Resource and Truth Searching Methods

Method 6	Find Resources. Search for what you know that likely relates to what you want or relates to some other important part of your problem.
Class of Method	A Primary Problem-Solving Strategy.
Use When	When the information you have obtained so far allows for little manipulation, or has proven difficult to successfully apply to your problem. You return to this routine whenever you have difficulties, but new information or some progress has been found during other routines. Return to this method when you have left this routine to use another routine with little success and had not finished all resource related routines or done all resource routines sufficiently.
How	Any time during this process that you find new truths, look at the new list of “known truths”, and re-analyze how elements in combination from your new list of “known information” could help you achieve a step that would lead to you want to overall achieve. Search for resources by executing the following steps relatively quickly and non-exhaustively by what seems to you to be most efficient. Then, you can redo these steps again later more exhaustively if it becomes necessary to do so: 1.† Search for resources already published. This is a top priority given that it is one of the 4 main keys to problem solving. In order to do this: a) Learn the branches of math that relate to your problem and familiarize yourself well with these concepts. () While doing this, if you obtain what seems may be sufficiently useful resources to solve your problem, then go to method 24 (Create and follow a plan), or even another method that may be more appropriate for the situation. If you are able to create a promising plan that does not seem relatively too time consuming to implement, then begin to implement your plan, otherwise continue this step (1) by studying to find resources that may help you create a plan that is easier to implement. b) Consider searching for a useful theorem that seems†† to likely have a relationship that would help to solve the problem or a part of the problem. Look for theorems that have relationships and parts that are similar‡ to those relationships and parts in your problem. As in the previous step, follow principle (). c) Consider what variations of those theorems would help solve your problem. Do this by considering what specification, generalization or other change in the theorem would allow you to fulfill a step of your plan or help to create a better plan for solving your problem. Be sure to do this when the variation of the theorem seems true. Then hypothesize the new theorem to prove in order to use it for your problem (unless the theorem is a specification of the original theorem). As in the previous step, follow principle (). If step (1) provides little success then: 2. Use method 31 (Break down and recombine). If seemingly necessary, you may need to return to step (1a) to search for resources related to the mathematical entities obtained by breaking down parts of your problem when using method 31 (break down and recombine). As previously done, follow principle () in step (1). If step (2) gives little success, then: 3. Try to find truths implied from the given information in the problem that might be helpful in solving the problem by searching for a pattern (method 12), or by experimenting (method 7). a) Ask questions first related to what you want. Find the correct questions to ask about your problem before experimenting or searching for a pattern using the data in your problem. You do not want to search for information unless it has high a likelihood of being both useful and sufficiently simple to obtain. Consider which questions are the right questions to ask that optimize the expectation of finding useful truths. -There are three types of questions as listed below: How does ____ vary as ___varies? (answered by using experimental design) What kind of relationship holds between ____ and _____? Is _____ true or false? b) Develop your list of all questions whose answer would most likely be useful to help solve your problem. Then experiment or search for a pattern involving the entities in those questions in a way that would most likely help you to answer the questions. As previously done, follow principle (**) in step (1). If step (3) provides little success towards helping you find a solution or promising plan then: 4. Try working forwards to find truths implied from the given information that might be helpful with solving the problem. Do this by repeating step (2) with the exception that you are asking questions about relationships without a direct focus on what you want. You are simply searching for any truths you can find. (Doing this is randomly searching for truths without proper guidance which sometimes creates some progress, but usually is inefficient.) (Step (4) is a desperate effort to find resources.)
Notes	√Implementing these steps quickly and efficiently on the first round of this method is important in order to avoid wasting time by excessively searching for resources. †The order of steps in this method are created with the purpose of learning first what resources already exist so that you can “see” the mathematical world that relates to your problem. Only then do you use the more desperate approaches of searching for resources involving the decomposed (broken down) entities in your problem. If the previous approaches didn’t provide results for your problem, then an even more desperate strategy of trying to construct the needed resources on your own through experimentation and searching for patterns to hypothesize needed truths. †† Whenever words like “seems” or “intuitively” are used, it means that you use your defeasible reasoning to make such judgments. ‡Two entities, relationships, or etc. being "similar" means that many defining parts of the entities, relationships, etc. are equivalent or that the defining parts of the entities have equivalencies in aspects of the definitions that apply to defining the entities, relationships, or etc. Any time you discover truths you know about the problem or that may relate to your problem, write them down in a list called your “list of known information”. Consider the implication of each part of the given information in the problem in every possible combination with other parts of the given information in the problem if doing so would require little time. If doing this requires much time, then carefully select to analyze those combinations that seem to likely provide insight into the problem. If your only tool is a hammer, then everything looks like a nail. Seek resources that better fit your circumstance than the hammer if you don’t have a nail.

