Tag Archive: algebra



The Suitability of Mycenaean Linear B, Classic & Acrophonic Greek, Hebrew and Latin Numeric Systems for Calculation

Here is the Mycenaean Linear B numeric system (A:) Click to ENLARGE

Mycenaean Linear B Numerics
Here are the 2 ancient Greek numeric systems, the so-called Classical (BA:) and the (CA:) Acrophonic, side by side: Click to ENLARGE

Classic Greek & Acrophonic Numerals
This table compares the relative numeric values of the so-called Classical Greek numeric (BA:) & the Hebrew numeric (BB:) systems, which are strikingly similar: Click to ENLARGE 

Greek & Hebrew Numerals

Finally, we have the Latin numeric system (CB:) Click to ENLARGE

L Latin Numerics

The question is, which of these 5 numeric systems is the the most practical in its application to the (a) basic process of counting numbers, (b) to accounting and inventory or (c) geometry & (d) algebra? Let's briefly examine each of them in turn for their relative merits based on these criteria. We can take the Classical Greek & Hebrew numeric systems together, since they are patently based on the same principle, the application of letters of the alphabet to counting. For the same reason, it is expedient to lump the Acrophonic Greek & Latin systems together. There are other ways of classifying each of these systems, but for our purposes, and for the sake of clarity and consistence, we have opted for this approach.

A: the Mycenaean Linear B numeric system:

Merits: well suited to accounting and inventory; possibly suited to geometry, but only in limited contexts, though never used for that purpose
Demerits: space-consuming, discursive; totally unsuitable for algebra. While their numeric system seems never to have been applied to geometry, the Minoans and Mycenaeans who relied on this system were, of course, not only familiar with but adept in geometry, as is attested by their elegant streamlined rectilinear & circular architecture. We must also keep firmly in mind the point that the Minoan scribes never intended to put the Mycenaean Linear B numeric accounting system to use for algebra, for the obvious reason that algebra as such had not yet been invented. But we mustn’t run away with ourselves on this account, either with the Mycenaean system or with any of the others which follow, because if we do, we seriously risk compromising ourselves in our own “modern” cultural biases & mind-sets. That is something I am unwilling to do.    

B = (BA:+BB:) the Classical Greek & Hebrew numeric systems: 

Merits: well-suited to both geometric and algebraic notation & possibly even to basic counting.
Demerits: possibly unsuitable for counting, but that depends entirely on one's cultural perspective or bias. Who is to say that the modern Arabic system of counting (0...9) is in any way inherently superior to either the Classical Greek or Hebrew numeric systems? Upon what theoretical or practical basis can such a claim be made? After all, the Arabic numerals, universally adopted for counting purposes in the modern world, were simply adopted in the Middle Ages as an expedient, since they fitted seamlessly with the Latin alphabet. Nowadays, regardless of script (alphabet, syllabary or oriental) everyone uses Arabic numerals for one obvious reason. It is expedient. But is it any better than the Classical Greek & Hebrew numeric systems? I am quite sure that any ancient Greek or Hebrew, if confronted with our modern Arabic system of numerics, would probably claim that ours is no better than theirs. Six of one, half a dozen of the other.

However, in one sense, the modern Arabic numeric notation is probably “superior”. It is far less discursive. While the ancient Greeks  & Hebrews applied their alphabets in their entirety to counting, geometry and algebra, the Arabic numerals require only 10 digits. On the other hand, modern Arabic numerals cannot strictly be used for algebra or geometry unless they are combined with alphabetic notation. The Classical Greek alphabetic numeric system has been universally adopted for these purposes, as well as for the ease of application they bring to calculus and other complex modern systems such as Linear A, B & C, which have nothing whatsoever in common with the ancient Minoan Linear A, Mycenaean Linear B or Arcado-Cypriot Linear C syllabaries, except their names.  Regardless, it is quite apparent at this point that the whole question of which numeric system is supposedly “superior” to the others is beginning to get mired down in academic quibbling over cultural assumptions and other such factors. So I shall let it rest.  

C = (CA:+CB:) the Acrophonic Greek & Latin numeric systems:

Before we can properly analyze the relative merits of these two systems, which in principle are based on the same approach, we are obliged to separate them from one another for the obvious reason that one (the Acrophonic Greek) is much less discursive than the other (the Latin). Looking back through the lens of history, it almost seems as if the Athenian Greeks took this approach just so far, and no further, for fear of it becoming much too cluttered for their taste. After all, the ancient Greeks, and especially the Athenians, were characterized by their all-but obsessive adherence to “the golden mean”. They did not like overdoing it. The Romans, however, did not seem much concerned at all with that guiding principle, taking their own numeric system to such lengths (and I mean this literally) that it became outrageously discursive and, in a nutshell, clumsy. Why the Romans, who were so eminently practical and such great engineers, would have adopted such a system, is quite beyond me. But then again, I am no Roman, and my own cultural bias has once again raised its ugly head.     

