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Dealing with negative results
Research methods and proposal writing
Dr M Benini
Università degli Studi dell’Insubria
kindl...
Science is positive
The vast majority of scientific results are stated as answers to
questions of the kind
why
how
The first...
Science is positive
Consider the negated questions:
why not
how not
The first question requires to provide a way to falsify...
Science is positive
Consider the negated questions:
why not
how not
The first question requires to provide a way to falsify...
Science is positive
When we do research, we collect many negative answers:
when an experiment fails, we have to understand...
Science is positive
When we do research, we collect many negative answers:
when an experiment fails, we have to understand...
Science is positive
When we do research, we collect many negative answers:
when an experiment fails, we have to understand...
Science is positive
When we do research, we collect many negative answers:
when an experiment fails, we have to understand...
Science is positive
When we reach the final stage of our research efforts, publishing
the results, we focus on the positive ...
Science is positive
When we reach the final stage of our research efforts, publishing
the results, we focus on the positive ...
Negative content, positively presented
Although some results we have found are negative, they can
often be positively pres...
Negative content, positively presented
Although some results we have found are negative, they can
often be positively pres...
Negative content, positively presented
Although some results we have found are negative, they can
often be positively pres...
Negative content, positively presented
When stating a result, use a positive language
Theorem [Gödel] Peano Arithmetic is ...
Negative content, positively presented
When stating a result, use a positive language
Theorem [Gödel] Peano Arithmetic is ...
Negative content, positively presented
When stating a result, use a positive language
Theorem [Gödel] Peano Arithmetic is ...
Negative content, positively presented
When stating a result, use a positive language
Theorem [Gödel] Peano Arithmetic is ...
Negative content, positively presented
When writing a statement, provide a positive content
Theorem An arbitrary list cann...
Negative content, positively presented
When writing a statement, provide a positive content
Theorem An arbitrary list cann...
Negative content, positively presented
When writing a statement, provide a positive content
Theorem An arbitrary list cann...
Negative content, positively presented
When writing a statement, provide a positive content
Theorem An arbitrary list cann...
Negative content, positively presented
When stating a counterexample, be positive
The real-valued function f (x) = |x| doe...
Negative content, positively presented
When stating a counterexample, be positive
The real-valued function f (x) = |x| doe...
Negative content, positively presented
When stating a counterexample, be positive
The real-valued function f (x) = |x| doe...
Negative content, positively presented
When stating a limit to a procedure or method, be positive
The goto statement of pr...
Negative content, positively presented
When stating a limit to a procedure or method, be positive
The goto statement of pr...
Negative content, positively presented
When stating a limit to a procedure or method, be positive
The goto statement of pr...
Negative content, positively presented
When stating a limit to a procedure or method, be positive
The goto statement of pr...
Impossibility results
Some results are inherently negative. If they are also significant,
it is worth stating them in a neg...
Impossibility results
Some results are inherently negative. If they are also significant,
it is worth stating them in a neg...
Impossibility results
Some results are inherently negative. If they are also significant,
it is worth stating them in a neg...
Dead ends
When we do research, experiments fail, proofs do not come
through, and procedures give crazy results
We use thes...
Dead ends
When we do research, experiments fail, proofs do not come
through, and procedures give crazy results
We use thes...
Dead ends
When you get into a dead end, you suffer from a lack of
alternatives. There are many ways to try to get out
Write...
Dead ends
When you get into a dead end, you suffer from a lack of
alternatives. There are many ways to try to get out
Write...
Dead ends
When you get into a dead end, you suffer from a lack of
alternatives. There are many ways to try to get out
Write...
Dead ends
When you get into a dead end, you suffer from a lack of
alternatives. There are many ways to try to get out
Write...
Dead ends
It is tempting to get some result out of a long period of hard
work. But, when the result is a dead end, do not ...
Dead ends
It is tempting to get some result out of a long period of hard
work. But, when the result is a dead end, do not ...
Dead ends
It is tempting to get some result out of a long period of hard
work. But, when the result is a dead end, do not ...
Publishing or discarding?
A basic rule in Economy says to buy when the market is low, and
to sell when the market is high
...
Publishing or discarding?
A positive result has bigger chances to get published than a
negative one. But negative results ...
Publishing or discarding?
A positive result has bigger chances to get published than a
negative one. But negative results ...
Publishing or discarding?