Method 7	Search for useful truths. Experiment by testing entities with powerful distinguishing properties.
Class of Method	A Primary Problem-Solving Strategy.
Use When	When what you want is defined on entities that can have distinguishing properties that may be useful in your problem. Also use this whenever given information in your problem is or can be applied to various distinguishing entities and if testing the implications of specifying those entities would likely help in achieving the overall goal of your problem.
How	Not only should you perform experiments in your problem, but you should also experiment with whatever you may be interested in that most likely relates to what you want.
Notes	The process of creating an explanation for observations takes enormous amounts of possible specifications and summarizes them with simple rules that would explain all the information. Model creation is a form of data compression. The more general a rule is, the more powerful it will be as it applies to more combinations of entities. When you are searching for truths using experimentation, you should try to find the rules that are as general as possible in order to have more mathematical power. Scientists try to justify their theories by carefully designing experiments that, if successful, have only one likely explanation—namely, their theory. However, hypothesis should not be created, but considered as possibilities. The better scientific method is to consider the most likely possibilities, then to set up optimal experiments to eliminate possible models to narrow the possible models.
Example	See example problem 8 of chapter 6 (Examples of Problem Solving).

Method 8	Search for counter-examples.
Use When	When searching for needed truths and you create a hypothesis that is complicated to prove, but has a good chance of being false.
How	If you have already worked hard on solving a problem using more than one plan to solve the problem, then: 1. Using the plans you tried applying to prove your hypothesis, do a root cause analysis (method 29) on why the best version of those plans cannot successfully be used to prove the hypothesis. Then if possible, create a counter-example that takes advantage of the root of the issue in proving the hypothesis. If the previous step becomes too time consuming or difficult to implement, then: 2. Search for examples that satisfy the conditions of the hypothesis, but don't satisfy the assumed implication (conclusion) of the hypothesis. In order to do this: a) Look for examples of your problem that are extreme in aspects of their specification. If this doesn't provide a quick counter-example, then: b) Consider what aspect(s) is/are needed for the implication to be violated, and form the extreme or other specifications to achieve one of the aspects that would cause the assumed implication (conclusion) to be violated. If this isn’t successful then: c) Restart the problem-solving process of solving the sub-problem of constructing something that satisfies the conditions of the hypothesis, but not the conclusion of the hypothesis. 3. Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.
Notes	If your hypothesis is a true statement, then it will be impossible to find a counter-example to it. If you have difficulty finding a counter-example, then that is a sign that your hypothesis might be true. In computer science, this method is known as a model checking algorithm. Sophisticated and useful algorithms have been developed recently for model checking.

Method 9	Explore the conditions of the problem holding all but one quantity fixed [7].
Class of Method	A type of experimentation (method 7).
Use When	When the problem has many complicated combinations of data to relate, or complicated combinations of data to show.
How	1. Assume one quantity, function, variable or other mathematical entity to vary and fix all others to a single value or entity type. 2. Identify all the distinguishing potentially useful types the allowed varying entity could be. 3. Begin to experiment using method 7 only allowing the one chosen entity to vary into its distinguishing entity types to search for truths/patterns given each such scenario. 4. Restart this process by allowing another entity in the problem to vary holding all others fixed. Do this until you have sufficient information to likely be able to solve the problem or until you feel that using this method is wasting time with little progress. 5. If you have not yet found your needed information, then consider repeating steps 1-3 with the exception of allowing two (or possibly more) entities in the problem to vary while holding all others fixed to a single value or entity type. However, do not waste time on this step if doing this step seems to provide information of little use. 6. Compile all likely useful information from this routine. 7. Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Method 10

Search for and exploit relationships to other known branches of mathematics.

Use When

When there are entities, relationships, or combinations of both in your problem that also occur together in theorems, problems or other parts of mathematics from another branch of mathematics.

How

1. Search for entities, relationships, or combinations of these in your problem that also show up together in theorems, problems or other parts of mathematics from another branch of mathematics.

2. Consider if the resources from the other mathematical branch would likely assist in solving your problem by experimenting to find a pattern by using those resources in how they relate to your problem.

3. Use the found information from the other branch of mathematics, and try to see how to make it relate to what you want, or consider how the information could help to simplify your plan for solving the problem.

4. If this method successfully gives you what you perceive to be a significant amount of useful information, then go to method 28 (vary your plan) in order to create a new plan that takes into account your new information.

5. Otherwise continue this method, and if searching becomes time consuming then choose to use another method (heuristic).

Method 11	Use relationships from other known branches of mathematics. Try to alter your problem to make it fit into that branch of mathematics. Consider the usefulness of the information provided under that model.
Class of Method	An extension of method 10.
Use When	When a subset of entities, parts, formulas or relationships in your problem strike you as similar‡ to problems, formulas, theorems, methods, etc. from another branch of mathematics.
How	1. Consider the entities and relationships in your problem and search for other problems, formulas, theorems, etc. from other branches of mathematics that have a significant amount of specific entities or relationships from at least one entity type or relationship type in your problem. For example, search according to specific entities such as numbers, corresponding to parts in identities, theorems, properties, or relationships in other branches of mathematics. Look for combinations that are very strongly correlated to parts in a specific field and have a small probability of not being related due to such strong entity or formula similarity. 2. Try to assign the remaining entities and relationships in your problem into the framework under the model from the associated field of mathematics. Try to relate other parts of the problem under the new framework by considering how those parts of the problem could be put into the framework of theorems, axioms, identities, or relationships available in that field of mathematics. 3. Take the information from the other branch of mathematics and its model. Add that information to list of known information and consider how it can help you to carry out your plan, create a better more promising plan, and how you can relate it to what you want. 4. Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.
Notes	‡Two entities, relationships, or etc. being "similar" means that many defining parts of the entities, relationships, etc. are equivalent or that the defining parts of the entities have equivalencies in aspects of the definitions that apply to defining the entities, relationships, or etc.