CA: Greek Acrophonic
Merits: well-suited to both geometric and algebraic notation & possibly even to basic counting.
Demerits: See alphabetic Classical Greek & Hebrew systems above (BA:+BB:)

CB: Latin
Merits: easy for a Roman to read, but probably for no one else.
Demerits: extremely discursive and awkward.  Useless for geometric or algebraic notation.

This cartoon composite neatly encapsulates the dazzling complexity of the Latin numeric system. Click to ENLARGE:

Composite 4 Cartoons Roman numerals 

Richard



Comparison of the Merits/Demerits of the Linear B, Greek & Latin Numeric Systems:

Linear B:

As can be readily discerned from the Mycenaean Linear B Numeric System, it was quite nicely suited for accounting purposes, which was the whole idea in the first place. We can see at once that it was a simple matter to count as far as 99,999. Click to ENLARGE:

Mycenaean Linear B Numeric System and Alpha

In the ancient world, such a number would have been considered enormous.  When you are counting sheep, you surely don't need to run into the millions (neither, I wager, would the sheep, or it would have been an all-out stampede off a cliff!)  It worked well for addition (a requisite accounting function), but not for subtraction, multiplication, division or any other mathematical formulae. Why not subtraction, you ask?  Subtraction is used in modern credit/deficit accounting,  but the Minoans and Mycenaeans took no account (pardon the pun) of deficit spending, as the notion was utterly unknown to them. Since Mycenaean accounting ran for the current fiscal year only, or as they called it, “weto” or “the running year”, and all tablets were erased once the “fiscal” year was over, then re-used all over for reasons of practicality and economy, this was just one more reason why credit/deficit accounting held no practical interest to them. Other than that, the Linear B numeric accounting system served its purpose very well indeed, being perhaps one of the most transparent and quite possibly the simplest, ancient numerical systems.

Of course, the Linear B numerical accounting system never survived antiquity, since its entire syllabary was literally buried and forgotten with the wholesale destruction of Mycenaean civilization around 1200 BCE (out of sight, out of mind) for some 3,100 years before Sir Arthur Evans excavated Knossos starting in early 1900, and successfully deciphered Linear B numerics shortly thereafter. This “inconvenient truth” does not mean, however, that it was all that deficient, especially for purposes of accounting, for which it was specifically designed in the first place. 

Greek:

Greek alpha-numeric
On the other hand, the Greek numeric system was purely alphabetic, as illustrated above. It was of course possible to count into the tens of thousands, using additional alphabetic symbols, as in the Mycenaean Linear B system, except that the Greeks were not anywhere near as obsessive over the picayune details of accounting, counting every single commodity, every bloody animal and every last person employed in any industry whatsoever.  The Minoan-Mycenaean economy was hierarchical, excruciatingly centralized and obsessive down to the very last minutiae. Not surprisingly, they shared this zealous, blinkered approach to accounting with their contemporaries, the Egyptians, with whom the Minoan-Mycenaean trade routes and economy were inextricably bound on a vast scale... much more  on this later in 2014 and 2015, when we come to translating a large number of Linear B transactional economic and trade records.

However, we must never forget that the Greek alphabetic system of numeric notation was the only one to survive antiquity, married as it is to the universal Arabic numeric system in use today, in the fields of geometry, theoretical and applied algebra, advanced calculus and physics applications. Click to ENLARGE:

geometry with Greek and English algebraic annotation 
It would have been impossible for us to have made such enormous technological strides ever since the Renaissance, were it not for the felicitous marriage of alphabetic Greek and Arabic numerics (0-10), which are universally applied to all fields, both theoretical and practical, of mathematics, physics and technology today. Never forget that the Arabians took the concept of nul or zero (0) to the limit, and that theirs is the decimal system applied the world over right on through to computer science and the Internet.

Latin (Click to ENLARGE):

Latin 1-1000

When we come to the Roman/Latin numeric system, we are at once faced with a byzantine complexity, which takes the alphabetic Greek numeric system to its most extreme. Even the ancient Greeks and Romans were well aware of the convolutions of the Latin numeric system, which made the Greek pale in comparison. And Roman numerics are notoriously clumsy for denoting very large figures into the hundreds of thousands. Beside the Roman system, the Linear B approach to numerics looks positively like child's play. Thus, while major elements of the alphabetic Greek numeric system are still in wide use today, the Roman system has practically fallen into obscurity, its applications being almost entirely esoteric, such as on clock faces or in dating books etc. And even here, while it was still common bibliographic practice to denote the year of publication in Roman numerals right on through most of the twentieth century, this practice has pretty much fallen into disuse, since scarcely anyone can be bothered to read Roman numerals anymore. How much easier it is to give the copyright year as @ 1998 than MCMXCVIII. Even I, who read Latin fluently, find the Arabic numeric notation simpler by far than the Latin. As for hard-nosed devotees of Latin notation, I fear that they are in a tiny minority, and that within a few decades, any practical application of Latin numeric notation will have faded to a historical memory.

Richard
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