Today, it is nearly impossible to imagine another paradigm of numbers than what we are taught at school and are experiencing in everyday life. But also academic mathematical studies of number theory, despite its enormous complexity, is based on the classic concept of natural numbers. It is not well known that even proper axiomatizations of the system of natural numbers (Peano Arithmetic) have failed to characterize the system of natural numbers up to concreteness. Despite of the conceptual gaps left, there are no serious attempts to liberate the concept of natural numbers from its modern historical determination as a series of linear ordered elements. There exist many logical systems, different to classic logic, but more or less no arithmetical systems which could be considered as non-classic (heterodox, deviant, alternative, etc.) like the logical systems.
Refutation by fear
There are only a very few attempts to develop radically new ways of thinking and computation. One of such an exception is given by the work of the philosopher and cybernetician Gotthard Gunther. He stated that his experiences shows him that there is a deep existential and emotional defence in Occidental thinking to accept an involvement into new thoughts concerning the foundations of logic and arithmetic and mathematics. Such new thoughts are not proven wrong but are simply rejected and denied by the fear to face and encounter new and unknown possibilities of thinking. Also such projects may be rooted in attempts of a new understanding and interpretation of Ancient experiences and knowledge, the emotional refutation is irreconcilable.
Thus, a historical understanding of the development and limits of our thinking in science and technology could be of help. World-views and paradigms of thinking appear to be enclosed by historical epochs. In his propaganda of global revolution, Joseph Stalin made it clear that their is a holy exception: mathematical sciences are neutral, i.e. class and history independent, especially mathematics of numbers. With logic, the case was more difficult because of the dialectics of Hegel/Marx. But there was a clear cleansing too: dialectics are beyond any formalism.
Today, there will even a math-gene be found and some math-neurons proving the inherent and innate natural human character of arithmetic and logic. Others will conceive a more spiritual explanation.
James R Hurford, The Neural Basis of Predicate-Argument Structure
http://www.ling.ed.ac.uk/~jim/newro.htm
Few new approaches
Nevertheless, new approaches are appearing, more or less despite academic prohibitions, in mathematical sciences, trying to surpass the monocontexturality of classical thinking: polycontexturality (Gunther), ultra-intuitionism (Yessenin-Volpin), poly-mathematics (Arnold) and the idea to a theory of n-categories (Baez). But all those attempts are grammatologically based on a classic understanding of semiotics and its alphabetism. The only exception I know, is proposed by the few sketches of Gunther’s kenogrammatics.
Transitions from Pythagoras to Aristotle and back?
"The access to pluri-dimensionality and to a delinearized temporality is not a simple regression toward the "mythogram"; on the contrary, it makes all the rationality subjected to the linear model appear as another form and another age of mythography.The ultimate cultural revolution which happened in Ancient Greece was the strict separation of the numbers as figurative, geometric, speculative, esoteric, thus Ancient, and numbers as ordinary objects of calculation and contemplation in science and the empirical, economic world, thus modern. The ancient Pythagorean approach was denounced as archaic and mythical. Not suitable for calculation. Aristotle
The meta-scientificity which are thus announced within the meditation upon writing can therefore be no more shut up within a science of man than conform to the traditional idea of science. In one and the same gesture, they leave man, science, and the line behind." Jaques Derrida, Of Grammatology, p.84
has done proper work in cleaning up the scenario. Even today it is dificult to understand his trick: his refutations are based on a logic he just introduced. A logic which in its restriction has no applicability and reasonability for the Ancient paradigm of thinking. Thus, to show a contradiction in Pythagorean number theory is simply a red herring. Figurative numbers have successors and – neighbors. But that is utter nonsense if we just have postulated the principle of linearity of natural numbers.There is not much written evidence from Pythagoras, he even cultivated "deep silence", meditation. In his powerful and detailed denial, Aristotle has drawn a picture of Pythagoreanism we can read now in two directions: pro-Pythagorean and pro-Aristotelian.
enter: Aristotle, Metaphysics, book XII, XIII.
http://www.britannica.com/eb/art-75247
Aristotle's trickery
Pythagorean aritmetic insisted on the systematic exclusivenes of the number four, used in their technique of the Tectractis ("counting by the principle of four"). It can only be a bad joke of Aristotle to presume that the Pythagoreans could not calculate further than to the number four. Because he has known this was wrong he constructed a logical contradiction in the Pythagorean number theory between the finitenes of the Tectractis and the infinitenes of counting numbers; simply by presuming that both number systems must be identical. He didn't accept the difference between counting numbers and Pythagorean conceptual category numbers as scientifically relevant. With this voluntary decision, for which there is no proof, he introduced the principle of the uniqueness of the serie of natural numbers. Today, we are restricted to the conceptual number two: dualism, binarism, digitalism, two-valuedness, etc. But this seems not to be a restriction but a taboo.