Method 12	Search for a pattern or create a model.
Class of Method	A Primary Problem-Solving Strategy.
Defining Terms	A Pattern is defined as a set of observations (or observations that have a way of being generated) where there is a method for generating those observations (or a set of axioms describing the observations in full), and the explanation for the observations requires far less information to state, than the information required to store the observations. The explanation of a pattern (such as axioms that define the pattern, or a method that generates the pattern) is called a Pattern Explanation. You may already have a pattern explanation, but it may not be a very convenient or useful explanation. Usually simpler and more concise pattern explanations are more useful for problem solving. Your goal is to find a pattern explanation that is likely to assist you to solve your problem. Partial (or incomplete) descriptions of patterns are called Lossy Pattern Explanations. Pattern explanations that contain sufficient information to precisely generate the pattern are called Lossless Pattern Explanations. A pattern explanation is called an Irreducible Pattern Explanation if none of its sub-explanations can together explain one of its other sub-explanations. nth order models: Let each observation be called yi where i is the index of an observed entity. Let xij be the jth property of observation yi (potentially, the only property you may be interested in is the value yi). An nth order model is a model that predicts xij for all i where for some g, xij = g(xi1(j1),xi2(j2), … xin(jn)). Common models include regression, functional equations, differential equations, etc. For example, a second order model could depend on x(n-1) and x(n-2) i.e. x(n) = x(n-1) + x(n-2) is used to describe a pattern called the Fibonacci sequence. An example of a general first order model is where you could solve the (potentially vector) functional equation g(f(x), f(h(x)))=0 for the function f(.), where g is some given model type, and h is an indexing for a next entity relative to x.
Use When	When there are entities that can likely be described with summarized rules where such rules would likely help solve an important problem.

How to Search for a Pattern

Pre Notes

Initially Complicated Pattern Explanations

If you use a method to create the observations, then you already have a pattern specified, however usually you are looking for simpler or more useful rules. If you already have a pattern explanation, but your pattern explanation is very cumbersome and of little use, then try to find a pattern explanation that is more simple and concise.

A complete pattern explanation is not always needed, but just enough of it to be sufficiently useful for your problem.

Usually when searching for patterns, you either have observations, or you have a way of generating observations that is slow in some way or not yet useful with respect to your problem-solving needs.

Steps

1. If observations and relations are not yet defined, then define the relations and entities involved by using method 3 sub-method I (create useful and meaningful definitions).

You may also define frequently occurring types of sub-patterns from sub-sections of the observations under some
seemingly useful indexing. This is often done by restarting this method on subsections of the data that have some
type of identified similarities in their sequences of observation.

-For entities involved in the problem, identify the entity classes and types of entities within those classes.

-Identify likely or potentially useful distinguishing properties of entity types to experiment with. This means that you search for a model based on those entity types that have those distinguishing defining properties. The idea is that usually pattern explanations will be different when different types of entities are involved. Entity types to experiment with could be defined entities that compose of several other observed primitive entities (such as a trills and arpeggios in a string of musical notes).

2. If no process is available for generating observations, then solve the sub-problem of creating some process or algorithm that can generate observations. Note what observations are costly to generate in order to guide experimentation later. Experimentation is the choice of which observations to observe.

3. Carefully consider what you value in a pattern explanation. In technical words, obtain some estimate of a utility function for attributes of pattern explanations.

You may value certain pattern explanation types that would be used for solving specific types of problems. For example, cumbersome and complicated pattern explanations may not be practical for your problem-solving needs. On the other hand, you may be willing to have a pattern explanation that explains observations correctly most of the time, or you may be willing to accept a pattern explanation that always provides errors where such error types are acceptable (such as an approximation of a number).

However, often you will need to have a pattern explanation that is completely error free.

4. Consider the set of potentially useful or natural indexing methods of potential (or already observed) observations. The method of indexing the entities will have a significant effect of finding useful pattern explanations with respect to that indexing (or potentially with respect to more than one method of indexing).

-If observations are already provided to you by some method, algorithm, or data stream, then the order that the observations are generated provides a natural indexing worthy of consideration.

-If entities you observe tend to have certain natural indices, then consider those indices.

-Later you may need to find a superior indexing as part of the process of finding specific pattern explanations. For example, Gauss found a clever method to add the numbers 1+2+…+n. He indexed the numbers by grouping the first and last number, the second and second to last, the ith and ith to last number which each had a sum of n+1.

-If the only natural indexing of the observations is the definition hierarchical indexing, then the definition hierarchy is the indexing you should use.

-Consider hierarchical types of indexing (such as that of a fractal where observations bifurcate or branch out).

5. Use Kernel #1 (below) for finding pattern explanations for observations that involve the following data types:

Data Streams: Such as musical patterns, computer data streams, observations with numerical indices, sequences of numbers, sequences of entities with various defining properties.

Bifurcating Data streams: Such as geometrical patterns that branch out from each new node created from the previous observation, or other data streams that bifurcate. For bifurcating data streams, you may need to find the pattern that defines when splits occur, the type of bifurcation split, and the properties of each entity from that bifurcation. Predicting at which entities bifurcations occur and how many splits there are at that entity can be considered a property of the entity where the split occurs.

