Polynomials!

Berkeley queered the pitch!

Well, he was not the only one. It became fashionable, after Kant published his works to attack Newton and Leibniz, principally on his method of fluxions, and Leibniz on his differential calculus. The reason was that the theological ministry, the state departments that conferred doctorates of Philosophy and Divinity alike, qualifying candidates to teach either academically or theologically perceived Kant as an antinomialist, and against theological tradition!

Nevertheless, during the ensuing debates and arguments Kant's approach, philosophy and rationalism increasingly won the peace. A growing number of intellectuals, intelligentsia or scientifically and academically minded people, tired of the hair splitting debates on the religious and spiritual front, agreed with the pragmatism he evinced.

http://muse.jhu.edu/login?auth=0&type=summary&url=/journals/journal_of_the_history_of_philosophy/v009/9.4wilson.pdf

Kant and "The Dogmatic Idealism of Berkeley"
Margaret Dauler Wilson

From: Journal of the History of Philosophy
Volume 9, Number 4, October 1971
pp. 459-475 | 10.1353/hph.2008.1561
In lieu of an abstract, here is a brief excerpt of the content:

Kant and "The Dogmatic Idealism of Berkeley" MARGARET D. WILSON IN TIlE "CRITIQUE OF PURE REASON" Kant maintains that space and time are (merely) a priori conditions of our perceptual experience — mere "forms" under which our sensible objects must appear. Thus space and time have no claim to reality independent of us, of our experience: they are "transcendentally ideal." Similarly, the objects we perceive in space and time are also said to be tran- scendentally ideal: since their character is determined by the spatial and temporal conditions of our experience, they have an intrinsic dependence on us. Kant contrasts these mind-dependent or conditional perceptual objects ("appearance") with the realm of the unconditioned, transcendentally real "thing in itself." Our knowledge is limited to sensible objects, to appearance. 1 One of Kant's most insistent claims is that his "transcendental" idealism differs radically from all previous idealisms, and indeed vindicates "empirical realism" against them. Whereas earlier idealisms deny or call into question the reality of the physical world, Kant contends that transcendental idealism provides a uniquely secure basis for the claim that we do have knowledge of real things in space. He holds that knowledge of spatial reality is possible if and only if space is regarded as a condition of our perception, and things in space are distinguished from things in themselves. Specifically, Kant represents himself as a defender of realism against two idealist positions. One is "problematic idealism"–defined as the doctrine that we can have no immediate knowledge of objects in space; that such objects can at best be inferred as causes of the immediately perceived ideas in our own mind. Kant claims that such inference can never lead to certainty; hence on this view the existence of outer objects would always remain problematic or doubtful. Not too surprisingly, Kant associates problematic idealism with Descartes and his i Kant characterizes his position as "transcendental idealism" at A 369 ft. Cf. A 28=B 44; A 36=B 52. The quotations in this paper are from Norman Kemp Smith's translation (Lon- don: Macmillan, 1958). [4591 460 HISTORY OF PHILOSOPHY followers. 2 He discusses this position in the first edition of the Critique ("Para- logisms" section), 3 in the Prolegomena, 4 and in a new section inserted in the second edition of the Critique, titled "Refutation of Idealism." s (As we shall have occasion to note below, there is a clear change in Kant's manner of replying to problematic idealism in the course of these three works.) The other idealist position to which Kant explicitly contrasts his own is called "dogmatic idealism." The dogmatic idealist is said to hold that there can be no real things in space, that space (and everything in it) is "false and impossible," or that spatial appearances are mere "illusion." Dogmatic idealism is mentioned only briefly in the first edition (in the course of the reply to Descartes); 6 it is not attributed to any particular philosopher. In the Prolegomena and second edition, however, Kant repeatedly associates this doctrine with the name of Berkeley. 7 And he seeks to emphasize the merits of his own position, and especially his conception of space, as an answer to the "dogmatic idealist." As Kant openly indicates, the new polemical interest in Berkeley resulted from the critical reception of the first edition: more than one reader, to Kant's displeasure, thought there were significant affinities between transcendental idealism and Berkeley's philosophy, a Kant's conception of problematic idealism is quite perspicuous. Further, while there are points of obscurity in all of Kant's anti-Cartesian passages the general line of attack is sufficiently clear in each case. His treatment of dogmatic or 2 The attribution of problematic idealism to Descartes does involve some license, of course, since Descartes himself concluded that the transcendental causal inference can be guaranteed. But Kant, like many post-Cartesian philosophers, evidently felt that the most…

