8...) in which each term is the sum of the previous two. But his greatest service to mathematics was his support of modern notation. In 1202 he put forth the Book of the Abacus, a manual of calculation (which also promoted the horizontal stroke - to separate the numerators and denominators of fractions, though his usage was, by modern standards, clumsy, and it took centuries for this notation to catch on). The use of Arabic numerals was further encouraged when the Yorkshireman John Holywood (died 1250) produced his own book on the subject, which was quote popular; Singer, p. 173, reports that Holywood "did more to introduce the Arabic notation than any other." Within a couple of centuries, they were commonly used. In Chaucer's Treatise on the Astrolabe I.7, for instance, addressed to his ten-year-old son, he simply refers to them as "noumbers of augrym" -- i.e., in effect, abacus numbers -- and then proceeds to scatter them all through the text.) If someone has determined the earliest instance of Arabic numbers in a Biblical manuscript, I confess I do not know what it is.
Most other modern mathematical symbols are even more recent. The symbols + and - for addition and subtraction, for instance, are first found in print in Johann Widman's 1489 publication Rechnung uff allen Kauffmanschafften. (Prior to that, it was typical to use the letters p and m.) The = sign seems to go back to England's Robert Recorde (died 1558), who published several works dating back to 1541 -- though Recorde's equality symbol was much wider than ours, looking more like ====. The notation became general about a century later. The modern notation of variables (and parameters) can be credited to François Viète (1540-1603), who also pushed for use of decimal notation in fractions and experimented with notations for the radix point (what we tend to call the "decimal point," but it's only a decimal point in Base 10; in Base 2, e.g., it's the binary point. In any case, it's the symbol for the division between whole number and fractional parts -- usually, in modern notation, either a point or a comma).
The table below briefly shows the forms of numerals in some of the languages in which New Testament versions exist. Some of these probably require comment -- e.g. Coptic numerals are theoretically based on the Greek, but they had a certain amount of time to diverge. Observe in particular the use of the chi-rho for 900; I assume this is primarily a Christian usage, but have not seen this documented. Many of the number systems (e.g. the Armenian) have symbols for numbers larger than 900, but I had enough trouble trying to draw these clearly!
Addendum: Textual Criticism of Mathematical Works
Most ancient mathematical documents exist in only a single copy (e.g. the Rhind Papyrus is unique), so any textual criticism must proceed by conjecture. And this is in fact trickier than it sounds. If an ancient document adds, say, 536 and 221 and reaches a total of 758 instead of the correct 757, can we automatically assume the document was copied incorrectly? Not really; while this is a trivial sum using Arabic numerals, there are no trivial sums in most ancient systems; they were just too hard to use!
But the real problems are much deeper. Copying a mathematical manuscript is a tricky proposition indeed. Mathematics has far less redundancy than words do. In words, we have "mis-spellings," e.g., which formally are errors but which usually are transparent. In mathematics -- it's right or it's wrong. And any copying error makes it wrong. And, frequently, you not only have to copy the text accurately, but any drawings. And labels to the drawings. And the text that describes those labels. To do this right requires several things not in the standard scribe's toolkit -- Greek mathematics was built around compass and straight edge, so you had to have a good one of each and the ability to use it. Plus the vocabulary was inevitably specialized.
The manuscripts of Euclid, incidentally, offer a fascinating parallel with the New Testament tradition, especially as the latter was seen by Westcott and Hort. The majority of manuscripts belong to a single type, which we know to be recensional: It was created by the editor Theon. Long after Euclid was rediscovered, a single manuscript was found in the Vatican, containing a text from a different recension. It is generally thought to be earlier. Such papyrus scraps as are available generally support the Vatican manuscript, without by any means agreeing with it completely. Still, it seems clear that the majority text found in Theon has been somewhat smoothed and prettied up, though few of the changes are radical and it sometimes seems to retain the correct text where the Vatican type has gone astray.
Bibliography to the section on Ancient Mathematics
The study of ancient mathematics is difficult; one has to understand language and mathematics, and have the ability to figure out completely alien ways of thinking. I've consulted quite a few books to compile the above (e.g. Chadwick's publications on Linear B for Mycenaean numerals), and read several others in a vain hope of learning something useful, but most of the debt is to five books (which took quite a bit of comparing!). The "select bibliography:"
In addition, if you're interested in textual criticism of mathematical works, you might want to check Thomas L. Heath's translation of Euclid (published by Dover), which includes an extensive discussion of Euclid's text and Theon's recension, as well as a pretty authoritative translation with extensive notes.
"Assuming the solution" is a mathematical term for a particularly vicious fallacy (which can easily occur in textual criticism) in which one assumes something to be true, operates on that basis, and then "proves" that (whatever one assumed) is actually the case. It's much like saying something like "because it is raining, it is raining." It's just fine as long as it is, in fact, actually raining -- but if it isn't, the statement is inaccurate. In any case, it doesn't have any logical value. It is, therefore, one of the most serious charges which can be levelled at a demonstration, because it says that the demonstration is not merely incomplete but is founded on error.
As examples of assuming the solution, we may offer either Von Soden's definition of the I text or Streeter's definition of the "Cæsarean" text. Both, particularly von Soden's, are based on the principle of "any non-Byzantine reading" -- that is, von Soden assumes that any reading which is not Byzantine must be part of the I text, and therefore the witness containing it must also be part of the I text.
The problem with this is that it means that everything can potentially be classified as an I manuscript, including (theoretically) manuscripts which have not a single reading in common at points of variation. It obviously can include manuscripts which agree only in Byzantine readings. This follows from the fact that most readings are binary (that is, only two readings are found in the tradition). One reading will necessarily be Byzantine. Therefore the other is not Byzantine. Therefore, to von Soden, it was an I reading. It doesn't matter where it actually came from, or what sort of reading it is; it's listed as characteristic of I.
This sort of error has been historically very common in textual criticism. Critics must strive vigorously to avoid it -- to be certain they do not take something on faith. Many results of past criticism were founded on assuming the solution (including, e.g., identifying the text of P46 and B with the Alexandrian text in Paul). All such results need to be re-verified using definitions which are not self-referencing.
