The Implications of the Linear B Geometric Syllabary for the Search for Extraterrestrial Intelligence: Part 1 — The Biggest Bang you will ever have seen from this blog!... so far... stay tuned!
Before I go any further, allow me to state categorically that this message the Voyager Space Capsules launched in 1977 with one of their missions being to search out suppositional extraterrestrials, is primitive at best, and ludicrous at its worst. Click to ENLARGE:
As far as I can figure it out (which isn’t very much at all - not that it matters), the message on this disc is difficult even for most humans to interpret, unless they happen to be astrophysicists, mathematicians or some sort of scientific geek. Unless the reader is human, it is probably impossible to make to make head or tails of it. And I for one, even though I am human and hopefully intelligent, cannot even begin to imagine how any target extraterrestrial civilization could even begin to out how to play the damn thing, unless they had a record player (ahem, as if!), a device already obsolete even to us! One of the fundamentally flawed assumptions of this analog device is that you have to play it on a device the human race alone has invented. The very concept of playing an analog recorded medium could very well be completely impenetrable to even the most advanced extraterrestrial civilizations, who might find the whole thing so laughable they would toss it out “the window”, assuming they even had windows, which is a helluva stretch in and of itself.
In the Wikipedia article on this mission, we read this:
Voyager 1 and 2 both carry with them a golden record that contains pictures and sounds of Earth, along with symbolic directions for playing the record and data detailing the location of Earth.
This patently assumes that whoever or whatever intelligence eventually (!) receives this message will look a great deal like us (i.e. be anthropomorphic) and will think almost exactly as we do, and so will understand human music, and will be able to interpret the capsule’s human historical, photographic archives & over a thousand human languages... probably so much gibberish to our poor benighted recipients some countless millennia hence, assuming it arrives in one piece, if at all. So as far as I am concerned, this mission is paramount to a futile exercise in pipe-dreaming. Even in 1977, when I was only 32 years old, I considered the whole thing a complete waste of time, money and human resources. If anything is a near-perfect example of “thinking inside the box, with the lid closed and sealed”, that project had to be it. This will all become all too painfully obvious as we proceed through our discussion of the truly formidable, quite possibly even insurmountable challenges of interstellar communication. Of course, since then, in the past 37 years, humankind has apparently begun to grow up from mid- to late-adolescence, to burst the chains of the outer limits of human consciousness as it then manifested itself, and quite literally gone cosmic. We appear to be on the cusp of our next leap in human consciousness, and if it is indeed transpiring at this very moment in our history, we are in for one helluva roller-coaster ride, the likes of which humankind has never come close to imagining in the past, right up to and including the twentieth century.
Richard Saint-Gelais’ Survey of the Potential Implications of the Application of the Linear B Syllabary as a Cipher for Extraterrestrial Communication:
In the first of our two previous posts we introduced the proposals that Richard Saint-Gelais of NASA set forth in the potentially theoretical, if not quite yet practical, application of the Mycenaean Greek Linear B Geometric Syllabary to the search for extraterrestrial intelligence. In the second of these posts, I myself posited some of the assumptions, principles and hypotheses underpinning a search of such tremendous magnitude that it stretches the powers of human reasoning practically beyond its outer limits.
Still, history has repeatedly demonstrated that our intellect and powers of reasoning can be, and at certain junctures in the timeline of human evolution, are stretched another notch up the ladder beyond the presumptive limits of our previously adduced levels of abstractive powers, finally allowing us today, for the first time in human history, to think more and more, and more and more swiftly “outside the box” than ever before prior to the twenty-first century.
The Ancient Greeks Take the First Great Leap of the Human Intellect onto the Higher Plane of Abstract Reasoning:
The first great leap onto the purely abstract plane of reasoning was taken by the ancient Greeks, in two discreet stages:
(A) the complete overhaul of the Minoan Linear A syllabary into the Mycenaean Greek Linear B syllabary, which swiftly and unceremoniously tossed overboard the most complicated and abstruse Linear B syllabograms, homophones & ideograms (some 1/4 of some 300), in less than 50 years, an incredibly rapid turnover in terms of socio-linguistic change, which otherwise nearly always occurred at a snail’s pace in the ancient world.
