Imre Lakatos and the philosophy of bad science
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You may think the abstract structures that you learn about are fixed fundamental truths, but just read Lakatos’s account of the multi-decade debugging of the Euler characteristic for polyhedra and you will realise that maths is both created and discovered.
How do ya like them apples for yer epistemology!
Another example would be limits vs infinitesimals - which is the “true” foundation of calculus? [1]
Or have you ever struggled with the Axiom of Choice?
[0]: https://en.m.wikipedia.org/wiki/Proofs_and_Refutations
[1]: And the correct answer, as all you Coventry City fans will know, is: trick question
Second, I don't know anything about the debugging of the Euler characteristic, but I suggest that the issue here is better characterized as taking decades to discover the correct representation. Sort of like Edison finding 1000 lightbulb ideas that didn't work, before finding one that did. (Don't take that analogy too far... I'm not suggesting that physical things are discovered in the same way that mathematical things are discovered. (But I'm also not suggesting that they are not.))
With respect to the debugging of the Euler characteristic, one might ask what truth value formulations during those decades of debugging had, from the perspective of those on the ground at the time. Are we so sure in our present age that there no bugs in Andrew Wiles’s work?
Even if you use automated machine proofs (Mathematica and friends) you still can’t be sure that stray cosmic rays don’t flip a bit every damn time.
Fun times.
I will say, though, that acknowledging that math exists independent of its observation makes it more difficult to argue that possible arrangements of atoms do not exist prior to their physical manifestation.
History is, of course, replete with examples of things that we got wrong for thousands of years - the existence of zero, the existence of negative numbers, "real" numbers, as distinct from "imaginary" numbers. These things are now considered trivial, and are taught to children.
I can't imagine Wiles's proof ever being understood by 1% of humanity. I hope, and believe, that there are no catastrophic errors in it, but I wouldn't bet more than $20 that there aren't.
> But, according to Lakatos, when the time comes, a revolution is driven by logic and method, not irrational mob psychology: ‘the Kuhnian “Gestalt-switch” can be performed without removing one’s Popperian spectacles’.
Even the consensus adoption of General Relativity took decades (long enough for an entire generation of older scientists to die), and the author provides no other examples to back up his claim that "results were mixed" in the history of science. I'd love to hear some counterexamples.
My take on this, largely from reading Kuhn, is that the picture of "scientific revolutions" that philosophers have doesn't really apply to anything that has happened in any particular field of science since that field became a science. That's not an accident: it's because the examples the philosophers were actually thinking of in a particular field happened before that field became a science.
For example, when Kuhn talks about paradigm shifts in physics, what he's actually thinking of is the shift from Aristotelian to Galilean/Newtonian physics. That was a truly revolutionary shift, but it was a different kind of shift from, say, the shift from Newtonian physics to relativity, or from classical physics to quantum physics. Steven Weinberg, in a review of Kuhn's book published years ago in the New York Review of Books, notes that, in the latter cases (i.e., in any transition to new paradigms in physics since physics really became a science), the old theory lives on as a special case or particular approximation within the new theory. Newtonian physics still gets used extensively today, because it's a valid approximation to relativity when gravity is weak and speeds are slow compared to the speed of light, and classical physics still gets used extensively today because it's a valid approximation to quantum mechanics when there are a large number of degrees of freedom and quantum interference effects are negligible.
By contrast, nobody does Aristotelian physics today, nor has anybody done it since Newtonian physics took over. Aristotelian physics is not a special case or an approximation to any physical theory we have today; it's a totally different theory of physics that simply is not useful any more, because we've found a much better and more accurate way of modeling the world. So comparing Aristotelian to modern physics is much, much more of a "gestalt switch" or "paradigm shift" than trying to compare Newtonian physics with relativity, or classical physics with quantum physics. And that kind of shift is what Kuhn was actually describing.
Lakatos seems to recognize that there are limitations to Kuhn's approach, but I don't think he really grasps the degree to which all modern science is one thing, and all fits together, and the sort of "revolution" that took us from Aristotelian to modern physics simply doesn't happen once a field becomes a science.
