If a watched pot doesn't boil, and you don't know if someone else is watching your teapot, you can only determine whether it’s boiling by pouring it out. Also: A teapot may or may not exist and may or may not contain boiling water, and it's up to the person that posits its existence to prove it. True story. Anyway, on the topic of functions and logic and philosophy, I thought you might appreciate a "computer science-y" perspective on the same question. (Thank you to my friend Eric deRiel for his helpful insight!) * * * * * * * Suppose you have an extremely sophisticated natural language processing function F (x) that, for any declarative natural language statement x , returns one of two values—TRUE or FALSE—depending on whether the assertion made by x is true or not. (For example, suppose you are running a social media site, and you are taking a stab at an algorithmic solution to the problem of “fake news.”) Some simple examples: F ("the moon is made of green cheese") => FALSE F ("water is wet") => TRUE We can even suppose that F is seriously sophisticated and can avoid some of the more obvious traps of undergraduate philosophy like: F ("the current king of France has a beard") => FALSE F ("Santa Claus is not real, and lives at the North Pole") => TRUE So far so good. But what do you get if you try: F ("This sentence is false") => ? How can you fit that into any kind of complete, holistic, function-based description of everything? In other words: Let R = { x │ x ∉ x} , then R ∈ R ⟺ R ∉ R This is a functionalized variation of Russell’s Paradox , which drove Russell to despair and caused some parts of mathematics to have to reinvent themselves—incidentally creating modern computer science in the process. A related problem arises if you use a very smart computer function (that is, a function capable of executing logical tests and making control flow decisions, like if/then). Consider the following: halt_test(other_func) In this example, other_func is some other computer function. So, for example, if the command is halt_test(print "Hello, world") …the program performs the finite function: it prints “Hello, world” and stops (TRUE). If the command is: halt_test(while [true] continue) …it goes on forever in an infinite loop (FALSE). But what happens if you pass halt_test to itself? In other words: halt_test(halt_test) => ?? This is the Turing Halting Problem and presents one example of a computer function that can never be written. There are an infinite number of such functions, just as there are an infinite number of variations on Russell’s Paradox. Poor Russell, I gather, tried to list all the cases he could think of, and kept thinking of more—that was part of what caused him so much anguish. [ Sort of like sitting around counting the pairs of tube socks until you realize that they’re infinite.—Ed. ] The answer to the riddle is that in any mathematical system sufficiently complex to be of any use to anyone, there are an infinite number of paradoxes like this, where you can find ways that functions or sets or definitions or whatever you please will defy evaluation. So even “everything is a function of everything else” has to come with an asterisk and a footnote about how that statement is provably incomplete. * * * * * * * Perhaps this is old hat to you computer programmers out there, but I missed that lesson in college. And high school. Such is the life of a Liberal Arts major with a focus on music, of all things. Applications to the Ineffable So I was thinking about all of this while listening to political commentary on the radio, and someone used the phrase, "The enemy of my enemy is my friend." If this is true—and negative × negative = positive—then it stands to reason that the friend of my friend is also my friend (positive × positive = positive). So is an enemy of a friend also my enemy? Yes, so it would seem. And a friend of an enemy is also my enemy; that makes sense. I can push it even further: The absolute value of friendship sure seems to be the measure of friend and enemy combined. [So what does that make Switzerland during World War II? —Ed. ] The function of x friend is to bring new people together. Set theory even seems to work. In spite of human relationships being irrational, they sure seem to act rationally! I was talking with my mother about this (I didn't realize that I would be writing so much about my family here!). She pointed out the following: "I think of dimensions and not binaries—people I feel closest to, people I don't know yet, people whom I have hurt, people who have hurt me, people I have forgiven, people I need to forgive. You're talking about relationships as if they were transitive, which makes no sense to me. Heck, even the concept of a personal "enemy", as opposed to a friend, doesn't make sense to me, outside of cartoons. [...] Even in politics, I don't think it's useful to talk about enemies . The way my mind works, the macro mirrors the micro." This is an enlightened point of view, I think. Just as "intuition" can't be relegated to a summing of all the available data, relationships cannot be isolated to simple binary applications. Or, perhaps, AI will have "relationships" based on all available data, but it will be unlike anything we humans know anything about. To Wrap Things Up So to sum up the last three posts on Cadence on the Beat: I started off by asking, will AI will have "intuition", at least in the human sense? The answer is: We don't know. If it will, it is both a long way off and will look nothing like what human intuition looks like. Context is more important than we can quantify. This led to a thought experiment about neural networks and the task of language translation , which led to exploring whether art and translation can really be possible if performed by a machine. The answer? We don't know. If it will, it is both a long way off and will look nothing like what we humans think art looks like. Context is more important than we can quantify. And while we're talking about turning computers into "intelligence" , my brother suggested that I do an infinite amount of laundry. [ Maybe I misunderstood his point? *checking* No, wait... —Ed. ] That is to say, I asked if everything were a function of everything else, how would you write it in mathematical form? He answered, essentially, that you can't. And my friend Eric (see above) said that any attempt to do so must acknowledge that the statement is unprovable. And context is more important than anything. And, while we’re at it, the cat is both alive and dead. — Meera![]()
![]()
↧