The revolution happend for general economic reasons. Its justification was given by its general success. Today we are trapped in this paradigm which has started to turn against its own legitimation. It's time to study the transition which established Aristotelian thinking. Its merits and its sacrifices.
Pluri-dimensionality
Figurative numbers are finite, qualitative, cosmic and related to the soul.
Mathematical numbers are infinite, formal and related to empirical reality. Figurative numbers are pluri-dimensional, arithmetic numbers are one-dimensional.
"Thus peculiar dialectic situation is produced for the earthly thinker. He has the choice of interpreting the Peano sequence of numbers as an ultimate dilution of the orders of esoteric numbers to a degree where they become unfit for the representation of philosophic problems and where they are only good for showing money amounts in cash registers or temperature grades on the scales of thermometers and for similar trivial tasks. But we can also look at them as the material from which we build up orders of esoteric numbers starting from systems with minimal complexity to ever increasing structures of higher order. This produces a scale that proceeds from finitude to finitude! An infinite system of esoteric numbers is inconceivable. If trying to think it we cannot help but apply the numbers of the Peano sequence - which means: we drop out of the realm of metaphysics." Gotthard Gunther, Number and Logos
Thus, in what exactly is such a profound fear rooted?
The deep fear, expressed by many scholars and politicians, is this: if we abandon the principles of linearity and hierarchy in arithmetic, logic, deduction and computation we end in chaos. That is, we naturally will lose our human dignity. Positively, this situation is collected in a beautiful text by Philip Wadler.
"Whether a visitor comes from another place, another planet, or another plane of being we can be sure that he, she, or it will count just as we do: though their symbols vary, the numbers are universal. The history of logic and computing suggests a programming language that is equally natural. The language, called lambda calculus, is in exact correspondence with a formulation of the laws of reason, called natural deduction. Lambda calculus and natural deduction were devised, independently of each other, around 1930, just before the development of the first stored program computer. Yet the correspondence between them was not recognized until decades later, and not published until 1980. Today, languages based on lambda calculus have a few thousand users. Tomorrow, reliable use of the Internet may depend on languages with logical foundations."Was Pythagoras and the Ancient Chinese mathematicians, to mention only this two, pre-humans? And would we fall back to a pre-human level of consciousness if we would give up to believe in the ultimate universality of mono-contexturality? Or do we have to transmute to trans-humans, or even to a Übermensch (super-human) like in Nietzsches Zarathustra? This, with involving all the theological hybris?
http://homepages.inf.ed.ac.uk/wadler/topics/history.html#drdobbs
More at: http://www.thinkartlab.com/pkl/media/SUSHIS_LOGICS.pdf
Whatever it may be.
Even if this world-view may be ultimate, universal and natural for human thinking, today, we are in big troubles with its narrowness and limited conception of hierarchic thinking and computing.
One serious fear to leave the paradigm of linearity and hierarchy for formal and opperative reasoning and computation is the believe that the only alternativity to linearity is circularity. And circularity is not only produc
ing antinomies and paradoxes but was the pattern of Ancient, non-operative thinking. The fear is well justified. "Circular" thinking brought it up to the informational feedback loops of Cybernetics and to its "post-modern" version of Second-Order Cybernetics with its metaphor of Uroboros and criculus creativus. But not to a working new paradigm of hard science. The opposite or complementarity to hierarchy is not given by a simple regression to circularity but needs a complex interplay between hierarchies and heterarchies. Such an interplay can not adequately be represented by a singular metaphor or model. Its realizations are embedded in the dynamics of an interacting complementarity of metaphors, models, paradigms of conceptualization and computation not accessible to classic scientific thinking.Computing paradigm for the 21st century
Problems of interactivity and reflectionality of computing systems are not covered by classic models of computation like Turing Machines. Such a paradigm is modeled along the line of calculation and algorithms. Both are closed systems without any interaction and reflection while computing. Obviously, they are based on the linearity of the arithmetic of natural numbers, or more generally on the linearity of algorithmic sign systems, i.e. formal (programming) languages.
