The Structure of Mathematics as a Well-Founded, Self-Similar Syntactic Fractal

1. The Structure of Mathematicsas a Well-Founded, Self-SimilarSyntactic Fractal Damon Scott Francis Marion University Spring 2010

2. The Sorry State of Prior Practice Prior formalizations of mathematics adopt different structures for different scales of endeavor. In the most standard formalization, statements are put together with the operators , , , and  and the quantifiers  and . Proofs have the structure of a string of statements, but lectures are completely unformalized and have no structure to call their own. Branches of mathematics are constructed differ-ently, with sets of axioms followed by sets of theorems.

3. The New Formalization In the new formalization, all mathematics has good structure and it has the same structure at all scales. This new formalization is called context-oriented mathematical logic.

4. How This Is Accomplished This is done by inventing new logical operators that ramify by scale of expression. Like the traditional logical operators, they show how statements are put together, but, unlike any logical operators seen so far, they ramify by scale of expression, and thus are able to describe how lectures and branches of mathematics are put together too. And this itself is done not by operators that con-struct mathematical statements per se, but rather by operators that build mathematical contexts. Hence the name, context-oriented mathematical logic.

5. Our New Operators are Written with Sans-Serif Letters Unfortunately, not all upside-down and backward letters are readily available for power-point presentations, and so we write all our operators as sans-serif letters, as in A and E to represent “for-all” and “there-exists”. This convention gives a unity of appearance for the new system.

6. The Basic Set of New Operators For any statements P and Q, make by definition G {P }, Q iff P  Q R {P }, Q iff P  Q Furthermore, for any statement Q and variable x, make by definition A {x }, Q iff  x, Q E {x }, Q iff  x, Q

7. Going Context-Oriented A radical change occurs when one separates the syntax displayed in the last slide and so creates “hungry functions” on the space of statements: “G {P },” “R {P },” “A {x },” “E {x },” Syntactically, they are clauses. Semantically, they construct contexts. And both semantically and syntactically they ramify by scale of expression. There are as well similar operators for negation and for performing substitutions, together with a few others. They are omitted from this lecture for want of time.

8. Ramifying the G Operator by Scale of Expression We now can ramify the G operator by scale of expression. In small scale expression (scale of statements), “G {P },” means “if P ,”. In middle scale expression (scale of lectures and proofs), “G {P },” means “Assume P.”. In large scale expression (scale of branches of mathe-matics), “G {P },” means “Let the following be postulated: P.”.

9. Ramifying the R Operator by Scale of Expression Wecan also ramify the R operator by scale of expression. Insmall scale expression (scale of statements), “R {P },” means “then P ,”. Inmiddle scale expression (scale of lectures and proofs), “R {P },” means “Thus, P.”. Inlarge scale expression (scale of branches of mathe-matics), “R {P },” means “Proposition: P.”.

10. Ramifying the Quantifer A by Scale of Expression As for the operator A, it is easier to see its ramification by using the qualified quantifier: Insmall scale expression (scale of statements), “A {x }, st {P },” means “for all x so that P,”. Inmiddle scale expression (scale of lectures and proofs), “A {x }, st {P },” means “Let x be so that P.”. Inlarge scale expression (scale of branches of mathematics), “A {x }, st {P },” means “Define x so that P.”.

11. Ramifying the Quantifer E by Scale of Expression The operator E does not ramify by scale of expres-sion nearly so well as does A in practice. This might indicate a limitation to how mathematicians conceive of mathematics at the present time. Alternatively, it might represent a limitation of ourselves to see the larger-scale E-clauses in the practice of mathematics. Perhaps mathematicians in the future will practice middle- and large-scale existential quantifications as readily as middle- and large-scale universal quanti-fications. At the moment, so far as we can tell, they don’t.

12. Small Scale Expression Inthe new language, the statement of a theorem might have the following form: G { A {x R}, st {a < x < b}, R {f is differentiable at x}, 1;} R { A {x R}, st {a < x < b}, R {f is continuous at x}, 1;} 1; The notation “1;” is the statement “boolean truth” and is necessary at the ends of context-oriented strings so that the syntax form statement structures.

13. Middle Scale Expression We can write a proof in the new language. Interior deductions are formalized with R-clauses: A { A, B, C  points}, A {a, b, c  lines}, st {a,b,c are opposite A, B, C}, G { the angle at C is a right angle}, A { D, E, F, G, H, K, L  points}, st { the famous windmill diagram is set up }, R { First interior deduction of Euclid’s proof }, . . .(other interior deductions as so many R-clauses) . . . R { a2 + b2 = c2}, 1;

14. Large Scale Expression We can write a branch of mathematics in the new language too. G { First Axiom }, G { Second Axiom }, A { 0  R }, st { A { x  R }, R { x + 0 = x }, 1;}, R { Proposition 1 }, R { Proposition 2 }, G { Definition 1 }, R { Proposition 3 }, 1;

15. Seeing the Self-Similarity Clearly the structure of expression was the same for the small-scale, the middle-scale, and the large scale examples. When it is all written in one, long expression, then we see all of mathematics forming a self-similar syntactic fractal. The structure is called “well-founded” because the scales do not go arbitarily small. At the level of atomic statements (or, for that matter, the individual symbols), the structure is not further divisible. But the scales can go arbitrarily large.

16. Example of Self-Similarity: Zooming In Here is a view when we zoom in on a piece of syntax. G { First Axiom }, G { Second Axiom }, A { 0  R }, st { A { x  R }, R { x + 0 = x }, 1;}, R { Proposition 1 }, R { A { f  R  R }, G { f is uniformly continuous }, R { A { x }, A { e }, st { e > 0 }, E { d }, st {d > 0}, A { y }, st { 0 < |y – x| < d }, R { |f (y) – f (x)| < e}, 1; } G { Definition 1 }, R { Proposition 3 }, 1;

17. Ramification of Calculus Rules Not only do the logical operators ramify by scale of expression, but all logic governing them also ramifies by scale of expression. Inthe new system, the logic governing state-ments is the logic governing lectures and proofs, which is the logic governing branches of mathematics. The new logical apparatus is thus remarkably coherent and powerful.

18. Read All About It Theentire system is explained in my book, The New Formal Linguistics and a User-Friendly Formalization of First-Order Logic. It should be coming to a bookstore near you once I can find a publisher for it.

19. The Structure of Mathematicsas a Well-Founded, Self-SimilarSyntactic Fractal Damon Scott Francis Marion University Spring 2010