Back ... ... ... ... ... ... ... ... ... ... ... Send comments to review committee. ... ... ... ... ... ... ... ... ... ... ... Forward

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = =

In 1949, German mathematician H.E.Richert proved the following inductive statement [1]: "IF there exists a natural number, say, n* such that Q(n*) is true THEN for any natural number n>n* P(n) is ture", or in a short symbolic notation:

[$ n*Q(n*)] ® [" n>n*P(n)], (1)

where P and Q=f(P) are two collections of number-theoretical properties of common finite natural numbers (or predicates given on the natural numbers set).

The Richert Theorem is an inductive statement of the form: "from a SINGLE statement, [$ n*Q(n*)], to a COMMON one, [" n>n*P(n)]".

In 1978, using Cognitive Computer Visualization of mathematical abstractions technology, A. Zenkin discovered two new classes of Induction Statement and formulated a general Super-Induction (SI) method. By means of the SI-method, conceptually new scientific results where obtained in Classical Number Theory

We list below some connections of SI-method with some basic logical conceptions.

1. According to inductive J.S.Mill's Logic, we always can formulate a common statement, say, H basing on a set of particular facts, but such H will always be only a plausible statement. The existence itself of SI-method show that the main inductive Logic paradigm is broken in some areas of discrete mathematics.

2. The "MODUS PONENS" RULE sounds so: [A&[A® B]] ® B. Mathematical Logic and meta-Mathematics consider the implication [A® B] as a deductive inference of a less common consequence B (e.g., a theorem) from a more common premise A (e.g., an axiom system). SI-method generalizes the "modus ponens" rule to the case when the premis A is a single statement, but the consequence B is a common one.

3. It can be easy shown, that the SI-method generalizes the classical complete mathematical induction B.Pascal's method. Moreover, SI-method works well there where B.Pascal's method simply does not work [5].

4. Cognitive Visualization of mathematical abstractions and SI-method allow, by certain conditions, to use corresponding cognitive images as quite legitimate arguments in rigorous mathematical proofs, i.e., they realize authentic proofs in, say, L.E.J.Brouower's sense.