Note: This is not a blanket condemnation of recognizing manuscripts based on agreements in non-Byzantine readings. That is, Streeter's method of finding the Cæsarean text is not automatically invalid if properly applied. Streeter simply applied it inaccurately -- in two particulars. First, he assumed the Textus Receptus was identical with the Byzantine text. Second, he assumed that any non-Textus Receptus reading was Cæsarean. The first assumption is demonstrably false, and the second too broad. To belong to a text-type, manuscripts must display significant kinship in readings not associated with the Byzantine text. This was not the case for Streeter's secondary and tertiary witnesses, which included everything from A to the purple uncials to 1424. The Cæsarean text must be sought in his primary witnesses (which would, be it noted, be regarded as secondary witnesses in any text-type which included a pure representative): Q 28 565 700 f1 f13 arm geo.
Probability is not a simple matter. The odds of a single event happening do not translate across multiple events. For instance, the fact that a coin has a 50% chance to land heads does not mean that two coins together have a 50% chance of both landing heads. Calculating the odds of such events requires the use of distributions.
The most common distribution in discrete events such as coin tosses or die rolls is the binomial distribution. This distribution allows us to calculate the odds of independent events occurring a fixed number of times. That is, suppose you try an operation n times. What are the odds that the "desired" outcome (call it o) will happen m and only m times? The answer is determined by the binomial distribution.
Observe that the binomial distribution applies only to events where there are two possible outcomes, o and not o. (It can be generalized to cover events with multiple outcomes, but only by clever definition of the event o). The binomial probabilities are calculated as follows:
If n is the number of times a trial is taken, and m is the number of successes, and p(o) is the probability of the event taking place in a single trial, then the probability p(m,n) is given by the formula
where
and where n! (read "n factorial") is defined as 1x2x3x...x(n-1)xn. So, e.g, 4! = 1x2x3x4 = 24, 5! = 1x2x3x4x5 = 120. (Note: For purposes of calculation, the value 0! is defined as 1.)
(Note further: The notation used here, especially the symbol P(m,n), is not universal. Other texts will use different symbols for the various terms.)
The various coefficients of P(m,n) are also those of the well-known "Pascal's Triangle""
0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1
where P(m,n) is item m+1 in row n. For n greater than about six or seven, however, it is usually easier to calculate the terms (known as the "binomial coefficients") using the formula above.
Example: What are the odds of rolling the value one exactly twice if you roll one die ten times? In this case, the odds of rolling a one (what we have called p(o)) are one in six, or about .166667. So we want to calculate
10! 2 (10-2)
P(2,10) = --------- * (.16667) * (1-.16667)
2!*(10-2)!
10*9*8*7*6*5*4*3*2*1 2 8
= ---------------------- * .16667 * .83333
(2*1)*(8*7*6*5*4*3*2*1)
10*9 2 8
= ---- * .16667 * .83333 = 45 * .02778 * .23249 = .2906
2*1
In other words, there is a 29% chance that you will get two ones if you roll the die ten times.
For an application of this to textual criticism, consider a manuscript with a mixed text. Assume (as a simplification) that we have determined (by whatever means) that the manuscript has a text that is two-thirds Alexandrian and one-third Byzantine (i.e., at a place where the Alexandrian and Byzantine text-types diverge, there are two chances in three, or .6667, that the manuscript will have the Alexandrian reading, and one chance in three, or .3333, that the reading will be Byzantine). We assume (an assumption that needs to be tested, of course) that mixture is random. In that case, what are the odds, if we test (say) eight readings, that exactly three will be Byzantine? The procedure is just as above: We calculate:
8! 3 5
P(3,8) = -------- * .3333 * .6667
3!*(8-3)!
8*7*6*5*4*3*2*1 3 5 8*7*6
= ------------------ *.3333 * .6667 = ----- * .0370 * .1317 = .2729
(3*2*1)*(5*4*3*2*1) 3*2*1
In other words, in a random sample of eight readings, there is just over a 27% chance that exactly three will be Byzantine.
We can also apply this over a range of values. For example, we can calculate the odds that, in a sample of eight readings, between two and four will be Byzantine. One way to do this is to calculate values of two, three, and four readings. We have already calculated the value for three. Doing the calculations (without belabouring them as above) gives us
P(2,8) = .2731
P(4,8) = .1701
So if we add these up, the probability of 2, 3, or 4 Byzantine readings is .2729+.2731+.1701 = .7161. In other words, there is nearly a 72% chance that, in our sample of eight readings, between two and four readings will be Byzantine. By symmetry, this means that there is just over a 29% chance that there will be fewer than two, or more than four, Byzantine readings.
We can, in fact, verify this and check our calculations by determining all values.
| Function | Value |
| P(0,8) | .0390 |
| P(1,8) | .1561 |
| P(2,8) | .2731 |
| P(3,8) | .2729 |
| P(4,8) | .1701 |
| P(5,8) | .0680 |
| P(6,8) | .0174 |
| P(7,8) | .0024 |
| P(8,8) | .0002 |
Observe that, if we add up all these terms, they sum to .9992 (which is as good an approximation of 1 as we can expect with these figures; the difference is roundoff and computational imperfection. Chances are that we don't have four significant digits of accuracy in our figures anyway; see the section on Accuracy and Precision.)
(It is perhaps worth noting that binomials do not have to use only two items, or only equal probabilities. All that is required is that the probabilities add up to 1. So if we were examining the so-called "Triple Readings" of Hutton, which are readings where Alexandrian, Byzantine, and "Western" texts have distinct readings, we might find that 90% of manuscripts have the Byzantine reading, 8% have the Alexandrian, and 2% the "Western." We could then apply binomials in this case, calculating the odds of a reading being Alexandrian or non-Alexandrian, Byzantine or non-Byzantine, "Western" or non-Western. We must, however, be very aware of the difficulties here. The key one is that the "triple readings" are both rare and insufficiently controlled. In other words, they do not constitute anything remotely resembling a random variable.)
The Binomial Distribution has other interesting properties. For instance, it can be shown that the Mean of the distribution is given by
m = np
(So, for instance, in our example above, where n=8 and p=.33333, the mean, or the average number of Byzantine readings we would expect if we took many, many tests of eight readings, is 8*.33333, or 2.6667.)
