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sn#157728 filedate 1975-05-06 generic text, type C, neo UTF8
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C00001 00001
C00002 00002 .DEVICE XGP
C00003 00003 .COMMENT TITLE PAGE
C00004 00004 .COMMENT OPENING DEFINITIONS
C00007 00005 .COMMENT BASIS INTERPED
C00008 00006 .COMMENT MATH IDEAS
C00009 00007 .COMMENT MOUNTAIN METAPHOR
C00010 00008 .COMMENT DRIVE/PRUNE FORCES
C00011 00009 .COMMENT understanding DEFINITIONS
C00013 00010 .COMMENT DIAGRAM OF MATH FIELDS' PREREQUISITES
C00015 00011 .COMMENT SYSTEM IDEAS
C00016 00012 .COMMENT USER'S ROLES DEFINITIONS
C00017 00013 .COMMENT GIVEN KNOWLEDGE
C00021 00014 .COMMENT SEESAW DIAGRAM
C00023 ENDMK
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.PREFACE 2
.NOFILL
.PREFACE 1
.!XGPLFTMAR←100
.MACRO B ⊂ BEGIN NOFILL SELECT 9 INDENT 0 GROUP PREFACE 0 MILLS TURN OFF "{↑↓}[]α" ⊃
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.INDENT 0
.SELECT 1
.COMMENT TITLE PAGE;
.BEGIN CENTER
.GROUP SKIP 2
⊗2The AUTOMATED MATHEMATICIAN⊗*
.END
.SELECT 5
OBJECTIVE
IDEAS
.SELECT 7
⊗7ABOUT HOW TO DO MATHEMATICAL RESEARCH⊗*
⊗7ABOUT HOW TO DESIGN A SYSTEM TO DO IT⊗*
.SELECT 5
A TYPICAL SESSION
KNOWLEDGE INITIALLY GIVEN
.SELECT 7
⊗7HOW IT IS ORGANIZED⊗*
⊗7EXACTLY WHAT IS GIVEN⊗*
⊗7THE CONTROL STRUCTURE THE SYSTEM⊗*
.SELECT 5
THE SESSION AGAIN: IN DEPTH
QUESTIONS, DISCUSSION
.SKIP TO COLUMN 1
.COMMENT OPENING DEFINITIONS;
.BEGIN FILL SELECT 7 SINGLE SPACE PREFACE 2 INDENT 0,6,0 GROUP SKIP 3
Propose Axiomatizations: Based on AM's earlier successes and/or on simulated
real-world situations
Propose and Prove conjectures: Using, e.g., the
heuristics discussed by Polya
INTUITION:
visual analogies to simulated real-world scenarios
AESTHETICS has components of:
Simplicity,
Harmony,
Relating entities which previously seemed disparate,
Elegance,
Promis of later usefulness,
Importance in completing some analogy,
Interrelated in many different ways,
short but theory-laden
.B
WEIGHING PILES OF PEBBLES\
αααααααα→ CARDINALITY αααααααα→ NUMBER THEORY
BUILDING TOWERS OF BLOCKS/ / ~ ~ \
.ONCE TURN ON "↑↓"
⊗4N ≡↓2 Z Q⊗*
\ ~ ~ /
GROUP AXIOMS αααααααα→ ABSTRACT ALGEBRA
.E
PRENUMERICAL Knowledge:
Elementary notions of sets, relations, properties, problems, problem-solving,
methods, proof techniques, analogy; and also some abilities
to evaluate interestingness, to locate relevant knowledge
IDEA really means "a ⊗4nonobvious opinion⊗*", and it is very
important to receive criticism of them before the system is actually programmed
.END
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.COMMENT BASIS INTERPED;
.BEGIN CENTER
⊗2Fundamental Concepts Noticed about
these Two Real-world Situations⊗*
.END
.BEGIN NOFILL SELECT 3 PREFACE 0 INDENT 0 TABS 41 TURN ON "\"
.ONCE CENTER
INTERPRETATION IN NUMBER THEORY
\ele
\set
\⊂
\C
\n
\S
\P
\=
\>
\+
\-
\pop
\0
\x
\1
\etc.
