9 ms·

> Now suppose that we construct the Gödel sentence for this formal system. Since the Gödel sentence cannot be proven in the system, the machine will be unable

by Udo 9y ago

> Now suppose that we construct the Gödel sentence for this formal system. Since the Gödel sentence cannot be proven in the system, the machine will be unable to produce this sentence as a truth of arithmetic. However, a human can look and see that the Gödel sentence is true. In other words, there is at least one thing that a human mind can do that no machine can. Therefore, “a machine cannot be a complete and adequate model of the mind” (Lucas 1961: 113). In short, the human mind is not a machine. That is not a difference between human and machine. It's a rhetorical trick at best, and an unreasonable appeal to mysticism at worst. A human mind looking at the Gödel sentence while strictly adhering to the same formal structure could also not see that the Gödel sentence is true. Conversely, a machine that is allowed to operate outside of this postulated formalism could perfectly well make an inference that the Gödel sentence is true. The Lucas-Penrose argument boils down to a questionable attempt at a logic bomb. So by first tying the machine's hands, they argue that the machine's hand's are indeed tied, and therefore humans - who in this example have been given free rein - can do something that you explicitly disallowed machines from doing.

scandox 9y ago
What formalism, if any, is the human mind subject to? Isn't our current conception of a machine that it will bound to some formalism in terms of reasoning? Are we so bound?
- naasking 9y ago
  > What formalism, if any, is the human mind subject to? There could be many that could answer some such questions, paraconsistent and inconsistent for example.
- Udo 9y ago
  That's a loaded argument. Human biology, and by extension human minds, follow a myriad of rules that are a result of biological processes being played out. Likewise "machines" also follow the rules laid out by their physical design. There is no fundamental difference in this regard, there is no new set of information-theoretical rules that comes into play when you switch to a carbon-based machinery. > What formalism, if any, is the human mind subject to? In the context of this argument, the mind of the human mathematician would be subject to the rules of the formal system. Since Gödel we know that such a formal system cannot be proven true inside the system itself. However, an entity that views the system from the outside can potentially see it's true. There is no fundamental difference between a human mind and a hypothetical machine mind as far as the capability to come to the same conclusion is concerned. That's why Lucas took special care to specify upfront that the machine mind is prohibited from coming to that conclusion, by confining it to live within the formal system only. Yes, their argument is really that circular. > Isn't our current conception of a machine that it will bound to some formalism in terms of reasoning? Are we so bound? We are not, and neither would a generally intelligent machine. The burden of proof is on Lucas/Penrose to show that a generally intelligent machine would still be bound to that formalism, at which point it would cease to be considered generally intelligent. They seem to argue that their postulate is true because the individual components of such a machine are bound by those rules, but then again so are the individual components of our brains.
  - scandox 9y ago
    Well I wasn't intending it as an argument. I was actually just interested to see what someone who had actually thought about it would say.
  - manyoso 9y ago
    So your claim is that the Human mind is a physical machine and that computers are also physical machines and thus both are bound to physical laws... and this negates Penrose how? Do you think Penrose disagrees that the Human mind is bound by physical laws??
    - Analemma_ 9y ago
      I think Penrose is handwaving to try and have it both ways: he's saying the brain obeys physical laws, but that it does not embed a formal computational model. But that's impossible: the second is a logical consequence of the first.
      - manyoso 9y ago
        "But that's impossible: the second is a logical consequence of the first." No, it is not. It is far from proven that universal physical law can everywhere be simulated by a Turing machine. If you had proof of that you'd be up for a Nobel. So publish your paper or perhaps consider that Roger Penrose is not an idiot and you might not know what you are talking about.
        deleted 9y ago
        [deleted]
        gpderetta 9y ago
        It is not proven and possibly it can never be proven. But it can be disproven, and given the success of the Church-Turing thesis, I would say that the burden of proof rest on those that claim that a system more powerful than a Turing machine exist.
- burkaman 9y ago
  > Isn't our current conception of a machine that it will bound to some formalism in terms of reasoning? No, that's just a current limitation of the kind of machines we know how to make.
- mcguire 9y ago
  > What formalism, if any, is the human mind subject to? Neurobiology. Chemistry. Physics. Of course, any machine will also be subject to the same formalism, so.... To start with, no physical entity has an unbounded memory available.
- danielam 9y ago
  Indeed. To make such a claim demonstrates a misunderstanding of what formalism is. When human beings develop formalisms, they do so on the basis of semantics. If I wish to formalize something, I must first have a relevant understanding of the nature of the thing, or things involved and the relations between them. The development of formalism is a translation of semantics into syntactic rules devoid of meaning. Doing so allows us to mechanize reasoning. Algebra is an example of an early move in this direction. Something we learn in mathematics is that two theories concerning different objects can be formally equivalent. Computers are strictly devoid of semantics. They are, at best, purely syntactic machines. There is no amount of syntax than can ever lead to semantics. Strictly speaking, computers don't even compute. Further reading: http://edwardfeser.blogspot.com/2013/10/do-machines-compute-functions.html http://edwardfeser.blogspot.com/2013/10/do-machines-compute-...
