1.  Also, Originally Posted by Phion Dimensions themselves are countable, are they not? not necessarily. One could create a space with dimensions indexed by the reals. For example, take the real line. Let's represent functions as vectors with one coordinate for each point on the real line. The value of that coordinate is the value of the function on the real number indexing that coordinate. So the coordinate x of the vector f is just f(x). This is a vector space with coordinate-wise operations and we have that f(x) + g(x) = (f + g)(x) and cf(x) = c(f(x)) so the vector operations are capturing the notion of adding and scaling functions according to the usual definitions. This space has a non-countable amount of dimensions. One doesn't usually need this much information. For special classes of functions one can make due with a countable basis. The basis for wave functions in QM is countable for instance.

2.  Originally Posted by PhilosopherStoned So what is this "two" that you claim to have shown me made out of? How would an electron behave in a collision with a two? You tell me. You just admitted that you believe it exists. Do you think it's an object? I don't. I think it's an amount.

3.  Originally Posted by PhilosopherStoned not necessarily. It's like saying the sky tends to be blue. You're defining vector operations in terms of single variable function notation, a non sequitur. Stating the fact that vectors are scalable does not equate to a two-dimension plane being infinitely dimensional, that's nonsense. If you want to define a vector in more than two dimensions, you need to state that implicitly, not substitute them with arbitrary function principles that don't apply to vectors. What I think you mean to say is that all terminal vectors are similar if their cosine or dot product are the same.

4.  Originally Posted by Universal Mind You tell me. You just admitted that you believe it exists. Do you think it's an object? I don't. I think it's an amount. Ok. So you have to go beyond the rational materialist model to claim that it exists, right? Or what material substance is an amount made out of? Originally Posted by Phion It's like saying the sky tends to be blue. You're defining vector operations in terms of single variable function notation, a non sequitur. This is similar to UM's rejection of the complex numbers. Who says it's a non-sequiter? A mathematician can define anything a mathematician could want. In particular, it's possible (and perhaps usual in set-theoretic treatments) to regard an n-dimensional vector space, V, on a field k, as the set of k-valued functions on the set S = {0, 1, ..., n-1}, if one wants to index starting from zero, or S = {1, 2, ... n} if one wants to index starting from one. This should be a trivial exercise, check it out. It essentially amounts to verifying that these functions constitute an abelian group under function addition and that multiplication by scalars distributes over addition of functions. Stating the fact that vectors are scalable does not equate to a two-dimension plane being infinitely dimensional, that's nonsense. If you want to define a vector in more than two dimensions, you need to state that implicitly, not substitute them with arbitrary function principles that don't apply to vectors. I never said that two is the same as infinity, that would be nonsense. You mentioned that dimensions had to be countable and that's not the case in priniciple. It just makes things easier. The vector stuff applies perfectly to functions as functions are perfect examples of vector spaces and vector spaces are interesting examples of functions. In particular, the best way to remember the vector space axioms is to remember that a vector space is just an embedding of a field into the endomorphism ring of an abelian group. That captures the whole tedious list. You just need to know what an embedding, endomorphism, field, ring, and abelian group are but you should know those things anyways. What I think you mean to say is that all terminal vectors are similar if their cosine or dot product are the same. Sounds old fashioned. What's a terminal vector? It doesn't make sense though because the dot product takes two vectors. But you've only referenced two vectors so what does it mean for them to have the same dot product? This seems like saying that 3 and 4 have the same product, 12. You can use a function of two arguments to define relationships on sets of two objects, not on individual objects. You want at least four vectors if you want to talk about doing this.

5.  Maybe Phion is visualising the space of single variable functions to be the plane because you can draw them on a plane (which is incorrect, of course).

7.  With respect I find it very hard to see how anything you're saying bears upon what Philosopher said. You made the statement 'dimensions are countable'... there was no analogising, Philosopher simply replied with a counterexample. And the counterexample is correct if dimension has its mathematical sense; the set of single-valued real functions with the operations specified do form a vector space, and that vector space does have uncountable dimension.

8.  Originally Posted by Xei With respect I find it very hard to see how anything you're saying bears upon what Philosopher said. He has got a nack for over-generalization to be sure. Originally Posted by Xei {snip}...there was no analogising,...{snip} Originally Posted by Wayfaerer electrical impulses? Originally Posted by PhilosopherStoned {snip}...How would an electron behave in a collision with a two?...{snip} That kind of thing. Originally Posted by Xei {snip}... And the counterexample is correct if dimension has its mathematical sense;... {snip} It looks to me like he's trying to clarify the affect dimensions have on algebraic structures. Which is cool and all, but I fail to see the relevance of dimension theorem for vector spaces in the discussion, aside from cardinality. Originally Posted by Xei {snip}...the set of single-valued real functions with the operations specified do form a vector space, and that vector space does have uncountable dimension. Maybe it's simply my taste for formalism, but that doesn't sit well with me.

