The Geometry of Understanding

May 19, 1998

Apparently, knowledge is to understanding as flat, uninvolved euclidian space is to curved, participating non-euclidian space. Once again, we see the distinctive geometry of yin-yang, the mutual enfoldment of opposites in a dipolar or complementary arrangement, creeping into issues of 'knowledge and understanding'.

The inferences are that there is opposition between the notions of 'knowledge' and 'understanding', and the prefix of 'non' in front of euclidian gives the deceptive impression that euclidian is the general case and non-euclidian the special case.

With respect to the latter, one could almost substitute the terms 'euclidian and non-euclidian' for 'linear and non-linear' into Ian Stewart's discussion in 'Does God Play Dice: The Mathematics of Chaos' on the topsy-turvyness of relativity in this language and how it distorts our thinking. Stewart enfolds his discussion within the self-similarly suggestive heading of 'nonpachdermology', the name he suggests might have evolved in place of 'Zoology' if we had first become besotted with the study of elephants and only later been forced to come up with a name for the rest of the animal kingdom.

Congruency between Stewart's geometry of how we regard 'linear and nonlinear', and how we regard euclidian and non-euclidian space include the following;

"Classical mathematics concentrated on linear equations for a sound pragmatic reason; it couldn't solve anything else. In comparison to the unruly hooligan antics of a typical differential equation, linear ones are a bunch of choirboys (Is it coincidence that 'rule' means both 'law' and 'straightedge'?"

In the case of euclidian knowledge and non-euclidian understanding, we could say; "Classical philosophy concentrated on euclidian knowledge for a sound pragmatic reason; it couldn't bring finality or closure to anything else. In comparison to the unruly hooligan antics of a typical curved-space dynamic, flat-space stasis is like a bunch of choir boys (Is it a coincidence that 'knowledge' means both 'recognition' (i.e. in the sense of 'recognizance' --- legal bond or obligation) and 'fact'?)

Stewart continues; "So ingrained became the linear habit that by the 1940's and 1950's many scientists and engineers knew little else [other than linear theory]. 'God would not be so unkind,' said a prominent engineer, 'as to make the equations of nature nonlinear.' Once more the Deity was carrying the can for humanity's obtuseness. The engineer meant he didn't know how to solve nonlinear equations, but wasn't honest enough to admit it."

It seems that the Greek rationalist philosophers, in trying to emancipate thought from myth, were equally the victims of their own obtuseness, insisting on a finality and closure in the 'subject of their enquiry', and suggesting that the Gods would never have had it any other way. Kirk et al, in 'The Presocratic Philosophers' discuss the critical point of bifurcation on this issue ca. 500 BC, via Parmenides arguments, encapsulated in a hexameter poem.

"After the proem, the poem falls into two parts. The first expounds 'tremorless heart of well-rounded Truth' [Parmenides was having none of those 'yin-yang' oscillations in his formulations!]. Its argument is radical and powerful. Parmenides claims that in any enquiry there are two and only two logically coherent possibilities, which are exclusive [sounds a bit like pre-conditioning our facts with theory at this point, or getting Santa Claus prepared in the wings, as Ian Stewart might say.] --- that the subject of the enquiry exists or that it does not exist. On epistemological grounds he rules out the second alternative as unintelligible [i.e. here Parmenides whips out his Ockam's razor quicker than Clint Eastwood draws a six-shooter in 'The Good, The Bad, and The Ugly', ensuring that the only good triadic or 'head-up-its-ass' unity is a dead one]. He then turns to abuse of ordinary mortals for showing by their beliefs that they never make the choice between the two ways 'is' and 'is not', but follow 'both' without discrimination. In the final section of this first part he explores the one secure path 'is' [the path of the Gods, to be sure] and proves in an astonishing deductive 'tour de force' that if something exists, it cannot come to be or perish, change or move, nor be subject to any imperfection. Parmenides' arguments and his paradoxical conclusions had an enormous influence on later Greek philosophy; his method and his impact alike have rightly been compared to those of Descartes 'cogito'."

Meanwhile, Parmenides' contemporary, Lao Tsu, the Imperial Archive Keeper over in China's Honan province, was taking a rather different tack which in no way subjected even God to such definitive existential questions as to the status of 'being' or 'not being' 'out there'. Instead, he could exist both out there and in here at the same time, an epistemology which can take us from a flat-space detached 'knowing' of what's 'out there' to a curved-space participative 'understanding' of us as being woven into the fabric we look out upon; an epistemology entirely lacking in the absolutist boundaries which Parmenides 'hung on' his Gods. One can imagine how Parmenides' stomach would have churned and his face contort had he heard Lao Tsu speak the following;

"The Tao that can be told is not the eternal Tao.

The name that can be named is not the eternal name.

The nameless is the beginning of heaven and earth.

The named is the mother of ten thousand things

Ever desireless, one can see the mystery.

Ever desiring, one can see the manifestations.

These two spring from the same source but differ in name;

this appears as darkness.