.E
.PREFACE 1
.SKIP TO COLUMN 1
.COMMENT MATH IDEAS;
.BEGIN NOFILL SELECT 7 PREFACE 1 INDENT 0
.ONCE CENTER
⊗2IDEAS: DOING MATH RESEARCH⊗*
1. MOTIVATED: By abstracting from what exists already, By analogy
2. DEVELOPMENT and EVALUATION: Some metaphors
3. DEVELOP and EVALUATE: By ⊗4↓_nonformal_↓⊗* methods, primarily
4. DEVELOPING ≡ EVALUATING
5. INDEPENDENT OF DOMAIN AND LEVEL
6. MULTIPLE FORMULATIONS: Understand each concept in several ways
7. ADDING NEW KNOWLEDGE NEVER DEGRADES PERFORMANCE
8. GENERAL BACKGROUND REQUIRED
9. WHAT LEVEL TO START AT
.END
.SKIP TO COLUMN 1
.COMMENT MOUNTAIN METAPHOR;
.ONCE CENTER
⊗2Metaphor: Theory Formation as Pathfinding⊗*
.B INDENT 10
⊂ααααααααααααααααααααααααααααααα⊃
~ Logic, Proof, ~
~ Naive Set Theory ~
%ααααααααααααααααααααααααααααααα$
.END
.COMMENT DRIVE/PRUNE FORCES;
.BEGIN NOFILL SELECT 7 PREFACE 1 INDENT 10
.ONCE CENTER
⊗2Judgmental Criteria and Driving Forces⊗*
Aesthetics, interestingness, and utility
Intuition, analogy, and inductive inference
Deductive inference
.END
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.COMMENT understanding DEFINITIONS;
.BEGIN NOFILL SELECT 7 PREFACE 1 INDENT 0
.ONCE CENTER
⊗2Terms related to Understanding⊗*
UNDERSTOOD means: internally accessable, manipulable, and represented
DECLARATIVE: definition, assertions
DEMONIC: recognizing when it is relevant
OPERATIONAL: knowledge of how to use it
EXEMPLARY: examples, non-examples, and boundary examples
INTUITIVE: abstract but powerful image, opaque simulation
.FILL SINGLE SPACE PREFACE 1 INDENT 0,4,0
PSYCHOLOGICALLY PREREQUISITE field: One which makes the new field easier
to learn, which contains more concrete analogues of the ideas of the new
field. For example, knowing about geometry makes it easier to learn about
topology, even though topology never formally uses any results or definitions
from geometry.
FORMALLY PREREQUISITE field:
For example, arithmetic is usually formally based upon set theory,
although most of us learn about them in the other order.
.END
.SKIP TO COLUMN 1
.COMMENT DIAGRAM OF MATH FIELDS' PREREQUISITES;
.BEGIN CENTER
⊗2Psychological and Formal Prerequisites
for Subfields of Mathematics⊗*
.END
.PREFACE 0 MILLS
.B
.INDENT 7
ELEMENTARY LOGIC ααα→ THEOREM-PROVING ααααααααααααααα⊃
↑ ~
~ ~
~ ~
εαααααααα→ GEOMETRY ααα→ TOPOLOGY ~
~ ~ ↑ ~ ~
~ ~ ~ ~ ~
~ ↓ ~ ↓ ~
~ ANALYTIC GEOMETRY ~ ALGEBRAIC TOPOLOGY ~
~ ↑ ↓ ↑ ~
~ ~ MEASURE THEORY ~ ~
↓ ~ ↑ ~ ~
BOOLEAN ALGEBRA αβαβαα→ ABSTRACT ALGEBRA ~
↑ ~ ~ ~ ↓
.ONCE TURN ON "α"
~ ~ ↓ ~ PROGRAM VERIFICATION
~ ANALYSIS ↓ ↑
~ ↑ CONCRETE ALGEBRA ~
~ ~ ↑ ~
~ ~ ~ ~
↓ ~ ~ ~
SET THEORY ααα→ ARITHMETIC ααα→ NUMBER THEORY ~
~ ~
~ ~
↓ ~
COMBINATORICS ←ααα→ GRAPH THEORY αααα$
.E
.SELECT 7
"A ⊗9αααα→ ⊗*B" means:
↓_⊗4Some⊗*_↓ knowledge about field A is crucial to working in field B
.PREFACE 1
.SKIP TO COLUMN 1
.COMMENT SYSTEM IDEAS;
