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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 presen

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.
  - guscost 6y ago
    Thanks, I think this makes sense, and it's quite a trip! But I'm a little foggy about any differences between "externally inconsistent", "indeed inconsistent", and "inconsistent overall". I'm also guessing at a few pieces like what you mean by "our theory" - I assume that is "our system of ZFC + ~Con(ZFC)". Does this corollary have a name? > This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background. I don't agree at all, and you have not addressed my other point. Formal systems which are used for science aren't designed to be paradox-systems, and truth in those systems is always intended to correspond with reality.
danharaj 6y ago
You can easily adapt the incompleteness theorem to empirical science: take its computational analogue, undecidability. There is no scientific method that can answer whether a computer emulating a turing machine, a physical phenomenon, will end with the computer accepting. If the fact that turing machines are infinite bothers you, then just restrict to some plausible but large amount of memory. It's still out of reach of science. We don't have to do much, we just have to show that there is a well posed physical question that no human scientific method could answer. But we go further than that, because incompleteness and its many guises say more: not only are there questions you can't reasonably answer, you can't separate the unreasonable questions from the reasonable ones. This applies to all science. So, all science which attempts to answer questions either is surefooted and facile or sophisticated and destined to make "mistakes" of pursuing questions that it won't be able to answer. But then what exactly counts as a "good" mistake which advances science and a "bad" one which is pseudoscience?
- guscost 6y ago
  Great arguments, thank you.
- dwohnitmok 6y ago
  There are two aspects I disagree with here. The first is formal relevancy. These questions only make sense if we already assume we have a theory of everything. We know we don't. From a formal perspective the totality of the formal theories behind modern science are inconsistent. So results like Godel's incompleteness theorems don't apply formally. Okay, but what if we do get a theory of everything? What happens if a string theorist comes out and says they've solved it? Well let's examine the Game of Life, where we do have its theory of everything (the game rules) and it happens to be undecidable. In particular, given an initial configuration, a pixel X might be eventually shaded in if and only if ZFC is inconsistent. Let's assume the game is playing in a screen. So is an investigation into whether pixel X on the screen will be shaded scientific? This brings me to my second point. Popper would probably say yes. This is a classic example of the problem of induction that is falsifiable. Kuhn would probably say he needs more information about how the study is going to be carried out and the underlying metatheory of the researchers. Lakatos probably would say something similar to Kuhn in this case. Yet I have a hard time believing any of these people would be hung up on the decidability part of the problem. As an empirical phenomenon whether pixel X on the screen will be shaded seems little different from whether a bowling ball on a plank will start spontaneously levitating. It probably won't, but oh boy will it be interesting if it does! And crucially in both cases I think, given the right setup, most everyone could agree that staring at the pixel or staring at the bowling ball for a long long time constitutes a scientific experiment, albeit a boring one. Perhaps the objection might be raised that these are different. We don't have a theory of everything behind the bowling ball, but we do have a theory of everything behind that pixel. Well if your theory is fundamental enough to completely obviate the need for empirical experimentation, I don't think many people would agree the subsequent logical work would qualify as science. To put it another way decidability and incompleteness are not very helpful sociological explanations of science. There's nothing stopping someone from saying that the definition of science is intrinsically tied to decidability. After all people can make whatever definitions they want. But it doesn't seem useful to me, because it doesn't explain what it is that people who call themselves scientists applaud and what they frown on. That is it's not a very useful descriptivist framework. That is whether a question is decidable or not does not seem to be the criteria of whether a question is reasonable or not. Indeed let's not talk about hypotheticals. What do scientists who are familiar with the incompleteness theorems have to say? Well Hawking talks about it: http://www.hawking.org.uk/godel-and-the-end-of-physics.html http://www.hawking.org.uk/godel-and-the-end-of-physics.html but doesn't use it to split the scientific from the unscientific. In fact he seems to derive joy from the incompleteness theorems. Indeed I have a hard time finding people who view logical completeness or decidability as the differentiating factor between pseudoscience and science as ways of thinking and acting, rather than as potential limiting results on what the theory of everything would look like.