A positive result has bigger chances to get published than a
negative one. But negative results ...
Publishing or discarding?
A positive result has bigger chances to get published than a
negative one. But negative results ...
Publishing or discarding?
Dead ends are the result of our efforts. What matters in Science
are our findings, not the effort y...
Experience
Get into the habit of writing down, as clearly as possible,
everything you find during your research
you can reu...
Experience
Get into the habit of writing down, as clearly as possible,
everything you find during your research
you can reu...
Experience
Get into the habit of writing down, as clearly as possible,
everything you find during your research
you can reu...
Experience
Get into the habit of writing down, as clearly as possible,
everything you find during your research
you can reu...
Experience
Get into the habit of writing down, as clearly as possible,
everything you find during your research
you can reu...
Experience
Get into the habit of writing down, as clearly as possible,
everything you find during your research
you can reu...
Experience
In the long term, negative results will form the fundamental part
of your experience
you will know what directi...
Experience
In the long term, negative results will form the fundamental part
of your experience
you will know what directi...
Experience
In the long term, negative results will form the fundamental part
of your experience
you will know what directi...
Experience
In the long term, negative results will form the fundamental part
of your experience
you will know what directi...
Experience
In the long term, negative results will form the fundamental part
of your experience
you will know what directi...
The end
( 20 of 19 )
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Dealing with negative results

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An invited lecture in the course "Research methods and proposal writing" by Dr Luca Longo

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Dealing with negative results

  1. 1. Dealing with negative results Research methods and proposal writing Dr M Benini Università degli Studi dell’Insubria kindly hosted by Dr L Longo marco.benini@uninsubria.it December 3rd, 2016
  2. 2. Science is positive The vast majority of scientific results are stated as answers to questions of the kind why how The first question asks for an explanation of some phenomenon; the second one requires to provide a justified method to achieve some result of interest ( 2 of 19 )
  3. 3. Science is positive Consider the negated questions: why not how not The first question requires to provide a way to falsify an explanation; the second question asks to show a faulty way to achieve a result, and to put in evidence its failure As a matter of fact, we have a huge number of negative answers in our experience, and we need them! ( 3 of 19 )
  4. 4. Science is positive Consider the negated questions: why not how not The first question requires to provide a way to falsify an explanation; the second question asks to show a faulty way to achieve a result, and to put in evidence its failure As a matter of fact, we have a huge number of negative answers in our experience, and we need them! ( 3 of 19 )
  5. 5. Science is positive When we do research, we collect many negative answers: when an experiment fails, we have to understand why it did not succeed when a (mathematical) proof fails to be found, we have to ask ourselves what prevents it, eventually searching for a counterexample when a procedure fails to achieve the desired result, we have to understand what went wrong, and why ( 4 of 19 )
  6. 6. Science is positive When we do research, we collect many negative answers: when an experiment fails, we have to understand why it did not succeed when a (mathematical) proof fails to be found, we have to ask ourselves what prevents it, eventually searching for a counterexample when a procedure fails to achieve the desired result, we have to understand what went wrong, and why ( 4 of 19 )
  7. 7. Science is positive When we do research, we collect many negative answers: when an experiment fails, we have to understand why it did not succeed when a (mathematical) proof fails to be found, we have to ask ourselves what prevents it, eventually searching for a counterexample when a procedure fails to achieve the desired result, we have to understand what went wrong, and why ( 4 of 19 )
  8. 8. Science is positive When we do research, we collect many negative answers: when an experiment fails, we have to understand why it did not succeed when a (mathematical) proof fails to be found, we have to ask ourselves what prevents it, eventually searching for a counterexample when a procedure fails to achieve the desired result, we have to understand what went wrong, and why ( 4 of 19 )
  9. 9. Science is positive When we reach the final stage of our research efforts, publishing the results, we focus on the positive achievements, And we discard the negative experience we have accumulated An experienced researcher, instead, should be more effective, trying to use the negative results and to maximise the outcomes of his efforts ( 5 of 19 )
  10. 10. Science is positive When we reach the final stage of our research efforts, publishing the results, we focus on the positive achievements, And we discard the negative experience we have accumulated An experienced researcher, instead, should be more effective, trying to use the negative results and to maximise the outcomes of his efforts ( 5 of 19 )