  - jamesrcole 9y ago
    For a bit of context on Feser and his views, see https://en.wikipedia.org/wiki/Edward_Feser https://en.wikipedia.org/wiki/Edward_Feser
  - simonh 9y ago
    > Something we learn in mathematics is that two theories concerning different objects can be formally equivalent. Can be, but do not have to be. A distinction through which a host of mistakes have fallen, including Donald Hoffman's 'disproof' of functionalism.
gue5t 9y ago
How does one create a machine with such untied hands? The problem I have here is that while you can run any formal system on a universal computer, you do so by embedding the semantics of your formal system into those of some machine, and you have to make changes from outside that embedding (writing a new interpreter) if you decide you want to shift formalisms. Meanwhile, humans can't step outside the computation their neurons run and reprogram their base method of thought (as far as we're aware, and distinguished from reprogramming one's set of premises), but can change the formal system they're simulating (as one does frequently in math and the sciences). In short, I agree you entirely with except that I don't see how we can design a machine with hands as untied as ours. Given a mechanistic view of the brain one concludes that it must be possible, but I still don't understand what formal trick would enable this difference in power. It does seem to give up consistency somehow (humans make mistakes) but not entirely (humans can detect mistakes they've made). I'd love to hear more thoughts on this matter.
- manyoso 9y ago
  Lots of folks missing the forest for the trees. Objecting to Penrose' argument on the grounds that it assumes Humans are logically consistent when that is no the case misses the point. The claim is that you can not adequately model the Human mind with a Turing machine. Pointing out that Human's are not logical only strengthens this claim and does not refute it. Further, the claim at this non-Turing machine-like ability of Humans is valuable and allows us insight that Turing machine is fundamentally incapable of.
  - mcguire 9y ago
    If you are inconsistent, you believe false statements. In fact you could be convinced to believe any statement. I'm unclear on how that is valuable.
    - manyoso 9y ago
      See Penrose argument. He argues that the fact that our minds are not Turing machines allows us knowledge above and beyond what Turing machines are capable of. I don't think this necessarily entails inconsistency, but replying to his argument that humans are inconsistent does not refute his argument. That is what I was trying to point out.
bordercases 9y ago
> A human mind looking at the Gödel sentence while strictly adhering to the same formal structure could also not see that the Gödel sentence is true. Conversely, a machine that is allowed to operate outside of this postulated formalism could perfectly well make an inference that the Gödel sentence is true. I think the key move you're making here is dictated by this. There isn't a largest model that will be able to capture all such truths (hence Gödel's incompleteness theorem) and what you would need is a system that is able to constantly build further systems else handle the ambiguity and do model-switching. Not that it's impossible to do, but have we proven that such a system is theoretically possible yet within conventional frameworks?
forgotpwtomain 9y ago
Thanks for this spark of a comment in an otherwise perturbed discussion. I cannot reply to all the child comments but hopefully I can help elucidate the misconception you are indicating. > It's a rhetorical trick at best This is precisely it. It works because 'true' is defined for the formal system but not for the human. It so happens that the humans in question say well 'of course we know what Godel's proposition means and what true is' but that's just because they are familiar with those words in an entirely unrelated sense to the system being described (akin to say the common speech use of 'energy'). Since you need a system at least as powerful to evaluate the propositions of an underlying system and in that more powerful system a similar proposition can be formed. Anologous to this is Wittgenstein's problem on the irreducibility of rules: that one can always ask for the explanation of a rule, explanation of an explanation an so on ad infinitum. So it is essentially impossible to determine whether a rule is being applied correctly or not. It so happens that we (humans) can stop asking and 'apply'; but I'm not convinced this is something machines cannot do.
- fiatjaf 9y ago
  Wittgenstein's "problem on the irreducibility of rules" is brainfart at best. Any rule that can be actually _followed_ cannot be logically precise, and vice-versa. If you try to reduce anything from the real-world to pure logic you'll surely end up crazy.
  - deleted 9y ago
    [deleted]
  - posterboy 9y ago
    there are abstract rules and practical laws, principles, whatchamacallit. Laws bind abstract rules to real thingies. The ground truth values are right and wrong en lieu of good and bad - basic emotions that are at the basis of experience. Trying to verbalize (the word logic is related to logos, old greek for tongue, language logic) every basic emotion you will surely go irrationaly crazy. Therefore, context is assumed and language is underspecific.
fiatjaf 9y ago
> A human mind looking at the Gödel sentence while strictly adhering to the same formal structure could also not see that the Gödel sentence is true. To say that you assume a human mind is a machine. Why isn't that begging the question?
- Udo 9y ago
  No, it means I think a human mathematician is capable of working within a formal structure by following the rules of that formalism. It's a way of saying that humans can do formal math. Of course you are correct in suspecting I do assume the human mind is a machine, because we have seen zero evidence that it's not the case, but that's not in any way part of the sentence you quoted.
  - fiatjaf 9y ago
    Ok, I get it. I had misunderstood your argument.

