Charles Saunders Peirce ("purse") is our greatest philosopher, our greatest 19th century mathematician and logician, the planner of The Bureau of Standards, and the creator of Semiotics: Theory of Signs. (Texts and reference books often write more about the independent semiotics ideas of French scholar, Ferdinand Saussure (x-y). I build upon Peirce's work since they give me a gnomonic basis and Saussure's does not.)

My MADMATH semiotic references derive from my acronym, "ISIS", for what I consider the four primary signs which Peirce taught us.

I(ndicator)S(ignal)I(con)S(ymbol).

INDICATOR Extending Peirce, I redefine
INDICATOR: ORDERED PAIR OF SIGNS: THE 1st IS HIGHLY VISIBLE, LOW IN INFORMATION CONTENT; THE 2nd LOW IN VISIBILITY, HIGH IN INFORMATION CONTENT.
(Examples: LIGHTNING INDICATING THUNDERSTORM; LITMUS PAPER TURNING RED, INDICATING LIQUID ACID IN TEST TUBE.

SIGNAL Extending Peirce, I redefine
SIGNAL: INDICATOR UNDER PHYSICAL AND LINGUISTIC CONTROL.
(Peirce mentioned only PHYSICAL CONTROL.) Example: TELEGRAPHIC SIGNAL -- under PHYSICAL CONTROL OF ELECTRIC CIRCUIT AND BREAKER KEY; under LINGUISTIC CONTROL OF MORSE CODE.)

INDICATOR-SIGNAL STRATEGY: THE GOAL OF SCIENCE IS TO SEARCH FOR SIGNALS AND TRY TO TRANSFORM INTO SIGNALS!

This can be seen in a Table of Indicators and SIGNALS.

ICON An ICON IS A SIGN THAT SUGGESTS OR INVOKES ITS MEANING OR REFERENCE.
Examples: "No Smoking!" ICONED as CIGARETTE WITH SLASH THROUGH IT; ONOMATOPOEIC WORDS; etc.

SYMBOL A SYMBOL IS A SIGN WITH ARBITRARILIY ASSIGNED MEANING OR REFERENCE.

(An ICON is HIGHLY VISIBLE, but with LIMITED MEANING OR REFERENCE. A SYMBOL has LOW 'VISIBILITY", BUT UNLIMITED INFORMATION CONTENT.)

ICON-SYMBOL STRATEGY: THE GOAL OF LITERACY IS TO MAKE THE ICON A BRIDGE TO THE SYMBOL.

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SEMSPACE: A METRIC SPACE FOR SEMIOTIC SIGNS

Many mathematicians and math students know that A TOPOLOGY PLUS A METRIC YIELDS A GEOMETRY AND GEOMETRIC SPACE, WHICH CAN BE ORGANIZED INTO DIMENSIONS.

And they know that A TOPOLOGY CAN BE CREATED FROM A SET OF OPEN SETS, OR FROM A SET OF CLOSED SETS. In Paul Halmos' Boolean Algebra they will read that ALL COMPLEMENTED ELEMENTS OF A LATTICE ARE CLOPEN: BOTH CLOSED AND OPEN.

And they can also read there that THE RANK MEASURE ON A LATTICE IS A METRIC. Hence, A DISTRIBUTIVE LATTICE -- WITH SOME COMPLEMENTED ELEMENTS -- PROVIDES FOR A TOPOLOGICAL SPACE AND A METRIC SPACE!

Let the SEMSPACE gnomons be SIGNS. Then we can form a TOPOLOGICAL SPACE OF SIGNS AND A (SEMIOTIC) METRIC DIMENSIONAL-SPACE OF SIGNS: SEMSPACE.

1. DIMENSION ZERO: SEMIOTIC SIGNS.
2. DIMENSION ONE: SYNTACTICS (SYNTAX) - THE RELATION BETWEEN SIGNS (WITHOUT REFERENCE OR MEANING).
3. DIMENSION ONE+ (HYPERSPACE): FORMALICS (SYNTAX WITH RECURSIVE CLOSURE).
4. DIMENSION TWO: SEMANTICS - THE RELATION BETWEEN SIGNS AND THEIR REFERENCES (MEANINGS).
5. DIMENSION THREE: PRAGMATICS - THE RELATION BETWEEN SIGNS AND REFERENCES AND A SINGLE SIGN USER.
6. DIMENSION FOUR: TRANSACTICS - THE RELATION BETWEEN VARIOUS SIGNS USERS.
7. DIMENSION FOUR+ (HYPERSPACE): CONSENSICS (EQUIVALENCE CLASSES OF TRANSACTIC CONSENSUS).