From MathWeb

Contents 1 Welcome 2 Your Experience 2.1 In which of the following settings have you ever managed mathematical knowledge? 2.2 What kind of mathematical knowledge base have you participated in? 2.3 How open were the mathematical knowledge base(s) you participated in? 2.4 What was the purpose of the mathematical knowledge base(s) you participated in? 2.5 What was the common granularity of knowledge items in the mathematical knowledge base(s) you participated in? 2.6 How were issues reported and tracked in the mathematical knowledge base(s) you participated in? 2.7 Were issues tied to individual knowledge items? 2.8 What support was there to solve issues? 3 Issues and Solutions 3.1 Proven Assertions 3.1.1 What kind of issues related to proven mathematical assertions have you ever experienced? 3.1.2 If a “proven” assertion turned out to be wrong, how was this issue solved? 3.1.3 If a proven assertion was incomprehensible, how was this issue solved? 3.1.4 If it was not clear how to apply the proven assertion to solve a problem, how was this issue solved? 3.1.5 What other issues with proven assertions have you experienced and how were they solved? 3.2 Unproven Assertions 3.2.1 What kind of issues related to unproven mathematical assertions have you ever experienced? 3.2.2 If an unproven assertion turned out to be wrong, how was this issue solved? 3.2.3 If it was uncertain whether an unproven assertion was true or false, how was this issue solved? 3.2.4 If an unproven assertion was incomprehensible, how was this issue solved? 3.2.5 If it was not clear whether an unproven assertion was useful, how was this issue solved? 3.2.6 What other issues have you experienced and how were they solved? 3.3 Examples 3.3.1 What kind of issues related to examples have you ever experienced? 3.3.2 If an example was not appropriate for a domain of interest, how was this issue solved? 3.3.3 If it was not clear why an example claimed to be an example for something, how was this issue solved? 3.3.4 If an example was incomprehensible, how was this issue solved? 3.4 Notations 3.4.1 What kind of issues related to notations have you ever experienced? 3.4.2 If a notation …, how was this issue solved? 3.4.3 If a notation …, how was this issue solved? 3.4.4 If a notation …, how was this issue solved? 3.4.5 If a notation …, how was this issue solved? 3.5 Definitions 3.6 Axioms 3.7 Proofs 3.8 Theories 4 Rest

Welcome

This survey deals with tracking issues with mathematical knowledge items in a collaborative environment, and with assistive technologies for resolving them. We assume that for different types of mathematical knowledge items there are different types of issues related to them, and finally different ways how these issues can be resolved. The survey addresses people who have dealt with a mathematical knowledge base before, be it by operating or developing such a system, or participating in such a system. Examples are diverse and range from informal knowledge bases like the mathematical section of Wikipedia to the libraries of automated theorem provers.

Your Experience

In which of the following settings have you ever managed mathematical knowledge?

• Operated/hosted/maintained a mathematical knowledge base
• Developed a mathematical knowledge base or a system for managing it
• Participated in an existing mathematical knowledge base
• Other (please specify)

What kind of mathematical knowledge base have you participated in?

• Website (e.g. Wikipedia, PlanetMath, MathWorld)
• Library of a software tool (e.g. of an automated theorem prover)
• Other (please specify)

How open were the mathematical knowledge base(s) you participated in?

• Open for any kind of contribution (e.g. Wikipedia)
• Writable by a group of editors/developers, other users could comment
• Only writable by editors/developers, read-only for others
• Completely closed, not public
• Other (please specify)

What was the purpose of the mathematical knowledge base(s) you participated in?

• General-purpose knowledge collection (e.g. Wikipedia)
• E-learning (e.g. ActiveMath)
• Automated theorem proving
• Scientific publishing
• Other (please specify)

What was the common granularity of knowledge items in the mathematical knowledge base(s) you participated in?

• Documents of the size of a course unit or lecture
• Documents of the size of one slide of a presentation
• One mathematical theory (e.g. the theory of groups, with a few definitions and axioms)
• One mathematical statement (e.g. one definition, one axiom, one assertion, one proof, one example)
• No uniform structure
• Other (please specify)

How were issues reported and tracked in the mathematical knowledge base(s) you participated in?

1. Built-in facility to submit structured issue reports where you could select from pre-defined types of issues
2. Built-in facility to submit unstructured issue reports or general comments
3. Mailing list, forum, newsgroup or any other place to report issues outside of the knowledge base
4. Contacting authors/editors personally (e.g. by e-mail)
5. If case 2-4 selected: Was guidance given how to report issues?
6. Other (please specify)

Were issues tied to individual knowledge items?

• Yes, issues could refer to one knowledge item
• Yes, issues could refer to multiple knowledge items
• No, issues were tracked on a general level (e.g. one forum for the whole site). Of course this includes the possibility for authors to name the subject of the issue in the text body of their reports.
• Other (please specify)

What support was there to solve issues?