Similarly, the variance is given by
s2 = np(1-p)
while the standard deviation s is, of course, the square root of the above.
Our next point is perhaps best made graphically. Let's make a plot of the values given above for P(n,8) in the case of a manuscript two-thirds Alexandrian, one-third Byzantine.
* *
* *
* *
* * *
* * * *
* * * *
* * * *
* * * * *
* * * * * * *
-------------------------
0 1 2 3 4 5 6 7 8
This graph is, obviously, not symmetric. But let's change things again. Suppose, instead of using p(o)=.3333, we use p(o)=.5. Then our table is as follows:
| Function | Value |
| P(0,8) | .0039 |
| P(1,8) | .0313 |
| P(2,8) | .1094 |
| P(3,8) | .2188 |
| P(4,8) | .2734 |
| P(5,8) | .2188 |
| P(6,8) | .1094 |
| P(7,8) | .0313 |
| P(8,8) | .0039 |
Our graph then becomes:
*
*
* * *
* * *
* * * * *
* * * * *
* * * * * * *
-------------------------
0 1 2 3 4 5 6 7 8
This graph is obviously symmetric. More importantly (though it is perhaps not obvious with such a crude graph and so few points), it resembles a sketch of the so-called "bell-shaped" or "normal" curve:
It can, in fact, be shown that the one is an approximation of the other. The proof is sufficiently complex, however, that even probability texts don't get into it; certainly we won't burden you with it here!
We should note at the outset that the "normal distribution" has no direct application to NT criticism. This is because the normal distribution is continuous rather than discrete. That is, it applies at any value at all -- you have a certain probability at 1, or, 2, or 3.8249246 or the square root of 3307 over pi. A discrete distribution applies only at fixed values, usually integers. But NT criticism deals with discrete units -- a variant here, a variant there. Although these variants are myriad, they are still countable and discrete.
But this is often the case in dealing with real-world distributions which approximate the normal distribution. Because the behavior of the normal distribution is known and well-defined, we can use it to model the behavior of a discrete distribution which approximates it.
The general formula for a normal distribution, centered around the mean m and with standard deviation s, is given by
This means that it is possible to approximate the value of the binomial distribution for a series of points by calculating the area of the equivalent normal distribution between corresponding points.
Unfortunately, this latter cannot be reduced to a simple formula (for those who care, it is an integral without a closed-form solution). The results generally have to be read from a table (unless one has a calculator with the appropriate statistical functions). Such tables, and information on how to use them, are found in all modern statistics books.
It's worth asking if textual distributions follow anything resembling a normal
curve. This, to my knowledge, has never been investigated in any way. And
this point becomes very important in assessing such things as the so-called
"Colwell rule" (see the section on E. C. Colwell & Ernest W. Tune:
"Method in Establishing Quantitative
Relationships Between Text-Types of New Testament Manuscripts.")
This is a perfectly reasonable dissertation for someone -- taking a significant
group of manuscripts and comparing their relationships over a number of samples.
We shall only do a handful, as an example. For this, we use the data from
Larry W. Hurtado, Text-Critical Methodology and the Pre-Caesarean Text: Codex W in
the Gospel of Mark. We'll take the three sets of texts which he finds clearly
related: 
and
B, A and the TR, Q and 565.
Summarizing Hurtado's data gives us the following (we omit Hurtado's decimal digit, as he does not have enough data to allow three significant digits):
| Chapter | % of with B
| % of A with TR | % of Q with 565 |
| 1 | 73 | 88 | 55 |
| 2 | 71 | 89 | 55 |
| 3 | 78 | 80 | 64 |
| 4 | 79 | 88 | 77 |
| 5 | 80 | 73 | 54 |
| 6 | 81 | 88 | 56 |
| 7 | 81 | 94 | 70 |
| 8 | 83 | 91 | 78 |
| 9 | 86 | 89 | 64 |
| 10 | 77 | 85 | 75 |
| 11 | 82 | 85 | 67 |
| 12 | 78 | 87 | 77 |
| 13 | 78 | 90 | 77 |
| 14 | 83 | 84 | 75 |
| 15-16:8 | 75 | 92 | 80 |
| MEAN | 79.0 | 86.9 | 68.3 |
| STD DEV | 4.0 | 5.2 | 9.6 |
| MEDIAN | 79 | 88 | 70 |
Let's graph each of these as variations around the mean. That is, let's count how many elements are within half a standard deviation (s) of the mean m, and how many are in the region one standard deviation beyond that, and so forth.
For and B,
m is 79 and s is 4.0. So:
%agree < m-1.5s, i. e. % < 73 |*
m-1.5s < %agree < m-.5s, i.e. 73 <= % < 77 |**
m-.5s < %agree < m+.5s, i.e. 77 <= % <= 81 |********
m+.5s < %agree < m+1.5s, i.e. 81 < % <= 85 |***
%agree > M+1.5s, i.e. % > 85 |*
For A and TR, m is 86.9 and s is 5.2. So:
%agree < m-1.5s, i. e. % < 80 |*
m-1.5s < %agree < m-.5s, i.e. 80 <= % < 85 |**
m-.5s < %agree < m+.5s, i.e. 85 <= % <= 90 |*********
m+.5s < %agree < m+1.5s, i.e. 90 < % <= 95 |***
%agree > M+1.5s, i.e. % > 90 |
For Q and 565, m is 70 and s is 9.6. So:
%agree < m-1.5s, i. e. % < 55 |*
m-1.5s < %agree < m-.5s, i.e. 55 <= % < 66 |*****
m-.5s < %agree < m+.5s, i.e. 66 <= % <= 74 |**
m+.5s < %agree < m+1.5s, i.e. 74 < % <= 84 |*******
%agree > M+1.5s, i.e. % > 84 |
With only very preliminary results, it's hard to draw conclusions. The first two graphs do look normal. The third looks just plain strange. This is not anything like a binomial/normal distribution. The strong implication is that one or the other of these manuscripts is block-mixed.
This hints that distribution analysis might be a useful tool in assessing textual kinship. But this is only a very tentative result; we must test it by, e.g., looking at manuscripts of different Byzantine subgroups.
