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A human only can fit only finitely much information in their memory. There are formal systems that no human could fit a description of into their mind. Also, t

by drdeca 7y ago

A human only can fit only finitely much information in their memory. There are formal systems that no human could fit a description of into their mind. Also, the way humans reason seems more like something paraconsistent than something truly self-consistent. A person can be convinced of something by some argument, and then convinced the other way. It is true that the human mind is not an example of [thing which the incompleteness theorems show cannot exist], but this is not surprising.

yters 7y ago
We can augment our memory with physical storage. We might not analyze within the lifetime or capacity of the universe (or multiverse), but that's a practical limitation, not a theoretical one. Humans are irrational and all, but we make surprisingly consistent formal systems. I don't think an inconsistent system could do this, so this suggests the human mind is something beyond finite formal systems. For example, we could model this ability as access to an infinite table of all possible Turing machines and their halting status, perhaps with some noise added to represent our irrationality. We would still have something beyond the limitations of goedel's theorem, as well as computation and randomness, which would account for our intuition in the matter. So there is no good reason to dismiss the idea there is something special about the human mind not captured by formal systems.
- drdeca 7y ago
  I do agree that there is something special about the human mind not captured by formal systems (in the "set of axioms or axiom schemas + inference rules" sense). (not so much for empirical external evidence reasons, so much as for like, what I want to call "teleological reasons"? Like, my goals and values are more coherent/less nonsensical assuming that there is something special, so it makes sense (as far as my goals are concerned) for me to assume that there is something special.) However, I'm not so sure that this something special includes anything Gödel related. As to whether an "inconsistent system" could "make" consistent formal systems, I'm not entirely sure what you mean. A formal system (in the sense of set of axioms + inference rules) is not an agent, so I'm not sure what you mean about it "making" things. We talk about a formal system prov"ing" something, but perhaps a more clear way of saying what we mean by that, is that the system admits a proof of the thing, or that there is a proof of the thing, for/within that system. If by a system "making" a system, we just mean that it "proves" (as in, "admits a proof") that some other system is consistent, then I see no reason that a paraconsistent system couldn't show that some other system is consistent. Have you seen the "Logical Induction" paper by MIRI ? If a system is inconsistent, then there is a proof that it is inconsistent. Logical induction assigns "probabilities" to different mathematical statements, and it can be shown that these probabilities converge in reasonable ways. If there is a proof of a statement in the system that the logical induction thing is working under, then the "probability" of that statement will converge to 1. Also, this statements that logical induction applies probabilities to include statements about itself. I think that, if some particular system is self consistent, then, well, for one thing, the probability for the statement "that system is inconsistent" will not go to 1, and furthermore, that there are statements of "the probability for 'that system is inconsistent' will be below c at step t" (for t much larger than the current time step) which will (I think) go to 1. In that sense, I think logical induction could probably do a good job at concluding that some systems are probably self consistent. Now, I'm not totally sure of that. I don't 100% understand the logical induction results. But, yeah. Seems relevant to me.
  - yters 7y ago
    Not looking for agreement. I think there is an empirical, mathematical case to be made for this 'specialness' of the mind that is beyond what computation and randomness can give us. Another way to state this is in terms of the halting problem. Halting oracles are possible and detectable. Halting oracles can authenticate the Gödel sentences for all finite, consistent formal systems. Looking at the logical induction page, I'm not sure it offers anything beyond Solomonoff. If they have a computable method, then it will be deficient compared to Solomonoff induction. Leonid Levin published something called randomness conservation in 1984, which implies computers cannot learn math, or any other consistent logic, beyond what they are initially fed with. So, given an initial set of consistent axioms, a computer cannot come up with anything new and consistent. Anyways, bottom line of what I'm saying is halting oracles are possible and empirically detectable. We should entertain the possibility that the human mind is a halting oracle, and set about researching that idea. We are making no progress with our assumption the mind is computational.
    - drdeca 7y ago
      Solomonoff induction can make eventually good predictions about any computable environment, but: While Solomonoff induction is limit computable, it isn’t computable, while logical induction is. Logical induction can reach conclusions about what a system using a computable approximation to solomonoff induction would conclude. Actually, it can also assign probabilities to what full solomonoff induction would conclude. Solomonoff induction cannot accurately model a world which contains agents who use solomonoff induction. Logical induction plays nicely with self reference. Importantly, with some naive ways of attempting to do probabilistic induction on mathematical statements, if you give the system different facts in an adversarial order, you can cause it to alternate between being highly convinced that something is true, and that it is false. Logical induction solves this problem. I don’t think solomonoff induction can be straightforwardly applied to this task. Solomonoff induction predicts what inputs it will receive, but this isn’t, I think, immediately well suited for assigning credence to mathematical statements. I’m not sure how a halting oracle could be empirically detected? Well, I guess if one believed that “this is like a halting oracle except only for the first 3^^3 Turing machines” and the like we’re substantially less likely than “this is a halting oracle in general”, which I guess seems reasonable. Ok, I guess I can see that, sorta. Not sure what practical tests you have in mind though.