Indexing by ordered pairs: Such as squares on a chessboard.

Use Kernel #2 (below) when there is a need to find pattern explanations that depend on the hierarchical definition indexing of entities involved.

Note that all defined entities are hierarchical where specified entities belong to the hierarchy of more generally defined entities. Pattern explanations found using Kernel #2 include: Real number axioms, set theory axioms, group theory, logic axioms, topology axioms, probability axioms, game theory axioms, choice axioms, etc. These models however require more complicated modeling and involve creating various axioms.

6. If no sufficiently simple axiom or model is found according to the type of pattern explanations that you would value, then use defeasible reasoning to decide weather to:

a) Go back to (1) to define sub-pattern sub-sections of entities under some index to create a definition hierarchy of such sub-patterns to search for a pattern that uses those sub-patterns as its primitive entities. If finding a zero order model is unpromising, try to define sub-pattern subsections, and restart step 1-2 on those.

Sometimes you have a pattern explanation for many things and you want a pattern explanation for a set of pattern explanations.

b) If steps 1-6 do not provide a pattern explanation, then search for even higher order models. Or you could also search for various possible methods of indexing the entities and restart the search for a pattern.

7. If your pattern explanation is supposed to be error free, then carefully but quickly consider if your explanation could be violated by other extreme or degenerate special observations. You may need to create an experiment to check these if necessary for generating such observations.

-If your observations came from a stream of data, then re-sample the stream of data to check the new data for consistency with your model.

-If your pattern explanation is supposed to allow for error types such as numerical approximation, then sample more data to check the consistency and accuracy of the model. A common mistake in statistical or other error allowing explanations is to over fit or over explain a specific data sample which creates explanations that really don’t exist for general data samples that are supposed to be explained by the model.

-Quickly try to check and verify that the axiom is consistent with already assumed axioms.

8. If you likely need more axioms, return to step (1) to choose another subset (of entity details) to find another axiom about those details being sure to choose that subset in a way that would likely be independent with currently found axioms. Otherwise if your pattern explanation seems sufficient for your problem-solving needs, consider checking consistency among your assumed axioms more carefully. Consider extreme cases that could violate some of the axioms in combination.

-If you find an example that violates your explanation, then go back to the necessary step in the appropriate kernel to incorporate the information from the extreme observation. Otherwise try to prove your hypotheses if necessary.

9. Perform quick searches of implications of the hypothesized truths and their possible usefulness to find what is "wanted". Usually you will do this using defeasible reasoning. Depending on the situation, you may not want to spend too much time on this, but just to quickly search for additional potentially useful properties.

10. Carefully try to prove some axioms by assuming the others in order to narrow down the axioms to a more concise collection of only necessary axioms. (Skip this step if it is not needed for your problem-solving needs. But if your pattern explanation would be used in a wide variety of circumstances, and would be commonly used, then it is best to try and simplify the system of axioms that describe your pattern explanation.)

After finding a collection of axioms that describe your observations, many of those axioms may be provable by assuming a subset of those axioms. There also may be axioms that are equivalent to axioms assuming other axioms. Three of the axioms together while assuming all other axioms may be equivalent to two other axioms combined thus allowing simplification of the set of assumed axioms. (Or more generally n to n-a axioms.)

Take those statements that have similarities, or are statements that involve similarities. Within those groups of statements with similarities:

-Solve the sub-problem of proving that needed truths are impossible to prove without certain axioms.

-Solve the sub-problem of considering which axioms might be provable from other axioms in all combinations.

11. Although you are now finished, you should however be willing to modify any axioms later if you discover one of the assumed axioms proves too strong, weak, contradictory, or has some other undesirable property.

Kernel #1 (For finding patterns involving data streams)

Try to find a sufficiently simple model that describes the observations. Here you are trying to find a model that would predict the observed properties from the properties of other observations relative to some index. Do this by following these steps:

1. Prepare to think inside the box as to potential pattern explanations by implementing preliminary tests of classes of n-order models to get an idea (to later guide defeasible reasoning as to what the potential pattern likely falls into.) Because you eliminate many possible pattern explanations by thinking inside the box, do the following steps until thinking outside the box likely becomes worthwhile for finding pattern explanations:

a) You first consider which of all observation output properties have a 0th order model (models that predict all of the entity properties as simple functions of the entity’s index). Possible strategies you can try:

Try to recognize the differences and similarities between different observations.

-Consider how each part of the differences between the observations are different and how they are the same. Consider how what stays the same changes, and how what changes again changes. Etc. You do this by considering sub-classes of the inputs that have zero order relationships (like being of the same entity types or some defining property of a class of input entities, consider all combinations of what would be defining properties of entity types) and denoting what properties are fixed about the outputs within each entity type.

b) For those properties that could not be predicted using 0 order models, search for 1st order models—starting with the simplest of such. You consider which have first order models (models where the function relates to the output of other entity types where the entity types can be specified in all possible ways entity types can differ such as n+1, n-1, etc.)

c) For those properties that could not be predicted using 1st order models search for 2nd order models—starting with the simplest of such.

d) Continue consecutively to nth order models until it becomes necessary to either think outside the box and consider lower order models not considered earlier in the process, or to proceed to step 2 to implement other strategies.

