Self-reference
Self-reference seems to me to be the fundamental problem of thought. It is essentially unavoidable, and yet its introduction seems to me to create infinity. Another way of thinking about this is that the way we define numbers is entirely self-referential. We start with zero or one, depending on mood - modern mathematics prefers zero, and we say the number that follows one is two, and then for any N in the set of integers, N + 1 is the next integer. Roger Penrose in the Emperor’s New Mind chooses to talk about the problem of intelligence in reference to the natural numbers because, and this surprising even to the non-mathematically inclined, it is the simplest system that we can talk about that involves ordinary human reasoning. I.e. all non-“mathematical human reasoning” is at least as intractable as are the natural numbers. Why intractable? Because some 80 years ago Gödel stunned the world of thinkers by showing that the theory of natural numbers must either be incomplete or inconsistent and that its damn hard to tell the difference.
It is easy for people to accept that the natural numbers constitute an infinite set. One just provides the simple definition above, and most fourth graders can readily accept the idea of infinity. It is also easy for most people to get that sets like the rational fractions are the same size as the natural numbers. That is easy, but most people are blown away when they find out that continuum is bigger, infinitely bigger that the natural numbers. I am not going to try to define the continuum very precisely, but the normal example is all of the points on the number line between zero and one. The rational numbers between zero and one, though infinite, being a vanishingly small part of this set! Surprisingly, the irrational numbers far outnumber the ones that we are used to talking about.
I spend a lot of time thinking about this problem, I think few people really get how profound this problem is and what it implies -- I think I’m starting to get it, but I might just be delusional. The fact that modern thought starts with the rational numbers rather than the continuum leads, I think, to a great confusion about the nature of reality. If we start with the continuum as our fundamental mathematical assumption then it would not seem strange that we can subtract out the rationales and loose nothing. OTOH, when we start with the natural numbers, 0, 1, …, we tend to get all hung-up on the nature of the universe and the nature of truth. When we start with the continuity assumption, though, I think something marvelous happens. The rational numbers become the holes in truth rather than the truth we are trying to stretch into the continuum. That is, these rational numbers are the special case, a case that the universe does not need and does not use! It is simply a mistake of the mind to imagine that there is a unit of anything. Units are just approximations of truth forced on us by language; they do not exist in reality. The Greek Geometricians may have understood this!
So, we have this situation where all of mathematics depends on self-reference, and self-reference leads to an unsolvable conundrum. Namely, that mathematics is incomplete if correct.
A thought about why self-reference gives us this conundrum. I don’t know how to prove this - or maybe how to state it precisely, but it seems to me that self-reference is a set operator that generates the power set of any set. The power set of a set is the set of all subsets of that set. The power set of a set can be shown to be bigger than the set, and furthermore, this holds even when the set is infinite. So self-reference causes minds to extrapolate out to infinity, giving us sets of ever increasing size and taking the mind into the infinite. Furthermore, all human thought involves self-reference. Giving us our conundrum - there is no tool by which we can categorize our own thought. We cannot speak about ourselves in any complete way.
Therefore, starting with the assumption of mind we cannot exclude God, any god, or even Gods. We can not know about what hides behind the veil - the dimensional gap.
I suspect that all of this changes when we start with the continuum as our fundamental assumption. I will write about that soon (I hope).
r.slime
It is easy for people to accept that the natural numbers constitute an infinite set. One just provides the simple definition above, and most fourth graders can readily accept the idea of infinity. It is also easy for most people to get that sets like the rational fractions are the same size as the natural numbers. That is easy, but most people are blown away when they find out that continuum is bigger, infinitely bigger that the natural numbers. I am not going to try to define the continuum very precisely, but the normal example is all of the points on the number line between zero and one. The rational numbers between zero and one, though infinite, being a vanishingly small part of this set! Surprisingly, the irrational numbers far outnumber the ones that we are used to talking about.
I spend a lot of time thinking about this problem, I think few people really get how profound this problem is and what it implies -- I think I’m starting to get it, but I might just be delusional. The fact that modern thought starts with the rational numbers rather than the continuum leads, I think, to a great confusion about the nature of reality. If we start with the continuum as our fundamental mathematical assumption then it would not seem strange that we can subtract out the rationales and loose nothing. OTOH, when we start with the natural numbers, 0, 1, …, we tend to get all hung-up on the nature of the universe and the nature of truth. When we start with the continuity assumption, though, I think something marvelous happens. The rational numbers become the holes in truth rather than the truth we are trying to stretch into the continuum. That is, these rational numbers are the special case, a case that the universe does not need and does not use! It is simply a mistake of the mind to imagine that there is a unit of anything. Units are just approximations of truth forced on us by language; they do not exist in reality. The Greek Geometricians may have understood this!
So, we have this situation where all of mathematics depends on self-reference, and self-reference leads to an unsolvable conundrum. Namely, that mathematics is incomplete if correct.
A thought about why self-reference gives us this conundrum. I don’t know how to prove this - or maybe how to state it precisely, but it seems to me that self-reference is a set operator that generates the power set of any set. The power set of a set is the set of all subsets of that set. The power set of a set can be shown to be bigger than the set, and furthermore, this holds even when the set is infinite. So self-reference causes minds to extrapolate out to infinity, giving us sets of ever increasing size and taking the mind into the infinite. Furthermore, all human thought involves self-reference. Giving us our conundrum - there is no tool by which we can categorize our own thought. We cannot speak about ourselves in any complete way.
Therefore, starting with the assumption of mind we cannot exclude God, any god, or even Gods. We can not know about what hides behind the veil - the dimensional gap.
I suspect that all of this changes when we start with the continuum as our fundamental assumption. I will write about that soon (I hope).
r.slime