But there is even more to this picture than we can possibly have imagined before the 1990s at the very earliest. Despite the proliferation of puissant supercomputers and the quasi-instantaneous communication afforded by the World Wide Web, a much better semiotic signifier for what it actually is than the word, “Internet”, which is significantly lamer, I say again, in spite of all these extremely recent massive technological advances at our disposal, the Minoan Linear A syllabary, which for a human language was already a quasi-geometric script complete with the base set of 5 vowels for the first time in history, has utterly defied any and all attempts whatsoever at decipherment since Sir Arthur Evans first excavated the ruins of Knossos in the spring of 1900. It just won’t budge a single centimetre. Now, if we are utterly incapable of deciphering a human language, Minoan in Linear A, even with all of our technological gadgets and goodies at our instant command, including The University of California Berkeley Campus’ newly conceived automated “time machine” to reconstruct ancient languages, Click to visit the site:
imagine how much more alarmingly daunting must be the gargantuan task of beginning to scratch even the surface of communicating anything sensible to any extraterrestrial civilization whatsoever. But is the task really all that hopeless?
Although the Linear B syllabary was used by the scribes at Knossos, Pylos, Mycenae, Phaistos, Thebes (in Greece) and in several other Mycenaean locales, almost solely for accounting and inventories, which function primarily on a concrete and semi-abstract level, the script itself, being fundamentally and almost exclusively geometric in nature, was by far the most abstract script ever developed in the ancient world until that time (ca. 1450 BCE). Geometric abstraction is also one of the outstanding characteristics of Minoan & Mycenaean architecture, as illustrated in these two examples:
Knossos: Click to ENLARGE
Here we can instantly isolate the perfectly Circular Frieze Motif shown here on one of the two buildings at Knossos, a motif which appears over and over on several Minoan and Mycenaean structures. Notice also that the other edifice is perfectly straight in every plane, including the then revolutionary liberal use of skylights for interior illumination. You can readily see that the building reminds us of the architecture of Frank Lloyd Wright (1867-1959), the first true pioneer in the advent of modern architecture. This is no accident. Lloyd Wright took much of his inspiration for the foundation of his architectural constructs from Japanese and, yes, Minoan architecture. Once again, this should not come as any surprise to anyone familiar with the amazing achievements of one of the most brilliant architects in the history of humankind, an architect whose applied principles fundamentally relied on the application of geometry to his buildings and structures.
Mycenae:
Even more astounding are the near perfect geometric proportions of the Mycenaean Tesoro Atreoyo (Treasury of Atreus), which the Mycenaeans constructed with astonishing mathematical accuracy hundreds of years before the great Greek mathematicians finally came round to working out the complex geometric and algebraic theorems underlying the elegant geometric proportions of this magnificent structure: Click to ENLARGE
SOURCE: Metron Ariston (Greek for “The Ideal Mean” (from: Liddell & Scott, 1986, pg. 442)
What can I say? The Mycenaeans were Greeks down to their very marrow.
As anyone with even a passing acquaintance with the Linear B syllabary can attest, its geometric elegance and economy is second-to-none.
Are Ancient Scripts Primitive?
When modern writers and the occasional deluded linguist refer to ancient scripts as “primitive”, as compared with so-called “modern” alphabets, which for Occidental languages (Greek & Latin) are ancient anyway, they do a great disservice to the former, propagating totally false misconceptions on that account alone. In point of fact, there is no such thing as a “primitive” script, which leads me almost inexorably to my next observation: if there are no primitive scripts, there are no modern, all scripts (ideographies, syllabaries & alphabets) ancient or modern being as sound as any other. It follows logically then that any and all future scripts as yet uninvented will also serve as well as, but no better than, the thousands of scripts humankind has dabbled in over the past 10 millennia at least, including any which we may devise for extraterrestrial communication. The implications of this factor alone are profound. They inform us that any language whatsoever we use for communication, terrestrial or otherwise is, and can only be, human, whoever tautological this may sound... or so it may appear.