2. If the currently best n-order models already considered are of poor utility (such as complicated to implement, sensitive to initial conditions, or impractical models), then use defeasible reasoning to decide weather to search for statistical pattern explanations (as some n-order model), or to search for sub-pattern subsections (like arpeggios can be subsections in music).

Sometimes several axioms (an nth order model involving vectors of properties) are needed for describing a data stream.

Notes:

No experimentation is needed when observations are in data stream because the data stream provides the observations.

If your observations come from a data stream that provides observations with properties of the entities that seemingly occur at random, then implement kernel #2.

For models of order N, use defeasible reasoning for considering the most likely classes of n-order models the pattern would fit into first. (usu. Lower order models)

Kernel #2 (For finding patterns by focusing on the definition hierarchical indexing of entities) The Scientific Method

1. Use the scientific method to find axioms that describe the pattern using experiments.

2. After discovering an axiom, try to discover why the axiom is true. In other words, your goal here is to try and find more primitive axioms that would imply the current axiom thus making the current axiom become a theorem of more primitive axioms you must find.

Do this by doing a root cause analysis (method 29) to find the more primitive axioms that explain why your currently assumed axiom is true. For the most likely root causes, do thought experiments (the steps below) to test their potential validity and state the root causes as generally as possible. Each time you discover a more primitive axiom, search for more primitive axioms that imply the previously assumed primitive axiom by again using root cause analysis and thought experiments.

Steps to do a Thought Experiment (Gedanke-experiment)

1. Consider the potential specified assumption (primitive axiom). (The potential root causes of the original axiom).

2. Consider the implications caused by assuming the specified assumption combined with already assumed axioms.

3. Consider if the implications in (2) are consistent with both observation and already assumed axioms. If needed, set up any necessary experiments to create any such needed observations. If the primitive axiom in question proves inconsistent, then either try to alter the axiom, or abandon the axiom and try to find another axiom that describes the needed relationships of entities properly.

4. Consider what specifications of the specified assumption in step (1) were not needed in the argument in step (2) to thus generalize the specified properties of the specified assumption. Also consider what parts of the argument in step (2) could be altered to further generalize the specified properties of the specified assumption in step (1).

When searching for the primitive axioms, you may unintentionally assume axioms that are not true or are inconsistent with other axioms that were assumed. In such a scenario:

1. Consider what part of an argument allowed the contradiction.

Consider what changes to each axiom would remove the contradiction.

Conduct experiments to consider which modifications to the axioms are legitimate.

Choose a modification to the appropriate axiom that removes the contradictions but is sufficiently satisfactory for problem-solving needs the axioms would be used for.

Method 12 Notes

Not Every Mathematical Circumstance That is Finitely Stated Has a Pattern

It is impossible to create a finite method (a method with a finite number of steps) that would decide if there is a pattern explanation for every possible mathematical circumstance that can be finitely stated. For example, consider the solution to Hilbert’s 10th problem which shows that no general finite algorithm exists to determine whether or not there is an integer solution to a Diophantine equation. Consider trying to find a pattern explanation for whether or not there is an integer solution to each given Diophantine equation. Because there is no finite algorithm that would determine whether a given Diophantine equation has a solution, this provides a counter-example to the notion that there is a finite method for creating pattern explanations for finitely stated mathematical circumstances. A pattern explanation for a finitely stated mathematical circumstance seems to always be determinable by creating a finite method for creating proper designs of experiments. It therefore seems that no finite method exists for creating a proper design of experiments to analyze every possible finitely stated circumstance.

Most mathematical circumstances of interest seem to have pattern explanations, and can be found through creating and implementing experiments where such experiments were generated using a finite method.

Often you don’t need to be able to generate the entire pattern of interest, but just need to know certain properties about the pattern in order to solve your problem. Pattern explanations such as invariants (see method 15a) may not be able to generate the pattern, but provide the needed information about the problem to allow for solving the problem. In fact, pay attention to when you don’t need all of the information from a potential pattern explanation to solve your problem. Do this, because recognizing what parts of a pattern explanation are not required to solve your problem allows you to solve a more generalized version of your problem by noticing these fewer requirements to solve your problem or prove your theorem.

Axiom Creation Notes

Motivation for creating an axiom is

To describe the variety of circumstances axioms would need to describe.

An axiom from a set of needed axioms, (partial pattern explanations) are often defined by the specific circumstances that pertain to that specific axiom. For example, in the real numbers, the circumstances may be addition of two numbers for one axiom, addition of three entities for another axiom, then of multiplication and addition together for even another axiom.

To specify models.

For example, the separation of possible remaining models for real numbers is done by stating the axiom that 1 ≠ 0.

To define the existence of needed entities.

An axiom is often simply stated by defining a needed entity, then stating the existence of that entity. For example, axioms that were created due to a needed property include the existence of multiplicative inverses, or defining the number zero then stating its existence. Usually such definitions are created when such properties seem to hold true, or as being useful properties when problem solving, and searching for when those properties would be true.

Proving that a system of axioms is consistent and complete is often extremely time consuming or sometimes is even impossible. Therefore, after creating a collection of axioms, often you should carefully check to see of you can find any contradictions among axioms. Defeasible reasoning guides your careful selection of axioms to check for potential contradictions and when checking for observed or needed mathematical properties.

