Having defined science, I should revisit first a prerequisite to applying this definition of "science" to the construction of problems and their solutions. It is necessary to understand what is implied by ex-post "certainty", which may be more accurately described as

Determinism simply describes the concept that a process (X) will lead to some outcome (Y). X may be concrete, such as in the case of an object striking another. X may also be abstract, such as the calculation of an arithmetic expression. In either of these cases, the result will be Y chosen from a range of all possible outcomes (the possibility set). Note that, insofar as it is a concept, Y may be "nothing" and still would constitute a valid outcome for X, as would the complete end of existence for all objects and concepts related to X. In short: Define a process X to be an event in time, space, or mental construct which

The possibility set for Y may be finite, as in the case of a deck of cards, where Y is a single card chosen from the deck. Another example is the result of a fair die thrown at random. However, the possibility set for Y may be infinite as well. If, instead of throwing a six-sided die, we define the possibility set for Y to be any real number on the range (0.5, 6.5), we would have an infinite possibility set, the uniform distribution of which would result in the same arithmetic mean. It is not difficult to see that the vast majority of problems involve infinite possibility sets for y. In the simple case of a billiard ball striking another (X), for instance, we can think of countless possible (if improbable) outcomes (Y): perhaps one or the other ball will explode, or perhaps they are made of liquid and will merge together, or perhaps the collision will expel their adherence to gravity and they will begin to float away. We could conceivably continue in this manner infinitely - there is no finite list of possible outcomes. (Note that, conceivably, one could argue that my examples of finite possibility sets could in fact be infinite possibility sets: perhaps some unforeseen phenomena will cause an outcome that appears highly implausible but is nonetheless conceivable; I would not argue this point, as it has no bearing on further topics.)

I have mentioned above that only one outcome Y is possible, such that X to Y is a one-to-one mapping. If you doubt this to be the case, consider that sets of outcomes {Y1, Y2} can be contained in the possibility set. Then Y1, Y2, and the set {Y1, Y2} would all be in the possibility set for Y and would thus be valid outcomes. So long as Y1 and Y2 are not mutually exclusive, their joint occurrence does not preclude us from stating that "only one outcome has occurred."

Determinism, then, is the idea that the single outcome Y

Taken in this way, if we are presented with a process X, we may use our knowledge of X to form expectations of some future state of related objects and concepts Y. Our confidence in forming these expectations depends on many things: our prior knowledge of the objects and concepts related to X, our experience with processes similar to X (including the observability of the results of those processes), the complexity of X, the consistency of the information related to X, the breadth of the possibility set for Y, and still other factors.

It will be important to firm up my definition of "knowledge" at this point to complete the groundwork for further examination of the nature of problem solving. I have used it here to mean "believing with certainty". If the thing to be known cannot be proven, as many things cannot be, then we will need to take care to ensure that true knowledge of the thing is not implied before continuing on.

*determinism*.Determinism simply describes the concept that a process (X) will lead to some outcome (Y). X may be concrete, such as in the case of an object striking another. X may also be abstract, such as the calculation of an arithmetic expression. In either of these cases, the result will be Y chosen from a range of all possible outcomes (the possibility set). Note that, insofar as it is a concept, Y may be "nothing" and still would constitute a valid outcome for X, as would the complete end of existence for all objects and concepts related to X. In short: Define a process X to be an event in time, space, or mental construct which

__must__result in an outcome Y. We can assume that only one outcome Y can possibly occur, creating a one-to-one mapping from X to Y (more on this below).The possibility set for Y may be finite, as in the case of a deck of cards, where Y is a single card chosen from the deck. Another example is the result of a fair die thrown at random. However, the possibility set for Y may be infinite as well. If, instead of throwing a six-sided die, we define the possibility set for Y to be any real number on the range (0.5, 6.5), we would have an infinite possibility set, the uniform distribution of which would result in the same arithmetic mean. It is not difficult to see that the vast majority of problems involve infinite possibility sets for y. In the simple case of a billiard ball striking another (X), for instance, we can think of countless possible (if improbable) outcomes (Y): perhaps one or the other ball will explode, or perhaps they are made of liquid and will merge together, or perhaps the collision will expel their adherence to gravity and they will begin to float away. We could conceivably continue in this manner infinitely - there is no finite list of possible outcomes. (Note that, conceivably, one could argue that my examples of finite possibility sets could in fact be infinite possibility sets: perhaps some unforeseen phenomena will cause an outcome that appears highly implausible but is nonetheless conceivable; I would not argue this point, as it has no bearing on further topics.)

I have mentioned above that only one outcome Y is possible, such that X to Y is a one-to-one mapping. If you doubt this to be the case, consider that sets of outcomes {Y1, Y2} can be contained in the possibility set. Then Y1, Y2, and the set {Y1, Y2} would all be in the possibility set for Y and would thus be valid outcomes. So long as Y1 and Y2 are not mutually exclusive, their joint occurrence does not preclude us from stating that "only one outcome has occurred."

Determinism, then, is the idea that the single outcome Y

*results from*X and comes from the possibility set for Y. This is subject to some interpretation because of the notion of causality. How can we know that Y*results from*X? For determinism to be useful as a concept to build upon, we should not require proof of causality between X and Y. Causality can be loosely inferred from the knowledge that X occurred and that the subsequent state of the objects and concepts related to X was Y. We cannot know with exactitude that Y resulted from X (in much the same way that we cannot know with exactitude X and Y themselves!), but we may take it as a hypothesis that this is so based on knowledge about X and Y and their ordering.Taken in this way, if we are presented with a process X, we may use our knowledge of X to form expectations of some future state of related objects and concepts Y. Our confidence in forming these expectations depends on many things: our prior knowledge of the objects and concepts related to X, our experience with processes similar to X (including the observability of the results of those processes), the complexity of X, the consistency of the information related to X, the breadth of the possibility set for Y, and still other factors.

It will be important to firm up my definition of "knowledge" at this point to complete the groundwork for further examination of the nature of problem solving. I have used it here to mean "believing with certainty". If the thing to be known cannot be proven, as many things cannot be, then we will need to take care to ensure that true knowledge of the thing is not implied before continuing on.

*Items in grey text above are topics of further consideration which will be linked back to this article (and this article updated to reflect the discussion of those topics as they are available here).*
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