by Udo 9y ago

scandox 9y ago

What formalism, if any, is the human mind subject to? Isn't our current conception of a machine that it will bound to some formalism in terms of reasoning? Are we so bound?

naasking 9y ago
> What formalism, if any, is the human mind subject to? There could be many that could answer some such questions, paraconsistent and inconsistent for example.
Udo 9y ago
That's a loaded argument. Human biology, and by extension human minds, follow a myriad of rules that are a result of biological processes being played out. Likewise "machines" also follow the rules laid out by their physical design. There is no fundamental difference in this regard, there is no new set of information-theoretical rules that comes into play when you switch to a carbon-based machinery. > What formalism, if any, is the human mind subject to? In the context of this argument, the mind of the human mathematician would be subject to the rules of the formal system. Since Gödel we know that such a formal system cannot be proven true inside the system itself. However, an entity that views the system from the outside can potentially see it's true. There is no fundamental difference between a human mind and a hypothetical machine mind as far as the capability to come to the same conclusion is concerned. That's why Lucas took special care to specify upfront that the machine mind is prohibited from coming to that conclusion, by confining it to live within the formal system only. Yes, their argument is really that circular. > Isn't our current conception of a machine that it will bound to some formalism in terms of reasoning? Are we so bound? We are not, and neither would a generally intelligent machine. The burden of proof is on Lucas/Penrose to show that a generally intelligent machine would still be bound to that formalism, at which point it would cease to be considered generally intelligent. They seem to argue that their postulate is true because the individual components of such a machine are bound by those rules, but then again so are the individual components of our brains.
- scandox 9y ago
  Well I wasn't intending it as an argument. I was actually just interested to see what someone who had actually thought about it would say.
- manyoso 9y ago
  So your claim is that the Human mind is a physical machine and that computers are also physical machines and thus both are bound to physical laws... and this negates Penrose how? Do you think Penrose disagrees that the Human mind is bound by physical laws??
  - Analemma_ 9y ago
    I think Penrose is handwaving to try and have it both ways: he's saying the brain obeys physical laws, but that it does not embed a formal computational model. But that's impossible: the second is a logical consequence of the first.
    - manyoso 9y ago
      "But that's impossible: the second is a logical consequence of the first." No, it is not. It is far from proven that universal physical law can everywhere be simulated by a Turing machine. If you had proof of that you'd be up for a Nobel. So publish your paper or perhaps consider that Roger Penrose is not an idiot and you might not know what you are talking about.
      - deleted 9y ago
        [deleted]
      - gpderetta 9y ago
        It is not proven and possibly it can never be proven. But it can be disproven, and given the success of the Church-Turing thesis, I would say that the burden of proof rest on those that claim that a system more powerful than a Turing machine exist.
burkaman 9y ago
> Isn't our current conception of a machine that it will bound to some formalism in terms of reasoning? No, that's just a current limitation of the kind of machines we know how to make.
mcguire 9y ago
> What formalism, if any, is the human mind subject to? Neurobiology. Chemistry. Physics. Of course, any machine will also be subject to the same formalism, so.... To start with, no physical entity has an unbounded memory available.
danielam 9y ago
Indeed. To make such a claim demonstrates a misunderstanding of what formalism is. When human beings develop formalisms, they do so on the basis of semantics. If I wish to formalize something, I must first have a relevant understanding of the nature of the thing, or things involved and the relations between them. The development of formalism is a translation of semantics into syntactic rules devoid of meaning. Doing so allows us to mechanize reasoning. Algebra is an example of an early move in this direction. Something we learn in mathematics is that two theories concerning different objects can be formally equivalent. Computers are strictly devoid of semantics. They are, at best, purely syntactic machines. There is no amount of syntax than can ever lead to semantics. Strictly speaking, computers don't even compute. Further reading: http://edwardfeser.blogspot.com/2013/10/do-machines-compute-functions.html http://edwardfeser.blogspot.com/2013/10/do-machines-compute-...
- jamesrcole 9y ago
  For a bit of context on Feser and his views, see https://en.wikipedia.org/wiki/Edward_Feser https://en.wikipedia.org/wiki/Edward_Feser
- simonh 9y ago
  > Something we learn in mathematics is that two theories concerning different objects can be formally equivalent. Can be, but do not have to be. A distinction through which a host of mistakes have fallen, including Donald Hoffman's 'disproof' of functionalism.

