Dear six,
You quote my submission:
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A problem for which there exist an algorithm to solve it (sic) is said to be solvable"
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The entire point of the sceptical argument is that ultimately we reach a level where we act without any reason in terms of which we can justify our action. We act unhesitatingly but blindly
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Then you ask:
:Please reconcile those two (1 statement + 1 more statement ;-) statements.
I hate to be thick, but what is the point?:
The point is that math is at the very least a form of symbolic manipulation. But we would be remiss to restrict it to the manipulation of arbitrary symbols. As we know, there are decidable or sovable propositions in mathematics, and we employ algorithms to solve such propositions. But our solutions are oftentimes so "loaded" and performed with so many preunderstandings that we begin to think math symbols inherently mean "plus" or "subtract." I am arguing that this supposition is basically erroneous.
Astrophysicist Paul Davies, in _The Mind of God_, admittedly shows that mathematics is more than symbol manipulation. Kurt Godel demonstrated this proposition in 1931. But Godel also showed that there are "undecidable propositions" in math. Davies thus writes: "In that year [1931] the Austrian mathematician and logician Kurt Godel proved a sweeping theorem to the effect that mathematical statements existed for which no systematic procedure could determine whether they are either true or false."
Godel essentially introduced the problem of self-referentiality and demonstrated the incomplete nature of mathematics.
The overall point I am trying to make is that thinking 2 + 2 = 4 is a BELIEF. We cannot indubitably prove that 2 + 2 actually equals 4.
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But we need not think these signs essentially "mean" anything.
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six: Well sure we do, don't we? It's important for communication. It's important for solving problems. Especially since it works.
Duns: When I say that + does not "essentially" mean anything, I am contending that + is an arbitrary sign. It works for the same reason that the term "cat" works. A certain speech community agrees on the use and "meaning" of a particular sign. But the community could just as well have chosen another symbol to represent addition or the creature we call a "cat".
I hope this post cleared things up for you.
Sincerely,
Dan
Duns the Scot