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Define "loose". Edit: Jokes aside, nothing is really "strictly-defined" as we would like it to be since definitions (even the most rudimentary ones) are based o

Define "loose". Edit: Jokes aside, nothing is really "strictly-defined" as we would like it to be since definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations. We come to the conclusion that an apple is a round object because we've seen an apple from multiple angles. Math accepts axioms, but those axioms are based on accepting definitions that are rudimentary observations. What is "1"? The idea of "1" can only be understood in terms of experience - hardly "rigorously defined".

catnaroek 10y ago
> definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations Mathematical definitions are usually abstractions of concrete observations, but they aren't the observations themselves. This is why coming up with a mathematical definition often requires more work than coming up with any other kind of definition. > What is "1"? The idea of "1" can only be understood in terms of experience 1 is very rigorously defined: As a natural number, it's the successor of 0. As an integer, rational, real or complex number, it's the result of mapping the natural number 1 into Z, Q, R or C in a suitable way (compatible with the semiring structure of N).
- nercht12 10y ago
  > Mathematical definitions are usually abstractions of concrete observations, but they aren't the observations themselves. This is why coming up with a mathematical definition often requires more work than coming up with any other kind of definition. ... 1 is very rigorously defined: As a natural number, it's the successor of 0. As an integer, rational, real or complex number, it's the result of mapping the natural number 1 into Z, Q, R or C in a suitable way (compatible with the semiring structure of N). That's all nice and dandy for mathematicians, but such is still meaningless (from an overall perspective) because ultimately you end up with recursive definitions. "The successor" of "0" means nothing because "successor" isn't defined, unless you define it via dimensions (or something else). Once you try to define dimension or otherwise, you have to define dimension, which leads to defining counting, which leads back to defining 1. Appealing to experience is the only way to end the cycle. That said, it's perfectly fine to talk about a system within itself, but it's applicability to anything real (which is what I'm concerned with - my apologies if that's not where you're going) is limited to whatever can be described in terms of reality.
  - catnaroek 10y ago
    > "The successor" of "0" means nothing because "successor" isn't defined The existence of a successor function is postulated, because it's an axiom of the theory of natural numbers. It's up to individual models of the theory to define this function concretely.

catnaroek 10y ago

> definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations Mathematical definitions are usually abstractions of concrete observations, but they aren't the observations themselves. This is why coming up with a mathematical definition often requires more work than coming up with any other kind of definition. > What is "1"? The idea of "1" can only be understood in terms of experience 1 is very rigorously defined: As a natural number, it's the successor of 0. As an integer, rational, real or complex number, it's the result of mapping the natural number 1 into Z, Q, R or C in a suitable way (compatible with the semiring structure of N).

nercht12 10y ago
> Mathematical definitions are usually abstractions of concrete observations, but they aren't the observations themselves. This is why coming up with a mathematical definition often requires more work than coming up with any other kind of definition. ... 1 is very rigorously defined: As a natural number, it's the successor of 0. As an integer, rational, real or complex number, it's the result of mapping the natural number 1 into Z, Q, R or C in a suitable way (compatible with the semiring structure of N). That's all nice and dandy for mathematicians, but such is still meaningless (from an overall perspective) because ultimately you end up with recursive definitions. "The successor" of "0" means nothing because "successor" isn't defined, unless you define it via dimensions (or something else). Once you try to define dimension or otherwise, you have to define dimension, which leads to defining counting, which leads back to defining 1. Appealing to experience is the only way to end the cycle. That said, it's perfectly fine to talk about a system within itself, but it's applicability to anything real (which is what I'm concerned with - my apologies if that's not where you're going) is limited to whatever can be described in terms of reality.
- catnaroek 10y ago
  > "The successor" of "0" means nothing because "successor" isn't defined The existence of a successor function is postulated, because it's an axiom of the theory of natural numbers. It's up to individual models of the theory to define this function concretely.