Sometimes you may consider carefully stating the axioms to be not only sufficient to prove needed truths, but to also be necessary conditions to prove needed truths.

When a specific model will be used in a variety of ways for a variety of applications, then it becomes more important to have a collection of axioms T for the model such that for every partition of axioms T into sub-collections G, A, B (potentially empty), the following happen:

G therefore (A implies not B) does not happen (no contradictions)
G therefore (A implies B) does not happen (no redundancies)
The total number of non-conjoined axioms is minimal, meaning that for any system of axioms S, then S iff T, but T has fewer axioms than S.

One way of trying to accomplish as much of this goal as possible is to implement a method that for each partition of the axioms into two subsets, and then for each axiom in the 2nd subset, you try to find if that axiom can be found to be dis-proven or proven redundant given the 1st subset of axioms. Such a method in some cases is impossible to fully execute due to Gödel's theorem—especially if the set of axioms can describe arithmetic, and is therefore not always practical. After satisfying as much of 1-3 that is practical, you may consider which needed statements or theorems depend on which subsets of the axioms in order to know which axioms are required to be satisfied in order to use certain statements.

Impractical Pattern Explanations

Some pattern explanations are not very useful for predicting or controlling circumstances which is usually the purpose of a pattern explanation—intelligent decision making. For example, dynamical systems in many instances are difficult to use because there is no way of identifying the exact initial conditions of the dynamical system thus making accurate predictions impossible to make. There may be extreme levels of complications in the system you try to model making needed pattern explanations so complicated that problem solving within that model becomes impractical. There may also be truly random events occurring thus requiring a statistical model. In fact, a way to overcome describing extremely complex situations is just to “wing it” and use a statistical pattern explanation (model) and accept some degree of error in prediction rather than having to completely do without some kind of a useful model.

Fuzzy Pattern Explanations are statistical pattern explanations, which predicts that a truth value under each specific circumstance would hold a certain percentage of the time while holding all other properties at random (or under some assumed random variable which is hopefully the random variable if selecting an entity were done at random). (In other words, under the assumed sample distribution for other properties of entities, and under the given fixed properties, there is a truth value of a percent T and a percent F) Truth tables where A is an observation and B is another observation, then to create the truth table of when 100% A with all other properties at random, what are the percent truth values of B? The closer a fuzzy truth table is to an actual truth table for A implies B, the stronger the relationship is that A causes B.

Zero Order Models by Testing Entity Types

In searching for possibly useful truths, zero order models include trying and testing (creating a model for) those entities that have specific powerful properties or specific different types of properties that distinguish those mathematical entities from other entities of its type. This is done after separating the entities into classes based upon their properties and powers to test elements from each class. For example, testing large positive numbers, small positive numbers, small negative numbers, large negative numbers, zero, and one. For another example, in functional equations, test f(0) because zero is powerful due to important properties of zero that can only be exploited by zero. You can also exploit properties of unique classes of entities such as zero by considering when the output of a function in question has that entity type as its output. For example, you may consider what values “a” cause the function f to satisfy f(a) = 0, or 1. It is important to test elements in various groups according to their different types of mathematically manipulative powers or unique properties. Then after testing 0 and 1, you may consider testing the rational numbers which are another specific class of numbers with distinguishing properties, etc. Then you may find the implications in your problem that result from testing those points. Another example of testing a class of mathematical entities that have specific distinguishing or powerful properties is testing the natural numbers for 1 then 2, and so on trying to see a pattern that you may be able to later prove by using induction. Classes of real numbers may include naturals, rationals, irrationals, big, small, negative, positive, zero, non-zero, one, between zero and one, or combinations of these classes. If the entities of interest are functions (instead of numbers from the previous example), then classes of entities to test in your problem could be step functions, the zero function, constant functions, the identity function, L2 functions, and many classes of entities similar to choosing from classes of numbers.

A sub-method to find a pattern whose statement requires use of propositional logic:

1. Consider what type of formula you are interested in. Truths should be sought that are potentially useful. For example, you may need to find when statement C is true (caused or implied) by combinations of truth values of other statements. You may need to find a fuzzy truth table using a fuzzy pattern explanation. You may interested in when A or B, or some other logical connective is true based on needs during problem solving.

2. If you are trying to find when an implication occurs for statement C, then consider all possible combinations of truth values for potentially affecting atoms A, B, D, etc. Then observe when C is true for each combination of truth value, and construct a formula that would have the associated truth table for the logic statement .

3. If you are interested in the truth value of a connective between two atoms, or a combination of connectives among atoms, then simply test the truth table by testing all possible combinations of truth values of atoms involved.

A sub-method to find a pattern whose statement requires use of first or second order logic:

1. Observe. Use defeasible reasoning to consider measures or properties that seem important, and list those that seem important. Observe or if needed, create experiments to observe when circumstances allow potentially useful properties to exist. This creates a statement using existential quantifiers (PROPERTY).

2. The order of the quantifiers makes for a different logical statement when changing some existential quantifiers to a universal. Choose several of the possible orderings of the quantifiers that would be useful for your needs to consider continuing experimentation for generalization of quantifiers. You can only interchange the order of two neighboring quantifiers if they are the same quantifier.