Kant not only advocated pragmatism, he also established the working principles of the empiricists. He advocated an mpirical basis to science, and helped codify the scientific method. In addition he made fundamental recommendations with regard to the synthesis of Knowledge based on empirical data, and the notions of procedural development and falsifiability of process.

http://plato.stanford.edu/entries/kant-spacetime/

http://en.wikipedia.org/wiki/George_Berkeley

Although he did not set out or work out an axiomatic basis for mathematics, he did endorse axioms as a fundamental logical necessity to the synthesis of sound logical arguments. The modularity of his notion was that should an axiom prove to be wrong then all derived logical statements from that axiom could be discarded, and the ground cleared to rebuild anew.

He however distinguished between mundane human knowledge, for which this approach was appropriate, and transcendental knowledge for which it was not. In his way he preserved the notion of Divine, inspirational knowledge emanating from god.

However, Kant was not a conformist theologian, and many of his views were at odds with the church, the state established church, and thus a ministerial department. Because Kant was Newton's champion in terms of the general scientific approach, and the efficacy of his Principia Mathematica in particular, championing it over Descartes philosophical method, and thus over Leibniz, he focused the narrow beams of religious apologetics onto Newton's method of fluxions, exposing it to some derisive criticism.

In the meantime , those in Leibniz school, mindful of this religious antagonism, sought to assuage the arguers by refuting their critiques. The critiques were rather simple minded but effective, because they raised the question," what is a number? "

Because" mathematicians", particularly those who revered the new calculus could not answer that simple question, that seemed to put the whole calculus on a foundation of shifting sands! And if the calculus was thus shown to be nonsense, well Newton's philosophy was thereby damaged, and Kant the main trunk they were seeking to chop down, was proved to be the Charlatan the religious establishment claimed he was!

As regards the foundations of mathematics, it seems that Kant was responsible for this shift toward setting up an axiomatic basis, but Fries is the one who first set mathematics out axiomatically . In this newly developing praxis, the ring theorists and the nascent group theorists began to take a leading research role. Amongst these, right at the forefront we find Justus Grassmann! Though he was initially known to a small group of collaborators, hs influence was felt and was extended by the work of his 2 sons Robert and Hermann. Principally Robert promoted his Fathers work, and got him known among a wider public. Robert also promoted hs brothers work, removing years of obscurity from his brothers masterpiece, nd helping to make the Grassmann's the second famous family from Stetin after Jakob Steiner.

I have dealt with the differences of opinion between the two brothers, so here I am looking at Justus contribution.

Justus contribution at first spears to be mysterious and new, even avante garde! But that is merely propaganda. In fact Justus is making a retrenchment back to the fundamentals of Newton's day. His contribution is a careful analysis of the algebra of polynomials!

When I first came upon polynomials it was at school in the traditional maths curriculum. Because of the lesson plans I was introduced to many confusing names starting with, unknowns, variables, expressions, equations formulae, quadratic cubic and linear, simultaneous, degrees and finally polynomials in one and 2 variables. When I went to university, analysis started with the axiomatic method, which confused the hell out of me! And then focused on polynomials on the interval [0,1).

That term on polynomials was all there was time for, and it was seen as old fashioned even then! It is only when I started to find solutions for a triple algebra in the fractalforums.com that the significance of polynomials started to dawn on me.

It is unspeakably profound how polynomials are the very heart of all algebra, and how they in fact encode all algbras in the most accessible way. It is because of polynomials that calculus is true and survived Berkleys attack. It is because of polynomials/multinomials that we have the most robust and effective computational machines that Turing could devise. It is because of polynomials that we have codified Gematria, that is combinatorial theory, and the resultants of ring, group and extensions Theories. Die Ausdehnungslehre of 1862 is principally Robert Grassmann's work on the back of his brothers slim metaphysical volume of the same name written in 1844. Roberts book ties straight into the traditions that lead to group and ring theory. However Hermann's book is something else almost entirely!

While I will lay out the fundamental simplicity that Justus exhorted, it must be born in mind that Hermann went fundamentally deeper and further back than even his father. Hermann felt he went back to touch the face of God!

There are 2 distinguished actions in combinatorice:sequencing and expanding brackets!

Of course one needs things to sequence or an experience of sequence such as progression or flow or ranking, layering folding, combining, factoring synthesising. You aso need bracketing to distinguish particular sequence relations.