WARNING: Cladistics is a mathematical discipline arising out of the needs of evolutionary biology. It should be recalled, however, that mathematics is independent of its uses. The fact that cladistics is useful in biology should not cause prejudice against it; it has since been applied to other fields. For purposes of illustration, however, I will use evolutionary examples because they're what is found in all the literature.
A further warning: I knew nothing about cladistics before Stephen C. Carlson began to discuss the matter with reference to textual criticism. I am still not expert. You will not learn cladistics from this article; the field is too broad. The goal of this article is not to teach cladistics but to explain generally what it does.
Consider a problem: Are dolphins and fish related?
At first glance, it would certainly seem so. After all, both are streamlined creatures, living in water, with fins, which use motions of their lower bodies to propel themselves.
And yet, fish reproduce by laying eggs, while dolphins produce live young. Fish breathe water through gills; dolphins breathe air through lungs. Fish are cold-blooded; dolphins are warm-blooded. Fish do not produce milk for their young; dolphins do.
Based on the latter characteristics, dolphins would seem to have more in common with rabbits or cattle or humans than with fish. So how do we decide if dolphins are fish-like or rabbit-like? This is the purpose of cladistics: Based on a variety of characteristics (be it the egg-laying habits of a species or the readings of a manuscript), to determine which populations are related, and how.
Biologists have long believed that dolphins are more closely related to the other mammals, not the fish. The characteristics shared with the mammals go back to the "ur-mammal"; the physical similarities to fish are incidental. (The technical term is an "analogous feature" or a "homoplasy." Cases of similar characteristics which derive from common ancestry are called "homologous features" or "homologies.")
This is the point at which textual critics become interested, because kinship based on homology is very similar to the stemmatic concept of agreement in error. Example: Turtles and lizards and horses all have four legs. Humans and chimpanzees have two arms and two legs -- and robins and crows also have only two legs. Are we more like robins or horses? Answer: Like horses. Four legs is the "default mode"; for amphibians, reptiles, and mammals; the separation into distinct arms and legs is a recent adaption -- not, in this case, an error, but a divergence from the original stock. This is true even though birds, like humans, also have two legs and two limbs which are not legs. Similarly, a text can develop homoplasies: assimilation of parallels, h.t. errors, and expansion of epithets are all cases where agreement in reading can be the result of coincidence rather than common origin.
Cladistics proceeds by examining each points of variation, and trying to find the "optimum tree." ("Optimum" meaning, more or less, "simplest.") For this we can take a New Testament example. Let's look at Mark 3:16 and the disciple called either Lebbaeus or Thaddaeus. Taking as our witnesses A B D E L, we find that D reads Lebbaeus, while A B E L read Thaddaeus. That gives us a nice simple tree (though this isn't the way you'll usually see it in a biological stemma):
-----------*----- | | | | | A B E L D
Which in context is equivalent to
Autograph
|
-----------*-----
| | | | |
A B E L D
The point shown by * is a node -- a point of divergence. At this point in the evolution of the manuscripts, something changed. In this case, this is the point at which D (or, perhaps, A B E L) split off from the main tree.
This, obviously, is very much like an ordinary stemma, which would express the same thing as
Autograph
|
--------------
| |
X Y
| |
---------- |
| | | | |
A B E L D
But now take the very next variant in the Nestle/Aland text: Canaanite vs. Canaanean. Here we find A and E reading Canaanite, while B D L have Canaanean. That produces a different view:
----------*------ | | | | | B D L A E
Now we know, informally, that the explanation for this is that B and L are Alexandrian, A and E Byzantine, and D "Western." But the idea is to verify that. And to extend it to larger data sets, and cases where the data is more mixed up. This is where cladistics comes in. Put very simply, it takes all the possible trees for a set of data, identifies possible nodes, and looks for the simplest tree capable of explaining the data. With only our two variants, it's not easy to demonstrate this concept -- but we'll try.
There are actually four possible trees capable of explaining the above data:
Autograph
:
----*----*---- i.e. ----*----*----
| | | | | | | | | |
B L D A E B L D A E
Autograph
:
--*---*----*---- i.e. --*---*----*----
| | | | | | | | | |
B L D A E B L D A E
Autograph
:
----*----*---*-- i.e. ----*----*---*--
| | | | | | | | | |
B L D A E B L D A E
Autograph
:
--*---*----*---*-- i.e. --*---*----*---*--
| | | | | | | | | |
B L D A E B L D A E
To explain: The first diagram, with two nodes, defines three families, B+L, D, and A+E. The second, with three nodes, defines four families: B, L, D, and A+E. The third, also with three nodes, has four families, but not the same four: B+L, D, A, E. The last, with four nodes, has five families: B, L, D, A, E.
In this case, it is obvious that the first design, with only two nodes, is the simplest. It also corresponds to our sense of what is actually happening. This is why people trust cladistics.
But while we could detect the simplest tree in this case by inspection, it's not that simple as the trees get more complex. There are two tasks: Creating the trees, and determining which is simplest.
This is where the math gets hairy. You can't just look at all the trees by brute force; it's difficult to generate them, and even harder to test them. (This is the real problem with classical stemmatics: It's not in any way exhaustive, even when it's objective. How do we know this? By the sheer number of possibilities. Suppose you have fifty manuscripts, and any one can be directly descended from two others -- an original and a corrector. Thus for any one manuscript, it can have any of 49 possible originals and, for each original, 49 possible correctors [the other 48 manuscripts plus no corrector at all]. That's 2401 linkages just for that manuscript. And we have fifty of them! An informal examination of one of Stephen C. Carlson's cladograms shows 49 actual manuscripts -- plus 27 hypothesized manuscripts and a total of 92 links between manuscripts!) So there is just too much data to assess to make "brute force" a workable method. And, other than brute force, there is no absolutely assured method for finding the best tree. This means that, in a situation like that for the New Testament, we simply don't have the computational power yet to guarantee the optimal tree.
Plus there is the possibility that multiple trees can satisfy the data, as we saw above. Cladistics cannot prove that its chosen tree is the correct tree, only that it is the simplest of those examined. It is, in a sense, Ockham's Razor turned into a mathematical tool.