3. Consider each example property found in 1. Create experiments to allow you to observe other cases to check for possible generalizations of the variables quantified by to . Find a pattern of the form (PROPERTY) over variations of observation types (such as sequences). Obviously you cannot check every w for generalization, but check many of them and assume quantifier generalization which can be checked later by proof (in the case of searching for a theorem), definition consistency (in the case of the pattern explanation being a definition), or to satisfy some need (such as when creating an axiom) depending on the circumstance of the pattern creation search.

4. Repeat step 3 to try and generalize other variable quantifiers from to by observations.

5. Test to observe more general classes of variables quantified by , to generalize their universes.

6. Search for any contradictions to generalizations made.

7. If the pattern explanation is a proposed theorem, then try to prove the theorem or some variation of it. If the pattern explanation is a definition or axiom, then verify that it will fulfill it’s intended need and be mathematically workable and practical.

An example of implementing the sub-method above is creation of the ε-N definition of the limit of a sequence.

After observing various sequences that “converge” to a point, (e.g. xn = 1/n), defeasible reasoning may suggest that distances are an important property to consider in describing sequences that “converge”. Defeasible reasoning may also suggest other important properties to consider. Then distances from the first point in the sequence to several other points in the sequence may be considered providing the statements

,,, and

,,, where d(.,.) is a distance measure for sequence points in sequences pn, or sn.

Observing several other examples allows creation of the statement

The statement above can be stated as

It can be easily recognized by observing “converging” sequences the following generalization

Now setting j to 2, 3, and so on making sure to test the validity of the statement above suggests the generalization to . On the other hand, it is easy to check that the quantifier for N cannot be generalized to a universal quantifier.

The next step is to generalize over potential universes. The variable j specifies a bound on a distance. That distance bound could be considered any positive distance making the new statement . Of course many other universe generalizations may have been considered with no useful consequence.

Now, the new statement above looks more tidy by simplifying the statement and instead saying. The commonly stated definition of a convergent sequence is a sequence where .

Theorems are pattern explanations. Less trivial theorems (pattern explanations) that are discovered are recognized by one of the following methods (Note that these are important methods for searching for pattern explanations not included in the general pattern explanation search method.):

1. Use defeasible reasoning to describe what occurs by what happens in similar circumstances, then try to prove the hypothesis.

2. Try to generalize conditions or consider variations of conditions or results of already known truths.

3. Use primitive pattern recognition searches as listed in previous pattern explanation methods.

4. Directly derive results implied from other known theorems or assumptions made.

5. Have a need for a given property and search for conditions necessary to satisfy that property. When you have a need during problem solving, you may need to purpose lemmas and variations of such lemmas, and then experiment to identify such a pattern. Then you should adjust the details and properties of the lemma to accommodate both the needs for solving your problem and the restrictions on the potential lemma that you recognize by information provided by your experimental results.

6. Recognize and remember the types of observations that relate to what you are interested in that your past experiences provide. Break down the properties of these observation into cases based upon the properties of the conditions or properties (inputs) of each such observation and generalize upon the experienced observations to create hypothesized pattern explanations of the various types of these remembered (or recognized) cases.

7. Find the condition for a desired property, generalize the condition, potentially generalizing by defeasible reasoning.

8. Experiment with types of results, then refine the conditions for such results.

9. When you have already used previous methods for finding a hypothesized or needed pattern explanation, but evidence supports the opposite or some other variant of the supposed statement, then propose the opposite of your original statement, or some other modification of the statement. For example, when there is a need to find two rationals such that no rationals are between them, another rational is always found, thus motivating the opposite hypothesis that rational numbers are potentially dense (which they are).

10. Potentially other methods exist that are not mentioned here.

When proving a theorem, do problem solving for creating the proof to require as little as possible of the properties required in the conditions of the theorem being proved in order to allow for potential generalizations of the theorem. You can also try to choose arguments that allow small variations in such arguments to allow for a variety of differing properties in the conditions of the theorem thus allowing various generalizations of the theorem.

Meta-Cognition and Meta-Pattern Explanations

Creating this method is an application of finding a pattern explanation of what would be a method to find pattern explanations. (Meta-pattern explanation). I look at various pattern explanations as observations, and try to create a pattern explanation in this method that would find pattern explanations. Some of my axioms are the needed steps to find those pattern explanations, and some of my other axioms explain the order in which to implement those steps. I use the axiom that if one step must be first for finding one pattern explanation, then it would probably be first for all pattern explanation discoveries. I also write the general steps of the method, then find a pattern explanation for each of the specific steps to accomplish the larger steps. My experiments are observing various pattern explanations.

Other Notes

Data compression is about finding patterns in data. Primitive definitions of pattern types that frequently occur in data sequences allow for higher level definitions in data sequences whose definitions can then be used to find a pattern involving those higher order defined entities. This method of data compression may be powerful in certain circumstances. This method is also very powerful for Artificial Intelligence. For example, Musical patterns can be created by defined entities such as trills, arpeggios, etc. This is done by defining primitive data sequences that frequently occur, then by defining more complex data structures that occur, to then find patterns on those defined sequences and structures. This example shows that powerful data compression can be done using this method.

Pattern searching can be considered classification of combinations of a group of a type of entity to a type of group. Tools such as Neural Networks, CART, Bayesian classification methods, Logistic regression, etc. find patterns in this way. For example, when classifying good moves from a given position in chess, the input is the position, and the classification (or pattern) is the set of “good” moves.

