\documentclass{article} \usepackage{amssymb} \usepackage[all]{xy} \begin{document} According to David H. Bailey and Jonathan M. Borwein in an article on experimental mathematics in the current issue of the Notices of the AMS, G. Polya quotes J. Hadamard, "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it." This predates my motto, "Rigor cleans the window through which intuition shines." I intuit that there is a Ground beneath all foundations of mathematics. A few details are in my book, ``Mathematical Mechanics: From Particle to Muscle," from which the following is derived. The primary distinction of the Ground of Mathematics is between Discourse and Surface. Every expression occurs in a region of the Surface. The Discourse may specify expressions and regions in the Surface. A Context is a specified region of the Surface within which smaller regions may or may not contain expressions. In any case, the extent of such a Context is clearly marked, for example, by Chapter, Section, Subsection, or Paragraph headings. Thus, Contexts may be nested. Sometimes a single expression is considered to be a Context. Care must be taken to observe Context boundaries. In a specified Context the choice of a symbol to represent an idea -- including all its copies in the Context -- may be replaced by some other symbol in all of its occurrences within the Context, provided the replacing symbol occurs nowhere else in the Context. In this sense the replaced symbol is called bound. For every symbol there is a sufficiently large Context in which it is bound. The Ground includes human cognitive ability capable of answering the following questions: What is the specified Context of the Surface? What is the specified region of the Surface? What is the specified expression? For a specified region of the Surface is there some expression occurring the region? Is a specified expression occurring in a specified region of the Surface? Of two specified regions is one left, right, above or below the other? Of two expressions in distinct regions, is one a copy of the other? What is the total count of expressions in a row, column, or other specified region? Is a Context nested within another Context? The Ground includes human muscle contraction capable of performing the following actions: Introduce an expression specified in Discourse into a specified region of Surface. For example, to introduce a copy of an expression of Discourse in a blank region to the right of a specified region. Repeat this action to yield a list expression on the Surface. Copy the expression in a specified region into a distinct specified region. Mark the start of a Context. Mark the end of a Context. Delete the expression -- if any -- occurring in a specified region. These capabilities are called the Ground Rules of Discourse. The book discusses Symbol \& Expression, Substitution \& Rearrangement, Dot \& Arrow, and that Diagrams Rule by Diagram Rules. Natural language locutions such as ``we write," ``we choose to write," ``we usually write," ``we sometimes simply write," and so on, are common in mathematical writing. A declaration in the Discourse that a described diagram ``exists" is equivalent to asserting the right but not the obligation to draw the diagram on the Surface. For example, assertion of the bounded existential quantifier formula $(\exists x\in A)P(x)$, where \[ \xymatrix{A\ar[r]^P&\Omega} \] \noindent is a diagram, corresponds to the existence on the Surface of a commutative diagram \[ \xymatrix{A\ar[rr]^P&&\Omega\\ &1\ar[ul]^a\ar[ur]_{\top}&\\ } \] \noindent such that $a$ does not occur unbound in the Context, and the assertion of the bounded universal quantifier formula $(\forall x\in A)(P(x))$ corresponds to the existence on the Surface of a commutative diagram \[ \xymatrix{ &1\ar[dr]^{\top}&\\ A\ar[rr]_P\ar[ur]^{\tau_A}&&\Omega\\ } \] \noindent In this Ground for foundations of mathematics, everything is a diagram. Ellis D. Cooper \end{document} [For admin and other information see: http://www.mta.ca/~cat-dist/ ]