Objectivist Mathematics.

[Starblade]

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?

[DavidOdden]

Why does 1 + 1 = 2?

It is implicit in the concepts "one", "two", "addition" and "equality".

Does 2 have an identity that is seperate from merely the additon of 1 + 1?

Yes, although it is true that there is a method of reducing "2" to "1" and "+". For "13" (as an example), that number is a higher-order abstraction, also almost certainly "9". The numbers 1-3 and perhaps up to 5 would be fairly low-order concepts. This is an issue for psychology, and is related to the "crow" problem.

The root of mathematics is the notion of "measurement", and is important in concept formation (since concept-formation is measurement-omission). "1" is what you have when you unify "1 dog", "1 frog", "1 sock", "1 clock" and omit the measurement "which thing". You can form the concept "2" the same way. Part of forming these concepts is having the ability to distinguish, for example you have to be able to tell that "1 dog" is not the same as "2 dogs". That's not hard to do (at all). OTOH is it perceptually almost impossible to directly distinguish a thousand dogs from a thousand and one dogs, in which case you have to rely on a method (of "counting"). By that method, you can derive a massively broad range of numbers.

[Starblade]

It is implicit in the concepts "one", "two", "addition" and "equality".Yes, although it is true that there is a method of reducing "2" to "1" and "+". For "13" (as an example), that number is a higher-order abstraction, also almost certainly "9". The numbers 1-3 and perhaps up to 5 would be fairly low-order concepts. This is an issue for psychology, and is related to the "crow" problem.

It seems that one necessary function of arithemetic is recurrency. That you can create higher and higher abstractions. But doesn't Godel's law say that we can't prove the natural numbers?

The root of mathematics is the notion of "measurement", and is important in concept formation (since concept-formation is measurement-omission). "1" is what you have when you unify "1 dog", "1 frog", "1 sock", "1 clock" and omit the measurement "which thing". You can form the concept "2" the same way. Part of forming these concepts is having the ability to distinguish, for example you have to be able to tell that "1 dog" is not the same as "2 dogs". That's not hard to do (at all). OTOH is it perceptually almost impossible to directly distinguish a thousand dogs from a thousand and one dogs, in which case you have to rely on a method (of "counting"). By that method, you can derive a massively broad range of numbers.

Yeah, I wonder if this is why higher numbers can't be proven, or something like that. Maybe it's because we have no proof that at a sufficiently high number it won't go in a circle?

BTW, I think it follows that if two number-objects exist completely independent of eachother, they can be added to create a higher number. That it's a simple of matter of having the right metaphysical axioms. Does anybody know specifically why Goedel requires us to assume higher numbers axiomatically?

Also, what attempts are there being made for a more Objective mathematics? How well will they be able to circumvent (not violate) Godel's Theorum? Probably not at all, but maybe Goedel was counting on traditional mathematics, so there is a chance.

I am trying to create such a mathematics. While I need no 'help', I do need criticism so that I can see where I'm going wrong. If someone would like to help, however, then that's all fine and well.

Anyway, I hope to get more responses on this!

Edited April 21, 2006 by Starblade

[Hal]

What is your understanding of Godel's Theorem? What do you think it's actually saying?

[EC]

It's saying mathematically that by starting with the arbitrary one can come up with any possible answer. It's why in reality context is so important and rationalism must be avoided at all costs.

[Hal]

It's saying mathematically that by starting with the arbitrary one can come up with any possible answer.

Huh? Godel's theorem applies to all sufficiently powerful axiomatic systems regardless of whether they are ''arbitrary', and it doesnt say that you can "come up with any possible answer" (I assume this is a way of saying that any statement would be provable within the system).

Edited April 21, 2006 by Hal

[DavidOdden]

Yeah, I wonder if this is why higher numbers can't be proven, or something like that. Maybe it's because we have no proof that at a sufficiently high number it won't go in a circle?

I don't know what that means, but I'll pretend I do and if I'm answering the wrong question you can correct me. This is a "focus" problem about perception, that you can't focus on a group of 20 dogs so that you can perceive that there are 20 and not 19 or 21. We overcome that by using an algorithm where we count the number of dogs, rather than directly seeing it. I don't see that there is an issue of circles -- I don't even know what that would mean, except the obviously wrong idea that at some point if you have, say, 1000 dogs and you add 1 then suddenly you have only 1 dog (I mean, that could be, depending on the dog, but that's not a general property of addition).

Also, what attempts are there being made for a more Objective mathematics? How well will they be able to circumvent (not violate) Godel's Theorum? Probably not at all, but maybe Goedel was counting on traditional mathematics, so there is a chance.

I don't know exactly what your objection to mathematics is, or what an Objectivist mathematics would be. I could suppose that you mean something like a mathematics which has only Euclidean geometry or just real numbers and no irrational or immoral numbers, but since mathematicians recognize that the product of their research has no necessary metaphysical status and it's just "the results of applying a method", then it's not clear to me what the distinction is that you're looking for.

Gödel's first incompleteness theory has a mildly Objectivist-like taste to it. Those "primacy of conscious" people think that all it takes is enough definitions, and you can reason yourself to knowledge of the universe. Gödel proved that arithmetic cannot be "self-proven" with no reference to something from outside the formal system of arithmetic. We recognise that what that "something" is, is existence.