First, I want to ask you what Objectivism has to say about Mathematics, and if anyone in the field of Objectivism is working on advanced mathematics. Second, what do you think of http://www.wolframscience.com/ which isn’t really science but a very narrow scope of mathematics? Third… do you think you have what it takes to tackle mathematics? Because that’s what we’re going to be doing in this thread.
I just realized that a theory of basic constructs would account for both mathematics at the most general levle and physics at the most fundamental level. This would not be a scientific theory, but would give rise to a whole range of mathematical possibilities, and define it well enough for us to be able to determine it once we see the world as it is.
First, for the science part, we need to break up the General Theory of Relativity and Quantum Field Theory into their mathematical components, and remove classical assumptions about the world. For the math part, we need to pretty much rely on logic, then widen its scope so that quantities and qualities are both dealt with. Then we merge the two. This is my strategy.
However, we also need to start at the basics. Why does 1 + 1 = 2? Because it is defined that way? Does 2 have an identity that is seperate from merely the additon of 1 + 1? If it does, then what is the definition of +? What is the defintion of 1? Can we define 1 as a multiplicative identity without defining multiplication? Does multiplication depend on addition? What is unique about 1?
I think that asking about the number 1 would be without context. 1 in what mathematical system? What are the axioms? In the real world, we’d be asking 1 what? In math, we have to ask what our axioms are. We cannot just have a naked 1. 1 must represent something. We may have to start with something more abstract, such as A.
Now as for +, that is a bit harder. We can ask the same questions, but we have to define it as an operator. Let’s call it a. Notice how I use letters for both the number and the operator, but the operator is lowercase whereas the number is upper case. We will assume for now that numbers cannot act on eachother directly, but through operators. In later sessions we may find out that this is false, or that it’s true, but we can interchange numbers and operators in the same way we can interchange dots and lines in geometry to get the same relations.
Now, = implies that if B = C then C = B. Otherwise, we’d use > and < signs. The basic way we do math is either AbC = D. Or aBc = d, but we can ignore the latter for now… but we shouldn’t ignore the possibility in the future. Now we have to define context. We could do it this way: (AbC=D)_“1” where 1 is axiom 1. The _ designates that AbC=D operates under an axiom. Is this too complicated already? Or are you following me?
At this point I really don’t know how to proceed without biasing myself. However, you see how I approach the subject, by using abstractions first then applying meaning to the abstractions based on context. Does anybody get it or am I being too vague? I’m basically using strings (not string theory type strings, but computer theory type strings) for lack of a better word. Would using propositional logic be easier for people or is this, what I’m doing now, a good idea?