Does this lack of absolute certainty render cladistics useless? By no means; it is the best available mathematical tool for assessing stemmatic data. But we need to understand what it is, and what it is not. Cladistics, as used in biology, applies to group characteristics (a large or a small beak, red or green skin color, etc.) and processes (the evolution of species). The history of the text applies to a very different set of data. Instead of species and groups of species, it deals with individual manuscripts. Instead of characteristics of large groups within a species, we are looking at particular readings. Evolution proceeds by groups, over many, many generations. Manuscript copying proceeds one manuscript at a time, and for all the tens of thousands of manuscripts and dozens of generations between surviving manuscripts, it is a smaller, more compact tradition than an evolutionary tree.
An important point, often made in the literature, is that the results of cladistics can prove non-intuitive. The entities which "seem" most closely related may not prove to be so. (This certainly has been the case with Stephen C. Carlson's preliminary attempts, which by and large confirm my own results on the lower levels of textual grouping -- including finding many groups not previously published by any other scholars. But Carlson's larger textual groupings, if validated by larger studies, will probably force a significant reevaluation of our assessments of text-types.) This should not raise objections among textual critics; the situation is analogous to one Colwell described (Studied in Methodology, p. 33): "Weak members of a Text-type may contain no more of the total content of a text-type than strong members of some other text-type may contain. The comparison in total agreements of one manuscript with another manuscript has little significance beyond that of confirmation, and then only if the agreement is large enough to be distinctive."
There are other complications, as well. A big one is mixture. You don't see hawks breeding with owls; once they developed into separate species, that was it. There are no joins, only splits. But manuscripts can join. One manuscript of one type can be corrected against another. This means that the tree doesn't just produce "splits" (A is the father of B and C, B is the father of D and E, etc.) but also "joins" (A is the offspring of a mixture of X and Y, etc.) This results in vastly more complicated linkages -- and this is an area mathematicians have not really explored in detail.
Another key point is that cladograms -- the diagrams produced by cladistics -- are not stemma. Above, I called them trees, but they aren't. They aren't "rooted" -- i.e. we don't know where things start. In the case of the trees I showed for Mark, we know that none of the manuscripts is the autograph, so they have to be descendant. But this is not generally true, and in fact we can't even assume it for a cladogram of the NT. A cladogram -- particularly one for something as interrelated as the NT -- is not really a "tree" but more of a web. It's a set of connections, but the connections don't have a direction or starting point. Think, by analogy, of the hexagon below:
If you think of the red dots at the vertices (nodes) as manuscripts, it's obvious what the relationship between each manuscript is: It's linked to three others. But how do you tell where the first manuscript is? Where do you start?
Cladistics can offer no answer to this. In the case of NT stemma, it appears that most of the earliest manuscripts are within a few nodes of each other, implying that the autograph is somewhere near there. But this is not proof.
Great care, in fact, must be taken to avoid reading too much into a cladogram. Take the example we used above, of A, B, D, E, L. A possible cladogram of this tree would look like
/\
/ \
/ \
/ /\
/ / \
/ \ / / \
B L D A E
This cladogram, if you just glance at it, would seem to imply that D (i.e. the "Western" text) falls much closer to A and E (the Byzantine text) than to B and L (the Alexandrian text), and that the original text is to be found by comparing the Alexandrian text to the consensus of the other two. However, this cladogram is exactly equivalent to
/\
/ \
/ \
/ \ \
/ \ \
/ \ \ / \
B L D A E
And this diagram would seem to imply that D goes more closely with the Alexandrian text. Neither (based on our data) is true; the three are, as best we can tell, completely independent. The key is not the shape of the diagram but the location of the nodes. In the first, our nodes are at
*\
/ \
/ \
/ /*
/ / \
/ \ / / \
B L D A E
In the second, it's
/*
/ \
/ \
* \ \
/ \ \
/ \ \ / \
B L D A E
But it's the same tree, differently drawn. The implications are false inferences based on an illusion in the way the trees are drawn.
We note, incidentally, that the relations we've drawn as trees
or stemmas can be drawn "inline," with a sort of a modified set
theory notation. In this notation, square brackets [] indicate a relation
or a branch point. For example, the above stemma would be
[ [ B L ] D [ A E ] ]
This shows, without ambiguity of branch points, that B and L go together, as do A and E, with D rather more distant from both.
This notation can be extended. For example, it is generally agreed
that, within the Byzantine text, the uncials E F G H are more closely
related to each other than they are to A; K and
P are closely
related to each other, less closely to A, less closely still to E F G H.
So, if we add F G H K
P
to the above notation, we get
[[B L] D [[A [K P]]
[E F G H]]]
It will be evident that this gets confusing fast. Although the notation is unequivocal, it's hard to convert it to a tree in one's mind. And, with this notation, there is no possibility of describing mixture, which can be shown with a stemmatic diagram, if sometimes a rather complex one.
Cladistics is a field that is evolving rapidly, and new methods and applications are being found regularly. I've made no attempt to outline the methods for this reason (well, that reason, and because I don't fully understand it myself, and because the subject really requires more space than I can reasonably devote). To this point, the leading exponent of cladistics in NT criticism is Stephen C. Carlson, who has been evolving new methods to adapt the discipline to TC circumstances. I cannot comprehensively assess his math, but I have seen his preliminary results, and am impressed.
In mathematical jargon, a corollary is a result that follows immediately from another result. Typically it is a more specific case of a general rule. An elementary example of this might be as follows:
Theorem: 0 is the "additive identity." That is, for any x, x+0=x.
Corollary: 1+0=1
This is a very obvious example, but the concept has value, as it allows logical simplification of the rules we use. For example, there are quite a few rules of internal criticism offered by textual critics. All of these, however, are special cases of the rule "That reading is best which best explains the others." That is, they are corollaries of this rule. Take, for example, the rule "Prefer the harder reading." Why should one prefer the harder reading? Because it is easier to assume that a scribe would change a hard reading to an easy one. In other words, the hard reading explains the easy. Thus we prove that the rule "Prefer the harder reading" is a corollary of "That reading is best which best explains the others." QED. (Yes, you just witnessed a logical proof. Of course, we did rather lightly glide by some underlying assumptions....)
Why do we care about what is and is not a corollary? Among other things, because it tells us when we should and should not apply rules. For example, in the case of "prefer the harder reading," the fact that it is a corollary reminds us that it applies only when we are looking at internal evidence. The rule does not apply to cases of clear errors in manuscripts (which are a province of external evidence).