9.  Originally Posted by PhilosopherStoned Ok. So you have to go beyond the rational materialist model to claim that it exists, right? Or what material substance is an amount made out of? No. It is a material property, not a material object. It is an amount... of material stuff. Amounts are not objects. They are of objects. What do you think 2 is?

10.  Well, I think at the very least he thought it relevant to where the discussion had reached, as he knew something which directly contradicted a statement in that discussion. My original post wasn't so much about the paradoxes themselves as it was intended to inspire these kind of talks about the epistemology of infinity and the like. With respect to formalism, I'd say that's the easiest philosophy of all to reconcile an uncountable dimensional vector space with. In fact I don't see any problems at all, it is all very well grounded and could be reduced without too much fuss to a set-theoretic foundation.

11.  With this basis, we could talk about damn near anything; lattices, tilings, etc. We might need a change of basis here...

12.  I'm pretty sure the failure of Rational Materialism to encompass 2 or indeed any mathamathical entity reflects an issue with Rational Materialism, not Maths. This belief of mine is solidified by the fact that Rational Materialism would be impossible to conceive of without Maths and the universal possible application of maths in the physical world. Of course I can't prove that it isn't just some enormous coincidence that maths sincs up so well with the world but I dare say the burden of proof is on the other party.

13.  Originally Posted by Phion He has got a nack for over-generalization to be sure. That kind of thing. None of the material that was quoted has anything to do with our conversation. In particular the question of if two is "real" is entirely orthogonal to the question of how many dimensions a vector space can have. A vector space can have any cardinal number of dimensions because one can always take a set of that cardnality and consider the set of all functions from that set into an arbitrary field to be a vector space over that field. You said something to the contrary and seem to like math so I thought you'd appreciate a cool trick that provides a counter-example for any statement along the lines of "A vector space can only be X big." Maybe it's simply my taste for formalism, but that doesn't sit well with me. It's precisely formalism that makes these kinds of things apparent. It's amazing how often we find ourselves talking about some number of apparently distinct things and really we're talking about the same thing. Functions are sets and sets are functions and vector spaces happen to be special cases of this. I just relied on that to correct an incorrect statement. I'm still waiting on somebody to say something intelligent in defense of the "reality" of numbers. PS: Thanks for the "over generalization" compliment. While I wouldn't have chosen that precise terminology, I do consider it the mark of a genius amd am deeply moved that you notice.

14.  Also, the fact that two doesn't exist by any meaningful sense of the word and is yet so objectively useful is pretty paradoxical if one stops to think about it for a minute.

15.  Phil, you said there are two heads in a set of two people. You tell me this time... What is "two?" Explain its meaning.

16.  I'd be glad to give you a full explaination of what I have to say when I'm done my paper on it, but it would be something along the lines of our conception and decision of sameness.

17.  Originally Posted by Wayfaerer I'd be glad to give you a full explaination of what I have to say when I'm done my paper on it, but it would be something along the lines of our conception and decision of sameness. So it depends on human conceptualization? If humans went extinct, there would no longer be two planets closer to the Sun than Earth?

18.  Originally Posted by Universal Mind So it depends on human conceptualization? If humans went extinct, there would no longer be two planets closer to the Sun than Earth? An excerpt taken from the Tao Teh Ching, as interpreted by Hua-Ching Ni, Chapter One (also one of my favorite translations): Tao, the subtle reality of the universe cannot be described. That which can be described in words is merely a conception of the mind. Although names and descriptions have been applied to it, the subtle reality is beyond the description.

19.  Originally Posted by Universal Mind What is "two?" Explain its meaning. The result of an unary operation with arity one.

20.  Imma come back and necro this thread in a about month, I'll see you guys later.

21.  Originally Posted by Phion The result of an unary operation with arity one. What is "one?" Phil, I'm glad you like the first page of Tao Te Ching, but you are dodging my question. You have already named the principle "two" by making it the answer to a question. What is "two?"

22.  Originally Posted by Universal Mind Phil, I'm glad you like the first page of Tao Te Ching, but you are dodging my question. You have already named the principle "two" by making it the answer to a question. What is "two?" It's a twol that we use to slice the universe into distinct pieces to make it easier to think about. Can I take it that you've accepted my point that postulating that everything is made out of matter cannot support thinking of "two" as being real?

23.  Originally Posted by Universal Mind What is "one?" Nullary operation with arity zero.

24.  Originally Posted by PhilosopherStoned It's a twol that we use to slice the universe into distinct pieces to make it easier to think about. Can I take it that you've accepted my point that postulating that everything is made out of matter cannot support thinking of "two" as being real? That says something about two, but it's not a distinct definition for the particular number. What is two, as opposed to three or nine? I think it's a principle, not just a way of slicing the universe. I agree that it's not made of matter, and I don't think matter is the only stuff that exists. There are principles, forces, energy, dimensions, space, etc. Originally Posted by Phion Nullary operation with arity zero. It's just an operation?

25.  Originally Posted by Universal Mind It's just an operation? It's all about context.

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