The notion of bracketing is quite general and though it refers to a particular printers markings it does not mean those marks or markings. it means "remarking" in the direct sense of calling attention to a particular sequence of things or processes.
The expansion of brackets is such a familiar algebraic operation that one misses its combinatorial significance. However, the whole notation of operations and actions is remiss in this exposition of mathematics. Even at the elementary level it cannot be justified! Particularly since there is no further development after this elementary introduction!

From the very outset, when a pupil is learning number bonds, they must also learn combinatorial terms reflecting those bonds in the most general algebraic settings. Numbers should no be anything more than just one of man examples of combinatorial principles of terminology, sequencing, bracketing and expansion of brackets. These should clearly be reinforced as examples of factorising based on distinguished analytical terminology and synthesis by appropriate sequencing and notation.

Inan earlier post I attempted to derive a link between the spatial dimension or orientation of sequencing objects and the dimensions of the concomitant array. This was a vague and not very successful attempt to exposit dimension as a function of sequence orientation. It is vague because we naturally collapse all sequencing to a few analogues that visually represent the flow or motion of an object. We further collapse this to a 2 dimensional mark system, and completely ignore 3d spatial orientation, replacing it with typographical concerns and page layout!

We have several sensory systems that mesh together so the notion of sequence is really pretty complex. The mst fruitful analogue of the general sequence notion is that of a freestyle dancer. In this exemplar we may track the notion of sequence in spatial and therefore orientation terms, but in addition or can track other sensory analogues of the same sequence from an objective and subjective standpoint.

One can also form he notion that ones internalisation of sequence casts a proprioceptive sequence map on ones experience of external sequence apprehension.." I perceive sequences because I process sequentially internally".

The origin of the subjective process sequence I have discussed in a previous post.

Thus sequencing as an action is very complex and involves great aesthetic sensibility when it comes to synthesis, and great systematic derivation when it comes to analysis. Bracketing of these sequence processes is a fundamental denotation of control, factorisation, aggregation etc. the conjugacy adjugacy analysis underlies the use of bracketing in sequence theory..

In algebraic history bracketing arose in relation to aggregation of unlike labels, but bracketing is an innate sequence process we have always used. Take for example the focus region, that is a conjugate bracket. The comparison of two adjugates is an adjugate bracket. The isolating of an analytical object is a singleton bracket, and the synthesis of a number of objects as building blocks in a larger synthesis is a synthetic bracket. It is the expansion of synthetic brackets that we call expanding brackets, and it is the heart of combinatorial process.

Combinatorial process processes sequences of things, each sequence is a distinct synthesis process with its own rules. Synthesis processes are not confined to one origin, and the same synthesis process may originate anywhere. It is our mental act of combination or synthesis that mentally initially and often subsequently physically brings these synthetic processes together. However the rules that govern this synthesis is that like attract and combine, unlike remain separate or even repel different synthetic processes.

We do not often consider a fundamental anomaly in physics likes repel and opposites attract!! We can see that algebraic rules and this supposed physical fact are at odds. This effects our mechanical description of reality. The point is that this physical description is no more true than the algebraic rule, and in fact may be a misunderstanding of the empirical data. The algebraic rule applies everywhere else, so it seems strange tht it does not apply to electricity and magnetism.

The distinction between conjugacy and adjugacy I have explained previously, and where conjugacy comes in as conjunction of factors or factorisation, while adjugates Coe in on the lower, dependent level as aggregate able pieces.. Thus within a bracket we may have a sequence synthesis that represents a conjugation or an adjugation. The distribution of conjugation onto aggregated adjugates is a remarkable thing taken for granted.

If a conjugation inheres a aggregated adjugate sequence, then that can be dualed with an aggregation of conjugates. The concept of aggregating conjugates. This only makes sense if the conjugates are in fact adjugates of a larger system..

This mental facility to flit between adjugate and conjugate status is the reason why there is no end to many things.

Not many know about adjugacy or conjugacy, because it is only taught in a mathematical version to and between higher mathematicians, physicists, theorists etc. otherwise it is the subject matter of esoteric teachings in meditation. However, it is not hard, just hard to remove the stunning mundaness of it. It is a function of my mental processing, therefore subjective . But ultimately it must derive from the ceaseless sequencing of Shunya. The way it does is through catalytic preference.

