What is Mathematical Conjecture?

Arash RastegarSharif University of Technology

Advices to a problem solver

• 1) Writing neat and clean• 2) Writing down the summary of arguments• 3) Clarifying the logical structure • 4) Drawing big and clean figures• 5) Recording the process of thinking• 6) Deleting irrelevant remarks and explanations• 7) Writing down side results• 8) Putting down the full proof after finishing the arguments• 9) Notifying important steps in form of lemmas• 10) Considering the mind of reader

Decisions to be made

• 11) Where to start• 12) Listing different strategies to attack the problem• 13) Mathematical modeling in different frameworks• 14) Using symbols or avoiding symbols• 15) Deciding what not to think about• 16) Organizing the process of coming to a solution• 17) How to put down the proof

Habits to find

• 18) Tasting the problem• 19) Gaining personal view towards the problem• 20) Talking to oneself• 21) Considering all the cases• 22) Checking special cases• 23) Performing a few steps mentally • 24) Thinking simple

Personality of good problem solvers

• 25) Patience• 26) Divergent thinking• 27) Criticizing conjectures• 28) Looking for equivalent formulations• 29) Fluency in working with ideas and concepts• 30) Looking for simpler models

Intuition

• 31) Geometric imagination• 32) Recognizing simple from difficult• 33) Decomposition and reduction to simpler problems• 34) Jumps of the mind• 35) Estimating how much progress has been made• 36) Finding the trivial propositions quickly• 37) Formulating good conjectures • 38) Being creative and directed in constructions• 39) Understanding an idea independent of the context • 40) Imagination and intuition come before arguments & computations

Conjecture for problem solvers• Guess the logical structure behind the problem• Guess important steps of the proof• Guess different strategies to attack the problem• Guess the process of coming to a solution • Guess mathematical models in different frameworks• Guess important special cases• Guess equivalent formulations• Guess reductions to simpler problems

The role of arguments inmaking assumptions

• Arguments test assumptions • Arguments generalize assumptions to wider scopes• There are natural barriers to generalization of assumptions revealed

by arguments. • Sometimes one can not unite two given theories.• Arguments do surgery on assumptions in order to repair

implications.• Surgery and repair performed by arguments could lead to

unification of assumptions. • Strength and weakness of assumptions are assessed by fluency and

naturality of implications.

The role of arguments indevelopment of theories

• Arguments test theories• Arguments generalize theories to wider scopes. By generalization

one can unite the realms of two theories.• Recognition of relations between assumptions via arguments

usually leads to unification of theories. • Recognition of relations between theories via arguments forms a

paradigm. One is interested to find relations between two theories for the further development of mathematics.

The role of arguments in search for the truth

• Truth in mathematics is understood by analogies. which are revealed by arguments.

• In mathematics one compares two or three theories and find dictionaries between them in order to look for background truth.

• In mathematics concepts are just a model of the truth. So arguments discuss relations between the models of truth.

The role of conjectures inmaking assumptions

• Nice conjectures approve assumptions • Nice conjectures generalize assumptions to wider scopes• There are natural barriers to generalization of assumptions revealed

by conjectures. Sometimes one can not unite two given theories.• Conjectures do surgery on assumptions in order to repair

implications.• Surgery and repair performed by conjectures could lead to

unification of assumptions. • Strength and weakness of assumptions are assessed by fluency and

naturality of conjectures.

The role of conjectures indevelopment of theories

• Nice conjectures approve theories• Nice conjectures generalize theories to wider scopes. By

generalization one can unite the realms of two theories.• Recognition of relations between assumptions via nice conjectures

usually leads to unification of theories. • Recognition of relations between theories via conjectures forms a

paradigm. One is interested to find relations between two theories for the further development of mathematics.

The role of conjectures in search for the truth

• Truth in mathematics is understood by analogies, which are revealed by nice conjectures .

• In mathematics one compares two or three theories and find dictionaries between them in order to look for background truth.

• In mathematics concepts are just a model of the truth. So conjectures discuss relations between the models of truth.

Theoriticians and problem solvers are pairs

• Role of father Theoritician:• Provides ideas and

intuitions and perspectives.• Management of relations

with other theories and other assumptions and provides the global perspectives.

• Determines how to generalize.

• Furnishes the soul.

• Role of mother problem solver:

• Provides appropriate formulation and language.

• Management of internal relations between sub-theories and provides the local perspectives.

• Determines how to specialize.

• Furnishes the body.

Fruit of theoretician-problem solver marriage

• Looking for background truth.• Understanding the relations between models of truth.• Approving theories.• Generalizing theories to wider scopes. • Uniting the realms of two theories.• Recognition of similarities leads to unification of theories. • Recognition of relations between theories via conjectures

and theorems forms a paradigm. One is interested to find relations between two theories for the further development of mathematics.

Conjecture for theoreticians

• Conjectures connect paradigms.• Every theorem was some day in form of a conjecture.• Conjectures give insight, the same way that arguments do.• Conjectures and arguments belong to different realms of

abstraction.• Conjectures are drived by intuition the same way that

arguments are in need of intuition.• Conjectures are hypothetical theorems in a more abstract

realm of knowledge.• Arguments are conjectural pathways to the truth.

Conjecture and argument are pairs• The role of father conjectures• Conjectures lead.• Conjecturing is an act of

vision.• We conjecture in light of

intuition.• Conjectures show the

connections.• Conjecture are visual.• Conjectures provide the soul.

• The role of mother arguments

• Arguments follow.• Arguing is an act of wisdom.• We argue in light of logical

understanding.• Arguments build the

connections.• Arguments are verbal.• Arguments provide the

body.

Fruit of conjecture-argument marriage

• Conjectures and arguments develop mathematics. • Conjectures and arguments develop science.• Conjectures and arguments develop the realm of

thought.• Conjectures and arguments reveal the truth.• Conjectures and arguments hide behind the truth.• Conjectures and arguments show us the path to

perfection.• Conjectures and arguments form the life of wisdom.

Intuition and logic are pairs

• The role of father intuition.

• Intuition incarnates in logic.

• Intuition goes up to truth.

• Intuition is timeless.

• Intuition provides the soul.

• The role of mother logic.

• Logic incarnates in thought.

• Logic goes up to intuition.

• Logic is governed by time.

• Logic provides the body.

Fruit of intuition-logic marriage

• Intuition and logic develop wisdom. • Intuition and logic develop knowledge.• Intuition and logic develop mind.• Intuition and logic reveal the truth.• Intuition and logic hide behind the truth.• Intuition and logic show us the path to

perfection.• Intuition and logic form the highest possible

indirect knowledge.