Gödel of the Gaps

So much of our attention in logic and math is focused on using processes to turn specific inputs into even more specific binary outputs. Very little attention is paid to what inputs and outputs are or to the understanding of what truth is in theoretical terms. The possibility of inputs is assumed from the start, since no program can exist without being ‘input’ into some kind of material substrate which has been selected or engineered for that purpose. You can’t program a device to be programmable if it isn’t already. Overlooking this is part of the gap between mathematics and reality which is overlooked by all forms of simulation theory and emergentism. Without some initial connection between sensitive agents which are concretely real and non-theoretical, there can be no storage or processing of information. Before we can input any definitions of logical functions, we have to find something which behaves logically and responds reliably to our manipulations of it.

The implications of binary logic, of making distinctions between true/go and false/stop are more far reaching than we might assume. I suggest that if a machine’s operations can be boiled down to true and false bits, then it can have no capacity to exercise intentionality. It has no freedom of action because freedom is a creative act, and creativity in turn entails questioning what is true and what is not. The creative impulse can drive us to attack the truth until it cracks and reveals how it is also false. Creativity also entails redeeming what has been seen as false so that it reveals a new truth. These capabilities and appreciation of them are well beyond the functional description of what a machine would do. Machine logic is, by contrast, the death of choice. To compute is to automate and reduce sense into an abstract sense-of-motion. Leibniz called his early computer a “Stepped Reckoner”, and that it very apt. The word reckon derives from etymological roots that are shared with ‘reg’, as in regal, ruler, and moving straight ahead. It is a straightener or comb of physically embodied rules. A computer functionalizes and conditions reality into rules, step by step, in a mindless imitation of mind. A program or a script is a frozen record of sense-making in retrospect. It is built of propositions defined in isolation rather than sensations which share the common history of all sensation.

The computing machine itself does not exist in the natural world, but rather is distilled from the world’s most mechanistic tendencies. All that does not fit into true or false is discarded. Although Gödel is famous for discovering the incompleteness of formal systems, that discovery itself exists within a formal context. The ideal machine, for example, which cannot prove anything that is false, subscribes to the view that truth and falsehood are categories which are true rather than truth and falsehood being possible qualities within a continuum of sense making. There is a Platonic metaphysics at work here, which conjures a block universe of forms which are eternally true and good. In fact, a casual inspection of our own experience reveals no such clear-cut categories, and the goodness and truth of the situations we encounter are often inseparable from their opposite. We seek sensory experiences for the sake of appreciating them directly, rather than only for their truth or functional benefits. Truth is only one of the qualities of sense which matters.

The way that a computer processes information is fundamentally different than the way that conscious thought works. Where a consistent machine cannot give a formal proof of its own consistency, a person can be certain of their own certainty without proof. That doesn’t always mean that the person’s feeling turns out to match what they or others will understand to be true later on, but unlike a computer, we have available to us an experience of a sense of certainty (especially a ‘common sense’) that is an informal feeling rather than a formal logical proof. A computer has neither certainty nor uncertainty, so it makes no difference to it whether a proof exists or not. The calculation procedure is run and the output is generated. It can be compared against the results of other calculators or to employ more calculations itself to assess a probability, but it has no sense of whether the results are certain or not. Our common sense is a feeling which can be proved wrong, but can also be proved right informally by other people. We can come to a consensus beyond rationality with trust and intuition, which is grounded the possibility of the real rather than the realization of the hypothetical. When we use computation and logic, we are extending our sense of certainty by consulting a neutral third party, but what Gödel shows is that there is a problem with measurement itself. It is not just the ruler that is incomplete, or the book of rules, but the expectation of regularity which is intrinsically unexpected.

One of the trickiest problems with the gap between the theoretical and the concrete us that the gap itself is real rather than theoretical. There can be no theory of why reality is not just information, since theory itself cannot access reality directly. Reality is not only formal. Formality is not real. There is a bias within formal logic which favors certainty. This is at the heart of the utility of logic. In mathematician Bruno Marchal’s book “The Amoeba’s Secret”, his view on dreams hints at what is beneath the surface of the psychology of mathematics. He writes

“What struck me was the asymmetry existing between the states of dreaming and of being awake: when you are awake, you can never be truly sure that you are. By contrast, when dreaming, you can sometimes perceive it as such.”

Surely most of us have no meaningful doubt that we are awake when we are awake. The addition of the qualification of being “truly sure” that we are awake seems to assume that there is a deeper epistemology which is possible – as if being awake required a true certainty on top of the mere fact of being awake. To set the feeling of certainty above the content of experience itself is an inversion; a mistake of privileging the expectations of the intellect over the very ground of being from which those expectations arise.

Likewise, to say that we can sometimes perceive our dreaming in a lucid dream is to hold the dream state to a different epistemological standard than we do of being awake. If we could be awake and not really be sure that we are, then certainly we could think that we are having a lucid dream, but could be similarly misinformed. We could be dead and living in an afterlife from which we will never return or some such goofy possibility. Mathematical views of reality seem to welcome a kind of escapist sophism which gives too much credence to rabbit holes and not enough to the whole rabbit.

That we sometimes tell when we are dreaming means only that we are more awake within our dream than usual – not that our usual awareness is any more true or sure than it ever is. If we are uncertain in waking life and certain in dreams, it is because our capacity to tell the difference is real and not a dream or theory. There is no way to prove that we are awake, but neither is there any need to prove it since it is self-evident. Any proof that we could have could theoretically be duplicated in a dream also, but that does not mean that there is no difference between dream and reality. The difference is more than can be learned by ‘proof’ alone.

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