#history of symbolic logic
Beyond Boole - Part 1
Symbolic Logic for the 21st Century
Boolean Algebra:
Fundamental Operations
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Looking back on how we arrived at this stage of reconstruction of Western thought, I see the difficulty arose in attempting to explain the “missing zero” of Taoism. Blame our troubles on Leibniz. It was he who introduced binary numbers to the West, and made the fateful choice of using zero(0) instead of -1 to counter with +1. Leibniz knew full well of the I Ching, but did not understand it well. He missed the point, seeing in it only a resemblance to his own newly devised system of numbers.
By Leibniz’s time negative numbers were firmly entrenched in the European mind. Why did Leibniz ignore them completely? In doing so he blazed a new trail that led eventually to the digital revolution of recent times. It also led to a dead end in the history of Western thought, one the West has not yet come fully face to face with. It will, though. Give it a few more years.[1]
George Boole, the inventor of what we know today as Boolean logic or Boolean algebra, was one of the thinkers who followed in the footsteps of Leibniz, building on the trail he blazed.[2] When he came to devise his truth tables, he also chose zero(0) as the counterpart to one(1). This led to certain resounding successes. And ultimately, to certain failures that introduced yet another layer to the blind spot of Western symbolic logic. Here we are, almost two centuries later,[3] saddled with and hampered by the unfortunate fallout of that eventful decision still.[4]
Most arguments in elementary algebra denote numbers. However, in Boolean algebra, they denote truth values falseandtrue. Convention has decreed these values are represented with the bits (or binary digits), namely 0 and 1. They do not behave like the integers 0 and 1 though, for which 1 + 1 = 2, but are identified with the elements of the two-element field GF(2), that is, integer arithmetic modulo 2, for which 1 + 1 = 0. (1,2) This causes a substantial problem when we attempt correlation of Taoist logic and Boolean logic. As we will soon discover, Taoist logic is a hybrid logic that is based on both vector inversion and arithmetic modulo 2. As such, it ought prove relatable to both Cartesian coordinates and Boolean algebra, though it may necessitate “forcing a larger foot in a smaller glass slipper.”
Taoism chose ages ago to use ‘yin’ and 'yang’ as its logical symbols. Although this appears, at first, to be a binary system, like those of Leibniz and Boole, on closer inspection it proves not to be. It is one of far greater logical complexity, alternatively binary or ternary with intermediate third element understood. This implied third element is able to bestow balance and equilibrium throughout all of the Taoist logical system. This is where the 'missing zero’ of Taoism went. Only it is a very different zero than the 'zero’ of Western thought. It is a zero of infinite potential rather than one of absolute emptiness. It is a zero of continual beginnings and endings, not of finality. It is one of the things that make the I Ching totally unique in the history of human cognition. All these hidden zeros are wormholes between dimensions and between different amplitudes of dimension.
So where does this all lead to, then? We’ve seen that the Taoist 'yin’ can readily be made commensurate with 'minus 1’ of Western arithmetic, the number line, and Cartesian coordinates.[5] But if it is to remain true to Taoist logic, it cannot be made commensurate with the Western 'zero’. We’ve found the Taoist number system and geometry to be Cartesian-like but not Cartesian. Now we discover them to be Boolean-like, not Boolean. Sorry, Leibniz, they are not so much as remotely like your binary system. You were far too quick to disesteem the unique qualities of the I Ching.[6]
This all has far-reaching consequences for Western thought in general. Especially though, for symbolic logic, mathematics, and physics. More specifically for our purposes here it means that when we create our Taoist notation transliteration of Cartesian coordinates, we will need also to translate Boolean logic into terms compatible with Taoist thought, that is to say, from a two-value system based on '1s’ and '0s’ into a three-value system based on '1s’, ’-1s’, and the ever-elusive invisible balancing-act '0s’ of Taoism.[7] We turn to that undertaking next.
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Image: Fundamental operations of Boolean algebra. Symbolic Logic, Boolean Algebra and the Design of Digital Systems. By the Technical Staff of Computer Control Company, Inc. Other logical operations exist and are found useful by non-engineer logicians. However, these can always be derived from the three shown. These three are most readily implementable by electronic means. The digital engineer, therefore, is usually concerned only with these fundamental operations of conjunction, disjunction, and negation.