Let's take another corollary of the rule "That reading is best which best explains the others." In this case, let's examine "Prefer the shorter reading." This rule is applied in all sorts of cases. It should only be applied when scribal error or simplification can be ruled out -- as would be obvious if we examine the situation in light of "That reading is best which best explains the others."
It may seem odd to discuss the word "definition" in a section on mathematics. After all, we all know what a definition is, right -- it's a way to tell what a word or term means.
Well, yes and no. That's the informal definition of definition. But that's not a sufficient description.
Consider this "definition": "The Byzantine text is the text typically found in New Testament manuscripts."
In a way, that's correct -- though it might serve better as a definition of the "Majority Text." But while, informally, it tells us what we're talking about, it's really not sufficient. How typical is "typical?" Does a reading supported by 95% of the tradition qualify? It certainly ought to. How about one supported by 75%? Probably, though it's less clear. 55%? By no means obvious. What about one supported by 40% when no other reading is supported by more than 30% of the tradition? Uh....
And how many manuscripts must we survey to decide what fraction of the tradition is involved, anyway? Are a few manuscripts sufficient, or must we survey dozens or hundreds?
To be usable in research settings, the first requirement for a definition is that it be precise. So, for instance, a precise definition of the Majority Text might be the text found in at least 50% plus one of all manuscripts of a particular passage. Alternately, and more practically, the Majority Text might be defined as In the gospels, the reading found in the most witnesses of the test group A E K M S V 876 1010 1424. This may not be "the" Majority reading, but it's likely that it is. And, of great importance, this definition can be applied without undue effort, and is absolutely precise: It always admits one and only one reading (though there will be passages where, due to lacunose or widely divergent witnesses, it will not define a particular reading).
But a definition may be precise without being useful. For example, we could define the Byzantine text as follows: The plurality reading of all manuscripts written after the year 325 C. E. within 125 kilometers of the present site of the Hagia Sophia in Constantinople. This definition is relentlessly precise: It defines one and only one reading everywhere in the New Testament (and, for that matter, in the Old, and in classical works such as the Iliad). The problem is, we can't tell what that reading is! Even among surviving manuscripts, we can't tell which were written within the specified distance of Constantinople, and of course the definition, as stated, also includes lost manuscripts! Thus this definition of the Byzantine text, while formally excellent, is something we can't work with in practice.
Thus a proper definition must always meet two criteria: It must be precise and it must be applicable.
I can hear you saying, Sure, in math, they need good definitions. But we're textual critics. Does this matter? That is, do we really care, in textual criticism, if a definition is precise and applicable?
The answer is assuredly yes. Failure to apply both precise and applicable definitions is almost certain to be fatal to good method. An example is the infamous "Cæsarean" text, Streeter's definition was, in simplest terms, any non-Textus Receptus reading found in two or more "Cæsarean" witnesses. This definition is adequately precise. It is nonetheless fatally flawed in context, for three reasons: First, it's circular; second, the TR is not the Byzantine text, so in fact many of Streeter's "Cæsarean" readings are in fact nothing more nor less than Byzantine readings; third, most readings are binary, so one reading will always agree with the TR and one will not, meaning that every manuscript except the TR will show up, by his method, as "Cæsarean!"
An example of a definition that isn't even precise is offered by Harry Sturz. He defined (or, rather, failed to define) the Byzantine text as being the same as individual Byzantine readings! In other words, Sturz showed that certain Byzantine readings were in existence before the alleged fourth century recension that produced the Byzantine text. (Which, be it noted, no one ever denied!) From this he alleged that the Byzantine text as a whole is old. This is purely fallacious (not wrong, necessarily, but fallacious; you can't make that step based on the data) -- but Sturz, because he didn't have a precise definition of the Byzantine text, thought he could do it.
The moral of the story is clear and undeniable: If you wish to work with factual data (i.e. if you want to produce statistics, or even just generalizations, about external evidence), you must start with precise and applicable definitions.
THIS MEANS YOU. Yes, YOU. (And me, and everyone else, of course. But the point is the basis of all scientific work: Definitions must be unequivocal.)
Also known as, Getting the units right!
Have you ever heard someone say something like "That's at least a light-year from now?" Such statements make physicists cringe. A light-year is a unit of distance (the distance light travels in a year), not of time.
Improper use of units leads to meaningless results, and correct use of units can be used to verify results.
As an example, consider this: The unit of mass is (mass). The unit of acceleration is (distance)/(time)/(time). The unit of force is (mass)(distance)/(time)/(time). So the product of mass times acceleration is (mass)(distance)/(time)/(time) -- which happens to be the same as the unit of force. And lo and behold, Newton's second law states that force equals mass times acceleration. And that means that if a results does not have the units of force (mass times distance divided by time squared, so for instance kilograms times metres divided by seconds squared, or slugs times feet divided by hours squared), it is not a force.
This may sound irrelevant to a textual critic, but it is not. Suppose you want to estimate, say, the number of letters in the extant New Testament portion of B. How are you going to do it? Presumably by estimating the amount of text per page, and then multiplying by the number of pages. But that, in fact, is dimensional analysis: letters per page times pages per volume equals letters per volume. We can express this as an equation to demonstrate the point:
letters pages letterspagesletters ------- * ------ = ------- * ------ = ------- pages volumepagesvolume volume
We can make things even simpler: Instead of counting letters per page, we can count letters per line, lines per column, and columns per page. This time let us work the actual example. B has the following characteristics:
So:
pages columns lines letters
142 ------ * 3 ------- * 42 ------ * 16 ------- =
volume page column line
pages columns lines letters
142*3*42*16 * ------ * ------- * ------ * ------- =
volume page column line
pages columns lines letters
286272 * ------ * ------- * ------ * ------- =
volume page column line
286272 letters/volume (approximately)
This, properly, is a rule of logic, not mathematics, but it is a source of many logical fallacies. The law of the excluded middle is a method of simplifying problems. It reduces problems to one of two possible "states." For example, the law of the excluded middle tells us that a reading is either original or not original; there are no "somewhat original" readings. (In actual fact, of course, there is some fuzziness here, as e.g. readings in the original collection of Paul's writings as opposed to the reading in the original separate epistles. But this is a matter of definition of the "original." A reading will either agree with that original, whatever it is, or will disagree.)
