What is Mathematical Conjecture? Arash Rastegar Sharif University of Technology Arash Rastegar Sharif 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 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 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 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 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 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 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 in making 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. 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 in development 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. 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. 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 in making 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. 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 in development 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. 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. 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 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. 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. 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. 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 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. 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. 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 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. 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. 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.