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Last updated: 2005-06-11 07:29 pm / Author: Victor Porton
21 Century Math Method (21MM) is a new method replacing older axiomatic method. 21MM is discovered by me, Victor Porton. I welcome you to participate in this new field of mathematical research.
Below are the possible research plans.
Conjecture. Any true expression can be proved. (I'm not sure whether the Godel theorems are applicable to 21 Century Math Method because the set of inference rules is not fixed. We need to look into the exact formulation of the Godel's theorems to check whether they require a fixed set of inference rules.)
A more strong conjecture:
Conjecture. Every expression which does not contradict to the classic logic and main axioms can be written as a theorem in 21MM formal system.
We probably can define applying substitutions inside our formal system.
Let's use only pairs [A B] instead of arbitrary lists (A0 A1 ... An) to express expressions. (Lists can be expressed through pairs using linked lists / binary trees.) (Alternatively we could introduce the concept of the first element of a list and the rest (n-1) elements.)
Then the following (wrong) equation would define substitution, except of the rule that the same variable should be substituted by the same expression (∧ is used here as a separator of set elements...):
(([A B] → [C D]) [E F]) = [(([A] → [C])E) (([B] → [D])F)] ∧ ([(([A B] → [C D])E) F] ∧ [E (([A B] → [C D])F)]).
Can we modify this equation to make it correct?
Further simplify the formal system.
Anything can be written using only substitutions.
(base X) and (base (derived Y)) can be used as type marks (instead of plain constants) to express base/derived class relation).
algorithms and software for 21MM:
algorithms of simplifications of equations in 21MM;
21MM based automatic provers of 2nd-predicates theorems;
theory of relations between axiomatic systems and classification of axiomatic systems (Isn't it already developed? Isn't so the entire math concluded in something like group theory?);
variations of set theory in 21MM, possible corresponding relations of 21MM with traditional axiomatic method, whether the old axiomatic method preserves its scientific value after discovery of 21MM;
we shall decline from the way of a formal system as a linear chain of formulas to some modular object oriented approach, centered on definitional constructs modifying meanings of symbols (a graph or tree of modules);
probably diagram based instead of formula based mathematics;
philosophy of 21MM, particularly regarding the principle of simplicity;
effective reformalizing known math in 21MM (see also about Algebraic General Topology below);
How 21MM relates with the Category Theory?
algorithmic methods for effective reformalizing known math in 21MM;
a map of of all published math knowledge formalized in 21MM, studying the properties of the map and its development;
about partial definitions (e.g. X is a real number) with later concretization (e.g. X=5);
I suspect that math can be split into three separate sciences: formal algebra, theory of definitions, and theory of math induction; how these may relate with each other?
self modeling of 21MM + models of infinite expressions;
Quantifiers could be be defined (check!) using constants as bound variable symbols. Which rules would disallow to use both quantifiers and any structures "isomorphic" to quantifiers? ("to evade the evil")
21st Century Math Method in pedagogy;
the next math method?
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