From Dakota

Contents 1 Mathematics 1.1 Notation 1.2 LaTeX 1.3 Minesweeper 1.3.1 Definitions 1.3.2 Same Difference Lemma 1.4 Zeno's Paradoxes 1.4.1 Zeno's Paradox 1.4.2 Zeno's Arrow Paradox

Mathematics

Notation

Unfortunately I do not have LaTeX set up for use on this wiki (yet). Therefore I have to resort to less than ideal notations. These are listed here.

• ℝ - The set of real numbers

LaTeX

Typesetting mathematics with LaTeX can be frustrating. The page on LaTeX contains commonly needed, but surprisingly hard to find, LaTeX commands and procedures.

Minesweeper

Definitions

• Let N(a) denote the collection of minesweeper spaces in the neighborhood of a. For the usual Moore neighborhood version of minesweeper, this will be the set of 8 minesweeper spaces adjacent to a.

Same Difference Lemma

The difference in the values of two adjacent Minesweeper spaces is the difference in the number of mines in the disjoint neighborhoods of those spaces.

Proof: Let M be a game of minesweeper. Consider two adjacent minesweeper spaces named a and b. Let the intersection N(a) and N(b) be F. Let number of mines in F be #F. Let the number of mines in N(a)\F be #A, similarly define #B. Then a - b = (#A + #F) - (#B + #F) = #A - #B. QED

Zeno's Paradoxes

• Assumptions
  • Assume time and space are (locally) Euclidean, i.e. a subset of ℝ⁴.
  • Assume events occur at a (locally) unique time and place, i.e a unique point in ℝ⁴.

Zeno's Paradox

Two essays I wrote on this subject for a class:

• The Problem of Zeno: Part I - Techtrifecta
• The Problem of Zeno: Part II - Techtrifecta

In too few words:

• Assume space is dense.
• Assume time is nowhere dense.
  • Motion is impossible.

Zeno's Arrow Paradox

In too few words:

• Assume events can be observed perfectly, i.e. located exactly in time and space.
• Assume a time coordinate is given.
  • Distinguishing an event in motion from a stationary event is impossible.