clouds said:
So if 1 + 1 = 2 is True and Real, then Truth exists and Reality can be.
When did 1 + 1 = 2 started to be True?
When we started to think? C'mon man.
I think this could be cleared up if you think of the statement "1+1=2" in terms of the definitions of 1,+,=, and 2. What exactly do you mean by 1 and 2? Clearly they're concepts that point to something beyond themselves - and it's hard to think of what they mean since we're so used to dealing with them. Bertrand Russell suggested the first number our species came up with was 2. He suggests that we noticed a certain similarity (or isomorphism) between, say, a man and his child, and a bird flying with another bird, and a stick with another stick, and so forth. This shared property we called "two" (or whatever grunt designated what we mean by 2). In Russell's thinking, "two" is the property shared by the set of all pairs of objects - that is the set of all sets whose elements can be matched up with the elements of the set {x,y}, where x and y are distinct objects. Similarly, "three" is the property shared by the set of all sets of the form {x,y,z}, where x, y, and z are distinct objects, and so forth. "One", then, is simply the property shared by sets of the form {x}, where x is any object. So 1+1=2 is a statement that simply means when we take any set that has the property of having one element, and combine that element with the element of another, distinct set containing one element, we get a set containing two elements. So 1+1=2 is always true, in any universe, when 1 and 2 are defined in this way. Perhaps in some universes, or on some planets, the properties of existence are somewhat different. Perhaps there are planets where everything is a gaseous muck, so there are no two distinct elements. To a conscious entity living on such a planet the statement "1+1=2" would not make sense; it would be meaningless. If this interests you I recommend checking out a very rich book by Roger Penrose called "The Emperor's New Mind".
gibran2 said:
I have trouble understanding this “abstract world” idea. Maybe it hasn’t been defined well enough, or maybe it’s just a concept beyond my grasp.
I think this "abstract world" is what Plato means by the "World of Forms". In "The Emperor's New Mind" Penrose gives two examples of mathematical objects that seem to exist quite apart from human invention, the Mandelbrot Set and the complex numbers. When Mandelbrot first outputted the images of his namesake set onto his computer screen, he thought there was some sort of machine error. However, when he kept refining the images (ie with further and more refined iterations of the algorithm) he saw a clear pattern emerging. All computer-based images of the Mandelbrot Set, however, are mere approximations to the Mandelbrot Set. The computer is limited in both computational capacity and image-producing ability, so we'll never be able to see a perfect picture of the Mandelbrot Set. So the Mandlebrot Set does seem to exist - we can view increasingly accurate images of it - but we can never actually produce a complete image of it here on Earth. So it appears that it actually does exist, but not in a physical sense, only in an abstract sense, ie: in an "abstract world", a "world of ideas". This may sound fluffy but there are plenty of concepts that seem to exist in the "abstract world", such as all sorts of mathematical properties/structures that have yet to be discovered, but that seem to exist on their own, awaiting discovery. The "abstract world" seems to have a structure to it, seems to have properties - such as being useful in modelling phenomena in the "physical world", so it makes since to describe it as a "world" or "level of reality" on its own.
Another point Penrose makes is that some mathematical concepts or ideas seem to be invented, while some seem to be discovered. The "complex numbers", for example, were "made up" by Cardano when he needed to describe a solution for the square root of a negative number. However, the complex numbers have all sorts of beautiful and fascinating properties that suggest they exist on their own, apart from human invention. They were, therefore, "discovered". On the other hand, in the course of mathematics, sometimes it's necessary to use a contrived example in the course of a line of argument, in order to, say, establish a theorem. Such contrived examples, unlike the complex numbers, often have no interesting properties of their own, so it makes sense to say that they are "inventions" of humans.
I don't know if this makes sense, but I think that's enough for now. I'm glad to see "the world of mathematics" continually being brought up in discussions about the "reality" of Hyperspace since the two debates are very similar.