Using a claim justified by observation and experiment to undermine the sufficiency of observation with regards to evaluating claims in general seems off to me.
Proof by contradition is a standard way to make proofs in mathematics.
true statements that are literally impossible to justify beyond a priori belief which requires that one must always to some extent appeal to a belief that is never further justified. I personally don’t accept this.
The fact that there are truths that one can’t verify, means that there are things one doesn’t know (and Gödel showed that on a very fundamental level). Being skeptic and not believing things without evidence means that there are things that one doesn’t know and can’t know.
>Heisenberg’s uncertainty principle clearly demostrates that there are claims about the physical world that we can’t evaluate as through or false through observation and science.
What you are saying implies, for example, that a particle’s momentum has a precise value but cannot be known by observation if the particle’s position is known with certainty. How do know this? It could just as well be that the particle’s position and momentum are mutually exclusive to a degree such that if the position is known with high certainty, then the momentum does not have a precise objective value. This is the standard view of most physicists. Rejecting this requires there to be non-local effects such as in Bohmian Mechanics.
Gödel proved that formal systems of sufficient expressive capabilities cannot prove all true statements regarding themselves. Relatedly, there are possible situations where people cannot know their futures actions because they could have the determination to do the opposite of what an oracle machine that analyzes their brain may say to guarantee that the oracle is wrong. This is a limitation or feature of systems with sufficient recursive capability. This says nothing of what can known in general. An outside observer could analyze the oracle machine and subject system and know the subject’s future action as long as the outside observer does not interfere with the system and become bound up in it. A personal knowledge limitation is not an absolute limitation, whether it be a formal system or person. What is unprovable in the domain of one formal system need not be unprovable by other formal systems.
Not all theorems can be proven with respect to a single formal system or person, but the key word here is “proven.” Any mathematical claim can be justified by observation. One can test a theorem by testing many cases. One can for example test whether the addition of even numbers always results in an even number by trying out many cases. With each case one’s probability belief of the theorem being true increases. The halting problem, related to Gödel’s incompleteness theorems, can be solved in the limit this way. A computer can run a program and assume that it does not halt. If the program does halt then the computer changes its claim. This way the computer is guaranteed to be right eventually, but it is unknown how long it will take for it to be correct. This corresponds to Bayesian updating where knowledge is increased throughout time with observation. One converges to correctness in the limit.
Besides, my argument was regarding claims of the universe and mind, not mathematics. If you have a better way than experimentation and observation to justify claims of consciousness and identity, then I would be ecstatic to hear it. Justifying worldly and consciousness claims with complexity and a priori probabilities is fine and even necessary as a starting point, but if there is no way even in principle to further justify them, then I am skeptical. Even math, which people say is above observation itself, can be justified in the limit by observation.
There are things we don’t know. There are questions where we don’t know the answer. Both saying “I know that identity survives cryonics” and saying “I know that identity survives cryonics” require justification. The position of not knowing doesn’t.
Gödel established fundamental limits on a very specific notion of “knowing”, a proof, that is, a sequence of statements that together justify a theorem to be true with absolute certainty.
If one relaxes the definition of knowing by removing the requirement of absolute certainty within a finite time, then one is not so restricted by Gödel’s theorem. Theorems regarding nonfractional numbers such as what Gödel used can be known to be true or false in the limit by checking each number to check whether the theorem holds.
Theorems of the nature “there exists a number x such that” can be initially set false. If such a theorem is true then one will know eventually by checking each case. If it is not true, then one is correct from the start. Theorems of the nature “for all numbers x P(x) holds” can be initially set true. If such a theorem is false then one will know eventually by checking the cases. If such a theorem is true then one is correct from the start.
The limitation here is absolute certainty within a finite time. One can be guaranteed to be correct eventually, but not know at which point in time correctness will occur.
Proof by contradition is a standard way to make proofs in mathematics.
The fact that there are truths that one can’t verify, means that there are things one doesn’t know (and Gödel showed that on a very fundamental level). Being skeptic and not believing things without evidence means that there are things that one doesn’t know and can’t know.
>Heisenberg’s uncertainty principle clearly demostrates that there are claims about the physical world that we can’t evaluate as through or false through observation and science.
What you are saying implies, for example, that a particle’s momentum has a precise value but cannot be known by observation if the particle’s position is known with certainty. How do know this? It could just as well be that the particle’s position and momentum are mutually exclusive to a degree such that if the position is known with high certainty, then the momentum does not have a precise objective value. This is the standard view of most physicists. Rejecting this requires there to be non-local effects such as in Bohmian Mechanics.
Gödel proved that formal systems of sufficient expressive capabilities cannot prove all true statements regarding themselves. Relatedly, there are possible situations where people cannot know their futures actions because they could have the determination to do the opposite of what an oracle machine that analyzes their brain may say to guarantee that the oracle is wrong. This is a limitation or feature of systems with sufficient recursive capability. This says nothing of what can known in general. An outside observer could analyze the oracle machine and subject system and know the subject’s future action as long as the outside observer does not interfere with the system and become bound up in it. A personal knowledge limitation is not an absolute limitation, whether it be a formal system or person. What is unprovable in the domain of one formal system need not be unprovable by other formal systems.
Not all theorems can be proven with respect to a single formal system or person, but the key word here is “proven.” Any mathematical claim can be justified by observation. One can test a theorem by testing many cases. One can for example test whether the addition of even numbers always results in an even number by trying out many cases. With each case one’s probability belief of the theorem being true increases. The halting problem, related to Gödel’s incompleteness theorems, can be solved in the limit this way. A computer can run a program and assume that it does not halt. If the program does halt then the computer changes its claim. This way the computer is guaranteed to be right eventually, but it is unknown how long it will take for it to be correct. This corresponds to Bayesian updating where knowledge is increased throughout time with observation. One converges to correctness in the limit.
Besides, my argument was regarding claims of the universe and mind, not mathematics. If you have a better way than experimentation and observation to justify claims of consciousness and identity, then I would be ecstatic to hear it. Justifying worldly and consciousness claims with complexity and a priori probabilities is fine and even necessary as a starting point, but if there is no way even in principle to further justify them, then I am skeptical. Even math, which people say is above observation itself, can be justified in the limit by observation.
There are things we don’t know. There are questions where we don’t know the answer. Both saying “I know that identity survives cryonics” and saying “I know that identity survives cryonics” require justification. The position of not knowing doesn’t.
How do you know there are things you cannot know eventually?
Gödel. And no the halting problem is separate from Gödels arguments.
Gödel established fundamental limits on a very specific notion of “knowing”, a proof, that is, a sequence of statements that together justify a theorem to be true with absolute certainty.
If one relaxes the definition of knowing by removing the requirement of absolute certainty within a finite time, then one is not so restricted by Gödel’s theorem. Theorems regarding nonfractional numbers such as what Gödel used can be known to be true or false in the limit by checking each number to check whether the theorem holds.
Theorems of the nature “there exists a number x such that” can be initially set false. If such a theorem is true then one will know eventually by checking each case. If it is not true, then one is correct from the start. Theorems of the nature “for all numbers x P(x) holds” can be initially set true. If such a theorem is false then one will know eventually by checking the cases. If such a theorem is true then one is correct from the start.
The limitation here is absolute certainty within a finite time. One can be guaranteed to be correct eventually, but not know at which point in time correctness will occur.