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Constructive Definitions

2Pattern

2Chris_Leong

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The tense of the last sentence is a little strange.

However, instead of becoming enmeshed in disputes like, "Are complex numbers even a number?" we instead end up in more productive conversations like, "Is this extension practically useful enough to be worth the increase in complexity?"

There's also the option of going for the biggest system, that includes all the others. (The effort involved in proving two systems "are equivalent"* where they overlap sounds non-trivial.

*In the sense that, anything that can be proved with one, can be proved with the other, and vice versa.)

In other words, there is a sense in which they are an answer and not a question.

A post just on that concept, would still be a good post. This post (overall) was great.

For some concepts, the most natural way for them to be defined is for the definition to first apply to a paradigmatic or natural domain which is then extended for reasons of convenience. We will call these constructive definitions.

Defining NumbersConsider the example of numbers. The most natural or paradigmatic case are the counting numbers 1, 2, 3... In many cases, you don't even need a number for zero, you can just say "There were no cats" instead of using the word zero.

Fractions allow us to represent divisible objects like pies or shares of a hunt.

The number zero becomes important as soon as you have a decimal system, but beyond this adding it in allows the sum operation to handle columns with zero items without it being a special case.

Once we allow borrowing negative numbers become convenient, especially if an account can be positive or negative.

Eventually we might reach the point where we find it useful to define complex numbers or even quaternians so as to produce neater proofs in some cases.

ReflectionsNotice how the definition of numbers was constructed incrementally.

There wasn't a single definition all the way through. As we progressed, we wanted numbers to be able to serve different functions, so we produced different formalisations or tools.

In other words, there is a sense in which they are an answer and not a question. If someone asks, "What are numbers?" and they haven't specified which kind of numbers they are referring to, then the question is essentially meaningless until clarified. It is only once a particular purpose has been chosen that we can answer with the appropriate kind of numbers for the task at hand. I suppose they could be looking for the paradigmatic case of numbers, but then they could have asked directly about that.

These kinds of definitions have a few properties. Firstly, while it is common to have a unique paradigmatic definition, this isn't always the case or undisputed. Some people might have included zero in the paradigmatic definition of numbers. Secondly, certain extensions might be controversial with people either rejecting an extension or proposing an alternative. However, instead of becoming enmeshed in disputes like, "Are complex numbers even a number?" we instead end up in more productive conversations like, "Is this extension practically useful enough to be worth the increase in complexity?". Thirdly, the final definition might be clearly an aggregate structure or be a natural way of carving reality at the joints. However, it's important to not assume that just because we have a term for a concept that the concept is ontologically basic.

Finally, I'm not making any claim about the existence or non-existence of abstract number objects. Constructive definitions are compatible with either theory.

Examples:I interpret Votja as suggesting building up counterfactuals in an aggregate fashion. This would start from cases where the model has sufficient uncertainty to allow multiple actions, then extend into cases where the model merely seems to have such uncertainty or where such uncertainty can be introduced by removing information or considering a slightly different past history.

My theory of truth is similar (I discuss this here, but this is a very old post that is probably embarrassingly in need of a rewrite). The paradigmatic case here is propositions that refer directly to the world. We can then build logical operations like AND, OR, THERE EXISTS and FOR ALL on top of this. Then we can start adding in propositions which refer to propositions, as long as we are carful to dodge the issue of self-reference. In this perspective, sentences like, "This sentence is false" are confusing because we mistakenly carry over the expectation of each proposition being true or false from a narrower domain to a broader domain. If we want a definition for a broader domain, then we need to co-ordinate on a new convention.

A third example would be probability. The paradigmatic case for defining probability tends to assume or be compatible with an observer standing outside the problem. However, this is broken in problems like the Absent-Minded Driver or the Sleeping Beauty Problem. My view is pretty much in agreement with that described in: If a tree fall on Sleeping Beauty - we are going outside the paradigmatic case, so the dispute is actually about how to extend the definition of probability to this new case rather than about anything else.

SummaryWe've identified a common way that definitions are formed in contrast to alternatives like merely noticing a natural structure and labelling it. This turned out to be important because being aware of constructive definitions avoided issues like asking whether complex numbers were really numbers or being confused over why we couldn't assign "This sentence is false" a truth value.