Darkness within darkness.

The gate to all mystery.

Clearly, we in the west have had a ravenous desire for the plain and simple; 'make mine manifest please and hold the mystery'.

Through this distinct choice (here we already 'succumbed' since this was a 'choice' which never even presented itself to Lao Tsu, suggesting that the very concept of 'choice' is bundled in with this bifurcation.), we locked onto the notion of seeing the world as a flat, inert, euclidian space of 'things' and 'void' as being the general case and turned our backs on the curved, participative non-euclidian space of the mythopoeic era peoples (ancient Egypt, Mesopatamia, the Celts and the East).

So, as Frankfort et al say in 'The Intellectual Adventure of Ancient Man'; "Heraclitus had declared 'being' a perpetual 'becoming' and had correlated the two concepts with his 'hidden attunement.' Now Parmenides declared the two to be mutually exclusive, and only 'being' to be real."

With appropriate follow-up by Plato and Aristotle and other Parmenidian fellow-travellers and with the purported proxy endorsement of the Gods, this binary cleaving of the inner and outer worlds cleared the path for gathering progressively more definitive 'knowledge' of things which were exclusively 'out there', fully detached from the cadre of disciplinary high priests who were observing them. Heraclitus' objection that; 'The learning of many things teacheth not understanding, else it would have taught Hesiod and Pythagorus.' was left behind in the dust. As Frankfort et al point out, "The question of how we can 'know' what is outside us was not raised. Heraclitus asserted that the unverse was intelligible because it was ruled by 'thought' or 'judgement' and that the same principle, therefore, governed both experience and knowledge."

Heraclitus, then, saw mind and matter as being bundled together in one unified system, a view which was not resurrected in western philosophy until Nietzsche, and a view which began to re-emerge through the field of psychology via Lev Semyonovich Vygotsky (1896 - 1934), who concluded through his psychology experiments that thought was ontogenous (see 'Thought and Language' by Vygotsky), and similarly via the psychologist R. D. Laing, whose work in treating the 'mentally ill' led him to the ontogenetic conclusion that 'The life we are reaching out to grasp is the we who are reaching out to grasp it."

General relativity theory and quantum physics, as described in our last note, the introduction to 'Geometry and Culture', lead us towards this same conclusion, that knowledge is an approximative view of natural systems, as a snake would see if he raised himself up and detachedly looked down upon his coiled body. The desire for ever more definitive knowledge, moreover, further detaches one from the system, making it more difficult to relax one's pursuit of a specialized line of thinking so that the mind can allow itself to be enveloped in and co-resonate with the natural resonances of the system which is not only 'out there' but 'in here'; a co-resonance which equates with 'understanding'.

In physics, this descent into the internals of the system has been called 'inverted perspective' and is what you do when you do the thought experiments of Einstein or Feynman and imagine yourself, for example, being the 'eyes' of a photon which turn back and look at a clock in a 'relative' reference frame (this mode of thinking is doubtless how the yin/yang geometry begins to re-emerge in theoretical physics).

Understanding, then, is what comes from 'entering into' the system, and this sets up a basic opposition between 'understanding' and 'knowledge'. That is, the pursuit of ever more detailed (final) knowledge can no more lead to understanding in an ontogenetic (space-time enfolded) sense than staring more intensively at the squiggles of a 3D hologram on a flat sheet of paper can allow you to enter into its three dimensional world fo virtual imagery; 'ever desiring one can see the manifestations, ever desireless, one can see the mystery.'

Like the snake, mesmerized by looking at the beatifully detailed embroidery of what he forgets is his own skin, we are mesmerized by the knowledge 'bites' of words. As Wittgenstein observed; "A 'picture' held us captive and we could not get outside it for it lay in our language and language seemed to repeat it to us inexorably." ... "Where does our investigation get its importance from, since it seems only to destroy everything interesting, that is, all that is great and important? (As it were all the buildings, leaving nothing behind only bits of stone and rubble.) What we are destroying is nothing but houses of cards and we are clearing up the ground of language on which they stand. (notes 114 and 118 from 'Philosophical Investigations').

This is the geometry of yin (integration) and yang (differentiation); the more detailed our 'yang' knowledge or 'house of cards' view of the phenomena becomes, the more integrative glue or threading of our 'yin' understanding we are dissolving.

The history of how language has evolved as a tool of knowledge, as an accomplice in the assassination of understanding, is nicely given by David Abram in 'The Spell of the Sensous', as we alluded to our introduction to 'Geometry and Culture'. The message is the same as in Charles Kahn's 'The Art and Thought of Heraclitus', that our language must be 'intentionally ambiguous' to open the door for us to enter into the resonance of ontogenetic thought (i.e. 'understanding').

