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You're gonna love incompleteness if you can grok it: https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems https://en.wikipedia.org/wiki/Gödel%27s_inc

You're gonna love incompleteness if you can grok it: https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

pfdietz 6y ago
Yes, yes, I know all about that theorem. How does it have any relevance to what I or this thread were talking about? Maybe because it stopped Hilbert's program?
- guscost 6y ago
  Kuhn's philosophy is a direct descendant of incompleteness. In a sense, all of postmodernism is too.
  - pfdietz 6y ago
    Ok, I don't see that at all.
    - guscost 6y ago
      I don't think you grok it yet ;) Hint: Every scientific theory is built on top of a formal system.
      - pfdietz 6y ago
        It has jack shit to do with the social aspects of what scientists actually do, though. And I am sure the vast majority of scientists do not care at all about the formal foundations of mathematics.
        guscost 6y ago
        It doesn't matter who cares about it, Godel proved that anything Kuhn calls a "paradigm" must be inconsistent and/or incomplete, and therefore must fail eventually. If that were not true, then some complete paradigm would arrive eventually and be dominant forever.
        pfdietz 6y ago
        I'm sorry, but from the point of view of a working scientist, or someone benefiting from science, or someone trying to understand how scienists work socially, this is just bafflegab and silliness. It's just irrelevant and of zero value, and has nothing to do with what Kuhn was trying to explain. Sure, science cannot explain everything. There was never any chance it was going to. Incompleteness has nothing to do with it.
        guscost 6y ago
        "What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted." - Stephen Hawking
        pfdietz 6y ago
        That doesn't contradict anything I wrote.
dwohnitmok 6y ago
At risk of being a bit uncharitable here, I think you may be overstretching the implications of the incompleteness theorems here. FWIW I think the usual presentations of the incompleteness theorems that talk about truth may it sound much more mysterious than it is (and lead people without a grounding in model theory very far astray). In fact I don't like to really talk to about truth or really even too much about proof since again they often lead people astray. Here's the informal-ish presentation I prefer: First incompleteness theorem: Given a formal system of axioms expressive enough to do arithmetic, there will always be theorems which are independent of these axioms, i.e. adding that theorem or its negation as a new axiom are both consistent with the other axioms (but of course not adding both at the same time). Second incompleteness theorem: Assuming that system of axioms is itself consistent, one of those aforementioned independent theorems will always be the theorem "all previous axioms listed are consistent with each other." Side note: a closer inspection of the second incompleteness theorem is perhaps slightly mind-bending. Note that we're talking independent here! That is the theorem "the previous axioms are inconsistent with each other" is also a valid axiom to add and results in a system of axioms that is consistent overall!
- guscost 6y ago
  In a profound sense, science is the application of formal systems to the natural world. Within those systems "truth" absolutely does intend to mean "truth". > the theorem "the previous axioms are inconsistent with each other" is also a valid axiom to add and results in a system of axioms that is consistent overall! Ehh, that's just another paradox. I don't think those are allowed in an overall-consistent system.
  - dwohnitmok 6y ago
    Nope they're perfectly allowed! ZFC + ~Con(ZFC) is a perfectly valid axiom system that is consistent if and only if ZFC is consistent. In fact as a result ZFC + ~Con(ZFC) cannot prove any theorem that would contradict ZFC. Indeed basically all of modern mathematics looks the same under ZFC + ~Con(ZFC) as it does under ZFC. But how can that be? What happened to the Principle of Explosion? How is this even coherent? The trick is that ~Con(ZFC) being true doesn't mean that... well it's true (in the usual way that we talk about it). This is why I find the discussion of truth in Godel's incompleteness theorems to be distractions without a backdrop of model theory. Here it is important to delineate between an "internal" and "external" perspective. For a formal system S, the statement Con(S) is at the end of the day an arithmetic statement postulating the existence of a natural number satisfying certain constraints (this is why Godel's incompleteness theorems require the ability to perform arithmetic). This is the internal perspective, asking about a natural number. Externally, we know of the correspondence between the natural numbers and proofs and sentences, but that is external to our theory. ZFC cannot use the statement ~Con(ZFC), which we externally know encodes a proof that 0 = 1, to prove internally that 0 in fact is equal to 1. Thus we avoid the Principle of Explosion within ZFC. Indeed in this case, it turns out that what ZFC + ~Con(ZFC) thinks is a natural number is not unique! Different universes of sets that agree on ZFC might have different notions of natural numbers. Indeed the element that ZFC + ~Con(ZFC) thinks is a natural number satisfying ~Con(ZFC) might not be something another version of ZFC thinks is a natural number. And so when externally we go to try to prove that ~Con(ZFC) shows that ZFC is indeed inconsistent our mapping breaks down because we disagree that the witness to ~Con(ZFC) is a natural number externally. This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background.