We have to attribute to Shunya every potential needed to get the job done. In this way I connect to space, that is Shunya, as a creative act of attribution. But I do this to orient the construction of my experiential continuum, and thus to provide tram lines and scaffolding for the creation of a self as an identity.. Thus the reality I accept is due to the processes I have accepted. Those processes are situated somewhere, and I have to accept where. It seems the healthiest disposition is to separate them out into internal and external with an intersection, and then to iterate this process as a lifelong process of apprehension and comprehension.

Thus the very foundation of my sanity and social adjustment is based on conjugating and adjugating in the most pragmatic way! It is the same for you.

So now, the attributes of conjugacy are : unless hindered conjugates are always associative and commutative;
Conjugates are always in pairs at the very least, but can be any number of factors;
Conjugates consist in adjugates;
Conjugacy is usually distributive across the other conjugates adjugates!

Let us look at this notion of distributivity.

Firstly, it is not an apprehensively property. If I conjugate Shunya into C.k and k happens to consist of 2 adjugates a,b I cannot conjugate Shunya into C.a +C.b because as I do the conjugations C changes!. In one case C is replaced by C+b, and in the other C+a
To make this distributive" law" work I actually have to say that anything conjugated with itself does not exist as a conjugate. So for example if my focus region is a it has a conjugate with a larger spatial region or a smaller spatial region may be conjugated in a but no conjugation can be done with itself. We meet these conditions when we come upon the proto Arithmoi in any serious study of them.

If I focus on Shunya, or analogously, fractally on a then that is Shunya, that is everything. I experience this as whole ness, or entirety and I usually denote this as 1 or a single experience whole and entire to itself, seamless and without distinctions, phases or transitions, the eternal dynamic now!

Therefore it makes no sense to assign conjugacy or comparison to this experience. However. That is not the same as saying it does not exist! Thus to equate it to 1 and to zero is a clear contradiction, but one we seem happy to make, or else it has gone unnoticed!

It has not gone unnoticed, for Brahmagupta remarked that the Greek monad was a departure from Brahma, the Entire! But I have explained that as a deliberate misunderstanding by Brahmagupta to preserve the superior Inian traditions.. Similarly the Greek loving west, and the Arabic mediators by hook or by crook converted the notion of Shunya into the notion of Sifre or Zephyr, a wind of nothingness! Later called Cipher, and then zero.

In the course of doing this, astute translators recognised the Indian tradition of encoding information in verses, rhymes, dance movements and numerals. This facilitated the military coding and decoding establishment in its proficiency in secreting information. The act of coding became known as creating a cipher! Decipherment was the act of decoding a coded message.

So it is as if the dichotomy has remained because the one encodes the other!

Distributivity can only work as a process if I accept that anything that conjugates itself can be ignored! Because it is ignored that is why it is left out of any notation. If ever it is included in notation it is usually assigned the notational label 0 , which as we have just seen is a "code" for "1", which is a label fr the experience of Shunya.

And Shunya is….everything!

There is a long tradition of returning everything to the background and calling it not counted, not countable, unaccounted. But this has never been confused with Nohingness until a particular Christian line of apologetics attempted to make the argument that god was so supreme he could call forth something from nothing! This is a logical nonsense and an example of extreme ascetic delusion, in my opinion! Fortunately the texts that underpin all religious wisdom do not support this modern delusional stance!

Now I have been pejorative in describing this particular view. I have a responsibility to be so, but that does not mean I am bothered or would deny you the right to accept this viewpoint. But it usually motivates extreme injurious action down the line and that I wouldn't counsel against. And although I have identified the Christian source of this thinking. It is not merely a western pattern of thought. Many unfortunate Muslims have blown themselves and others to pieces in the pursuit of A misunderstanding of Jihad, and the operation of Allah in his creation! Allah. Blessed be his name, is all merciful and Wise! Therefore He is merciful to the infidel, whereas some of those who claim to serve Allah, blessed be his name, are not!

Continuing:
Thus C.k = C.(a,b) where comma means an undefined adjugate relationship
Can be dualed with C.a,C.b providing a.a and b.b are viewed as returning to Shunya . In so doing, we slightly adjust the underlying Euler/Venn diagrams to reflect this change in the relationship tp Shunya. It is like a bubble adjustment: a bubble of a.a and b.b gas pops out leaving C , a, b the same metrically.

Straight away, that last comment shows that we want this behaviour to simplify our calculuus when metrics are involved.

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