Now, the implications of this scenario for the potential transmission of some sort of set of signals susceptible to possible decipherment by extraterrestrial intelligences are profound. My point is simply this: if the historical timeline in the (apparent) “evolution” of human scripts is not sufficiently impressive even for us to make a big deal out of it, and if the transmission of any one or more of humankind’s most mathematically elegant scripts, past or present – and eventually future – are deemed by some to be just the right recipe, then why not try them? What have we got to lose? Nothing... to gain? - cosmic communication = cosmic consciousness. Now there’s something to put in your pipe & smoke.
(B) Then the very same people, the Greeks, went plunging ahead, completely abandoning the Mycenaean Linear B syllabary for the even more elegant Greek alphabet, but significantly not casting aside the Arcado-Cypriot Linear C syllabary (even more geometrically economic than Linear B), which held its own right down to 400 BCE! No use re-inventing the wheel, or so the Arcadians and Cypriots believed. But, and here is the wringer. Now get this! The Linear C syllabary was no longer used merely for record keeping and inventory purposes, in fact, far from it. Its primary use was for publication of much more abstract legal and constitutional documents. Abstract geometric syllabary, abstract thought. That’s the next big leap forward. And the next: abstract geometric syllabary --> abstract communication --> abstract extraterrestrial communication.
What about the Greek Alphabet, and its Widespread Use for Algebraic Notation?
Now, of course, the Greek alphabet itself is not characteristically geometric, so we can pretty much eliminate it, and for that matter any other Occidental alphabet (Latin or Cyrillic) as suitable for interstellar communication. This includes our Arabic numerals, which you can be pretty much sure no extraterrestrial civilization would be able to distinguish from letters in an alpha-numeric system, since all characters in such a system would look the same to them, and almost certainly far too complex for them to take seriously.
We can also be pretty well assured that no extraterrestrial civilization, even if they too used alphabets, would have the faintest idea what human alphabets were supposed to be signifiers of. But does that really matter? My short answer is simply, not at all. If we were to transmit from the source (ourselves) for instance just these rectilinear & circular 10 Linear B symbols &/or 10 Linear C symbols – for a potential total of 20 — as simple signals and nothing more (10 supposedly being a universally recognizable number), all kinds of wacky scenarios are likely to transpire at the target (them, whoever or whatever they are).
Now, of course, since our target extraterrestrial civilization will not have the faintest idea what these symbols mean to us, as humans – if they see them as symbols as such at all – or whether or not they simply see them as geometric signals, the latter will do the trick just fine, thank you very much. So in this case, it does not matter a hill of beans which syllabograms from which syllabary we as the source civilization transmit to them, the target civilization, since they are going to interpret these 20 signals – if we decide to send that many – whatever damn well way suits them just fine, regardless of who we are, since they could care less anyway. All that would matter to them is that someone or some entity or entities from somewhere in our (meaning, their) galactic neighbourhood sent them a signal that meant something significant to themselves (the targets), though God only knows what. And why should we care any more than they do anyway? Come to think of it, they do not even have to live on a planet such as we construe it. If they do not, they might just as easily assume that whoever or whatever sent the signal would not live on a “planet” either. Any scenario is possible. So for this reason alone, if it were up to me to send the signal, I would simply mix-and-match Linear B & Linear C geometric signifiers any old way I felt like, and be damned the consequences... well, that might be a bit of an overstatement in case they turn out to be hostiles, we piss them rightly off and they invade us! But the chances of that ever happening are so extremely remote as to approach quantum zero.
Still, we have to admit that the Linear B & Linear C syllabaries have a helluva lot going for them. If anything, both are eminently suited for extraterrestrial communication, for the following reasons (as I see it):
1. What is the “Message” in the Extraterrestrial Communication Medium? What does it signify? Does it matter to “them”? Should it matter to us? Whose “Message” is it anyway? Woah!
As Richard Saint-Gelais correctly points out, any attempt on our part to communicate with extraterrestrial intelligences cannot, and must not, be based on what we as humans understand as being signifier(s) and signified, but rather on (hopefully) recognizable patterned sequences, by which I mean either digital (0 1), decimal or geometric, but not algebraic (see above). In fact, I posit that it does not matter a hill of beans whether signals of these three mathematical orders mean anything at all like what they clearly signify to us, but not clearly at all to our extraterrestrial compeers, other than what they signify to them, and in that light, applying reverse logic, almost certainly not to us. All that matters is that they, our extraterrestrial buddies, understand that the constructs mean whatever the hell they mean to them. If they do meaning anything, anything at all, then we will have established communication.
