5 ms·

I really don't think that this is the case. Godel proved that some truths cannot be proven. That seems pretty fundamental, and I think that's what made him so i

by trgn 3y ago

I really don't think that this is the case. Godel proved that some truths cannot be proven. That seems pretty fundamental, and I think that's what made him so impactful, e.g. much more than e.g. the halting problem, which is about computation solely. What Godel also showed was that we can expand our a prioris, e.g. add another truth that "feels" intuitive to the axioms, and then we can understand some theorem or truth (ie. look under the hood, understand its mechanics). But that means expanding the system by some sort of oracle, some sort of "genius intuition". Fair enough, mathematicians will agree by consensus if it's the right think to do, but it's also infinitely regressive. So I'd agree with corysama here "we won't be able to get past regardless of how much time and cleverness we apply to them".

auggierose 3y ago
> Godel proved that some truths cannot be proven. If by proof you mean using a fixed formal axiom system, then yes, you are right. But let's look at one particular such thing that cannot be proven formally within itself: The consistency of Peano arithmetic. But, I don't need a formal proof for that: It's obviously consistent, because the natural numbers form a model for Peano arithmetic. So I just gave you a proof, but it is not a formal one. Yet, I don't see how anyone of sound mind can reject this proof. So Peano arithmetic is obviously something we can assume when proving other, more complicated truths. And nobody can say, not even Gödel, that we will not keep finding similarly obvious truths for the more complicated truth we otherwise care about, but which might not be provable from the currently established axioms. Given that Gödel is a Platonist, he would probably agree with me on that.
- hnfong 3y ago
  I'm not sure you're responding to the same point by the GP. GP claims some truths cannot be proven. You took one claim that could be proven ( https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof ) and tried to proclaim its truth without referring to the proof. Even the GP's claim is problematic. The Gödel's theorems don't prove that some truths cannot be proven. It only proves either ZF is inconsistent, or some truths in it cannot be proven. Ironically GP's implicit assumption that ZF is indeed consistent mirrors your response in which you claim Peanno is consistent... :D That said, if you somehow proved that "some truths cannot be proven within a fixed formal axiom system (that is sufficiently powerful)", and then you also take into account the Church-Turing thesis, i.e. that Turing machines can fully model our thoughts, then the proof of the claim about "some truths cannot be proven" is more or less complete. Of course you can always posit the existence of some "unknown unknowns" that are outside of our wildest imaginations and thus invalidating whatever we thought we have known. But that's outside of our current ability to reason. I could go on and talk about what I understand about divine inspiration and knowledge, but I suppose those topics are a bit too speculative for the crowd here...
  - trgn 3y ago
    > I could go on and talk about what I understand about divine inspiration and knowledge, but I suppose those topics are a bit too speculative for the crowd here... I'm listening :) !
    - hnfong 3y ago
      Here you go (at least on the knowledge part) - https://hnfong.github.io/public-crap/writings/2023/00-Knowledge_vs_Belief.html https://hnfong.github.io/public-crap/writings/2023/00-Knowle... [Read at your own peril(?). Please don't debate me on this.]
- kragen 3y ago
  you're appealing to common sense to demonstrate the consistency of peano arithmetic. this has two main problems: 1. it's subjective and culturally dependent. most people intuitively reject peano arithmetic in their childhood; until they have been repeatedly told otherwise, they intuitively believe that there must be some biggest number. often in english-speaking countries they think that number is called 'infinity' 2. common sense is also what tells us that the earth is flat, the sun goes around the earth, it's impossible to create a vacuum, things only keep moving as long as there's some force impelling their continued motion, people can't fly, if a vacuum did exist rockets wouldn't work in it, and so on it turns out that once people stopped relying on common sense so much and started using formal axiom systems, they were able to learn a lot of things in only a few centuries that not even their smartest ancestors had ever been able to figure out during thousands of centuries
  - trgn 3y ago
    > stopped relying on common sense This I learned from GEB too I think, the aha moment around 200 years ago when people understood formal axiom system can be arbitrary, and even counterintuitive. Rather than creating a crisis of confidence, and the collapse of the field, it was the start of a whole new era for mathematics (and eventually physics, e.g. it's hard to imagine something like, I don't know, relativity, if math/physics had remained rooted in "common sense").
    - kragen 3y ago
      note that this isn't the first time; you can see it clearly in eudoxus, euclid, archimedes, and the antikythera mechanism, and lucio russo's book tells the story of how science was rapidly developing until the roman conquest brought it to a crashing halt
  - auggierose 3y ago
    So you are saying Kepler and Galileo used formal axiom systems? Give me a break. 1. needs a citation. I never believed such a nonsense. Common sense with respect to the physical world is different to common sense with respect to the world of mathematical objects. In fact I don't like the word "common sense" when applied to mathematics, I prefer the expression "mathematical reasoning". In fact we model the physical world by representing it via mathematical objects, because mathematical reasoning is so much more reliable than common sense with respect to the physical world. In fact the advantage of such a mathematical model which we can subject to mathematical reasoning are so big that we are OK with the fact that the model might only be approximating physical reality. In the mathematical world formal logic is a great tool for mathematical reasoning. Mathematical reasoning tells us that if we can represent our reasoning via formal logic, then we can be sure that it is correct. But formal logic has its limits, and in the end mathematical reasoning is the final arbiter, not formal logic: The peano axioms are consistent, because the natural numbers exist. Which is obvious and doesn't require a formal proof (which does not exist, by the way, and will never exist; but that we can know this, is a testament to the power of mathematical reasoning).