by drdeca 7y ago

yters 7y ago

We can augment our memory with physical storage. We might not analyze within the lifetime or capacity of the universe (or multiverse), but that's a practical limitation, not a theoretical one. Humans are irrational and all, but we make surprisingly consistent formal systems. I don't think an inconsistent system could do this, so this suggests the human mind is something beyond finite formal systems. For example, we could model this ability as access to an infinite table of all possible Turing machines and their halting status, perhaps with some noise added to represent our irrationality. We would still have something beyond the limitations of goedel's theorem, as well as computation and randomness, which would account for our intuition in the matter. So there is no good reason to dismiss the idea there is something special about the human mind not captured by formal systems.

drdeca 7y ago
I do agree that there is something special about the human mind not captured by formal systems (in the "set of axioms or axiom schemas + inference rules" sense). (not so much for empirical external evidence reasons, so much as for like, what I want to call "teleological reasons"? Like, my goals and values are more coherent/less nonsensical assuming that there is something special, so it makes sense (as far as my goals are concerned) for me to assume that there is something special.) However, I'm not so sure that this something special includes anything Gödel related. As to whether an "inconsistent system" could "make" consistent formal systems, I'm not entirely sure what you mean. A formal system (in the sense of set of axioms + inference rules) is not an agent, so I'm not sure what you mean about it "making" things. We talk about a formal system prov"ing" something, but perhaps a more clear way of saying what we mean by that, is that the system admits a proof of the thing, or that there is a proof of the thing, for/within that system. If by a system "making" a system, we just mean that it "proves" (as in, "admits a proof") that some other system is consistent, then I see no reason that a paraconsistent system couldn't show that some other system is consistent. Have you seen the "Logical Induction" paper by MIRI ? If a system is inconsistent, then there is a proof that it is inconsistent. Logical induction assigns "probabilities" to different mathematical statements, and it can be shown that these probabilities converge in reasonable ways. If there is a proof of a statement in the system that the logical induction thing is working under, then the "probability" of that statement will converge to 1. Also, this statements that logical induction applies probabilities to include statements about itself. I think that, if some particular system is self consistent, then, well, for one thing, the probability for the statement "that system is inconsistent" will not go to 1, and furthermore, that there are statements of "the probability for 'that system is inconsistent' will be below c at step t" (for t much larger than the current time step) which will (I think) go to 1. In that sense, I think logical induction could probably do a good job at concluding that some systems are probably self consistent. Now, I'm not totally sure of that. I don't 100% understand the logical induction results. But, yeah. Seems relevant to me.
- yters 7y ago
  Not looking for agreement. I think there is an empirical, mathematical case to be made for this 'specialness' of the mind that is beyond what computation and randomness can give us. Another way to state this is in terms of the halting problem. Halting oracles are possible and detectable. Halting oracles can authenticate the Gödel sentences for all finite, consistent formal systems. Looking at the logical induction page, I'm not sure it offers anything beyond Solomonoff. If they have a computable method, then it will be deficient compared to Solomonoff induction. Leonid Levin published something called randomness conservation in 1984, which implies computers cannot learn math, or any other consistent logic, beyond what they are initially fed with. So, given an initial set of consistent axioms, a computer cannot come up with anything new and consistent. Anyways, bottom line of what I'm saying is halting oracles are possible and empirically detectable. We should entertain the possibility that the human mind is a halting oracle, and set about researching that idea. We are making no progress with our assumption the mind is computational.
  - drdeca 7y ago
    Solomonoff induction can make eventually good predictions about any computable environment, but: While Solomonoff induction is limit computable, it isn’t computable, while logical induction is. Logical induction can reach conclusions about what a system using a computable approximation to solomonoff induction would conclude. Actually, it can also assign probabilities to what full solomonoff induction would conclude. Solomonoff induction cannot accurately model a world which contains agents who use solomonoff induction. Logical induction plays nicely with self reference. Importantly, with some naive ways of attempting to do probabilistic induction on mathematical statements, if you give the system different facts in an adversarial order, you can cause it to alternate between being highly convinced that something is true, and that it is false. Logical induction solves this problem. I don’t think solomonoff induction can be straightforwardly applied to this task. Solomonoff induction predicts what inputs it will receive, but this isn’t, I think, immediately well suited for assigning credence to mathematical statements. I’m not sure how a halting oracle could be empirically detected? Well, I guess if one believed that “this is like a halting oracle except only for the first 3^^3 Turing machines” and the like we’re substantially less likely than “this is a halting oracle in general”, which I guess seems reasonable. Ok, I guess I can see that, sorta. Not sure what practical tests you have in mind though.