gue5t 9y ago

How does one create a machine with such untied hands? The problem I have here is that while you can run any formal system on a universal computer, you do so by embedding the semantics of your formal system into those of some machine, and you have to make changes from outside that embedding (writing a new interpreter) if you decide you want to shift formalisms. Meanwhile, humans can't step outside the computation their neurons run and reprogram their base method of thought (as far as we're aware, and distinguished from reprogramming one's set of premises), but can change the formal system they're simulating (as one does frequently in math and the sciences). In short, I agree you entirely with except that I don't see how we can design a machine with hands as untied as ours. Given a mechanistic view of the brain one concludes that it must be possible, but I still don't understand what formal trick would enable this difference in power. It does seem to give up consistency somehow (humans make mistakes) but not entirely (humans can detect mistakes they've made). I'd love to hear more thoughts on this matter.

manyoso 9y ago
Lots of folks missing the forest for the trees. Objecting to Penrose' argument on the grounds that it assumes Humans are logically consistent when that is no the case misses the point. The claim is that you can not adequately model the Human mind with a Turing machine. Pointing out that Human's are not logical only strengthens this claim and does not refute it. Further, the claim at this non-Turing machine-like ability of Humans is valuable and allows us insight that Turing machine is fundamentally incapable of.
- mcguire 9y ago
  If you are inconsistent, you believe false statements. In fact you could be convinced to believe any statement. I'm unclear on how that is valuable.
  - manyoso 9y ago
    See Penrose argument. He argues that the fact that our minds are not Turing machines allows us knowledge above and beyond what Turing machines are capable of. I don't think this necessarily entails inconsistency, but replying to his argument that humans are inconsistent does not refute his argument. That is what I was trying to point out.

bordercases 9y ago

> A human mind looking at the Gödel sentence while strictly adhering to the same formal structure could also not see that the Gödel sentence is true. Conversely, a machine that is allowed to operate outside of this postulated formalism could perfectly well make an inference that the Gödel sentence is true. I think the key move you're making here is dictated by this. There isn't a largest model that will be able to capture all such truths (hence Gödel's incompleteness theorem) and what you would need is a system that is able to constantly build further systems else handle the ambiguity and do model-switching. Not that it's impossible to do, but have we proven that such a system is theoretically possible yet within conventional frameworks?

forgotpwtomain 9y ago

Thanks for this spark of a comment in an otherwise perturbed discussion. I cannot reply to all the child comments but hopefully I can help elucidate the misconception you are indicating. > It's a rhetorical trick at best This is precisely it. It works because 'true' is defined for the formal system but not for the human. It so happens that the humans in question say well 'of course we know what Godel's proposition means and what true is' but that's just because they are familiar with those words in an entirely unrelated sense to the system being described (akin to say the common speech use of 'energy'). Since you need a system at least as powerful to evaluate the propositions of an underlying system and in that more powerful system a similar proposition can be formed. Anologous to this is Wittgenstein's problem on the irreducibility of rules: that one can always ask for the explanation of a rule, explanation of an explanation an so on ad infinitum. So it is essentially impossible to determine whether a rule is being applied correctly or not. It so happens that we (humans) can stop asking and 'apply'; but I'm not convinced this is something machines cannot do.

fiatjaf 9y ago
Wittgenstein's "problem on the irreducibility of rules" is brainfart at best. Any rule that can be actually _followed_ cannot be logically precise, and vice-versa. If you try to reduce anything from the real-world to pure logic you'll surely end up crazy.
- deleted 9y ago
  [deleted]
- posterboy 9y ago
  there are abstract rules and practical laws, principles, whatchamacallit. Laws bind abstract rules to real thingies. The ground truth values are right and wrong en lieu of good and bad - basic emotions that are at the basis of experience. Trying to verbalize (the word logic is related to logos, old greek for tongue, language logic) every basic emotion you will surely go irrationaly crazy. Therefore, context is assumed and language is underspecific.

fiatjaf 9y ago

> A human mind looking at the Gödel sentence while strictly adhering to the same formal structure could also not see that the Gödel sentence is true. To say that you assume a human mind is a machine. Why isn't that begging the question?

Udo 9y ago
No, it means I think a human mathematician is capable of working within a formal structure by following the rules of that formalism. It's a way of saying that humans can do formal math. Of course you are correct in suspecting I do assume the human mind is a machine, because we have seen zero evidence that it's not the case, but that's not in any way part of the sentence you quoted.
- fiatjaf 9y ago
  Ok, I get it. I had misunderstood your argument.