Joseph at Rationalizing the Universe explores the modern formulation, rooted in Goedel’s Theorem of logical incompleteness, of living with uncertainty. The following discussion ensued:
You point out correctly that Goedel’s theorem is restricted to a fairly narrow problem: proving that a proof system is “correct” – i.e. – that its axioms and operations are consistent. In other words, we can’t take a set of axioms and apply the operations to disprove any other axiom.
This seems to lead to the conclusion that we can’t trust our proofs of anything, which means that there are no guarantees that our expectations will be met. Unfortunately, expectations are undermined by many other problems, among them determination of initial conditions, noise and adaptation. The last is the special bete noir of sociology, as people will often violate social norms in order to assuage primitive drives.
At this point in my life, I am actually not at all troubled by these problems. Satisfaction is not found in knowing the truth, it is found in realizing creative possibilities. If we could use mathematics to optimize the outcome of social and economic systems, we would have no choices left. Life would become terribly boring. So what is interesting to me is to apply understanding of the world to imagine new possibilities. Mathematics is a useful tool in that process, particularly when dealing with dumb matter.
This brings me back to the beginning of the post: you state that “mathematics is the unspoken language of nature.” If there is anything that Goedel’s theorem disproves, it is precisely that statement. Mathematics is a tool, just as poetry and music are tools. At times, both of the latter have transported my mind to unseen vistas; mathematics has never had that effect.
You raise a very interesting point; if we could optimise everything then would we take all of the joy out of being…. you may well be right. I know I get a lot of my satisfaction from the quest to know more. Although I disagree that Godel’s theorems disprove my original statement in this sense; language is essentially about describing things. That is why you can have different languages but they are easily translatable…. bread/pan/brot etc…. we all know what they mean because they all describe the same thing. In exactly the same way, mathematics describes things that actually exist; that isn’t to say nature is mathematics at all – mathematics is the language of nature but it is just as human in its construction as the spoken word. But is matter not matter because a human invented the label? Matter is matter.
To be, these theorems don’t break down all of our proofs; but what they do show is a vital point about logic. One which I think is going to become and increasingly big issue as the quest to understand and build artificial intelligence increases – can we every build a mind as intelligent as a humans when a human can know the answer to a non-programmable result? We hope so! Or rather I do – I do appreciate it’s not for everyone
I appreciate your enthusiasm, but I must caution that the mathematical analogies in classical physics cannot be extended in the same way to the quantum realm. Richard Feynman warned us that there is no coherent philosophy of quantum mechanics – it is just a mathematical formulation that produces accurate predictions. Ascribing physical analogies to the elements of the formulation has always caused confusion. An extreme example was found in the procedure of renormalization, in which observable physical properties such as mass and charge are produced as the finite ratio of divergent integrals.
Regarding human and digital intelligence: one of the desirable characteristics of digital electronics is its determinism. The behavior of transistor gates is rigidly predictable, as is the timing of clock signals that controls that propagation of signals through logic arrays. This makes the technology a powerful tool to us in implementing our intentions.
But true creativity does not arise from personal control, which only makes me loom bigger in the horizon of others’ lives, threatening (as the internet troll or Facebook post-oholic) to erase their sense of self. Rather, creativity in its deepest sense arises in relation, in the consensual intermingling of my uniqueness with the uniqueness of others.
Is that “intelligence?” Perhaps not – the concept itself is difficult to define, and I believe that it arises as a synthesis of more primitive mental capacities, just as consciousness does. But I doubt very much that Artificial Intelligence is capable of manifestations of creativity, because fundamentally it has no desires. It is a made thing, not a thing that has evolved out of a struggle, spanning billions of years, for realization. Our creativity arises out of factors over which we have no control: meeting a spouse-to-be, witnessing an accident, or suffering a debilitating disease. We have complex and subtle biochemical feedback systems which evolved to recognize and adjust to the opportunities and imperatives of living. We are a long way from being able to recreate that subtlety in digital form, and without those signals, meaningful relation cannot evolve, and thus creativity is still-born.