  11. 11. Negative content, positively presented Although some results we have found are negative, they can often be positively presented when a procedure fails, precisely delimit the domain of applicability, excluding the cases we know to be bad: the limit becomes a necessary condition, and the failure, a counterexample to a direct generalisation when a mathematical proof fails, find a counterexample, and use it to add a condition that prevents its applicability: the additional hypothesis becomes necessary because of the counterexample ( 6 of 19 )
  12. 12. Negative content, positively presented Although some results we have found are negative, they can often be positively presented when a procedure fails, precisely delimit the domain of applicability, excluding the cases we know to be bad: the limit becomes a necessary condition, and the failure, a counterexample to a direct generalisation when a mathematical proof fails, find a counterexample, and use it to add a condition that prevents its applicability: the additional hypothesis becomes necessary because of the counterexample ( 6 of 19 )
  13. 13. Negative content, positively presented Although some results we have found are negative, they can often be positively presented when a procedure fails, precisely delimit the domain of applicability, excluding the cases we know to be bad: the limit becomes a necessary condition, and the failure, a counterexample to a direct generalisation when a mathematical proof fails, find a counterexample, and use it to add a condition that prevents its applicability: the additional hypothesis becomes necessary because of the counterexample ( 6 of 19 )
  14. 14. Negative content, positively presented When stating a result, use a positive language Theorem [Gödel] Peano Arithmetic is not complete. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that both G and ¬G are not provable. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that G is true on natural numbers, but G is not provable. ( 7 of 19 )
  15. 15. Negative content, positively presented When stating a result, use a positive language Theorem [Gödel] Peano Arithmetic is not complete. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that both G and ¬G are not provable. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that G is true on natural numbers, but G is not provable. ( 7 of 19 )
  16. 16. Negative content, positively presented When stating a result, use a positive language Theorem [Gödel] Peano Arithmetic is not complete. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that both G and ¬G are not provable. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that G is true on natural numbers, but G is not provable. ( 7 of 19 )
  17. 17. Negative content, positively presented When stating a result, use a positive language Theorem [Gödel] Peano Arithmetic is not complete. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that both G and ¬G are not provable. Theorem [Gödel] There is a sentence G in Peano Arithmetic such that G is true on natural numbers, but G is not provable. ( 7 of 19 )
  18. 18. Negative content, positively presented When writing a statement, provide a positive content Theorem An arbitrary list cannot be sorted in linear time. Theorem An arbitrary list can be sorted in O(nlogn) steps. Theorem The heap sort algorithm sorts any list in at most O(nlogn) steps. ( 8 of 19 )
  19. 19. Negative content, positively presented When writing a statement, provide a positive content Theorem An arbitrary list cannot be sorted in linear time. Theorem An arbitrary list can be sorted in O(nlogn) steps. Theorem The heap sort algorithm sorts any list in at most O(nlogn) steps. ( 8 of 19 )
  20. 20. Negative content, positively presented When writing a statement, provide a positive content Theorem An arbitrary list cannot be sorted in linear time. Theorem An arbitrary list can be sorted in O(nlogn) steps. Theorem The heap sort algorithm sorts any list in at most O(nlogn) steps. ( 8 of 19 )
  21. 21. Negative content, positively presented When writing a statement, provide a positive content Theorem An arbitrary list cannot be sorted in linear time. Theorem An arbitrary list can be sorted in O(nlogn) steps. Theorem The heap sort algorithm sorts any list in at most O(nlogn) steps. ( 8 of 19 )
  22. 22. Negative content, positively presented When stating a counterexample, be positive The real-valued function f (x) = |x| does not admit a derivative in x = 0. The real-valued function f (x) = |x| admits f (x) = −1 as its derivative on the interval (−∞,0), and f (x) = 1 on the interval (0,+∞). Thus, f (0) is not defined. ( 9 of 19 )
  23. 23. Negative content, positively presented When stating a counterexample, be positive The real-valued function f (x) = |x| does not admit a derivative in x = 0. The real-valued function f (x) = |x| admits f (x) = −1 as its derivative on the interval (−∞,0), and f (x) = 1 on the interval (0,+∞). Thus, f (0) is not defined. ( 9 of 19 )
  24. 24. Negative content, positively presented When stating a counterexample, be positive The real-valued function f (x) = |x| does not admit a derivative in x = 0. The real-valued function f (x) = |x| admits f (x) = −1 as its derivative on the interval (−∞,0), and f (x) = 1 on the interval (0,+∞). Thus, f (0) is not defined. ( 9 of 19 )
