The Common Inferences(ennoia) in the Stoikeioon are principally about the aplication of the notion of duality. There is one unvoiced inference which it is clear is adduced to the others as one goes through the Stoikeion: The exact same process as a sequence of actions, inferences ,assignments and definitions produces a dual result.
One must not confuse the notion of equality , which is based on this notion and set of common inferences, with duality. Whereas we may assign the one notion to the other we may not do so equally. Thus historically duality precedes equality, and in fact may be applied to a prior notion oneness(monos/monas)
Thus it is allowable to apply equality to many groupd of quantific magnitudes which possess this common identity, the indistinguishable nature of each member of the group from any other; and thus quadrallity or quintallity is also the basis of equality should we so wish it.
Probability arises exactly when this last common inference is no longer held as certain: that is when the process of identical actions does not produce the identical result. Such a circumstance is in fact an everyday occurence, and thus it reveals the "ideal" ideomatic basis of these common inerences. We in fact constantly shore them up as true, or in fact hold to them as a useful tool to guide our process sequence, and to move the sequence through its steps.
However, the experience is that everything is approximate to these notions, because the quantitative magnitudes, whether objects or motion traces, are not under our control. We in fact infer their quantitative magnitude , that is their form from extensive interaction with the objects or traces. Thus any magnitude or quantitative information about a cube is derived from direct observation many times or from the witness statement s of many people who have interacted with a cube many times.
When you look at such evidence one immediately is struck by how much faith is required to express the attributes of a cube exactly as they are "commonly" held to be. In fact that word commonly speaks to the "uncommon" views of a cubes attributes.
Thus we exercise judgement when it comes to the inference of what a cube is, and in judging we must allow probabilities. Thus we may accurately say that of 6 billion individual surveyed only 1000 held a different view of the cubes attributes, for example. We may be able to statistically differentiate the survey in other ways. We may in fact only arrive at a majority if we select those attributes to which a majority assent, leaving open the possibility that a cube may have more attributes but that there is no majority for any particular other attribute.
Now we have a tool for changing the statistical results of a survey into a measure called a probability, by which we may say that the probability that a cube has the common properties is 99/100, for example. Thus we now have to say that any cube having this probability will be dual to another cube having this same probability. It will not be dual to the "ideal" cube.
The upshot of this is that it is not wise to mix ideal processes with statistical processes, even though the formal processes may be written down in similar language, up to the term "probability of"
Scientific methods and conclusions are usually written in this statistical or probabilistic way.
This highlights the difference between a formal analysis and an empirical one. We may infer from many such empirical analyses the formal analysis, but not the other way round. Thus though formal analyses may seem useful, they are in fact only useful in guiding the empirical analysis. Thus Theoretical Physics provides only tools not "laws". Laws of "Nature" are derived from empirical analyses and are therefore probabilistic, cultural inferences.
Kepler's "laws" and Newton's "laws" are both inferred from empirical data from observations, but Kepler culturally was looking for a "geometrical" and Pythagorean Harmony of the data, while Newton was looking for a Fluxionic interpretation of the data. Of the two, Newton's Theology was more radical and Cromwellian(puritan), and thus he looked for a mechanical explanation that linked time, God's Time to the ordinance of every motion. In this he was truly relativistic, Unlike Descartes who fundamentally believed God ordained every action by his "manifest" will.
Today many scientists divorce the cultural and religious expressions behind these "laws", replacing them with the probabilistic nature of the empirical data, However, this is precisely where they mix probabilistic data with idealised processes. The formal idealisation of Newton's "laws" for example lead people to forget that they were inferences form the data at the time, and thus probabilistic. Thus data was fed into the ideal las with the expectation that god's truth would be discerned out of the equations! Problems quickly arose.
When Einstein proposed some radical solutions , nobody cared about his mistakes! They were just glad to be free of the claustrophobic atmosphere that the old school had clamped down on the new questing physicists who wanted to overturn everything because they did not understand the cultural aspects of these "laws".
Behind Einsteins incorrect formulations a whole new school of dogmatists rose up, gradually replacing the old guard with a new , "anti establishment" dogma, replacing "ordinance" with probability.
It is only now, that once again the problems of mixing formal , ideal laws with probabilistic empirical inferences has come back out of hiding to the fore that we may say: They are at it again!
Probability in fact may remove modern concepts of god, but does not remove the ancient concept of the Monad. In fact it places the Monad right at the centre of Modern scientific investigation, interpretation and understanding.
Thus through probability the monad has returned in mathematical guise as the principle interpreter of empirical data.
Naturally, from the monad, under this theurgical, pythagorean system flows the Duad. The duality of the Stoikeioon has always been at the heart of all Philosophy, but now it is put to the use of the monad, that is probability.
"Mathematical" reasoning has arrived at us by a process of wresting cultural and religious practices from their original purpose and putting them to some other analogous use. Quite often this has been done by plagiarists, but those who were forced to be so "unscrupulous" on pain of death by the prevailing religious bigotry, or cultural inertia, or commercial necessity. Whatever the reason, it cannot be denied that the mish mash of methods and notion in what is glibly called mathematic is a confusing mess that needs clearing up.
Apparently this is nothing new, and this is what Euclid attempted to do using the principle of Duality, the cultue of the goddess Isis. Euclid selected from older notions of magnitude and quantitative form, and the attributes between and within such forms to establish his basic teaching material and course material. He kept it current, and revised the format occasionally to reflect new insights. The Theory of the circle and the Eudoxian theory of proportion based on it were his chief fundamentals. That we have lost sight of the fundamental role of the circle and sphere with deductions goin back to the point at its centre and inferences going forward to the notions ot the Arithmoi based on the idea, the fractal notion, of a net of forms (schema,sterea), is due to the rise of Descartes and DeFermat's algebraisation of rhetoric; the development of notation to represent whole sentences of rhetoric.