Funny Things Happened on the Way to the Extraterrestrial Conference:
Let us imagine a few ludicrous sounding examples. Say, for instance, we transmit a circle in the source signal, and our extraterrestrial friends at the target “read it”. Well, what if the circle we send is not abstract at all to them, but concrete only? What if they cannot even think on the abstract, connotative plane? Don’t laugh. Maybe to them the circle is just one of a thousand polka dots on one of their pet five-legged orgathonics with two heads and four arms, but no legs, just flippers instead. Again, take the straight line. Same scenario. If the language of that particular extraterrestrial civilization is concrete and denotative only, it would not matter how many straight lines we transmitted to them. They simply would not recognize them as such. But they would recognize them as something concrete, such as, for instance, a pole sticking in the ground.
2. The exact reverse scenario may just as easily obtain, namely that a particular extraterrestrial intelligence we sloppily target and by sheer accident hit (there is after all no other way we would hit them, if we ever could... imagine trying to hit the Earth with a pin-pong ball from 1,000 light years away!) uses a language or languages which are absolutely abstract and connotative, and not concrete and denotative at all. I hear someone shouting, “Eureka! We’re in luck!” Not so fast. To such a civilization a circle may be far more than just a circle or a straight line as we envision them. To them, a circle might automatically mean a sphere, if even their language is entirely three-dimensional on the abstract plane. Woops! As for a straight line, God forbid! It would at the very least probably be that naughty old straight line drawn out to infinity, and looping back in a circle to the point where it started to bite them in the conjectural ass. Then they would really get confused!
To them, a circle and a straight line might even be paramount to one & the same phenomenon, so I can hear them asking themselves, “Why would anyone or any entity such as ourselves bother sending the same exact symbol as two discrete symbols – unless of course they were stupid?” If that were the case, I suspect that they would not even bother communicating with us, targeting us with their far more intelligent signals, because they would (rightly) see us as utterly incapable of interpreting them, not having even the minimal intellectual resources to tackle their “message”, or rather I should say, the signals in their “medium”, whatever that happens to be. Your guess is as good as mine.
So we end up with at least two scenarios, and plenty more besides, I strongly suspect. Either our abstract geometric symbols are interpreted as signals of concrete objects alone or they are considered to be far too primitive for our hyper-intelligence recipients, who would probably just laugh them off as some sort of hopelessly dumb joke from the equivalent of what we would generously refer to as apes!
The Enigma Code:
3. There is yet another highly fruitful source for food for thought in the massively daunting challenge facing us in the apparently Quixotic search for potential solutions to the problem of extraterrestrial communication. This is, leaving aside the absolutely monumental achievement of the decipherment of Linear B by Michael Ventris, the astonishing work of another genius of decipherment in the mid-twentieth century. I speak of course of Alan Turing (1912-1954), who not only was the first person in history to actually correctly conceptualize the theoretical base of the digital computer, based on the 0-1 binary construct, but who successfully cracked both versions of the German Enigma Code in World War II (the earlier easier & later more difficult one). Click on his photo for his biography:
Now there is a term I can latch onto, Enigma Code. In fact, I fairly burst to leap on it, because I can think of no other term that more aptly exemplifies the fundamental precepts and hypotheses underlying the search for some way, any way, to communicate with any kind of extraterrestrial intelligence. It is no longer a question of us, or to put it bluntly, of the nature of our own human intelligence.
Speaking frankly, I for one do not believe it matters one jot what kind of intelligence is at the source and the target of extraterrestrial communication, provided that there is at least some common universal signal substrate which may (or may not) be susceptible to an interpretation, any interpretation of the source message by the target recipient, even if their understanding of what the “message” actually says (to them) differs drastically from what it means to us.
The only thing that matters at all is that the extraterrestrial target recipients of the signals we transmit are able to recognize a clearly repetitive pattern of sufficient variations on a “theme” to the point that it is intelligible to them (not us), in the fundamental framework of their own intelligence (not ours), however much it differs from our own human paradigm(s) for what we ourselves call “intelligence”. That is what I mean by a potentially universal signal, an Enigma Code which, although it remains an Enigma Code to our target recipients, is at least an enigma with a clearly recognizable pattern.