  25. 25. Negative content, positively presented When stating a limit to a procedure or method, be positive The goto statement of procedural computer programs should be avoided whenever possible. The goto statement of procedural computer programs should be limited; use if ...then ...else ... or loops, instead. The use of goto statement of procedural computer programs should be limited to treat exceptions, when a specific statement is not available; use if ...then ...else ... or loops in the normal practice. ( 10 of 19 )
  26. 26. Negative content, positively presented When stating a limit to a procedure or method, be positive The goto statement of procedural computer programs should be avoided whenever possible. The goto statement of procedural computer programs should be limited; use if ...then ...else ... or loops, instead. The use of goto statement of procedural computer programs should be limited to treat exceptions, when a specific statement is not available; use if ...then ...else ... or loops in the normal practice. ( 10 of 19 )
  27. 27. Negative content, positively presented When stating a limit to a procedure or method, be positive The goto statement of procedural computer programs should be avoided whenever possible. The goto statement of procedural computer programs should be limited; use if ...then ...else ... or loops, instead. The use of goto statement of procedural computer programs should be limited to treat exceptions, when a specific statement is not available; use if ...then ...else ... or loops in the normal practice. ( 10 of 19 )
  28. 28. Negative content, positively presented When stating a limit to a procedure or method, be positive The goto statement of procedural computer programs should be avoided whenever possible. The goto statement of procedural computer programs should be limited; use if ...then ...else ... or loops, instead. The use of goto statement of procedural computer programs should be limited to treat exceptions, when a specific statement is not available; use if ...then ...else ... or loops in the normal practice. ( 10 of 19 )
  29. 29. Impossibility results Some results are inherently negative. If they are also significant, it is worth stating them in a negative way Theorem [Turing] Deciding whether a program terminates on all inputs, is not a computable problem. Theorem [Turing] There is no computer program that, taking as input a computer program, decides whether it terminates on all inputs. ( 11 of 19 )
  30. 30. Impossibility results Some results are inherently negative. If they are also significant, it is worth stating them in a negative way Theorem [Turing] Deciding whether a program terminates on all inputs, is not a computable problem. Theorem [Turing] There is no computer program that, taking as input a computer program, decides whether it terminates on all inputs. ( 11 of 19 )
  31. 31. Impossibility results Some results are inherently negative. If they are also significant, it is worth stating them in a negative way Theorem [Turing] Deciding whether a program terminates on all inputs, is not a computable problem. Theorem [Turing] There is no computer program that, taking as input a computer program, decides whether it terminates on all inputs. ( 11 of 19 )
  32. 32. Dead ends When we do research, experiments fail, proofs do not come through, and procedures give crazy results We use these failures to better understand our problems, and eventually to fix what did not work properly Sometimes, we get in a situation where there seem to be no way out, in which we are unable to find a way to fix the incorrect behaviours we encounter. We call these situation dead ends ( 12 of 19 )
  33. 33. Dead ends When we do research, experiments fail, proofs do not come through, and procedures give crazy results We use these failures to better understand our problems, and eventually to fix what did not work properly Sometimes, we get in a situation where there seem to be no way out, in which we are unable to find a way to fix the incorrect behaviours we encounter. We call these situation dead ends ( 12 of 19 )
  34. 34. Dead ends When you get into a dead end, you suffer from a lack of alternatives. There are many ways to try to get out Write down what you got as clearly as possible. The effect to organise the knowledge collected so far often shows alternatives Show your attempts to a colleague or to your supervisor. A fresh look often provides a new point of view Leave the problem apart for a while. When we do not think consciously to a problem, our mind follows different paths, and we leave room for new intuitions ( 13 of 19 )
  35. 35. Dead ends When you get into a dead end, you suffer from a lack of alternatives. There are many ways to try to get out Write down what you got as clearly as possible. The effect to organise the knowledge collected so far often shows alternatives Show your attempts to a colleague or to your supervisor. A fresh look often provides a new point of view Leave the problem apart for a while. When we do not think consciously to a problem, our mind follows different paths, and we leave room for new intuitions ( 13 of 19 )
  36. 36. Dead ends When you get into a dead end, you suffer from a lack of alternatives. There are many ways to try to get out Write down what you got as clearly as possible. The effect to organise the knowledge collected so far often shows alternatives Show your attempts to a colleague or to your supervisor. A fresh look often provides a new point of view Leave the problem apart for a while. When we do not think consciously to a problem, our mind follows different paths, and we leave room for new intuitions ( 13 of 19 )
