What is the Meaning of Deep for a Mathematician

1. What is the Meaning of Deep for a Mathematician Arash Rastegar Department of Math. Sciences Sharif University of Technology

2. Mathematician is a • Problem solver • Theoretician • Scientist • Artist • Arguer • Conjecturer • Coach • Teacher • Educator • Evaluator • Physician • Philosopher

3. Mathematician is a problem solver • As a problem solver lists strategies to attack the problem and chooses the most appropriate one. • As a problem solver reduces the problem to simpler components and tries to solve each of them separately. • As a problem solver estimates how much progress has been made in the process of coming to a solution. • As a problem solver criticizes conjectures and conjectural approaches to solve the problem. • As a problem solver tries to recognize easy from difficult and simple from complicated in the process of coming to a solution.

4. 1. Deep for a problem solver • Powerful strategies for solving problems are deep. • Important lemmas which widely appear in the structure of arguments are deep. • Those steps in the argument which make much progress are deep. • Conjectures could be deep the same way that arguments could be deep. • Deep is simple not complicated; because it can combine with many ideas. • Deep is not always easy. Deep could be difficult.

5. Mathematician is a theoretician • As a theoretician makes assumptions and does repair and surgery of assumptions and theories based on experience. • As a theoretician Tests assumptions and theories. • As a theoretician generalizes of approved assumptions and theories to wider scopes. • As a theoretician recognizes the relation between two assumptions or two theories. • As a theoretician compares the strength and weakness of different assumptions and theories. • As a theoretician is searching for the truth.

6. 2. Deep for a theoretician • Deep are assumptions and phenomena that widely appear. • Deep are a series of related concepts, assumptions or phenomena. • Deep assumptions and theories are appropriate for generalizations. • Deep assumptions and theories are strong assumptions and theories. They have many implications. • Deep is close to the truth.

7. Mathematician is a scientist • As a scientist communicates and consults with other scientists. • As a scientist criticizes scientific assumptions and theories and tries to think divergent. • As a scientist relates the previous knowledge to the new problem considering • As a scientist tries to control the nature. • As a scientist tries to explain and describe the nature. • As a scientist tries to solve everyday problems.

8. 3. Deep for a scientist • It is important for the deep to be easily and fluently communicatable. Deep must be easily discribable. • It is important for the deep to be criticizable. • Deep must be divergent. • Deep is related to several assumptions, theories and phenomena. • Deep is helpful to control the nature. Deep is related to nature. • Deep is helpful to solve everyday problems.

9. Mathematician is an artist • As an artist looks for beauty. • As an artist looks for simplicity. • As an artist looks for coherence. • As an artist borrows many ideas from nature and also from metaphysics. • As an artist has a hidden message and meaning. • As an artist is the creator of the art work. • As an artist uses what is available to create what is new. • As an artists creates new needs and therefore new problems to solve.

10. 4. Deep for an artist • Deep is beautiful. • Deep is simple. • Deep is coherently present everywhere. • Deep could be natural or metaphysical. • Deep has a hidden message and meaning. • Deep is created by the mathematician. • Deep is made of what is available and consists of what is new. • Deep creates new needs and therefore new problems to solve.

11. Mathematician is an arguer • As an arguer evaluates the given reasoning. • As an arguer tries to correlate given information. • As an arguer controls and evaluates the process of discovery from outside the process of solution. • As an arguer tries to make conclusions presentable. • As an arguer correlates local to global, or parts to the whole. • As an arguer argues by means of postulates and their natural implications.

12. 5. Deep for an arguer • Deep is highly evaluated among other reasonings. • Deep correlates given information. • Deep is recognizable from outside the process of solution. • It is important for deep to be presentable. • In recognition of deep one correlates local to global, or parts to the whole. • Postulates and their natural implications reveal depth of arguments.

13. Mathematician is a conjecturer • As a conjecturer approve assumptions and theories. • As a conjecturer generalize assumptions and 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. • As a conjecturer does surgery on assumptions in order to repair implications. • As a conjecturer could unify assumptions and theories by surgery and repair performed by conjectures. • As a conjecturer assesses strength and weakness of assumptions by fluency and naturality of conjectures.

14. 6. Deep for a conjecturer • Deep approves assumptions and theories. • Deep generalizes assumptions and theories to wider scopes. • It is important to do surgery on deep in order to repair implications. • Deep is surgery and repair which could unify assumptions and theories. • Deep assesses strength and weakness of assumptions by fluency and naturality.

15. Mathematician is a coach • As a coach tries to teach doing mathematics by imitation. • As a coach tries to correct the students according to their behavior. • As a coach teaches skills and considers pre-skills to teach a given skill. • As a coach designespractical exercises to prepare students to be taught particular pre-skills. • As a coach tries to communicate with students in a practical level.

16. 7. Deep for a coach • Deep could be taught by imitation. • Deep is described according to behaviors. • Teachinddeep skills needs considering pre-skills. • One should designe practical excercizes to prepare for teaching deep. • Deep could be communicated with students in a practical level. • Deep could be a skill.

17. Mathematician is a teacher • As a teacher tries to teach doing mathematics by conceptualization. • As a teacher tries to correct the students according to their thought and perspectives. • As a teacher teaches concepts and considers pre-sconcepts to teach a given concept. • As a teacher designesmental exercises to prepare students to be taught particular pre-concept. • As a teacher tries to communicate with students in a conceptual level.

