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I recently read in a book (a science textbook if that's relevant) that the difference between such-and-such and so-and-so is not very clear (in a sense, as if the definitions were overlapping).

But this is utter nonsense, is it not? What is a good definition of definition? I think that logically a definition must be able to unambiguously determine which group to place a given undetermined object in.

If definitions overlap, their very purpose is defeated. They are no more definitions and must be revised.

So why do science textbooks continue to maintain that definitions can overlap? Why aren't these definitions revised?

  • It is a repeated trend that it seems to me that there are is some people, prominently textbook writers, who think that the overlap of two "defintions" can overlap in a logically coherent system. – BlowMaMind Apr 10 '17 at 18:39
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It sounds to me like you have too strict of a definition of definitions =)

I think that logically a definition must be able to unambiguously determine which group to place a given undetermined object in.

Consider this example:

Natural numbers are defined to be 0, 1, 2, 3, 4,... (formally, using the Peano axioms, but this informal definition is sufficient) Even numbers are defined to be 2, 4, 6, 8, 10....

So if I take the number 4, what should it be defined to be? Is it a natural number or an even number?

Surely the mere fact that there is overlap does not oblige us to create a new definition?

But you might say "ah, but even numbers are a subset of natural numbers, so it's okay for them to overlap." To which, I say fine. Take a look at OEIS, the Online Encyclopedia of Integer Sequences. They define sequences such as

  • A000001 - Number of groups of order n.
  • A000003 - Number of classes of primitive positive definite binary quadratic forms of discriminant D = -4n; or equivalently the class number of the quadratic order of discriminant D = -4n.

And so forth. Now if I defined a number as being "of A#######" as a phrase which means that number appears in that particular sequence, I could run into a bit of a problem. There's an awful lot of overlap. For example, the number 1 appears in both sequences.

To meet your "logical" expectation, I would be required to discuss 2^N different definitions, where N is the number of sequences in OEIS, which is over 280,000. I'm going to need an awful lot of definitions!

Definitions obviously don't work that way. The purpose of definitions is to anchor the symbols you are using in a conversation (or proof, or debate, etc.) so that others may understand how and why you are using them. If it is inconvenient for such symbols to overlap, it's a good idea for the definitions to have no overlap. However, if overlapping symbols is reasonable (and it almost always is), then it's okay for the definitions to have overlap.

In other works, definitions uniquely map a symbol within a discourse to reality, but there is no guarantee that the mapping is bijective.

A key benefit about this is being able to discuss concepts within a framework. Consider that sometimes it's effective to think of sound in waves. Other times it's effective to think in phonons, packets of sound. The definitions absolutely overlap -- they're talking about the exact same thing! However, those overlapping definitions are a very powerful tool which lets you apply multiple frameworks to the same problem.

It's also why we can talk about how light acts like a wave, or acts like a particle, without always being obliged to dip into the quantum physics world and admit that it really is neither, and then not always being obliged to point out that what the scientific community calls light may not actually exist, but the empirical observations of our universe suggest it's a good model for the observations we've seen.

My kid just wants to make sure that when they switch the flashlight switch on, the monsters go away. I don't need a solid unambiguous definition of light for that!

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