  37. 37. Dead ends When you get into a dead end, you suffer from a lack of alternatives. There are many ways to try to get out Write down what you got as clearly as possible. The effect to organise the knowledge collected so far often shows alternatives Show your attempts to a colleague or to your supervisor. A fresh look often provides a new point of view Leave the problem apart for a while. When we do not think consciously to a problem, our mind follows different paths, and we leave room for new intuitions ( 13 of 19 )
  38. 38. Dead ends It is tempting to get some result out of a long period of hard work. But, when the result is a dead end, do not publish it! A dead end leads to nowhere, by definition. You have few chances to motivate your partial achievements You will spend time writing, submitting the paper, and receiving comments. You know in advance that the results are not complete, and that the direction leads nowhere: honestly, any reviewer will recognise these weaknesses, lowering your chances. Are you sure you are not wasting more time? ( 14 of 19 )
  39. 39. Dead ends It is tempting to get some result out of a long period of hard work. But, when the result is a dead end, do not publish it! A dead end leads to nowhere, by definition. You have few chances to motivate your partial achievements You will spend time writing, submitting the paper, and receiving comments. You know in advance that the results are not complete, and that the direction leads nowhere: honestly, any reviewer will recognise these weaknesses, lowering your chances. Are you sure you are not wasting more time? ( 14 of 19 )
  40. 40. Dead ends It is tempting to get some result out of a long period of hard work. But, when the result is a dead end, do not publish it! A dead end leads to nowhere, by definition. You have few chances to motivate your partial achievements You will spend time writing, submitting the paper, and receiving comments. You know in advance that the results are not complete, and that the direction leads nowhere: honestly, any reviewer will recognise these weaknesses, lowering your chances. Are you sure you are not wasting more time? ( 14 of 19 )
  41. 41. Publishing or discarding? A basic rule in Economy says to buy when the market is low, and to sell when the market is high Consider pursuing a scientific result as an investment. Publishing is the act which provides you with a revenue. It makes no sense to submit an article when you know in advance that it will not get published. You spend time and effort with no revenue ( 15 of 19 )
  42. 42. Publishing or discarding? A positive result has bigger chances to get published than a negative one. But negative results have some chances, too Try to positively present your negative outcomes Strong impossibility results are landmarks. Prepare yourself to defend them, but be sure they will be important in the long term Weak impossibility results are interesting only for specialists. You should carefully evaluate if your audience is wide enough to invest time and effort to try to publish such a negative achievement ( 16 of 19 )
  43. 43. Publishing or discarding? A positive result has bigger chances to get published than a negative one. But negative results have some chances, too Try to positively present your negative outcomes Strong impossibility results are landmarks. Prepare yourself to defend them, but be sure they will be important in the long term Weak impossibility results are interesting only for specialists. You should carefully evaluate if your audience is wide enough to invest time and effort to try to publish such a negative achievement ( 16 of 19 )
  44. 44. Publishing or discarding? A positive result has bigger chances to get published than a negative one. But negative results have some chances, too Try to positively present your negative outcomes Strong impossibility results are landmarks. Prepare yourself to defend them, but be sure they will be important in the long term Weak impossibility results are interesting only for specialists. You should carefully evaluate if your audience is wide enough to invest time and effort to try to publish such a negative achievement ( 16 of 19 )
  45. 45. Publishing or discarding? A positive result has bigger chances to get published than a negative one. But negative results have some chances, too Try to positively present your negative outcomes Strong impossibility results are landmarks. Prepare yourself to defend them, but be sure they will be important in the long term Weak impossibility results are interesting only for specialists. You should carefully evaluate if your audience is wide enough to invest time and effort to try to publish such a negative achievement ( 16 of 19 )
  46. 46. Publishing or discarding? Dead ends are the result of our efforts. What matters in Science are our findings, not the effort you made You should carefully judge what parts of your efforts are significant, and what are irrelevant to other people, with no regards for the time you invested Do not try to publish what you think is irrelevant! If you think so, how could you possibly convince a reader? You have to believe in the value of your results before trying to explain them ( 17 of 19 )
  47. 47. Experience Get into the habit of writing down, as clearly as possible, everything you find during your research you can reuse this pieces when writing the final paper you can use them to help a new collaborator to enter in your team you have an evidence of your work you can use them to illustrate the point you reached to, e.g., a colleague or to your supervisor you keep track of what you have understood. In the future, you may want to reuse these partial results elsewhere, and you do not want to rework all the details from scratch ( 18 of 19 )
