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Rev

Here is a problem. Your definitions are not REQUIRED to reflect reality. I don't know what strong axioms are. All scientific axioms can be falsified so this does not guarantee solid proofs in mathematics and in reality simultaneously.
– Logikal
Commented Jul 15, 2019 at 16:51
Euclid didn’t have strong definitions of “point,” “line” or ”number.” His axioms only defined what operations you are allowed to perform on them (although a few of his proofs end up handwaving). Later mathematicians were able to define non-Euclidian geometries where a “point” or “line” mean something different but (most of) the same axioms hold. A lot of mathematics looks for similar structure of very different things, and ends up defining relations between abstract objects that don’t presuppose what those objects are. For example, theorems about vectors can apply to function spaces.
– Davislor
Commented Jul 15, 2019 at 18:08
2

@Davislor I disagree. Euclid's definitions of point and line (not sure where he defined "number") were strong because the objects themselves were abstracted away to the point where a list of operations completely determined the relevant characteristics of those objects. That, if I recall, was the whole point of considering abstractions in the first place. Maybe you have a different definition of "strong" than I do, though (in that case, you shouldn't make such unequivocal claims like "Euclid didn't have strong defintions" without first making clear what you think a strong definition is).
– probably_someone
Commented Jul 15, 2019 at 20:21
1

@Davislor I would still argue that "imprecise" and "abstract" are completely different. In my experience, a definition is "precise" if its application completely specifies all relevant properties of an object, whereas a definition is "abstract" if it does not specify all properties of an object. When not all of the properties of an object are relevant (i.e. almost all of the time), you can have definitions that are simultaneously precise and abstract.
– probably_someone
Commented Jul 15, 2019 at 20:29
1

Let us continue this discussion in chat.
– probably_someone
Commented Jul 16, 2019 at 8:20

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