They certainly do not need to decipher it as we understand the principle of “decipherment” in human terms, any more than we need to actually decipher the Minoan language in Linear A to recognize highly repetitive morphemic and semiotic patterns and even oblique declensions, which we in fact do recognize as essential markers of human languages. But even a partial decipherment can serve well enough to convince us that we are on the right track. We know this because signifiers-signified are universal in human languages. Moreover, the entire Linear A numeric system has been successfully deciphered, and a great many toponyms we know in Linear B have (nearly) exact counterparts in Linear A.
Yet even if the fundamental construct of the intelligence of our extraterrestrial buddies contains neither the signifier “language” nor “decipherment”, their intelligence, if at least as advanced as ours (and that is not very advanced) will be able to derive some sort of “sense” from our “signal”, because for them, just as for us, the medium would be the message. The clue would be McLuhanesque, even if they could never have a clue what a McLuhan is. So the situation is far from hopeless.
The Enigma Machine:
At the crux of the problem, however, there is this: what is universal to human language constructs is almost certainly bound to be far from being universal even for any single target extraterrestrial “language”, let alone any number of them, whatever their intrinsic nature, it being almost certainly equally enigmatic to us. Ah the old double-blind scenario.
The Germans knew what their Enigma Codes meant, because they could decipher them by reverse extrapolation at the source. But until Bletchley Park and its brightest star, Alan Turing, could get a grip on it – and it took years of the most backbreaking analysis – it remained just what it was to the Allies, an Enigma Code. Still, they knew perfectly well that the code itself, however massively complex it was (and it was!) overlay relatively simple original military messages in perfectly intelligible German. They new it was an artificial human means of communication. And that was all they needed to know. Let us never forget that those clever bastards at Bletchley Park cracked the Enigma Code without the benefit of computers, which says far more for them than it does for us today!
A Universal Enigma Code for Extraterrestrial Communication?
“Are you completely bonkers?” I hear you protest. Not so fast. Yes, the irksome question still remains, and refuses to just go away in a puff of smoke: would any extraterrestrial communication system or “language”, if we must insist on calling it that, even be able to begin to crack a human Extraterrestrial Enigma Code we so blithely sent buzzing off into interstellar space at the speed of light, unless their communication system were in fact a “language” something along the lines of what we understand a language to be? Conceivably they might, but their “language” would have to be a language fairly approximating the universal construct of what we call human language for them to be able to do so. Otherwise... fill in the blanks. Rather, do not fill in the blanks. Firing off blanks does not kill anyone. Firing off blank “blank” messages does not “mean” anything to any higher intelligence which has no need of language as we understand it. In fact, they might even toss our medium, forget the “message” into the “garbage”, considering it as nothing more significant than “dog poop” or whatever they call “it”.
One thing is pretty obvious to me at least: sending a code which would be interpreted as an Enigma Code by some extraterrestrial civilization would probably be more like child’s play to them than vainly struggling trying to decipher what the silly messages on the Voyager spacecraft mean, simply because the latter are plainly and solely human, nothing more or less & next nothing else at all. But as I have said over and over, the “message” or more properly the signals we transmit cannot & must not be simply human in nature, they must at least make a stab at being cosmically universal, at least to one extraterrestrial civilization whose communication system bears attributes roughly equivalent to what we deem to call language – excuse me, human language. Oh and by the way, good luck finding it, because the odds are almost certainly stacked trillions to one against us.
4. The problem gets far more complex, if we just pause for even a moment and allow the scary realization to sink in that any signals we send at the source, particularly geometric, even if they are entirely abstract to us, may run the full gamut from concrete to semi-abstract to abstract and, yes, even beyond abstract and consequently beyond our ken. Just stop and consider for a second what would happen if we sent our silly geometric symbols to a four-dimensional extraterrestrial civilization? I cringe to think of it. And let’s not forget what I just said above: what if another three-dimensional extraterrestrial civilization interpreted absolutely all of our signals, even the two-dimensional, as three-dimensional only? Then there are nuances within nuances within nuances of every shade between these extremes. Beyond these scenarios I have just outlined, my mind simply explodes.