  48. 48. Experience Get into the habit of writing down, as clearly as possible, everything you find during your research you can reuse this pieces when writing the final paper you can use them to help a new collaborator to enter in your team you have an evidence of your work you can use them to illustrate the point you reached to, e.g., a colleague or to your supervisor you keep track of what you have understood. In the future, you may want to reuse these partial results elsewhere, and you do not want to rework all the details from scratch ( 18 of 19 )
  49. 49. Experience Get into the habit of writing down, as clearly as possible, everything you find during your research you can reuse this pieces when writing the final paper you can use them to help a new collaborator to enter in your team you have an evidence of your work you can use them to illustrate the point you reached to, e.g., a colleague or to your supervisor you keep track of what you have understood. In the future, you may want to reuse these partial results elsewhere, and you do not want to rework all the details from scratch ( 18 of 19 )
  50. 50. Experience Get into the habit of writing down, as clearly as possible, everything you find during your research you can reuse this pieces when writing the final paper you can use them to help a new collaborator to enter in your team you have an evidence of your work you can use them to illustrate the point you reached to, e.g., a colleague or to your supervisor you keep track of what you have understood. In the future, you may want to reuse these partial results elsewhere, and you do not want to rework all the details from scratch ( 18 of 19 )
  51. 51. Experience Get into the habit of writing down, as clearly as possible, everything you find during your research you can reuse this pieces when writing the final paper you can use them to help a new collaborator to enter in your team you have an evidence of your work you can use them to illustrate the point you reached to, e.g., a colleague or to your supervisor you keep track of what you have understood. In the future, you may want to reuse these partial results elsewhere, and you do not want to rework all the details from scratch ( 18 of 19 )
  52. 52. Experience Get into the habit of writing down, as clearly as possible, everything you find during your research you can reuse this pieces when writing the final paper you can use them to help a new collaborator to enter in your team you have an evidence of your work you can use them to illustrate the point you reached to, e.g., a colleague or to your supervisor you keep track of what you have understood. In the future, you may want to reuse these partial results elsewhere, and you do not want to rework all the details from scratch ( 18 of 19 )
  53. 53. Experience In the long term, negative results will form the fundamental part of your experience you will know what directions and tools have few hopes to lead to a result you have a track of open problems, not-yet-attempted alternatives, variants to explore negative results, well documented, can speed up your achievements when you hear about a new result that overcomes a difficulty which blocked you in the past All these aspects of experience qualify you as an expert ( 19 of 19 )
  54. 54. Experience In the long term, negative results will form the fundamental part of your experience you will know what directions and tools have few hopes to lead to a result you have a track of open problems, not-yet-attempted alternatives, variants to explore negative results, well documented, can speed up your achievements when you hear about a new result that overcomes a difficulty which blocked you in the past All these aspects of experience qualify you as an expert ( 19 of 19 )
  55. 55. Experience In the long term, negative results will form the fundamental part of your experience you will know what directions and tools have few hopes to lead to a result you have a track of open problems, not-yet-attempted alternatives, variants to explore negative results, well documented, can speed up your achievements when you hear about a new result that overcomes a difficulty which blocked you in the past All these aspects of experience qualify you as an expert ( 19 of 19 )
  56. 56. Experience In the long term, negative results will form the fundamental part of your experience you will know what directions and tools have few hopes to lead to a result you have a track of open problems, not-yet-attempted alternatives, variants to explore negative results, well documented, can speed up your achievements when you hear about a new result that overcomes a difficulty which blocked you in the past All these aspects of experience qualify you as an expert ( 19 of 19 )
  57. 57. Experience In the long term, negative results will form the fundamental part of your experience you will know what directions and tools have few hopes to lead to a result you have a track of open problems, not-yet-attempted alternatives, variants to explore negative results, well documented, can speed up your achievements when you hear about a new result that overcomes a difficulty which blocked you in the past All these aspects of experience qualify you as an expert ( 19 of 19 )
  58. 58. The end ( 20 of 19 )

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