by trgn 3y ago

auggierose 3y ago

> Godel proved that some truths cannot be proven. If by proof you mean using a fixed formal axiom system, then yes, you are right. But let's look at one particular such thing that cannot be proven formally within itself: The consistency of Peano arithmetic. But, I don't need a formal proof for that: It's obviously consistent, because the natural numbers form a model for Peano arithmetic. So I just gave you a proof, but it is not a formal one. Yet, I don't see how anyone of sound mind can reject this proof. So Peano arithmetic is obviously something we can assume when proving other, more complicated truths. And nobody can say, not even Gödel, that we will not keep finding similarly obvious truths for the more complicated truth we otherwise care about, but which might not be provable from the currently established axioms. Given that Gödel is a Platonist, he would probably agree with me on that.

hnfong 3y ago
I'm not sure you're responding to the same point by the GP. GP claims some truths cannot be proven. You took one claim that could be proven ( https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof ) and tried to proclaim its truth without referring to the proof. Even the GP's claim is problematic. The Gödel's theorems don't prove that some truths cannot be proven. It only proves either ZF is inconsistent, or some truths in it cannot be proven. Ironically GP's implicit assumption that ZF is indeed consistent mirrors your response in which you claim Peanno is consistent... :D That said, if you somehow proved that "some truths cannot be proven within a fixed formal axiom system (that is sufficiently powerful)", and then you also take into account the Church-Turing thesis, i.e. that Turing machines can fully model our thoughts, then the proof of the claim about "some truths cannot be proven" is more or less complete. Of course you can always posit the existence of some "unknown unknowns" that are outside of our wildest imaginations and thus invalidating whatever we thought we have known. But that's outside of our current ability to reason. I could go on and talk about what I understand about divine inspiration and knowledge, but I suppose those topics are a bit too speculative for the crowd here...
- trgn 3y ago
  > I could go on and talk about what I understand about divine inspiration and knowledge, but I suppose those topics are a bit too speculative for the crowd here... I'm listening :) !
  - hnfong 3y ago
    Here you go (at least on the knowledge part) - https://hnfong.github.io/public-crap/writings/2023/00-Knowledge_vs_Belief.html https://hnfong.github.io/public-crap/writings/2023/00-Knowle... [Read at your own peril(?). Please don't debate me on this.]
kragen 3y ago
you're appealing to common sense to demonstrate the consistency of peano arithmetic. this has two main problems: 1. it's subjective and culturally dependent. most people intuitively reject peano arithmetic in their childhood; until they have been repeatedly told otherwise, they intuitively believe that there must be some biggest number. often in english-speaking countries they think that number is called 'infinity' 2. common sense is also what tells us that the earth is flat, the sun goes around the earth, it's impossible to create a vacuum, things only keep moving as long as there's some force impelling their continued motion, people can't fly, if a vacuum did exist rockets wouldn't work in it, and so on it turns out that once people stopped relying on common sense so much and started using formal axiom systems, they were able to learn a lot of things in only a few centuries that not even their smartest ancestors had ever been able to figure out during thousands of centuries
- trgn 3y ago
  > stopped relying on common sense This I learned from GEB too I think, the aha moment around 200 years ago when people understood formal axiom system can be arbitrary, and even counterintuitive. Rather than creating a crisis of confidence, and the collapse of the field, it was the start of a whole new era for mathematics (and eventually physics, e.g. it's hard to imagine something like, I don't know, relativity, if math/physics had remained rooted in "common sense").
  - kragen 3y ago
    note that this isn't the first time; you can see it clearly in eudoxus, euclid, archimedes, and the antikythera mechanism, and lucio russo's book tells the story of how science was rapidly developing until the roman conquest brought it to a crashing halt
- auggierose 3y ago
  So you are saying Kepler and Galileo used formal axiom systems? Give me a break. 1. needs a citation. I never believed such a nonsense. Common sense with respect to the physical world is different to common sense with respect to the world of mathematical objects. In fact I don't like the word "common sense" when applied to mathematics, I prefer the expression "mathematical reasoning". In fact we model the physical world by representing it via mathematical objects, because mathematical reasoning is so much more reliable than common sense with respect to the physical world. In fact the advantage of such a mathematical model which we can subject to mathematical reasoning are so big that we are OK with the fact that the model might only be approximating physical reality. In the mathematical world formal logic is a great tool for mathematical reasoning. Mathematical reasoning tells us that if we can represent our reasoning via formal logic, then we can be sure that it is correct. But formal logic has its limits, and in the end mathematical reasoning is the final arbiter, not formal logic: The peano axioms are consistent, because the natural numbers exist. Which is obvious and doesn't require a formal proof (which does not exist, by the way, and will never exist; but that we can know this, is a testament to the power of mathematical reasoning).