So I will end it there before it does.
However, stay tuned. There’s more, a lot more. I have scarcely begun. Stay tuned for more on extraterrestrial communication. And stay tuned for a possible breakthrough on an entirely new approach to the first baby steps in deciphering Linear A. We’re taking the ball where it wants to take us.
Richard
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This chart illustrates both the ancient Greek acrophonic and alphabetical numeric systems. However, the acrophonic system, used primarily in Classical Athens ca. 500 – 400 BCE, came much later than the alphabetical system. So in effect we must resort to the only Greek numeric system we can use to represent numbers in Mycenaean Greek numbers, i.e. the alphabetical system. The alphabetical numbers are displayed in the second column after the modern numbers, 1 – 100,000 in the following chart. Here are some examples of alphabetic numbers representing Mycenaean numbers:
My decipherment is partial. The only candidate for Mycenaean derived vocabulary is the word uro = entire, whole, i.e. total, a synonym of kuro = reaching, attaining, i.e. total. The word jaku obviously refers to the cargo.
Upon close examination of the syllabogram WE in the context of dry weight in Mycenaean Linear B, in this particular instance, dry weight of saffron, I have come to the conclusion that the line(s) transversing the syllabogram WE at an approximate angle of 105 - 110 º are actually equivalent to the tens (10 & 20), while the black circles in the upper and lower portions of WE are equivalent to the 100s (100 & 200) in the Linear B numeric system. Once again, the scribes would never had added these lines and circles to the syllabogram, unless they had good reason to. And they surely did. There is a striking resemblance between the approximately horizontal lines to the 10s, and of the black circles to the 100s in that system, as can be seen from the actual placement values for 10s and 100s immediately above the syllabogram WE. As if this is not impressive enough, there is even more to this syllabogram. It is in fact a supersyllabogram. Its meaning is identical to the same SSYL for crops in the agricultural sector, namely; WE is the first syllable of the Mycenaean Linear B word weto, which literally means “the running year”, in other words “the current fiscal year”. This makes perfect sense, since the scribes at Knossos, Phaistos, Mycenae, Pylos, Thebes and other Mycenaean locales only kept records for the current fiscal year, never any longer. The most astonishing feature of this supersyllabogram is that it combines itself as a SSYL with the Linear B numeric system, meaning that it alone of all the SSYLS refers to both the number of minusucle items (in this case, saffron, but it could just as easily refer to coriander or other spices) and the total production output of the same items for the current fiscal year. The Linear B scribes have truly outdone themselves in this unique application of the supersyllabogram, distilling it down to the most microscopic level of shorthand, thereby eliminating much more running text from the tablet we see here than they ever did from any other tablet, including all of those sporting “regular” supersyllabograms. In this instance alone (on this and the few other tablets on which it appears), this unique “special” SSYL is a supersyllabogram with a specific numeric measurement value at the minuscule level, something entirely new, and seen nowhere else in all of the extant Linear B literature. Quite amazing, if you ask me. NOTE: the assignment of a value approximating 1 gram for the single unit, i.e. the simple syllabogram WE with no traversing lines or black circles, is just that, nothing more than an approximation. I had to correlate the single unit with something we can relate to in the twenty-first century, so I chose the gram as an approximate equivalent. One thing is certain: the unit WE is very small, indicating as it does minuscule dry measurement weight. Richard
Aside from the fact that we cannot be at all sure how much each of these units of measurement is supposed to represent, I am still operating on the premise that the Mycenaean system of measurement is 10-based or decimal, hence, something along the lines of the modern metric system. However the units are configured, it is quite certain that in the case of these two tablets, the units must be small, because the items measured, saffron (on the left) and olive oil (on the right) are usually dispensed in small amounts. Since saffron is very light, I assume that the weight is something like 10 grams, while the liquid measurement for the olive oil is in the range of about 2 litres, or whatever amount the Minoans & Mycenaeans used to house these commodities. Richard
Here are the 2 ancient Greek numeric systems, the so-called Classical (BA:) and the (CA:) Acrophonic, side by side: Click to ENLARGE
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
Finally, we have the Latin numeric system (CB:) Click to ENLARGE
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:
Richard
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