• Manual editing and restructuring of knowledge items in a rather formal language (something like a programming language)
• Manual editing and restructuring of knowledge items in a representation that looked like mathematics written on paper (e.g. WYSIWYG editing of formulæ)
• Assistance with restructuring (like refactoring support in software development environments)
• Semi-automated solution of common issues, e.g. by software assistance
• Automated solution of common issues
• Other (please specify)

Issues and Solutions

The following questions deal with issues you have ever experienced with different types of mathematical knowledge, and how they were solved. This refers either to experiences you made yourself, or to others' experiences you closely observed in the mathematical knowledge base where you participated.

Proven Assertions

By “proven assertions” we mean theorems, lemmas, corollaries, etc. which have to be proven. Here, we are referring to the assertion itself, not to its proof. Proofs will be subject of a later question.

What kind of issues related to proven mathematical assertions have you ever experienced?

• It turned out to be wrong
• It was incomprehensible
• It was not clear how to apply it to solve a problem
• Other (please specify)

If a “proven” assertion turned out to be wrong, how was this issue solved?

• Found out what was wrong and fixed it
• Kept it (probably together with a counter-example) as an instructive example of a wrong assertion
• Deleted it from the knowledge base
• Issue was not solved at all
• Other (please specify)

If a proven assertion was incomprehensible, how was this issue solved?

• Gave an instructive example for the assertion to make it comprehensible without requiring an understanding of the proof
• Improved the wording/structure/presentation of the assertion itself
• Improved the wording/structure/presentation of the proof
• Issue was not solved at all
• Other (please specify)

If it was not clear how to apply the proven assertion to solve a problem, how was this issue solved?

• Gave an example for how to apply the assertion to solve a problem
• Improved the wording/structure/presentation of the assertion itself
• Issue was not solved at all
• Other (please specify)

What other issues with proven assertions have you experienced and how were they solved?

(free entry)

Unproven Assertions

By “unproven assertions” we mean hypotheses, conjectures, postulates, etc. for which there is no proof (yet).

What kind of issues related to unproven mathematical assertions have you ever experienced?

• It turned out to be wrong
• It was uncertain whether it was true or false
• It was incomprehensible
• It was not clear whether it was useful
• Other (please specify)

If an unproven assertion turned out to be wrong, how was this issue solved?

• Found out what was wrong and fixed it
• Kept it (probably together with a counter-example) as an instructive example of a wrong assertion
• Deleted it from the knowledge base
• Issue was not solved at all
• Other (please specify)

If it was uncertain whether an unproven assertion was true or false, how was this issue solved?

• Improved the wording/structure/presentation of the assertion
• (Eventually) proved or refuted it
• Issue was not solved at all
• Other (please specify)

If an unproven assertion was incomprehensible, how was this issue solved?

• Gave an instructive example for the assertion
• Improved the wording/structure/presentation of the assertion itself
• Improved the wording/structure/presentation of the proof
• Issue was not solved at all
• Other (please specify)

If it was not clear whether an unproven assertion was useful, how was this issue solved?

• Gave an example how the assertion could be applied to solve a problem (if it were true)
• Issue was not solved at all
• Other (please specify)

What other issues have you experienced and how were they solved?

(free entry)

Examples

We assume that examples can be given for … (TODO: list)

What kind of issues related to examples have you ever experienced?

• It was not appropriate for the domain of interest
• It was not clear why it claimed to be an example for something
• It was incomprehensible
• Other (please specify)

If an example was not appropriate for a domain of interest, how was this issue solved?

• TODO: complete
• Issue was not solved at all
• Other (please specify)

If it was not clear why an example claimed to be an example for something, how was this issue solved?

• TODO: complete
• Issue was not solved at all
• Other (please specify)

If an example was incomprehensible, how was this issue solved?

• TODO: complete
• Issue was not solved at all
• Other (please specify)

Notations

We assume that notations of mathematical symbols are defined separately from the symbols. This is particularly useful if there are multiple notations for a symbol that can be used for different audiences. E.g. the second derivative of a function f can be written as f'', f⁽²⁾, , or D²f.

What kind of issues related to notations have you ever experienced?

• TODO: complete
• Other (please specify)

If a notation …, how was this issue solved?

• TODO: complete
• Issue was not solved at all
• Other (please specify)

If a notation …, how was this issue solved?

• TODO: complete
• Issue was not solved at all
• Other (please specify)

If a notation …, how was this issue solved?

• TODO: complete
• Issue was not solved at all
• Other (please specify)

If a notation …, how was this issue solved?

• TODO: complete
• Issue was not solved at all
• Other (please specify)

Definitions

Axioms

Proofs

Theories

Rest

Why did issues remain unsolved (poor editing/refactoring support? poor social interaction?)