18. 8. Deep for a teacher • Deep could be taught by conceptualization. • Deep is described according to thoughts and perspectives. • Teachinddeep concepts needs considering pre-concepts. • One should designe mental excercizes to prepare for teaching deep. • Deep could be communicated with students in a conceptual level. • Deep could be a concept.

19. Mathematician is an educator • As an educator mathematician educates the patterns and structures of cognition. • As an educator mathematician manages the relations between different abstract layers of cognition. • As an educator studies the scientific personality of the student. • As an educator correlates the knowledge of the student in different disciplines. • As an educator correlates personality of the student with his knowledge.

20. 9. Deep for an educator • Deep is defined in terms of the patterns and structures of cognition. • Deep appears in and between different abstract layers of cognition. • Deep is independent of the scientific personality of the student. • Deep correlates the knowledge of the student in different disciplines.

21. Mathematician is an evaluator • As an evaluator mathematician evaluates the patterns and structures of cognition. • As an evaluator mathematician evaluates the relations between different abstract layers of cognition. • As an evaluator mathematician evaluates the correlation of the knowledge of the student in different disciplines. • As an evaluator mathematician evaluates the correlation of the knowledge of the student with his personality. • As an evaluator evaluates the capacity of student in attending group work.

22. 10. Deep for an evaluator • Depth can be evaluated by the patterns and structures of cognition. • Fluency of appearance of depth in different abstract layers of cognition can be evaluated. • Independent of depth from the scientific personality of the student can be evaluated. • Depth has the capacity to engage students in group work.

23. Mathematician as a physician • As a physician or as a pathologist recognizes diseases in different scales, such as organic, molecular, etc. • As a physician provides a language to explain the disease which is fluent for its analysis. • As a physician studies the causal relations between different diseases. • As a physician controls the side effects of diseases and drugs used for curing them. • As a physician gives time to the body to be cured naturally.

24. 11. Deep for a physician • Smaller the scale of pathology, deeper the disease. • A language which explains the disease more fluently, is deeper. • The deeper the disease is, causal affects are more serious. • Deep disease has more side effects. • Deep disease takes more time to be cured naturally.

25. Mathematician is a philosopher • As a philosopher looks for hierarchies. • As a philosopher tries to create hierarchies. • As a philosopher tries to create a new language to study a package of concepts. • As a philosopher tries to create new concepts in order to correlate unrelated concepts. • As a philosopher tries to understand theories in light of new postulates or new assumptions. • As a philosopher tries to discover the language of truth.

26. 12. Deep for a philosopher • Deepstands in a hierarchy. • Creating depth needs creation of hierarchies. • Deep is creating a new language to study a package of concepts. • Deep is creating new concepts in order to correlate unrelated concepts. • Deeps is understanding theories in light of new postulates or new assumptions. • Deep isthe language of truth.

27. Depth is the main concern of a mathematician • I. Understanding depth • II. Evaluatingdepth • III. Discoveringdepth • IV. Communicatingdepth • V. Teaching depth • VI. Creating depth • VII. Designing depth

28. I. Understanding depth • Deep is an object. • Deep is a theorem. • Deep is a proof. • Deep is a strategy. • Deep is a concept. • Deep is a language. • Deep is a theory. • Deep is a phenomena. • Deep is truth not a model of truth.

29. II. Evaluating depth • Evaluate by a hierarchy. • Evaluate by comparison of concept relations. • Evaluate by comparison of influence. • Evaluate by simplicity. • Evaluate by computational fluency. • Evaluate by correlation fluency. • Evaluate by comparison of appearance. • Evaluate by generalizability. • Evaluate by beauty.

30. III. Discovering depth • Look for central facts in a theory. • Look for powerful strategies for solving problems. • Look for dictionaries between theories. • Look for hierarchies. • Look for facts interpreted in several languages. • Look for concepts related with several concepts. • Look for generalizable facts. • Look for analogies between theories. • Look for phenomena.

31. IV. Communicating depth • Communicate hierarchies. • Communicate generalizations. • Communicate a package of correlated concepts. • Communicate analogous theories. • Communicate dictionaries. • Communicate phenomena.

32. V. Teaching depth • Teach in several languages. • Teach several generalizations. • Teach correlated concepts. • Teach hierarchies and incarnation of truth in several layers of abstractness. • Teach analogous theories. • Teach dictionaries between several theories. • Teach phenomena and their several incarnations.

33. VI. Creating depth • Create hierarchies. • Translate to several languages. • Correlate to several concepts. • Create several analogous theories. • Create several generalizations. • Incarnate truth is several layers of abstractness.

34. VII. Designing depth • Depth is not designed. • Depth is predetermined. • Depth is close to the truth. • Truth is created by God, not by human. • Truth could incarnate in manmade creatures. • Truth can be found in humanities, nature and metaphysics.

35. Knowledge in the light of depth • Knowledge in the light of depth is knowledge in the light of truth. • Knowledge is a model of truth. • Scientific languages are created to communicate models of truth. • Mathematics is a model for the truth. • Only part of the truth could incarnate in mathematics. • Evaluating depth of knowledge is evaluation of closeness to the truth.