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.
- guscost 6y ago
  Thanks, I think this makes sense, and it's quite a trip! But I'm a little foggy about any differences between "externally inconsistent", "indeed inconsistent", and "inconsistent overall". I'm also guessing at a few pieces like what you mean by "our theory" - I assume that is "our system of ZFC + ~Con(ZFC)". Does this corollary have a name? > This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background. I don't agree at all, and you have not addressed my other point. Formal systems which are used for science aren't designed to be paradox-systems, and truth in those systems is always intended to correspond with reality.

danharaj 6y ago

You can easily adapt the incompleteness theorem to empirical science: take its computational analogue, undecidability. There is no scientific method that can answer whether a computer emulating a turing machine, a physical phenomenon, will end with the computer accepting. If the fact that turing machines are infinite bothers you, then just restrict to some plausible but large amount of memory. It's still out of reach of science. We don't have to do much, we just have to show that there is a well posed physical question that no human scientific method could answer. But we go further than that, because incompleteness and its many guises say more: not only are there questions you can't reasonably answer, you can't separate the unreasonable questions from the reasonable ones. This applies to all science. So, all science which attempts to answer questions either is surefooted and facile or sophisticated and destined to make "mistakes" of pursuing questions that it won't be able to answer. But then what exactly counts as a "good" mistake which advances science and a "bad" one which is pseudoscience?

guscost 6y ago
Great arguments, thank you.
dwohnitmok 6y ago
There are two aspects I disagree with here. The first is formal relevancy. These questions only make sense if we already assume we have a theory of everything. We know we don't. From a formal perspective the totality of the formal theories behind modern science are inconsistent. So results like Godel's incompleteness theorems don't apply formally. Okay, but what if we do get a theory of everything? What happens if a string theorist comes out and says they've solved it? Well let's examine the Game of Life, where we do have its theory of everything (the game rules) and it happens to be undecidable. In particular, given an initial configuration, a pixel X might be eventually shaded in if and only if ZFC is inconsistent. Let's assume the game is playing in a screen. So is an investigation into whether pixel X on the screen will be shaded scientific? This brings me to my second point. Popper would probably say yes. This is a classic example of the problem of induction that is falsifiable. Kuhn would probably say he needs more information about how the study is going to be carried out and the underlying metatheory of the researchers. Lakatos probably would say something similar to Kuhn in this case. Yet I have a hard time believing any of these people would be hung up on the decidability part of the problem. As an empirical phenomenon whether pixel X on the screen will be shaded seems little different from whether a bowling ball on a plank will start spontaneously levitating. It probably won't, but oh boy will it be interesting if it does! And crucially in both cases I think, given the right setup, most everyone could agree that staring at the pixel or staring at the bowling ball for a long long time constitutes a scientific experiment, albeit a boring one. Perhaps the objection might be raised that these are different. We don't have a theory of everything behind the bowling ball, but we do have a theory of everything behind that pixel. Well if your theory is fundamental enough to completely obviate the need for empirical experimentation, I don't think many people would agree the subsequent logical work would qualify as science. To put it another way decidability and incompleteness are not very helpful sociological explanations of science. There's nothing stopping someone from saying that the definition of science is intrinsically tied to decidability. After all people can make whatever definitions they want. But it doesn't seem useful to me, because it doesn't explain what it is that people who call themselves scientists applaud and what they frown on. That is it's not a very useful descriptivist framework. That is whether a question is decidable or not does not seem to be the criteria of whether a question is reasonable or not. Indeed let's not talk about hypotheticals. What do scientists who are familiar with the incompleteness theorems have to say? Well Hawking talks about it: http://www.hawking.org.uk/godel-and-the-end-of-physics.html http://www.hawking.org.uk/godel-and-the-end-of-physics.html but doesn't use it to split the scientific from the unscientific. In fact he seems to derive joy from the incompleteness theorems. Indeed I have a hard time finding people who view logical completeness or decidability as the differentiating factor between pseudoscience and science as ways of thinking and acting, rather than as potential limiting results on what the theory of everything would look like.