For a great example outside of physics, I can't imagine how anyone could read the story of plate tectonics and not understand it to be exactly what Kuhn was talking about: https://en.wikipedia.org/wiki/Plate_tectonics#Development_of...
I think you're confusing "paradigm" with "model". See below.
> QM has been in conflict with GR since its invention.
"Conflict" in the sense that neither one can possibly be a final fundamental theory, yes.
But whatever final fundamental theory we come up with is going to have our current QM and our current GR as special cases/approximations in the relevant regimes. In terms of any such final fundamental theory, current QM and current GR would simply be particular models built for particular applications.
So current QM and current GR are not different or conflicting "paradigms" in the sense that Aristotelian and Newtonian physics are. They're just our best current models.
> Lambda-CDM is in conflict with MOND and other cosmological paradigms.
These aren't paradigms, they're models. Both of them accept the same underlying framework for model building. They just make different claims about which models are more accurate.
> String theory is a paradigm.
String theorists like to make it seem that way, but it's really not any different from any other quantum field theory in the fundamentals. It's just QFT using strings instead of point particles. The original justification for it as a fundamental theory candidate was that gravity just pops out (since the simplest closed string mode looks like a massless spin-2 particle at low energies, and that's the graviton), so it seemed to be an easy way to get a quantum theory of gravity that would reduce to GR in the classical limit. (I think it's fair to say that the jury is still out on whether this promise has been realized.)
About string theory, "really not any different" and "strings instead of point particles" don't seem compatible. I agree that it is largely an attempt to explain the fine-tuning in the Standard Model, not to overthrow it. But it is in conflict with other paradigms that attempt to "complete" the Standard Model in other ways.
First, why do you need to?
Second, renormalization only has to be done at all in perturbation theory, and perturbation theory is just an approximation anyway. There is plenty of work done in non-perturbative QFT, in which renormalization doesn't even arise at all.
> Are those not incompatible paradigms to the best of our knowledge?
I'm not sure what you mean. Different approximations to the same underlying theory don't have to be compatible unless their domains of validity overlap, and even then the only compatibility they have to have is that they have to make predictions that are consistent with each other.
More generally, s/approximations/models/ in the above, since approximations are one kind of model.
> "really not any different" and "strings instead of point particles" don't seem compatible
Why not? There's nothing about the fundamentals of QFT that says it can only apply to point particles, or that applying it to point particles and applying it to strings are inconsistent. They're different models.
> it is in conflict with other paradigms that attempt to "complete" the Standard Model in other ways.
To the extent that the different attempts at a fundamental theory are inconsistent with each other (e.g., string theory vs. loop quantum gravity, which AFAICT are not consistent with each other), yes, I suppose they could be considered different paradigms. But there will never be a transition from one of them to the other. The most that could happen is that we transition from our current theories as the best we have, to our current theories being approximations to whichever more fundamental theory wins out. (Or it's possible that none of the current candidates will win out.)
What does renormalization actually represent? Why does nature do that? Those rules seem kinda tacked-on after the fact, to me...
> There's nothing about the fundamentals of QFT that says it can only apply to point particles
String Theory does not present a serious challenge to QFT or the Standard Model yet, but it is a massive set of additional ideas that are completely unnecessary for either one, and it is a paradigm in vigorous conflict with all of the other proposed completions to the Standard Model right now. If any of these completions ever wins conclusively, it will also "win" against the Standard Model, in the sense that the Standard Model will turn out to be a higher-level approximation of some truer lower-level theory, one which will never explain any of its fine-tuning, and therefore one which will fail.
It's very similar to GR vs. Newtonian Dynamics, and I would argue that Newtonian Dynamics is in fact a failed paradigm, and it has been replaced by a better paradigm, no matter how useful it may remain.
> There is plenty of work done in non-perturbative QFT, in which renormalization doesn't even arise at all.
Yes, because that stuff doesn't even attempt to be compatible with GR. There is no conflict between the probably-incompatible paradigms of GR and non-perturbative QFT because their respective predictions have zero overlap.
Sure it does. You can do non-perturbative QFT in curved spacetime.