Notes
[1] It is at times like this that I am thankful I am not a member of Academia. Were I so, I could not afford, from a practical standpoint, to make claims such as this. Tenure notwithstanding.
[2] A knowledge of the binary number system is an important adjunct to an understanding of the fundamentals of Symbolic Logic.
[3] If we look back far enough in time, it was the introduction of “zero” as a number and a philosophical concept that led us down this tangled garden path, though the history of human thought is nothing if not interesting.
[4] Far out speculative thought here: Were binary numbers and Boolean logic based on +1s and -1s instead of +1s and 0s, might it not be possible to construct today a software-based quantum computer requiring no fancy juxtapositions and superpositions of subatomic particles? Think on it for a while before dismissing the thought as irrational folly.
[5] More correctly expressed, it can be made commensurate with the domain of negative numbers, since it is a vector symbol, properly speaking, concerned only with direction, not magnitude.
[6] Unfortunately there is still little understanding of the true nature of the symbolic logic encoded in the I Ching, as exemplified by this quote:
The I Ching dates from the 9th century BC in China. The binary notation in the
I Ching is used to interpret its quaternary divination technique.
It is based on taoistic duality of yin and yang.Eight trigrams (Bagua) and a set of 64 hexagrams (“sixty-four” gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.
The contemporary scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically. Viewing the least significant bit on top of single hexagrams in Shao Yong’s square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines
as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63.
[Wikipedia]
It was this Shao Yong sequence of hexagrams (Before Heaven sequence) that Leibniz viewed six centuries after the Chinese scholar created it, so maybe he can be forgiven his error after all.
The more significant point here might be that an important Neo-Confucian philosopher, cosmologist, poet, and historian of the 11th century either was no longer able to access the original logic and meaning of the I Ching or, at the very least, was hellbent on reinterpreting it in a manner contradictory to its original intent. The latter is a distinct possibility, as Neo-Confucianism was an attempt to create a more rationalist secular form of Confucianism by rejecting superstitious and mystical elements of Taoism and Buddhism that had influenced Confucianism since the Han Dynasty (206 BC–220 AD).
[7] Taoist logic and mandalic geometry share some of the characteristics of both Cartesian coordinates and Boolean logic, but not all of either. Descartes’ system is indeed a ternary one when viewed in terms of vector direction rather than scalar magnitude. That fits with the requirements of Taoist logic. It is, on the other hand, dimension-poor, as Taoist logic and geometry require a full six independent dimensions for execution. Boolean logic lacks the necessary third logical element -1, which causes inversion through a central point of mediation. But we shall see, it does bestow the ability to enter and exit a greater number of dimensional levels by means of its logical gates. Used together in an appropriate manner, these two can provide a key to understanding Taoist logic and geometry. Speculating even further, Taoist thought might provide a key to interpretation of quantum mechanics, the same quantum mechanics devised in the early twentieth century that no one can yet explain. Well, I mean, actually, Taoist thought in the formulation given it by mandalic geometry. Why feign modesty, when this work will likely linger in near-total obscurity for the next hundred years gathering dust or whatever it is that pixels gather in darkness undisturbed.
© 2015 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
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Bootstrapping Neo-Boolean - I
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So yes. This is very much a work in progress. And we have strayed now as it happens into unfamiliar territory. Terra incognita. Therewill be dragons.[1] Dragons are errors. Errors are dangerous, and we must slay them. But all in good time. First, we should scout out the terrain. That would be prudent.
Descartes in constructing his system of coordinates built upon the bedrockofelementary algebra and the number line. We’vepreviously called attention to the important but mostly overlooked issue of the 1:1 congruence between number and geometric/spatial position he incorporated implicitly in the logic of his coordinates and questioned the validity of such correspondence, at least with respect to subatomic scales.
