Is Extension continuous or discrete?
The question arises in the context of the Cartesian Theory of Extension, and took sevral centuries for a view to be formed on the answer. the answer is that extension is continuous unless it is dicretThis very m, very much depends on a concept called Scale, and the microscopic scale was only just beginning to be investigated by the time of Hooe and Newton. Nevertheless Newton proposed a calculus that dealt with observables pramatically at any scale, and he did it on the basis of fluents, fluxions and the notion of exhaustion.
In one very real sense Newton proposes that everything is discrete up to the metron used. When it becomes tiresome to assign a distinctive faractal pattern, to count or account anymore, then pragmatically the object is continuous that is less than the metron. Newton carefully constructed an "infinite" series which made that pragmatism secure, and it is the very basis of number theoretical approaches or numerical solutions.
However, on the continent, this clear pramatism was not adopted, and the Leibnizian calculus suffered from logical errors because of this . Dedekinds cut , a discretization of otherwise continuous extensive magnitude into a quantity was necessary to rescue Leibniz calculus from its faults, but not so for Newtons methods of Fluents. However, Berkely ensured Newton was tarred with the same brush as Leibniz.
The concern was the loss of the notion and experience of "spirit". This was in facr a medieval notion much honed to the theology of the Church at the time. Whereas ghost, djin, wind, and essence have pragmatic sensory sources, the medieval notion of spirit was made immaterial and insubstantial not consubstantiable. Thus it was a wonder how ot was perceivable, and the answer was tort: spirit perceives spirit. To argue otherwise put one on a very dangerous ground in European Christendom.
The notion of extension was therefore not the extension of Gods omnipresent spirit, but the extension of his separate material creation, into which he placed his spirit . Poor , sinful humans were bound in such a material body by gods decree, for no material could interact with spirit it was taught. Thus how spirit controlled the body was an endless debate, with one side saying there must be a link, and another side saying there was not, the purpose of being trapped was to learn to kill the body and its temptations by clinging to Gods moral precepts.
Thus Monism and Dualism became ahot debate that fracturd into many camps. The scientific camp was also fractured along religious lines. Those that attempted to cause spiritual "mirac;es" like transmuting material were clearly in league with the "Devil" or occult spiritual powers!
The reason Kant was so important was because he said to those who were of scientific and mechanistic bent seek empirical evidence! similarly he said to those of spiritual immaterial bent let God demonstrate his power to bring about the down fall of error! Thus on a very contentious issue he was able to bring about relative peace, with all waitin to see the outcome of the "trial"!
Well the trial is not over, but over time more have switched over to the scientific , pragmstic, mechanical view by sheer empirical demonstration, and even some religious leaders bow to the lack of divine intervention as indicating they got it wrong, but not all.
It is probably fair to say that the modern western Scientific view is down to Newton and Kant. Howeever, that does not make it the only view of reality, and many eastern philosophical views are just as apt. However the technological advance achieved through the engine of imperial appropriation meant that the "best" of the worlds technological wisdom came to be concentrated in the west. To then pout out ones chest in triumphalism claiming superiority or manifest destiny is a perversion of the trial by the gods system so prevalent in the medieval world.
I watched richard Hall inventing the Indian, and would have been shocked, were these practices not a common occurence of imperial vanquishing of subjugated peoples. Morally it appears reprehensible, biologically it is totally in keeping with the concepts of dominance , symbiosis and antibiosis, including apophysis. We are what we are made of!
The current data sets support the new knowledge we have acquired by better sensors. The theory behind these new tools may not always be sound, but the pragmatic data is invaluable.
Turings unique access to Babbage engines, and rotating calculating deices that fullexploited the p-adic, Indian aabic numeral structures, which were thouroughly investigated in Indian Vedic mathematics as well as by Euler, meant he had a unique opportunity to meditate on the confluence between motion, rotation, computation and encoding. That he designed or glimpsed a Universal Calculating and Encoding machine is a measure of the enuring atmospere he was forced to work under during those hard difficult war days. No other creative atnosphere would hae lead to the rapid development of the Babbage engine, The Leibniz calculating macine, and many othermechanical devices that preceded the creation of the Enigma coding machine or the Colossus Decoding devices.
What all this innovation demonstrated was that Newton was correct to approach quantisation discretely. Approximations of real solutions could be computationally achieved by discrete methods!
The snobbery of the Mathematical board, delayed the implementation of this technology into the education system by decades, but today we could not live without it, or so they say!
Thus, though the jury is still out in some peoples minds, culturally discrete computation is well accepted and inherently understood by all who see it in action. The continuous spiritual world is not so accessible and is rightly consigned to mythological duties. This is not dissing mythology, as i believe mythology has a profoundly important role in the human Psyche
Ray Bradbury famously said when Voyager reached Jupiter that it required us as humans to develop New
Mythologies! this i concur.
Extension, wen discretized underpins all the sciences and Mechanical engineering and especially the biological and chemical, and the computational. For this reason the Grassmann Analytical method is an important theoretical tool.
If i have some extended material and i cut away some of it leaving a hole. where does that "hole" come from? was that space there before the material extended into and through it , or does it onl exist within the material as a space created and sustained by the materials extension? Of course, as usual the answer is a continuum of both! And so it is , for continua and discrete, that both attributes i may apply to the same form in the same reference.