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I definitely agree that much of this argument boils down to a set of poorly fleshed out definitions. Though I would argue that you’ve misrepresented the article

by stult 7y ago

I definitely agree that much of this argument boils down to a set of poorly fleshed out definitions. Though I would argue that you’ve misrepresented the article’s definitions a bit. The article is asking whether the formal system in question can fully explain human intelligence. Which we all agree it can’t. It’s not arguing that no machine in the sense you or I would understand that word can explain human intelligence, the author just assumes implicitly that any machine will consist of a formal symbolic system of the type Gödel studied. Obviously if you challenge that assumption, the argument loses its potency. But in defense of the OP, if we discard the perhaps unhelpful term “machine,” the argument does raise an interesting question: what distinguishes the formal, mechanistic logic of pure mathematics from what you describe as “intuitive reasoning?” I think a worse flaw in the argument is the assumption that formal symbolic completeness is somehow required for human intelligence. I’d say the opposite if anything is true. If modern psychological research has provided anything of value, it’s pretty convincing evidence that human brains don’t require any sort of logical completeness or even internal consistency to function.

pdonis 7y ago
> The article is asking whether the formal system in question can fully explain human intelligence. No, it's asking whether any formal system whatever can fully explain human intelligence. And it gives the wrong answer; see below. > Which we all agree it can’t. No, we don't, because the article misrepresents what "fully explain" means. The article assumes that, for a formal system to "fully explain" something, it must produce proofs of all sentences that that something could produce. But that condition is too strong. All the formal system needs to do is produce, by hook or by crook, the same outputs as the something (a human, or whatever) does. It doesn't have to produce proofs of all those outputs as theorems. Humans certainly don't produce formal proofs of all the things we claim we can intuitively see are true.
pdonis 7y ago
> the author just assumes implicitly that any machine will consist of a formal symbolic system of the type Gödel studied. Obviously if you challenge that assumption, the argument loses its potency Daniel Dennett, in his refutation of Lucas' argument (which I think is in his book Brainstorms; I have not been able to find a version of it online), makes a point related to your point here in a very stark manner: by showing that, in fact, a physical realization of a formal system can produce a proof of that sytem's own Godel sentence! Dennett's argument makes use of a critical observation: in order to view any physical system as a realization of a formal system, we have to interpret the inputs and outputs of the physical system in a particular way. But there is nothing in the laws of physics that rules out the possibility of multiple interpretations of the same inputs and outputs. So it is perfectly possible for a given set of outputs of some physical system to have one interpretation as sentences of formal system A, but also another interpretation as sentences of formal system B, with the latter sentences including a proof of the Godel sentence of formal system A. Put abstractly this way, this seems unlikely, but in fact we experience this sort of thing all the time with computers. As I am typing these words, my computer is taking a series of inputs and producing a series of outputs which have multiple interpretations: as sequences of bytes, as machine instructions, as typed characters from the keyboard and displayed characters on the screen, as changes in voltage across transistors, etc. And there is nothing in principle that prevents, for example, writing a program that runs on my computer's CPU and proves the Godel sentence of that CPU, considered as a formal system. Or writing a program in, say, Python that proves the Godel sentence of the gcc compiler used to compile that instance of the Python interpreter, considered as a formal system. So in fact the author of this article is not just assuming that any machine will consist of a formal symbolic system, he is assuming that there is one unique interpretation of any physical machine as a formal symbolic system. And we now have abundant evidence that that assumption is false.
- neerajk 7y ago
  Somewhat tangentially, from Dennett's review of Lucas' "The Freedom of the Will" "He obtains some of his pet conclusions by patent non sequiturs, aided by a great deal of hand waving, and not a few elementary logical confusions and slips (e.g., pp. 39, 61, 72, MO). But he gives us a splendid display of his command of Latin, Greek, and Middle English, and, except for failing to give any reference for a crucial argument of Wolfgang Pauli's that he mentions (p. 112), his footnotes are erudite."

by stult 7y ago

pdonis 7y ago

> The article is asking whether the formal system in question can fully explain human intelligence. No, it's asking whether any formal system whatever can fully explain human intelligence. And it gives the wrong answer; see below. > Which we all agree it can’t. No, we don't, because the article misrepresents what "fully explain" means. The article assumes that, for a formal system to "fully explain" something, it must produce proofs of all sentences that that something could produce. But that condition is too strong. All the formal system needs to do is produce, by hook or by crook, the same outputs as the something (a human, or whatever) does. It doesn't have to produce proofs of all those outputs as theorems. Humans certainly don't produce formal proofs of all the things we claim we can intuitively see are true.

pdonis 7y ago

> the author just assumes implicitly that any machine will consist of a formal symbolic system of the type Gödel studied. Obviously if you challenge that assumption, the argument loses its potency Daniel Dennett, in his refutation of Lucas' argument (which I think is in his book Brainstorms; I have not been able to find a version of it online), makes a point related to your point here in a very stark manner: by showing that, in fact, a physical realization of a formal system can produce a proof of that sytem's own Godel sentence! Dennett's argument makes use of a critical observation: in order to view any physical system as a realization of a formal system, we have to interpret the inputs and outputs of the physical system in a particular way. But there is nothing in the laws of physics that rules out the possibility of multiple interpretations of the same inputs and outputs. So it is perfectly possible for a given set of outputs of some physical system to have one interpretation as sentences of formal system A, but also another interpretation as sentences of formal system B, with the latter sentences including a proof of the Godel sentence of formal system A. Put abstractly this way, this seems unlikely, but in fact we experience this sort of thing all the time with computers. As I am typing these words, my computer is taking a series of inputs and producing a series of outputs which have multiple interpretations: as sequences of bytes, as machine instructions, as typed characters from the keyboard and displayed characters on the screen, as changes in voltage across transistors, etc. And there is nothing in principle that prevents, for example, writing a program that runs on my computer's CPU and proves the Godel sentence of that CPU, considered as a formal system. Or writing a program in, say, Python that proves the Godel sentence of the gcc compiler used to compile that instance of the Python interpreter, considered as a formal system. So in fact the author of this article is not just assuming that any machine will consist of a formal symbolic system, he is assuming that there is one unique interpretation of any physical machine as a formal symbolic system. And we now have abundant evidence that that assumption is false.

neerajk 7y ago
Somewhat tangentially, from Dennett's review of Lucas' "The Freedom of the Will" "He obtains some of his pet conclusions by patent non sequiturs, aided by a great deal of hand waving, and not a few elementary logical confusions and slips (e.g., pp. 39, 61, 72, MO). But he gives us a splendid display of his command of Latin, Greek, and Middle English, and, except for failing to give any reference for a crucial argument of Wolfgang Pauli's that he mentions (p. 112), his footnotes are erudite."