The problem with the law of the excluded middle lies in applying it too strongly. Very many fallacies occur in pairs, in cases where there are two polar opposites and the truth falls somewhere in between. An obvious example is the fallacy of number. Since it has repeatedly been shown that you can't "count noses" -- i.e. that the majority is not automatically right -- there are some who go to the opposite extreme and claim that numbers mean nothing. This extreme may be worse than the other, as it means one can simply ignore the manuscripts. Any reading in any manuscript -- or even a conjecture, found in none -- may be correct. This is the logical converse of the majority position.
The truth unquestionably lies somewhere in between. Counting noses -- even counting noses of text-types -- is not the whole answer. But counting does have value, especially at higher levels of abstraction such as text-types or sub-text-types. All other things being equal, the reading found in the majority of text-types must surely be considered more probable than the one in the minority. And within text-types, the reading found within the most sub-text-types will be original. And so on, down the line. One must weight manuscripts, not count them -- but once they are weighed, their numbers have meaning.
Other paired fallacies include excessive stress on internal evidence (which, if taken to its extreme, allows the critic to simply write his own text) or external evidence (which, taken to its extreme, would include clear errors in the text) and over/under-reliance on certain forms of evidence (e.g. Boismard would adopt readings solely based on silence in fathers, clearly placing too much emphasis on the fathers, while others ignore their evidence entirely. We see much the same range of attitude toward the versions. Some would adopt readings based solely on versional evidence, while others will not even accept evidence from so-called secondary versions such as Armenian and Georgian).
Much of the material in this article parallels that in the section on Arithmetic, Exponential, and Geometric Progressions, but perhaps it should be given its own section to demonstrate the power of exponential growth.
The technical definition of an exponential curve is a function of the form
y=ax
where a is a positive constant. When a is greater than one, the result is exponential growth.
To show you how fast exponential growth can grow, here are some results of the function for various values of a
| a=2 | a=3 | a=5 | a=10 | |
| x=1 | 2 | 3 | 5 | 10 |
| x=2 | 4 | 9 | 25 | 100 |
| x=3 | 8 | 27 | 125 | 1000 |
| x=4 | 16 | 81 | 625 | 10000 |
| x=5 | 32 | 243 | 3125 | 100000 |
| x=6 | 64 | 729 | 15625 | 1000000 |
| x=7 | 128 | 2187 | 78125 | 10000000 |
| x=8 | 256 | 6561 | 390625 | 100000000 |
It will be seen that an exponential growth curve can grow very quickly!
This is what makes exponential growth potentially of significance for textual critics: It represents one possible model of manuscript reproduction. The model is to assume each manuscript is copied a certain number of times in a generation, then destroyed. In that case, the constant a above represents the number of copies made. x represents the number of generations. y represents the number of surviving copies.
Why does this matter? Because a small change in the value of the constant a can have dramatic effects. Let's demonstrate this by demolishing the argument of the Byzantine Prioritists that numeric preponderance means something. The only thing it necessarily means is that the Byzantine text had a constant a that is large enough to keep it alive.
For these purposes, let us assume that the Alexandrian text is the original, in circulation by 100 C.E.. Assume it has a reproductive constant of 1.2. (I'm pulling these numbers out of my head, be it noted; I have no evidence that this resembles the actual situation. This is a demonstration, not an actual model.) We'll assume a manuscript "generation" of 25 years. So in the year 100 x=0. The year 125 corresponds to x=1, etc. Our second assumption is that the Byzantine text came into existence in the year 350 (x=10), but that it has a reproductive constant of 1.4.
If we make those assumptions, we get these results for the number of manuscripts at each given date:
| generation | year | Alexandrian manuscripts | Byzantine manuscripts | ratio, Byzantine to Alexandrian mss. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 0 | 100 | 1.2 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1 | 125 | 1.4 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2 | 150 | 1.7 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 3 | 175 | 2.1 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4 | 200 | 2.5 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 5 | 225 | 3.0 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 6 | 250 | 3.6 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 7 | 275 | 4.3 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 8 | 300 | 5.2 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 9 | 325 | 6.2 | -- | 0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 10 | 350 | 7.4 | 1.4 | 0.2:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 11 | 375 | 8.9 | 2.0 | 0.2:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 12 | 400 | 10.7 | 2.7 | 0.3:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 13 | 425 | 12.8 | 3.8 | 0.3:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 14 | 450 | 15.4 | 5.4 | 0.3:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 15 | 475 | 18.5 | 7.5 | 0.4:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16 | 500 | 22.2 | 10.5 | 0.5:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17 | 525 | 26.6 | 14.8 | 0.6:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 18 | 550 | 31.9 | 20.7 | 0.6:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19 | 575 | 38.3 | 28.9 | 0.8:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 20 | 600 | 46 | 40.5 | 0.9:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 21 | 625 | 55.2 | 56.7 | 1.0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 22 | 650 | 66.2 | 79.4 | 1.2:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23 | 675 | 79.5 | 111.1 | 1.4:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 24 | 700 | 95.4 | 155.6 | 1.6:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25 | 725 | 114.5 | 217.8 | 1.9:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26 | 750 | 137.4 | 304.9 | 2.2:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 27 | 775 | 164.8 | 426.9 | 2.6:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28 | 800 | 197.8 | 597.6 | 3.0:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 29 | 825 | 237.4 | 836.7 | 3.5:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 30 | 850 | 284.9 | 1171.4 | 4.1:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31 | 875 | 341.8 | 1639.9 | 4.8:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 32 | 900 | 410.2 | 2295.9 | 5.6:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33 | 925 | 492.2 | 3214.2 | 6.5:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 34 | 950 | 590.7 | 4499.9 | 7.6:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 35 | 975 | 708.8 | 6299.8 | 8.9:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 36 | 1000 | 850.6 | 8819.8 | 10.4:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 37 | 1025 | 1020.7 | 12347.7 | 12.1:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 38 | 1050 | 1224.8 | 17286.7 | 14.1:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 39 | 1075 | 1469.8 | 24201.4 | 16.5:1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 40 | 1100 | 1763.7 | 33882.0 | 19.2:1
