12:12 AM
It seems that we are viewing "context-dependent" in two fundamentally different ways, so let me give more explicit examples to illustrate my intended denotation.
Proposition: There is no [real] number x such that x^2=-1.
Proof: If x>0, then x^2=x\cdot x>x\cdot 0=0. [The first equality is by definition, the inequality is axiomatic, and the second equality is easily proved from axioms.]
If x=0, then x^2=0^2=0\cdot 0=0=1+-1>0+-1=-1. [The first equality is by substitution, the second is by definition, the third is from the easily-proved result mentioned before, the fourth is by definition/axiom, the inequality is by axiom, and the last equality is by axiom.]
Proof: If x>0, then x^2=x\cdot x>x\cdot 0=0. [The first equality is by definition, the inequality is axiomatic, and the second equality is easily proved from axioms.]
If x=0, then x^2=0^2=0\cdot 0=0=1+-1>0+-1=-1. [The first equality is by substitution, the second is by definition, the third is from the easily-proved result mentioned before, the fourth is by definition/axiom, the inequality is by axiom, and the last equality is by axiom.]
Proposition: There is a [complex] number x such that x^2=-1.
Proof: By definition, i is a number such that i^2=-1, so x=i satisfies the conditions we set out to demonstrate. //
Proof: By definition, i is a number such that i^2=-1, so x=i satisfies the conditions we set out to demonstrate. //
Hopefully, that helped to illustrate what I mean by "context-dependence."
Next, let me use your earlier (now deleted) example of a "context-dependent" result: P\vee Q\Longrightarrow P.
Next, let me use your earlier (now deleted) example of a "context-dependent" result: P\vee Q\Longrightarrow P.
Here, "context-dependent" is according with your definition, or at least it was when you posted it.
I can no longer recall your exact wording, but the tone was that this claim was preposterous. And this is rightly so...in context.
The erroneous (though understandable) inference that you were making is that, by "context-dependent reason," I was suggesting that one might show particular examples, and from those examples infer a general principle. However, this is not what I meant to suggest.
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I can no longer recall your exact wording, but the tone was that this claim was preposterous. And this is rightly so...in context.
The erroneous (though understandable) inference that you were making is that, by "context-dependent reason," I was suggesting that one might show particular examples, and from those examples infer a general principle. However, this is not what I meant to suggest.
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12:30 AM
Given two statements [that is, two sentences that must be true or false, and cannot be both true and false] P and Q, we define the symbolic notation P\vee Q to be another statement, true precisely when both P and Q are true. Given two statements R and Q, we define the symbolic notation R\Longrightarrow P to be another statement, true unless R is true and P is false.
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Proposition: For all statements P,Q, we have that (P\vee Q)\Longrightarrow P.
Proof: Take P to be a true statement. Then (P\vee Q)\Longrightarrow P is true. [By definition of \Longrightarrow.]
Take P to be a false statement. Then P\vee Q is false. [By definition of \vee.] Hence, (P\vee Q)\Longrightarrow P is true. [By definition of \Longrightarrow.]
Thus, we're done with the proof. [By definition of "statement."] //
Proof: Take P to be a true statement. Then (P\vee Q)\Longrightarrow P is true. [By definition of \Longrightarrow.]
Take P to be a false statement. Then P\vee Q is false. [By definition of \vee.] Hence, (P\vee Q)\Longrightarrow P is true. [By definition of \Longrightarrow.]
Thus, we're done with the proof. [By definition of "statement."] //
2 hours later…
2:44 AM
The concept "context-dependent" is not well defined. There are more precise ways to say things. In one of the comments you wrote "in the context of real-valued functions on the reals, it wouldn't make sense to say that $i$ is a solution to the equation $x^2=-1$. Saying that "$i$ isn't a solution of $x^2=-1$" is wrong, while saying that "$i^2 = -1$ is meaningless in the context of the reals" is also a strange statement. The more precise way to say it is that $i \not\in \mathbb{R}$.
If we work in a context where we assume a rule that is not considered valid, then we're essentially working…
If we work in a context where we assume a rule that is not considered valid, then we're essentially working…
2 hours later…
4:33 AM
I must wonder what the following means, though: "If we work in a context where we assume a rule that is not considered valid, then we're essentially working in a context that isn't math."
I readily submit that "context-dependent" isn't well-defined--I never claimed it to be a rigorous mathematical concept, nor will I--but by the same token, one may as easily criticise your use of the word "context." One may even more easily criticise your use of the words "rule," "valid," and "math." However, this is irrelevant to my point, which you still seem to be missing.
7 hours later…
12:21 PM
"I must wonder what the following means": What it means is that these context-dependent arguments are not rigorous well-defined math. For every accepted theorem T in math you could cook up some context where it'd be wrong. That doesn't mean there's anything wrong with T, all it means is that there's something wrong with the context. Express things in terms of better defined things like sets, and the issues disappear.
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Transcript for
Feb '1723
Feb24
re: context dependence
discussing context dependence as it relates to mathematical pr...
