Superconmathical Definition of Mathematics:a.Superconmathical definition of pure mathematics – Pure mathematics is a system of 100% precise and 99.99…% necessary propositions. (The term precise means that every term in a proposition is absolutely clarified, and every non-axiomatic proposition is supported on the basis of axiomatic one/s, leaving no doubt, except the 0.00…1% superhyperbolic doubt, the principle that ‘anything may be possible’.)
b.Superconmathical definition of applied mathematics – Applied mathematics is a system of propositions constructed by applying some or up to all of the pure mathematical propositions to explain and/or predict unnecessary phenomena. In other words, applied mathematics is a system of 100% precise and 99.99…% unnecessary propositions.2.Philosophy as Mathematics:According to the superconmathical definition of mathematics, the core ideas in superultramodern science and philosophy, though appearing to be philosophical, are, in fact, mathematical. For example, the axiomatic component of the NSTP (Non – Spatial Thinking Process) theory is pure mathematical, while its hypothetical component is applied mathematical.3.Superconmathical Foundations of Pure Mathematics:These are in contrast with the symbolic or, in particular, set theoretic foundations ofpure mathematics (as laid out in Bertrand Russell’s Principia Mathematica). Thesuperconmathical foundations are conceptual (though symbolism itself is a concept)which attempt to define number, for example, as a symbolic representation ofquantity and justify the equality a + b = b + a on the reason that in scalar additionorder is irrelevant (and, if possible, to decompose this concept or a group of conceptsfurther).4.Superconmathical Reconstruction of Pure Mathematics:It entails some flaws in modern/ultramodern pure mathematics, and presents the superultramodern reconstruction of pure mathematics, free of those flaws. One of the flaws is mentioned below.Flaw in the concept of hyperspace – The Joshian conjecture of three – dimensional space [Space, whether appearance or reality, can have three and only three dimensions (The conjecture if based on two grounds: a. The NSTP theory implies falsehood of the ontology of general relativity. b. Four or higher dimensional space cannot justifiably be imagined.) ] implies that the concept of hyperspace is invalid. And the flaw in the concept of hyperspace further implies that the Poincare conjecture [if three - dimensional sphere (the set of points in four - dimensional space at unit distance from the origin) is simply connected] shall neither be proved nor be disproved, as it is based on the concept of four – dimensional space.5.Superconmathical Resolution of Modern/Ultramodern Mathematical Problems:Superconmathical Resolution of Russell’s Paradox -Russell’s Paradox -’A paradox uncovered by Bertrand Russell in 1901 that forced a reformulation of set theory. One version of Russell’s paradox, known as the barber paradox, considers a town with a male barber who, every day, shaves every man who doesn’t shave himself, and no one else. Does the barber shave himself? The scenario as described requires that the barber shave himself if and only if he does not! Russell’s paradox, in its original form considers the set of all sets that aren’t members of themselves. Most sets, it would seem, aren’t members of themselves – for example, the set of elephants is not an elephant – and so could be said to be “run-of-the-mill”. However, some “self-swallowing” sets do contain themselves as members, such as the set of all sets, or the set of all things except Julius Caesar, and so on. Clearly, every set is either run-of-the-mill or self-swallowing, and no set can be both. But then, asked Russell, what about the set S of all sets that aren’t members of themselves? Somehow, S is neither a member of itself nor not a member of itself.’ (See David Darling: The Universal Book of Mathematics, 2004)Superconmathical Resolution -Superconmathically Russell’s paradox is quite easy to resolve. The superconmathical resolution could be stated in just one sentence: As there is no barber who shaves every man who doesn’t shave himself, and no one else, likewise there is no set of all sets that aren’t members of themselves.This sentence is justified or explained below.Suppose there is a barber who shaves every man who doesn’t shave himself, and no one else. Now the barber himself is a man and the supposition requires that the barber shave himself if and only if he does not! This contradiction straightaway implies that the supposition is false. That is, there is no barber who shaves every man who doesn’t shave himself, and no one else.The justification of the sentence ‘there is no set of all sets that aren’t members of themselves’ goes on similar lines.Superconmathical approach to mathematics, being more conceptual than the traditional symbolic or set theoretic approach, makes it possible to resolve such fuzzy problems or paradoxes.