You're gonna love incompleteness if you can grok it: https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

pfdietz 6y ago

Yes, yes, I know all about that theorem. How does it have any relevance to what I or this thread were talking about? Maybe because it stopped Hilbert's program?

guscost 6y ago
Kuhn's philosophy is a direct descendant of incompleteness. In a sense, all of postmodernism is too.
- pfdietz 6y ago
  Ok, I don't see that at all.
  - guscost 6y ago
    I don't think you grok it yet ;) Hint: Every scientific theory is built on top of a formal system.
    - pfdietz 6y ago
      It has jack shit to do with the social aspects of what scientists actually do, though. And I am sure the vast majority of scientists do not care at all about the formal foundations of mathematics.
      - guscost 6y ago
        It doesn't matter who cares about it, Godel proved that anything Kuhn calls a "paradigm" must be inconsistent and/or incomplete, and therefore must fail eventually. If that were not true, then some complete paradigm would arrive eventually and be dominant forever.
        pfdietz 6y ago
        I'm sorry, but from the point of view of a working scientist, or someone benefiting from science, or someone trying to understand how scienists work socially, this is just bafflegab and silliness. It's just irrelevant and of zero value, and has nothing to do with what Kuhn was trying to explain. Sure, science cannot explain everything. There was never any chance it was going to. Incompleteness has nothing to do with it.
        guscost 6y ago
        "What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted." - Stephen Hawking
        pfdietz 6y ago
        That doesn't contradict anything I wrote.

dwohnitmok 6y ago

At risk of being a bit uncharitable here, I think you may be overstretching the implications of the incompleteness theorems here. FWIW I think the usual presentations of the incompleteness theorems that talk about truth may it sound much more mysterious than it is (and lead people without a grounding in model theory very far astray). In fact I don't like to really talk to about truth or really even too much about proof since again they often lead people astray. Here's the informal-ish presentation I prefer: First incompleteness theorem: Given a formal system of axioms expressive enough to do arithmetic, there will always be theorems which are independent of these axioms, i.e. adding that theorem or its negation as a new axiom are both consistent with the other axioms (but of course not adding both at the same time). Second incompleteness theorem: Assuming that system of axioms is itself consistent, one of those aforementioned independent theorems will always be the theorem "all previous axioms listed are consistent with each other." Side note: a closer inspection of the second incompleteness theorem is perhaps slightly mind-bending. Note that we're talking independent here! That is the theorem "the previous axioms are inconsistent with each other" is also a valid axiom to add and results in a system of axioms that is consistent overall!

guscost 6y ago
In a profound sense, science is the application of formal systems to the natural world. Within those systems "truth" absolutely does intend to mean "truth". > the theorem "the previous axioms are inconsistent with each other" is also a valid axiom to add and results in a system of axioms that is consistent overall! Ehh, that's just another paradox. I don't think those are allowed in an overall-consistent system.
- dwohnitmok 6y ago
  Nope they're perfectly allowed! ZFC + ~Con(ZFC) is a perfectly valid axiom system that is consistent if and only if ZFC is consistent. In fact as a result ZFC + ~Con(ZFC) cannot prove any theorem that would contradict ZFC. Indeed basically all of modern mathematics looks the same under ZFC + ~Con(ZFC) as it does under ZFC. But how can that be? What happened to the Principle of Explosion? How is this even coherent? The trick is that ~Con(ZFC) being true doesn't mean that... well it's true (in the usual way that we talk about it). This is why I find the discussion of truth in Godel's incompleteness theorems to be distractions without a backdrop of model theory. Here it is important to delineate between an "internal" and "external" perspective. For a formal system S, the statement Con(S) is at the end of the day an arithmetic statement postulating the existence of a natural number satisfying certain constraints (this is why Godel's incompleteness theorems require the ability to perform arithmetic). This is the internal perspective, asking about a natural number. Externally, we know of the correspondence between the natural numbers and proofs and sentences, but that is external to our theory. ZFC cannot use the statement ~Con(ZFC), which we externally know encodes a proof that 0 = 1, to prove internally that 0 in fact is equal to 1. Thus we avoid the Principle of Explosion within ZFC. Indeed in this case, it turns out that what ZFC + ~Con(ZFC) thinks is a natural number is not unique! Different universes of sets that agree on ZFC might have different notions of natural numbers. Indeed the element that ZFC + ~Con(ZFC) thinks is a natural number satisfying ~Con(ZFC) might not be something another version of ZFC thinks is a natural number. And so when externally we go to try to prove that ~Con(ZFC) shows that ZFC is indeed inconsistent our mapping breaks down because we disagree that the witness to ~Con(ZFC) is a natural number externally. This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background.