| | I wrote that the proposition "2 + 2 = 4" refers to the fact that any two things plus any two other things equal four things. All abstract propositions that have factual referents refer to the real world. 2 + 2 = 4 refers to the real world, because it refers to the fact that two things plus two things equal four things.
Cal replied: No, no, no, mathematics does not refer to reality, it is purely abstract reasoning.
Cal, what do you think the abstraction "person" refers to? It doesn't refer to itself, right? So what is it's referent? The abstraction "person" refers to any and every person. The same is true of the abstraction "society," which refers to any and every society. So, the abstraction "two" would refer to what? It would refer to any and every instance of two--to two apples, two oranges, two persons, etc. There is no such thing as pure "two," that is not two something, any more than there is such a thing as a pure person, who is not someone in particular.
That doesn't mean that mathematical concepts may not have originally been derived from empirical observations, but those concepts are abstractions that no longer refer to their empirical origins.
They don't refer just to their empirical origins. If I observe that two people, John and Mary, have something in common as against two dogs, say, and from that observation form the abstraction "person," that abstraction doesn't refer just to John and Mary; it refers to every other person as well. But the referent of the abstraction "person" is still particular persons; it is not the abstraction "person." The referent of an abstraction is not the abstraction itself; it is the various instances of the abstraction.
Further, mathematics is of course an indispensable tool for physics, but one shouldn't confuse the abstract theory with its applications.
But a theory is a theory about something--about what actually happens under a given set of conditions. The theory pertains to concrete reality--to the theory's various applications.
What do you mean by "independently verified"? Independently of what?
Independently of that particular observer.
Okay, but then it's also possible to verify something non-independently right? In other words, the particular observer himself can verify it, just as well as someone else can.
One can also make errors in addition, but that doesn't mean that the laws of arithmetic are not objective, i.e., demonstrably true.
True, but not relevant. The laws of arithmetic are demonstrably true while they don't refer to reality. But someone who uses these laws in a correct way to describe something in the real world may still arrive at an incorrect conclusion while his perception of reality is subjective and may not be correct, and that is the point of this whole discussion: the logic may be correct and it is universal, but it doesn't guarantee that the conclusion is correct.
Okay, but you're not saying, are you, that this leads to epistemological skepticism? You would still grant that we can have certain knowledge of empirical facts, just as well as we can have certain knowledge of abstract relationships, right?
Subjective knowledge may in many cases be good enough - we don't need constant proof that everything in our direct environment is really there and is what it seems to be. But in some cases the distinction becomes important, for example in witness testimony, especially when the stakes are high, and in science, where in general independent verification of the outcome of an experiment is considered to be necessary before the results may be accepted, especially when those results are unexpected. Do we readily accept the conclusion that some experiment shows the existence of a paranormal phenomenon? Extraordinary claims require extraordinary evidence, and the evidence of one particular researcher (or even a small group of researchers) is not deemed enough, independently obtained evidence is an absolute necessity in this case to determine whether the subjective knowledge is also objective knowledge.
Knowledge is knowledge, regardless of how many or how few people hold it. Fifty million Frenchmen can be wrong, and one genius can be right as he stands against a multitude of true believers, who "independently" condemn him as a heretic. Yes, extraordinary claims require extraordinary evidence, but that evidence can be available to a single individual; it doesn't need to be duplicated or verified "independently" by others in order to qualify as knowledge. The distinction that you're making between "subjective" and "objective" knowledge is bogus. Knowledge is the correct identification of the facts of reality, period. It does not require independent verification by others.
Now, obviously, if someone says that he ran an experiment and proved X, then unless I, as a fellow scientist, can duplicate the results, I'm not going to believe him. Nor should a layman who has no specialized knowledge of the subject, if other reputable scientists cannot duplicate it. But that's a different issue; it's an issue, not of what can count as knowledge, but of what can reasonably be believed by those with no specialized knowledge or understanding of the issue.
- Bill
|
|