The first column, "generation," counts the generations from the year 100. The second column, "year," gives the year. The next two columns, "Alexandrian manuscripts" and " Byzantine manuscripts," give the number of manuscripts of each type we could expect at that particular time. (Yes, we get fractions of manuscripts. Again, this is a model!) The final column, the "ratio," tells us how many Byzantine manuscripts there are for each Alexandrian manuscript. For the first 250 years, there are no Byzantine manuscripts. For a couple of centuries after that, Byzantine manuscripts start to exist, but are outnumbered. But by 625 -- a mere 275 years after the type came into existence -- they are as numerous (in fact, slightly more numerous) than Alexandrian manuscripts. By the year 800, when the type is only 450 years old, constitutes three-quarters of the manuscripts. By the year 1000, it has more than a 10:1 dominance, and it just keeps growing. This doesn't prove that the Byzantine type came to dominate by means of being faster to breed. All the numbers above are made up. The point is, exponential growth -- which is the model for populations allowed to reproduce without constraint -- can allow a fast-breeding population to overtake a slower-breeding population even if the slow-breeding population has a head start. We can show this another way, by modelling extinction. Suppose we start with a population of 1000 (be it manuscripts or members of a species or speakers of a language). We'll divide them into two camps. Call them "A" and "B" for Alexandrian and Byzantine -- but it could just as well be Neandertals and modern humans, or Russian and non-Russian speakers in one of the boundary areas of Russia. We'll start with 500 of A and 500 of B, but give A a reproductive rate of 1.1 and B a reproductive rate of 1.15. And remember, we're constraining the population. That is, at the end of each generation, there can still only be 1000 individuals. All that changes is the ratio of individuals. We will also assume that there must be at least 100 individuals to be sustainable. In other words, once one or the other population falls below 100, it goes extinct and the other text-type/species/language takes over. So here are the numbers:
Observe that it takes only 26 generations for Population A to die out. How fast the die-off takes depends of course on the difference in breeding rates. But 26 generations of (say) dodos is only 26 years, and for people it's only 500-800 years. It may be argued that a difference in breeding rate of 1.1 versus 1.2 is large. This is true. But exponential growth will always dominate in the end. Let's take a few other numbers to show this point. If we hold B's rate of increase to 1.2, and set various values for A's rate of population increase, the table below shows how many generations it takes for A to go extinct.
Note the first column, comparing a reproductive rate for A of 1.19 with a rate of 1.2 for B. That's only a 5% difference. Population A still goes extinct in 264 generations -- if this were a human population, that would be about 6000 years. (There is an interesting point here: Right now, as best I can tell looking at the statistics, liberals have about half as many children as conservatives. And there is evidence that political leanings are inherite. If that trend continues, pure demographics means that liberals will be extinct in about the year 2450! Of course, if conservatives keep breeding at their present rate, the whole human race will likely be extinct by 2450, because the earth simply doesn't have the available calories to feed that large a population....) In any case, to return to something less controversial than political genetics, the power of exponential growth cannot be denied. Any population with a high growth rate can outpace any population with a slow growth rate, no matter how big the initial advantage of the former. One cannot look at current numbers of a population and predict past numbers, unless one knows the growth factor. Game TheoryAs far as I know, there is no working connection between game theory and textual criticism. I do not think there can be one. But I know someone who hoped to find one, so I suppose I should discuss the topic here. And I find it very interesting, so I'm going to cover it in enough depth to let you perhaps do some useful work -- or at least realize why it's useless for textual criticism. Although this field of mathematics is called "game theory," a better name might be something like "strategy theory." The purpose is to examine strategies and outcomes under situations with rigid rules. These situations may be genuine games, such as tic-tac-toe -- but they may equally be real-world situations such as buying and selling stocks, or even deciding whether to launch a nuclear war. The rules apply in all cases. Indeed, the economics case is arguably the most important; several Nobel prizes have been awarded for applications of game theory to market situations. Game theory is a relatively new field in mathematics; it first came into being in the works of John von Neumann, whose proof of the minimax theorem in 1926 gave the field its first foundations; von Neumann's 1944 Theory of Games and Economic Behavior is considered the foundation of the field. (There are mentions of "game theory" before that, and even some French research in the field, but it was von Neumann who really founded it as a discipline. For the record, an informal statement of the minimax theorem is that, if two "players" have completely opposed interests -- that is, if they're in a situation where one wins if and only if the other loses -- then there is always a rational course of action for both players: A best strategy. Not all games meet this standard -- e.g. if two competing companies are trying to bolster their stock prices, a rising stock market can allow them both to win -- but those that do can often illustrate even the cases that don't meet the criterion. The minimax theorem doesn't say those other games don't have best strategies, after all -- it's just that it isn't guaranteed.) To try to give an idea of what game theory is like, let's look at a problem I first met in The Lonely Monk and Other Puzzles. It shows up in many forms, so I'll tell this my own way. A mafia boss suspects that one of his hit men, Alonzo, may have been cheating him, and puts him under close guard. A week later, he discovers that Bertrand might have been in the plot, and hands him over to the guard also. Finally, evidence turns up against Cesar. At this point, the boss decides it's time to make an example. He decides to stage a Trial by Ordeal, with the three fighting to the finish. Alonzo, however, has been in custody for two weeks, and has been severely debilitated; once a crack shot, he now can hit a target only one time in three. Bertrand too has suffered, though not quite as much; he can hit one time in two. Cesar, newly placed in detention, is still able to hit every time. So the boss chains the three to three equidistant stakes, and gives each one in turn a single-shot pistol. Alonzo is granted the first shot, then Bertrand, then Cesar, and repeat until two are dead. There are two questions here: First, at whom should Alonzo shoot, and second, what are his odds of survival in each case? Assume first that Alonzo shoots at Bertrand. If he hits Bertrand (33% chance), Bertrand dies, and Cesar instantly shoots Alonzo dead. Not such a good choice. But if Alonzo shoots at Bertrand and misses, then Bertrand, knowing Cesar to be the greater threat, shoots at Cesar. If he misses (50% chance), then Cesar shoots Bertrand, and Alonzo has one chance in three to kill Cesar before b |