I agree with Spiritus Mundi, there’s way too much to catch up on…fortunately I also agree with everything Arnold Winkelried, panamajack, and Cabbage said, so that simplifies things.
Whoa there. I’m a mathematician, and I think that mathematics is more than a language. See Cabbage’s post. (Note that he’s a mathematician too.)
Here’s an interesting point where we disagree. The formal systems of mathematics do not, at their heart, “define” numbers. The first of Peano’s axioms is “There exists a natural number 0”. That’s not defining “0”, in my mind: that’s defining a symbol to refer to a pre-existing thing.
Cardinality is more than a “perceived property”. Take two protons and two anti-protons, toss them in a magnetic bottle, and wait for them to annihilate each other. There won’t be any protons or anti-protons left over, because the two sets have the same cardinality. If the two sets had different cardinalities, then some protons or some anti-protons would not get annihilated. So how can cardinality be just a “perceived property” of the two sets of particles? Certainly whether or not a particle remains after the experiment is a lot more than a “perceived property”!
Excuse me? If a language didn’t describe pre-existing phenomena, how the heck would anyone ever learn it? How would anyone ever share a language with anyone else?
How many fingers do you have? Seriously, you seem to be suggesting that math is entirely invented because it’s something we think about rather than something that just happens, like chemical reactions with our taste buds or apples hitting the ground. The problem is, that’s not entirely true. Math happens all the time in the physical world without any concious thought occuring. In the example of protons and anti-protons above, subtraction is occuring. Whenever an enzyme docks with the appropriate molecule, a geometrical operation is performed, namely a comparison of two shapes.
These mathematical operations may not get interpreted until an observer comes along and thinks in the language of math – the operations may or may not be given meaning – but they’re still occuring. And the person describing the phenomenon is not inventing the phenomenon being described.
Your point #5 is exactly what I’m trying to say. I just claim that some of those properties are also part of “mathematics”…that math is more than the language. Again, see Cabbage’s post. And as for #8, just substitute “math” for “physics” and you have my position again. (Nen, this applies to your latest post as well.)
Cabbage, FriendRob: I forgot that there were two “Godel’s Theorems”, thanks for clearing that up. It’s interesting that both sides in this debate are using them.
Okay, back to work for me…