This is actually quite simple to transfer to law (later mathematically systems where built to be resistant to self-reference, and it took some serious effort to force it (in particular Godel’s incompleteness theorem doesn’t apply to just a single formal system, but to *every* formal system that can encode arithmetic). Take the statement A: “A is not the conclusion of any valid legal argument.” This statement is true statement about our legal system, but yet our legal system is not capable of proving it. If A was referred to in a contract, the court could not rely on the fact that it was true.

This is where the “human” element comes in: The judge can simply throw out any case involving statement A.

]]>But as you observe, mechanistic/computational metaphors for human thought remain very strongly embedded in discussions on these issues, despite Gödel and Turing having shown their limitations before electronic computers even existed.

]]>in this regard i like the line of luitzen brouwer who stated somewhere/somehow that freedom of thought means more than following the construction of a given rule set. it also means to question the rules set and to feel free to add to it.

this is one reason why i am triggered by this discussion as i take it as a nice exercise in exactly that: the freedom to combine, to pick one line, but not necessarily all.

however, i used the trigger to go back to the archives and papers collected by the gödel society. my thoughts are influenced, the interpretations are my own.

as we know already from lawrence lessig: “code is law”. so i personally don’t make that distinction between different rule sets any longer. more importantly i am eager to learn how to make progress in spite of laws, which i like to understand as a set of agreement within a given living society.

gödel brings the inside that derivability is not as strong as hilbert wanted it to be. neither law nor the arithmetic of natural numbers can be seen as an end in itself: a formula is true if it can’t proof that it can be derived from that formula. the decision if that makes a formula true is imho a different story. nobody can proof it. however, i found an interesting hint how to deal with that inside as there is much more to be found in the work of gödel in order to question mechanical thinking. thanks to this discussion, i found that interesting source: a script written by the gödel society for a tv show about gödel. (google books: kurt gödel und die mathematische logik, engl translation starts at bookpage 111, spanish at p160)

in the script peter weibel and werner depauli-schimanovich set a pointer to the question: is it possible to write a program which is able to check the correctness of an arbitrary number of programs without ending in an infinite loop. gödels answer is no. ergo human mind can’t be mechanised. turing, they argue, took gödels problem with formal systems and changed provability with calculability.

in the script they quote robin gandy: “gödel had shown that a particular form of this problem (about what can be done by routines) could not be done by routines. and i think that was the starting point, that suggested to turing one should be able to characterise what can be done and then to show that there are these things that can’t be done.”

i think in the current debate about what technology and automation/sensors can or should not do, we lost this inside from the early days of computing that human are capable to think bigger than words or rule sets allow. yet this implies that thoughts are not just there to be taken and implemented in a program but can be used to grasp what should not be done.

there is this tendency in computing the last couple of years to re-inventing ideas that have already been here since the 1950s, at least. (mechanical thinking as a desirable goal is with us much longer). the difference: i don’t assume that back in the 1950s the ideas were discussed in order to re-install the rule-set of the middle ages within society but to build something new. today we should be able to judge the early enthusiasm: the world of computers is not a perfect world. it’s not even provable rigid, but it follows today a tendency to ignore new combinations which could be seen as new possibilities/extensions for society. instead of making use of recursive functions we ignore in both worlds – law (ie civil rights, geneva conventions) and computing (useful extensions) – what we have achieved -in regard of differences in mindsets – since then.

]]>That, perhaps, is where the true analogy between maths/logic and the law can be found: that both are refreshingly open-ended and impossible to summarise finally and entirely by a single set of rules – as that wonderful quotation from Freeman Dyson in the second of your posts which I linked expresses so well.

]]>First of all, thank you for your post. I’m glad to see someone interested in this debate.

Second, clearly as you well notice, the comparison with Godel is metaphorical, not formal. I am not a lawyer myself, but to the best of my knowledge there is no point in a formal comparison with law given that our laws do not emanate from an axiomatic system. Furthermore Jay is not using “law” meaning a part of a legal body, but “a general relation proved or assumed to hold between mathematical or logical expressions”. When computers and machines enter into the equation, under our current computing paradigm, there is a stronger basis to pursue the comparison further, but even in that case I don’t think Jay is intending any formal comparison, neither am I, just making it clear that we must be cautious about the implicit assumptions we do about the behaviour of computer based systems. Very easily we can run into trouble. There are many examples of problematic (paradoxical) use cases, not necessarily Godel propositions.

Third, regarding your comments on Godel theorem itself, an undecidable proposition is neither true nor false (within the given axiomatic system). If you add the proposition as a new axiom to the system, of course, as you say, the system will continue to express other undecidable propositions. My remark about adding new axioms to the systems does not imply I think it is the way to close the gap, it would be a way to evolve the system. And yes, it seems to me also similar (metaphorically) as how the legal system expands through new court decisions.

Sorry for the poetic license.