Working two centuries later but very much under the influence of Descartes’ thought, George Boole introduced his own unique brand of algebra. A second major influence on the development of his symbolic logic was the binary number system of Leibniz, himself influenced to a large degree by Descartes. We need to carefully follow and connect the dots here. Great advances in human cognition rarely, if ever, occur in isolation and seclusion. There is a fine line to tread though. If progress requires the shoulders of giants to stand on, it is still difficult at times not to be overly influenced by those who came before.
Boole’s new logic, constructed in the wake of what by his time were firmly entrenched systematizations of thought by two of the most highly regarded philosopher mathematicians, was devised in such a manner as to conform to both of these conventions of system design. Significant to our purposes here are the facts that first, Boolean logic echoes Cartesian convention of attributing to each and every location in geometric space a single unique number, and second, it adheres to Leibniz’s convention of using a modulo-2 number system based on binary elements 1 and 0.[2]
The symbolic logic systems of mandalic geometry and the I Ching do not abide by either of these conventions. Instead they are based on what is best described as composite dimensions with four unique truth values (or vector directions) each, ranging from -1 through two distinctive zeros (0a; 0b) to +1, and assignment of numbers to spatial locations through all dimensions by means of probability distributions in place of a simple and simplistic 1:1 distribution. To accommodate these alternative conceptual concepts, we will need to expand and modify traditional Boolean logic as we have already done as regards Cartesian coordinate theory.
For starters here we should doubtless add, the mandalic form is the probability distribution through all dimensions, and the probability distributions are the mandalas. And movement through either or both can only be accomplished by discretized stepwise maneuvers between different amplitudes of dimension separated by obscure quantum leaps of endless being and becoming and being and unbecoming, toward and away from the centers and subcenters of holistic systems, the parts of which are always aiming towards some kind of equilibrium never quite within reach. Which then makes error also a necessary aspect of reality and not simply the fearful monster we imagined. It is error that makes achievement possible.[3][4]
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Image:Here Be Dragons Map. Detail of he Carta marina (Latin “map of the sea” or “sea map”), drawn by Olaus Magnus in 1527-39. This is the earliest map of the Nordic countries that gives details and place names, by Olaus Magnus [Public domain], via Wikimedia Commons. The map was in production for 12 years. The first copies were printed in 1539 in Venice. [Wikipedia]
Notes
[1] Mapmakers during the Age of Exploration sometimes placed the phrase “here be dragons” at the edges of their known world, presumably to warn of the dangers lying in wait for sailing vessels and travelers by land who strayed too far from well-traveled routes. Here is a list of all known historical maps on which these words appear.
[2] Or in Boole’s case, we might say, attributing to each proposition in concept space a single truth value: TRUE or FALSE (var YES or NO; or, in electronics applications, ON or OFF.) What we have here, I believe, is in many instances a false dilemma or the old Aristotelian dichotomy of either/or. Quantum physics demands and deserves better. OK, true enough, Boole gets around to extending possibilities by means of multi-term propositions, which his system can readily handle. The question here, though, is whether nature can or does handle such similarly. I think not. I think it approaches the question at a more fundamental level of reasoning and reality: at the most basic level of spacetime itself.
[3] This echoes the view of cybernetics, a transdisciplinary approach for exploring regulatory systems, their structures, constraints, and possibilities.
Cybernetics is relevant to the study of systems, such as mechanical, physical, biological, cognitive, and social systems. Cybernetics is applicable when a system being analyzed incorporates a closed signaling loop; that is, where action by the system generates some change in its environment and that change is reflected in that system in some manner (feedback) that triggers a system change, originally referred to as a “circular causal” relationship. [Wikipedia]
[4] This entire blog and its predecessor are in some sense the chronology of a journey from the familiar shoreline into largely uncharted waters. Hesitant at first, increasingly more daring as time has gone on and I’ve come to see errors to be stepping stones along the way. And there have beenmanyerrors along the way. Some I am not yet cognizant of. But of those I am aware, I have left most intact in spite of since being superseded by ideas superior, more correct or better formulated. I’ve done this because I think it important to map the course of a conceptual journey, how the ideas evolved from A to B to C to D. It also allows readers to participate, to a degree, in the thrill of an exciting adventure of mind, should they so choose. Happy travels.
© 2016 Martin Hauser
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where
n = x + 1 - p. :)
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