We have ideas about what we want, but we don’t really know what’s possible at the beginning.
Dilemma
A way to define our scope
Problem-solving starts vaguely
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To solve a problem mathematically, we need to make assumptions until we have something specific to go solve.
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Each assumption is an invitation to question “what if this wasn’t so”.
Clarify the scope
Beginning with a question helps you clarify the outcome you want, and how you might get there. It also gives you a clear understanding of what is in scope and what is not, an important consideration for all problem solving.
The Dilemma is just one part of the science of the Meta-Problem Method. To see illustrations of this idea in action, turn to the Examples section.
Read on for a more technical explanation.
Incompletely defined problems
Using the Meta-Problem Method starts by defining an incompletely defined problem.
An incompletely defined problem is one where we have defined some of the criteria, but we may not know how to trade off one variable versus another.
To solve an incompletely defined problem, we can make assumptions about anything that we do not know. For example, we could assume some tradeoff between criteria and see what the result is.
Completely defined problems
While incompletely defined problems are best explained by showing what they are not, a completely defined problem is one where there is a right answer.
You have some flexibility in how you solve a completely defined problem, but the value of the answer will be the same no matter how you get there. Incompletely defined problems in a classroom setting often become completely defined through the power of the teacher telling students what assumptions to make.
For example, if the teacher just taught a class how to do division with a “remainder,” the students assume they should not use fractions to answer the question. If I ask what 3 divided by 2 is and 1 with a remainder of 1 is correct but 3/2 is not, then this is a completely defined problem.
Assumptions
Assumptions fill the gap between an incompletely defined problem and a specific completely defined problem. They determine some unknown in an incompletely defined problem.
With enough assumptions, we can get a version of a particular incompletely defined problem which is completely defined. Those assumptions could include simplifications of the problem, models of uncertainty, predictions of the future, estimates of preferences, and so on.
The term assumption as we use it here comes from engineering, where we acknowledge whether something is known for certain or not. Models require a lot of elements to be known, many of which we do not in fact know. By explicitly stating those unknowns engineers can discuss whether the assumption is appropriate or not. If not, we often need to consider different methods.
Check out the other pages on the science of the method
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