Definition Smells

Signs that a definition probably needs work.


I have a feeling that Top is making crap up again. He makes several claims above he only wishes had some validity. What... he does this about once a month? creates a new page of bullshit to direct people to when he wishes to defend his fallacy? Last time, IIRC, it was ObjectivityIsAnIllusion, and before that was TautologyMachine, I think. I should start keeping a list... or, on second thought, perhaps not. Tracking top's crap isn't exactly a good use of my time.

(Moved reply to ObjectivityIsAnIllusionContinued?)


I'm adding this back in, because I'm going to disagree with both. It's not unusual to define something in order to show that it doesn't exist (E.g. "Largest Natural Number", "Ideal Voting System"). So even definitions that are "tautological" (as top would put it) are perfectly acceptable.

In the end, a definition only has to do two things, Usually, though we also want it to By this, we can see that poorly defined words are usually a problem (but, not always. Set theory never defines what a set is. Euclidean geometry never defines what points, lines, and planes are. Etc.). Length would be a problem only if an equivalent shorter one was available. Requiring a specific language, math; not being falsifiable; or depending on abstraction/intent are definitely not problems.

I think everyone agrees that "points" are a UsefulLie. Poor definitions tend to be useless lies or a UselessTruth. A useful truth would be the ideal, but is probably a rare beast.

{Every set theory individually defines what a 'set' is, usually constructively through a finite collection of axioms. And Euclidean geometry defines points, lines, and planes in a similar manner - they gain existence and meaning based upon the axioms utilized to describe them. And Top is wrong: like all definitions, points aren't propositions and therefore cannot be "lies". That leaves them at just being "Useful". Top, I'll say it again: definitions are not propositions. Definitions, no matter how good or how poor, are neither lies NOR truths.}

Every set theory does not define what a 'set' is. Let's say you have two models for your set theory (and I don't know of any interesting set theories that don't have at least two), the axioms can't be used tell which one of the two are the "real" sets. If they are both sets, then you should be able to take a set from one model, a set from the other, and take their union. You can't. Likewise with 'points', 'lines', and 'planes' in Euclidian Geometry.

{Individual set theories are mathematical models already. There is little need to model the set theories. A single set theory IS a set of axioms, and the 'primitive' in a set theory tends to be 'element'; 'set' is defined 'implicitly and constructively by virtue of axioms over collections elements. If you discuss taking the union of sets from two different set theories, that can often be done (if there is a homomorphism from one set theory to another), but it doesn't change that each set-theory implicitly defines 'set'. And 'points', 'lines', and 'planes' in Euclidean Geometry are defined by reference-identity and the axioms that relate them.}

A model is something that allows you to map all the statements in a language to the truth values. Individual set theories are not models, and there is just as much need to model set theories as there is to model any other part of mathematics. To hopefully make it clearer, I'm not talking about finding the equivalent set in the first model of the set from the second, taking its union with the set from the first. I'm talking about directly taking the union of the set from the first model and the set from the second. The result of such a union is (most likely) not a member of either model.

{Can you reference a set-theory that is not a model? All those set theories I've a recollection of reading ARE mathematical models by virtue of listing their axioms and formation rules, which map the set-descriptions to legal/illegal (and thus, implicitly, define 'set'). These axioms generally provide a specific rule for unions in terms of the underlying elements.}

None of them are models. A theory (in this sense) is a subset of all the statements in a language. A model is a mapping from all the statements in a language to the truth values that satisfies a theory (maps to true for every statement in the theory and preserves the meanings of the logical connectors). Less formally, a theory is what is provable, while a model is what is true. Any "interesting" set theory will have models that disagree about the truth of some statements. For example, there are models of ZFC where the continuum hypothesis is true, and models where it isn't.

While there is some truth to the idea that 'set' is implicitly defined by set theory, I find it unsatisfying. It's kind of like defining 'red' as 'a color'. It's incomplete. We can get away with this incompleteness because nothing in set theory actually depends on what a 'set' is.

{On review, you are correct about the models... though, trivially, every theory comes with a free model called 'is-a-legal-sentence'.}

{The truth that 'set' is implicitly defined by a set theory is of import. In discussions I've often found people assuming that 'set theory' depends upon the definition of 'set' simply because it has 'set' in its name. Same with 'type theory' and 'type system' with 'type'. This assumption is often so deeply ingrained that they have difficulty grasping any other possibility. Recognizing that it is often the other way around, that a 'set theory' is essentially a theory that implicitly defines a 'set' and that a 'type system' implicitly defines 'type', and that there is no dependency requiring that 'set' or 'type' be defined first, can be a rather profound revelation. Thus, I emphasize it a bit here.}

{And as far as defining 'red' as 'a color' - that's not too far from complete. Physiologically, red is just 'a color'. At least in the sense of human vision, you could 'define base_color = red | green | blue', then define images as being patterned collections of colors based upon spatial triggering of the rods and cones in our eyes. All other colors are mixtures of the three.}


See also: LaynesLaw ItDepends
Definition is a valuation

"To Describe is perhaps to value" Huh?

{Basically, we don't bother to define things if we don't value them in some manner.}


AprilEleven

CategoryDefinition

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