Most often the indexing is already set for a pattern search problem, such as indexing over the counting numbers. In this case then you search for a formula that describes the observations as a function of (n), or more generally as a function of (n) and previous f(n-1, n-2), etc.

Each axiom specifies possible models further (e.g. the axiom:1≠0). Each experiment splits possible axioms with respect to entity types or defining properties involved. Axioms such as 1≠0 are discovered through either quick observation, or often through a process of axiom reduction to simpler aesthetic axioms. However, some axioms like 1≠0 could be found during problem solving by having a need for separation of possible models.

In the process of searching for a property or axiom, you may discover another needed property or axiom.

Relativity is a pattern explanation (model) that explains the large extreme cases of the physical universe, whereas quantum mechanics is a pattern explanation of the physical universe for the small extreme cases. Newtonian physics is a pattern explanation of the physical universe for non-extreme cases. Superstring and other such theories are an attempt to create a pattern explanation of the physical universe in a way that would describe both physical extremes and all phenomenon in between those extremes.

New pattern explanations that describe extremes should approximately be the same as the pattern explanations that are relatively accurate when observations are not extreme. This principle is a commonly used principle in theoretical physics and was used to test the validity of quantum mechanics and relativity.

Examining special cases of a problem to find a solution of the original problem is really just looking for a pattern in the method of solution for the original problem.

Finding an invariant is a sub-method of finding a pattern, and creating an invariant is a sub-method of creating a pattern.

The essence of learning how to find patterns is in code breaking. Thus the NSA likely holds many secrets about this method.

There may be innumerable truths that could be discovered by experimentation. However, you are searching for those truths that have utility in some way with respect to your needs in order to avoid wasting time with truths of little use.

Examples

Sub-Method 12a	Create a pattern—instead of just finding one.
Use When	When your problem allows great freedom of choosing a specific element of a class that would be used to control or specify some difficult aspect of your problem. Especially do this if there are commonly known patterns that are used for solving similar problems in the field of mathematics related to your problem.
How	1. Consider what patterns (or partial pattern explanations) that if you did have (combined with other representations or steps of solving your problem) would solve your problem. 2. Solve the sub-problem of considering which types of those patterns (or partial pattern explanations) you could create given your required restrictions in the problem. 3. Solve the sub-problem of creating those combinations of needed pattern(s) that are most feasible to create. 4. Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.
Example problems	In control theory, creating a control that makes a certain representation of the dynamics have a Lyapunov function (an invariant which is a partial pattern explanation), and therefore control the dynamics to the desired state.

Method 13	Search for an auxiliary† element that would help you to solve the problem.
Use When	When resource search routines provide little useful information.
How	Sub-Method I) 1. Find a similar‡ problem already solved that has a similar or same type of unknown (or for a theorem with a similar or same type of conclusion or implication). 2. Consider how you might construct an auxiliary element that makes your problem become the similar problem or satisfy the conditions of the problem you found. You do this by restarting the problem-solving process on this new sub-problem. 3. Maybe there are no available similar problems you can use to create auxiliary elements. Consider what variations of the problem that if you did solve, would allow you to solve the original problem by starting at the beginning of (Sub-Method I). Restart the problem-solving process on the similar simpler sub-problems that would allow you to use an auxiliary element to solve your original problem until you solve one of them. (Start with the most promising sub-problems). For example, consider what change in the data would allow you to both solve the new problem and allow you to introduce an auxiliary element to make the data fit the way you need the data to be in order to use the solution to the problem with the varied data and/or problem. You can also consider other variations not just restricted to changes in the data. You may also consider if changing the unknown a little bit could help you solve the problem in a way that you could create an auxiliary element that would take you from the new unknown to the original unknown. Consider searching for auxiliary elements to solve the problem using a combination of changing the unknown and the data. 4. Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100. Sub-Method II) 1. Consider what mathematical entity would allow you to control the available information in the problem. For example, using the extremal and invariance principle (methods 15a and 15b). 2. Solve the sub-problem of constructing such an entity. 3. Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100. Sub-Method III) 1. Consider what small change in the unknown could help you create a plan to solve the problem. (See vary the problem—methods 20 and 21). 2. If you find these small changes, then consider if an auxiliary element or process could take you from the new changed unknown to the original unknown. 3. Now switch to considering what alteration of the data/given information in the problem would help you to create a plan to solve the problem. (See vary the problem—methods 20 and 21). 4. Try to construct an auxiliary element to make the data/given information fit the way you need the data to be in order to use the solution to the problem with the varied data. 5. Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.
Notes	†An auxiliary element is a type of created or engineered resource. Auxiliary elements are mathematical entities such as specified types of sets, functions, variables or other such mathematical entities that are created for the purpose of assisting in steps of a problem-solving process. ‡Two entities, relationships, or etc. being "similar" means that many defining parts of the entities, relationships, etc. are equivalent or that the defining parts of the entities have equivalencies in aspects of the definitions that apply to defining the entities, relationships, or etc. Do not introduce auxiliary elements without a good reason, because otherwise you are likely wasting your time throwing nonsense ideas into the air. Everything you introduce should have a motive behind its introduction [7] pg 46-50.
Example	See example problem 10 of chapter 6 (Examples of Problem Solving)

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