.BEGIN NOFILL SELECT 7 PREFACE 1 INDENT 0
.ONCE CENTER
⊗2IDEAS: AUTOMATING MATH RESEARCH⊗*
1. INTERACTIVE: Roles of the user
2. COMMUNICATION: Suitable languages
3. BELIEF: Primarily nonformal
4. TRANSPARENT KNOWLEDGE: Facts, strategies
5. OPAQUE KNOWLEDGE: Intuition, Control
6. KNOWLEDGE IN THE SYSTEM: BEINGs
7. CONTROL IN THE SYSTEM: Primitive Heterarchy
8. WHY STANDARDIZE THE STRUCTURE?
9. BEATING THE "CRITICAL MASS" CONSTRAINT
10. WHAT A.M. ⊗4WON'T⊗* BE ABLE TO DO
.END
.SKIP TO COLUMN 1
.COMMENT USER'S ROLES DEFINITIONS;
.BEGIN NOFILL SELECT 7 PREFACE 1 INDENT 0
.ONCE CENTER
⊗2Various Roles of the User⊗*
CREATOR
ADVISER
AUTHORITY
COLLEAGUE
⊗2Various User-System Languages⊗*
Formal: like predicate calculus
Traditional: textbook phrasings
Pictorial: using the intuition functions
Examples: encode into examples, decode by inference
English: not anticipated for this system!!
.END
.SKIP TO COLUMN 1
.COMMENT GIVEN KNOWLEDGE;
.ONCE CENTER
⊗2KNOWLEDGE ORIGINALLY SUPPLIED⊗*
.BEGIN FILL SELECT 7 SINGLE SPACE PREFACE 0 TURN OFF "{}-"
.ONCE PREFACE 1
⊗2Objects⊗*
Ordered-pair,
Variable,
Propositional-constant,
Structure,
List-structure,
Bag-structure,
Set-structure,
Assertion,
Axiom
.ONCE PREFACE 1
⊗2Actives⊗*
.INDENT 0,5,0
Operations: Compose, Insert, Delete, Convert-structure, Substitute, Assign,
Map-structure,
Undo,
Reverse-ordered-pair, Rule-of-inference, Disjoin, Conjoin, Negate,
Imply, Unite, Set-union, Cross-product, Common-parts, Set-intersection,
Set-difference, Put-in-order, Abstract-out-similarities, Contrast
Relations: Equality, Membership, Containment, Equivalence,
Equipollence,
Scope,
Quantification, ⊗1For-all, There-exists, Not-for-all, Never, Only-sometimes⊗*
Properties: Ordered, Extreme, Properties-of-Activities
.ONCE PREFACE 1
⊗2Higher-order Actives⊗*
Find, Select, Guess, Ellipsis, Analogize, Conserve, Approximate
Examine, Test, Assume, Judge,
Define,
Prove, ⊗1Logically-deduce, Prove-directly, Cases,
Working-backwards, Prove-indirectly, Prove-universal-claims, Mathematical-induction,
Prove-existential-claims,
Prove-existence-constructively, Prove-existence-deductively,⊗*
Disprove, ⊗1Disprove-constructively, Disprove-indirectly,⊗*
Solve-problem, Debug, Trial-and-error, Hill-climb, Subgoal, Work-indirectly,
Relations-between-problems
Communicate, Communicate-with-user, Translate-into-English-for-User,
Translate-from-English-for-BEINGs, User-model, Communicate-among-BEINGs,
Infer-from-examples, Select-representation
Isomorphism, Categoricity, Canonical, Interpretation-of-theory
.ONCE PREFACE 1
⊗2Higher-order Objects⊗*
Statement, Conjecture, ⊗1Statement-of-generality, Statement-of-existence,⊗*
Theorem, Proof, Counterexample, Contradiction, Analogy, Assumption
Problem, Problem-to-find, Problem-to-prove, Problem-to-creatively-define,
Action-problem, Inference-problem, Bug
Representation, Symbolic-representation, Diagram, Scenario
Mathematical-theory, Foundation, Basis, Formal-system
.END
.SKIP TO COLUMN 1
.COMMENT SEESAW DIAGRAM;
.BEGIN CENTER
⊗2Diagrammatic Representation of the
SEESAW Intuition Scenario⊗*
.END
.PREFACE 0 MILLS
.B
.INDENT 7
/
\\/
\\
\\
\\
\\
\\ For each factor below, whatever the caller
\\ omits will be filled in by SEESAW function
\\
\\
\\ Provide, for each player:
\\
\\ Name
\\ Age
\\ Sex
\\ Side sitting on
\\/ Distance from center
\\ Weight
~\\
~ \\ Also provided:
~ \\
~ \\ Initial position of board
~ \\ Final position of board
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\
~ \\/
~ \\/
~
αααααααααααααααα∀ααααααααααααααααααααααα
.E
.PREFACE 1
.SKIP TO COLUMN 1