This "codification" developed the fundamental notation and symbols we use today, but not systematically, but rather ad hoc and driven by convention. Thus we have "confusion" of tongues!
The different uses of the same symbol derives from its different language based origins, its incorrect translation, its opportunistic adoption, and the convenience and art of multilingual translators. Thus, those to whom it was addressed were by convention assumed to have been thoroughly trained as apprentices in the use of such terminologies in such contexts. By this means an unlearned person distinguished themselves!.
It probably causes more harm than good using = to mean "dual" as well as "assign the "value" of"
Thus we have received confusion as knowledge for centuries and that has not always turned out bad for us, as one might expect, even though a general principle of bad siring bad may be put forward. This is because all principles inferred from empirical data are probable!
The best that one can say is that idealism is probably bad. We might even quantify it roughly by adding "in the long term". Such a statement , taken as a principle enshrines the fact that idealism may have some short term good effects, even some longterm good effects, but we doubt that this will be the case as time progresses. Such a sense of foreboding, in fact becomes a self fulfilling prophecy for the pessimist while for the optimist every down turn is a bli in a longterm rosy picture!
However, i may add that such outlooks are predicated on cultural models of time and space, and indeed, those who have a vorticular or "circular" philosophy tend to be more equanimitous in the face of the inevitable change. Thus those who truly hold to a probailistic view must also have a vorticular or circular view since any probability will reset once the event has happened. We currently couch forecasts in this probabilistic way, and in the case of weather we reset the probability every day or cycle.
We have an intrinsic sensibility to probability, and especially as "luck" or chance. In some cultures it has been deified as a separate godlike entity, in others it is a common attribute of any or all gods.
Those that seek to replace the gods in our theurgical philosophy and especially with probability have the problem of causality. The Pythagorean Theurgy that underpins all modern knowledge production,as redacted by Plato, avoids such difficulties,because gods are causal as well as formal. Thus the rock is caused by the god of rocks, for example. And what form would such a god take! why the form of a rock!.
That we have anthropomorphic attributions to such cods is also explained by the power of the god being expressed in terms we might apprehend. Thus kairos governs how such an entity is portrayed in its essential nature.
Those who make probability the god of the dualistic possibility, the binary outcome, do not acknowledge where there idealisation of natural law springs from , nor indeed their incipient idealism! It arises fom the ancient concerns, cultures and philosophies which they choose to ignore.
Thus the goddess of "luck",and the god of bad luck Loki in norse traditions, is the model on which they are building the notion of probabilistic causality. But even in this there is a choice, as luck may be distributed amongst all the gods, and thus causality may be distributed amongst the many differing but related forms , the diffeomorphisms, of current philosophical/mathematical theory. Whether causality is located in one "god" of probability, or spread out statistically and thence probablistically amongst the many related forms in our experience does not really matter. The point is that the ideas are not new but ancient; and have been worked out before modern thinkers in analogous terms in the various forms of the Pythagorean/Platonic philosophical theurgy.
I come now to the notion of hidden parameters, and note initially that the use of the term parameters is subject to scrutiny, especially as it like dimension has taken on a popular use in scientific and philosophical debate that extends it beyond its basic meaning. IN this form it is quite capable of being used in 2 o3 senses even within the same sentence! For the parameters of may parameters are not a well defined parameter!
Thus to search for hidden parameters in causal process inheres this definitional difficulty. I can characterise it as trying to find within a god what actually gives it power to do what it does. I see, and hope it is clear, that essentially we come to the limit and want to go beyond. In this example we come to the idea that there is a god within the god which gives it its power! This is essentially Fractal!
Thus the fractal paradigm becomes the best paradigm we have yet invented to couch these questions and notions in.
The so called hidden parameters may usefully be seen as inner fractal structure within the probability interval (0,1) and this definitely exists as a by product of fractally seequencing the interval. To attribute causality to these "hidden parameters" therefore is no different to attributing causality to the gods. No matter how narrowly we may define the parameters we still will leave deep unnacounted for fractal structure in the probability dscription to which we may ascribe cause.
Those who have replaced the gods as cause with probability distributions are then able , with a bit of deliberate mystification, o ascribe "real" life interpretations to statistical and therefore probabilistic distributions tp link up with our common notions of causality namely :"life force". This notion (vis vitae) we have distributed among the physical sciences as momentum, energy and force after years of ignoring philosophers who have probed these things to alert anyone who would listen or care to read, what it is exactly we are doing.
We cannot escape the limitations we posses vis a vis the universe. We do change our stance, but ultimately we cycle back to the same thing, either directly or by analogy. The ghosts that are in our machines are the spirits and the philosophies of our ancient forebears, who had time to think deeply about these things and to pass on to us genetically as well as culturally the fruit of their ruminations. Thus we may, probably, colonize the solar system and the calaxy as time rounds on us but the probability that we will do it increases with the long term prospects of technological and environmental development, just as at the same time in the short term the probability of some worldwide catastrophic disaster gradually increases. We will probably survive this in one form or another and go on to probably colonize several moons of the solar system as safe harbours in the planetary seas. That being said, the risks of significant harm or failure remain extant, and as always if we survive it will probably be by the skin of our teeth!
The atrraction of duality is clear. Certainty is high, the probability of following a process that gives predictable outcomes is high, and our false sense of knowing ow the universe works is high. Probably most of us would find this a more attractive set of cultural notions to adhere to. The probability of salvation through these means is empirically not any higher than any other. However the quality of insight and knowledge produced by this duality method is better than confusion and being ignorant. That is what the Kohelth says, and i tend to agree.
Ignorance is Bliss? I probably ought to rephrase that aphorism as "there is a probability that ignorance is bliss!"