To allow our mind to enter into an ontogenic thought which encompasses us at the same time it encompasses our 'subject of enquiry', we need to allow the resonances to build and not to 'cut them off' via words or phrases designed to 'stand on their own'. The imagery is one of a juggler progressively adding more items to the juggle until the individual items involved cease to be the 'story'; the overall multivalent harmony (the space-time dynamical envelope) subsuming them and becoming the bigger, or 'transcendent' meta-story. Catherine Atherton in 'The Stoics on Ambiguity' cites Cicero as follows; "First, if it can be done, it must be proved that what is written is not ambiguous, because in everyday discourse everyone habitually uses this single word, or (these) several words, in the sense in which the pleader will prove that it should be understood. Next, it must be shown that the question is cleared up by the text which precedes and follows. For were words looked at in themselves, in isolation, all or most would seem ambiguous; but words which become clear when the whole of what is written is looked at should not be thought ambiguous."

Of course, it is not the 'words' which become clear, but the resonant whole, supporting Epicurus' "rejection of definition and division 'en masse' as uninformative and redundant." as also cited in Atherton's work.

Lao Tsu's legacy, of course, was to put us into the peaceful state which transcends the anxieties of 'choice', since choice does not emerge when we entertain both alternatives at once. Parmenides, who was apparently disgusted with his own birthing process, apparently cooked up all this anxiety about 'unGodly' ambiguity arbitrarily, as current day mathematician/philosopher Bart Kosko delights in pointing out via his 'yin-yang' equation in 'Fuzzy Thinking'.

Had Lao Tsu had a few beers with Cicero, he might well have succeeded in inverting Cicero's perspective and they might have jointly decided that the yin-integrative approach to understanding PULLS meaning into the yang knowledge-words, that 'resonant understanding' is the mother of 'static or steady-state knowledge'. In any case, had Einstein joined the duo, he would surely have quoted from his writings on 'Geometry and Experience' and showed that ultimately, one can not build upwards from defined concepts (the 'children' of understanding) but must first 'bootstrap' an understanding by, as Einstein put it; 'bringing a multiplicity of real or imaginary sensory experiences into connection in the mind'

Ambiguity in language, then, is essential for building resonant understanding, the mother of knowledge. If our bar tab can stand pulling Wittgenstein over to the table for a few minutes, we can listen to him talk about how to use language to develop the synoptic (ontogenetic) thought; "So I suggest repetition as a means of surveying the connections." The word 'connections' here is key as the only way to get our word structures of chinese boxes within boxes into ontogenetic (self-referential space-time enfolded) form is to allow ambiguity within each of the inner folds which slowly resolves itself via the containing layers. In this way, we can get back to the resonant yin (integrative) understanding which contains the yang (differentiating) knowledge.

Clearly (as Vygotsky has pointed out), we do not build understanding, 'bottom-up' from building blocks of knowledge, but rather, we first 'understand' in a resonant, integrative sense and subsequently precipitate knowledge from this understanding. There is clear self-similarity here to the yin-yang (latent integrative energy, manifest differentiating material) geometry of nature itself where, as Einstein says in 'Ether and the Theory of Relativity', "...the elementary particles of matter are also, in their essence, nothing else than condensations of the electromagnetic field [i.e. 'space']."

Of course, if the 'snake' mentioned above, persists in looking at its coiled body as a detached object, and continuing to operate upon it (solving problems) from that perspective without ever correlating the pain it feels with its seemingly detached system interventions (though such an expedient can be very profitable in a world where 'time is money'), the result could be painful, if not fatal, and that is what we in the western world of euclidian flat-space vision seem to be experiencing.

What we appear to have lost, as discussed in the historical accounting in 'Culture and Geometry' is the ancient sense that language is first and foremost about sharing understanding rather than documenting knowledge. Understanding through language means taking the words as musical notes and trying to reconstitute the ontogenetic melody or resonances, as David Abram has so well described it. Such a process demands the soft oscillatory tones of ambiguous connection rather than the harsh clang of logical gates banging shut (as Wittgenstein says; such "Philosophical 'clarity' will have the same affect on the growth of mathematics [understanding thereof], as sunlight does on potato-sprouts. (In the dark cellar they grow meter long.)"

Of such stuff were the 'Orphic Rhapsodies' and the epistemological paradigm of the East, the Celts, the Native Americans and the mythopoeical peoples of the west (Egyptians, Mesopotamians) made of, prior to; the Greek rationalist philosophies rallied by Parmenides, and the conversion of aramaic and other early western written languages from their ambiguous form (e.g. lacking vowels so that one could breath the resonances of understanding into them as one spoke) to their current 'stand-alone' or explicit definition-oriented static forms.

These alternative views on how we know and interact with the world through language, are intimately entwined with our view of 'being' (as static, detached from nature) or 'being-as-becoming' (as dynamic participants in nature), and these views are, in turn, entwined in our notions of space-time as being either euclidian (flat, uninvolved and time independent) or non-euclidian (a curved, participative space-time continuum). The mutually exclusive 'either or' operation in the last sentence doesn't really come into play if one starts with the latter option since it 'contains' the former. What Lao Tsu was trying to tell us was that we could 'have our cake and eat it too'.

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