But, say, the Internet is not an algorithmic problem solving system. Locally, there are many programs solving specific problems, but the system as a whole, globally, it is not solving a single problem. It is an interactional service system – without beginning nor ending.
Peter Wegner is emphazising this new situation and has developed important work to its conceptionalization. A new distinction is introduced: abstract algorithmical computation vs. empirical situational interaction.
"The interaction paradigm provides a new conceptualization of computational phenomena that emphasizes interaction rather than algorithms. The recognition that these characteristics are inherently outside the traditional conceptualization of computation is the basis for this new paradigm for computing, built around the unifying concept of interaction. Concurrent, distributed, reactive, embedded, component-oriented, agent-oriented and service-oriented systems all exploit interaction as a fundamental paradigm.With such an interactional approach, referring to situational real world events, like "driving home from work", problems of formalization beyond classic abstract algorithms are arising. But modeling the intuition of the new situation is not yet delivering a working formalism for computation. Interactional computation as a new empirical paradigm needs a mathematical framework which is surpassing the limits of encapsulated linearity. Maybe we should understand that Ancient number theory, Pythagorean and Chinese, is positioned before the distinction of formal/material, abstract/empirical, subjective/objective and computable/non-computable. Such a pre-/trans-scientific paradigm would involve, from the very beginning, interactional subjectivity into the game of its formalisms and operativity.
Peter Wegner’s claim (CACM, May 1997) that “interaction is more powerful than algorithms” challenges our fundamental assumptions about the nature of computation and the notion of computational problems, reinterpreting the Church-Turing thesis without attacking it directly. This claim is an open invitation to researchers to develop models, tools, and methods that can lend credence to it. Since then, pervasive/ubiquitous computing – which epitomizes interaction – has been proposed as the leading computing paradigm for the 21st century."
http://www.cse.uconn.edu/cse/finco05/#panel
Chances to learn from the past
"The author himself confesses that if somebody - before he had the good fortune of knowing McCulloch - had suggested that in Metaphysics we require numbers in order to understand ideas instead of saying that ideas are necessary to understand numbers he would have more or less politely changed the topic.Was Pythagoras Chinese?
It took a McCulloch to show him that it had been the tragic fate of Western civilization to permit the concept of the idea to gain metaphysical precedence before number and that from this very choice the fateful split between sciences and the humanities had resulted." Gotthard Gunther, Number and Logos
Citations from: Jinmei Yuan, Exploring the logical space in the patterns of classical Chinese mathematical art.
"The beauty of Chinese mathematical arts is, to some degree, similar to the beauty of poetry; it requires the participation of subjects."
"Chinese people are only concerned with the logical relations that exist in the present practice, not something beyond the present time, such as “universal truth.”"
"The logical reasoning of kind (lei) can be described as a net, which represents the main characteristic of Chinese logic."
"My standpoint is that Chinese mathematicians’ reasoning was based on a very different presumption. The presumption in the Euclidean tradition is that there is a fixed order in this world, and the goal of doing mathematics is to represent the beauty of this rational order. The presumption of the Chinese mathematicians is that there is no fixed order in this world. For them, things are changing all the time. Following this presumption, any universal rule, which aims to represent the fixed order in the world, is not important, or for that matter, even impossible. The mathematical art in Chinese culture is akin to conversational reasoning."
"The logic that Chinese mathematicians followed in this kind of conversational reasoning deals with the relations among particulars in present practice. The aim of this kind of reasoning is to represent the harmony of relations among particulars at the moment." Jinmei Yuan
http://ccbs.ntu.edu.tw/FULLTEXT/JR-JOCP/jc106031.pdf
To teach Chinese students Aristotle can have a double function: to learn about the world-view of Western logic, ontology, semiotics, etc. and to learn against which Ancient thinking it was established and which means had been developed to do it. It could be an exiting possibility to compare Pythagoreanism with Ancient Chinese thinking. Grammatologically, both are not based in the medium of alphabetism.
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