Apr 06, 2011 | By George Gantz
Blind Spots – Reflection and Recursion
In the previous discussion of solipsism, we avoided the abyss of pure skepticism by being willing to accept evidence that other people exist, even without an “absolute” proof. Where do we derive our awareness of others and the distinction between “self” and “other” – and what are the implications of this self-reflection? This is a really big question with implications for sentience, consciousness, and even the foundations of logic.
One of the curious insanities of solipsism is that it rejects the reality of others – yet it does so with the tools of language. Language has no reality and would not exist except through interactions with many other people. Articulating the concept of self, as Descartes did, is impossible without the context provided by the others that taught us the language we need to reason in abstract terms.
The development of language and the abstract reasoning it permits may, in fact, be what define sentience and distinguishes human intelligence from that of other species. Inherent in this language is the learning process by which we gain our personal identity and self-reflective capacity through communication with others – beginning with parents, from whom we learn to separate very early in life. Our self-awareness and self-consciousness is really a reflection of, and entirely dependent on, our relationship with others.
There are a number of big topics that follow from this: How does the quest for consciousness in brain science factor in the reality of self and other?; What does theology have to say about human consciousness of God, and man as a reflection of God?
But today I want to talk about the implications of self-reflection in logic and math. Mathematical self-reflection, and specifically the topic known as recursiveness, may not seem like a big deal, but in fact, major developments in the past century in set theory and symbolic logic have pointed to some unexpected and profound results.
It makes perfect sense to talk about mathematical or logical statements that refer to themselves. For example, we all know what it means when we say something like “This statement contains five words” or “This sentence is true”. But knowing the meaning of a sentence becomes more difficult when you say “This statement is false.” If true, the statement contradicts itself. If false, it curls back and bites its own tail.
This curious linguistic sample is known as the liar’s paradox, and while it may seem no more than a parlour trick, among mathematicians since Frege and Russell, it has led to no end of difficulties. The definitive statement of those difficulties came some eighty years ago, when renowned logician Kurt Godel proved his incompleteness theorems. The proofs address two key characteristics we would hope for in any formal system of logic, the qualities of consistency and completeness. Note that among the systems of logic we are concerned about here is the very helpful system of numbers and arithmetic.
Consistency means that we can prove things that are true, but we cannot prove things that are false. Imagine how disconcerting that would be – to prove a statement and its negation!
Completeness means that we can determine whether any particular statement is true or false. “Undecidable” statements, after all, are a significant nuisance if we really want to know the whole truth.
Godel’s theorems prove, in formal terms, that even in a system as straightforward as arithmetic, it is impossible to achieve both consistency and completeness. There are statements that are true but unprovable within the system. Moreover, if you design a system that is provably complete (no indeterminate statements), then it must be inconsistent, which effectively means you can prove anything.
To put it simply, there is a blind-spot, a serious blind-spot, in every logical system of any complexity. The implications to mathematics are significant. Many thinkers, over the millennia, have vested mathematics with an absolute and inviolable perfection that in some cases rises to the level of mysticism or religion. But now we know that math has blind-spots. The most we can hope for is consistency – but not completeness. Some true conjectures can never be proved.
What does this mean? Most of the time, we are not going to be troubled by the blind-spots. But their existence does leave a queasy feeling – like a hole at the center of what has always seemed to be a reliable, dependable and complete standard of truth.
Notably, this hole at the center seems to arise only in the context of recursiveness – the exercise of self-reference. Unprovable truths arise in logic – just as an unprovable truth, the existence of others, arises in our personal self-reflection.
In both cases, we are faced with blind-spots that require us to move beyond the inherent limits of our internal perception and understanding in order to grasp the full truth of reality.
3 Responses to “Blind Spots – Reflection and Recursion”
Join the Discussion