I don't understand what you mean. If I make a prediction in curved spacetime, it's a prediction about what will happen in curved spacetime, not what will happen in flat spacetime. So you can't falsify it by doing an experiment in flat spacetime. You have to do an experiment in curved spacetime. Non-perturbative QFT can certainly make falsifiable predictions about the results of experiments done in curved spacetime, just as it can do so about experiments done in flat spacetime. But the predictions won't be the same, because they are about different experiments under different conditions.
> As I understand it the whole purpose of perturbation is to approximate QFT so that it can predict macro-scale things.
I have no idea why you would think that. Perturbation theory is applicable to any QFT problem; its first major successes were predicting the Lamb shift and the shift in the magnetic moment of the electron due to quantum field effects. Neither of those are "macro-scale" things. The reason not to do perturbation theory, in cases where it is possible to find a non-perturbative (i.e., exact) solution, is that perturbation theory is inherently an approximation, and why would you settle for an approximation if you can compute the exact answer?
Perhaps "make a risky prediction that could be falsified if spacetime was in fact not curved" is a better phrasing. Spacetime is spacetime. It's curved.
> Perturbation theory is applicable to any QFT problem; its first major successes were predicting the Lamb shift and the shift in the magnetic moment of the electron due to quantum field effects.
Oof, looks like I badly underestimated the extremes of scale that are involved here. It turns out you need to use perturbation theory to approximate the solution for something as complex as a single atom of hydrogen. I guess "macro" was an awful choice of word, but that means non-perturbative QFT is even less feasible than I thought for stuff at macro (to say nothing of relativistic) scales.
That doesn't seem to me to be any better. If spacetime is curved, and you make a prediction assuming curved spacetime, you can test that prediction in curved spacetime. The prediction isn't assuming flat spacetime to begin with, so what would it even mean to say it could be falsified if spacetime were not curved?
> that means non-perturbative QFT is even less feasible than I thought for stuff at macro (to say nothing of relativistic) scales
It means no such thing. Non-perturbative QFT is commonly used in condensed matter physics, which deals with objects of macroscopic size. In fact, IIRC it was condensed matter physicists who first developed non-perturbative QFT methods; the particle physicists adopted them later.
As for "relativistic scales", you are aware that relativity has measurable effects even on atomic scales, right? Look up why gold has the color it does, for example. For that matter, our best current understanding of why nucleon masses are what they are has to take relativistic effects into account.
> The prediction isn't assuming flat spacetime to begin with, so what would it even mean to say it could be falsified if spacetime were not curved?
That's how the whole process works, your theory makes a risky prediction (spacetime is curved, therefore we should see X) and then you go check. If spacetime is actually flat, you're gonna see Y instead!
> Non-perturbative QFT is commonly used in condensed matter physics, which deals with objects of macroscopic size. In fact, IIRC it was condensed matter physicists who first developed non-perturbative QFT methods; the particle physicists adopted them later.
Pretty cool, when was that? Do you need a fast computer to do that? But it's not relevant to the rest of this, unless this is a good example where GR also predicts something that those objects do (riskily, anything that simplifies to Newtonian dynamics doesn't count).
> As for "relativistic scales", you are aware that relativity has measurable effects even on atomic scales, right? Look up why gold has the color it does, for example. For that matter, our best current understanding of why nucleon masses are what they are has to take relativistic effects into account.
Yes, I've heard of Dirac. I think you're confusing special and general relativity, and/or I should have used "gravitational scales" instead.
The most famous example is of course Hawking's prediction that black holes emit thermal radiation and will eventually evaporate. This prediction has not been tested yet, but it is testable.
Another example would be the recent test of whether the Newtonian gravitational potential affects the quantum phase of neutrons (it does). This doesn't test any relativistic effects of gravity, but it does provide evidence that gravity is an interaction in the quantum sense.
> which doesn't rely on a kludge like the renormalization
AFAIK Hawking's prediction did not make any use of renormalization, nor do the more recent treatments of the same thing.
> when was that?
The 1960s, IIRC.
> I should have used "gravitational scales" instead
That makes it clearer what you meant, yes.
Ahhh you're right, that's a gravitational-scale thing! But even a rotating black hole has a tiny state vector, which is the only reason that model has any chance of working. Good luck extending this approach to an ordinary pebble, asteroid, or planet. Also the compression-of-quantum-state contributes to the black hole information paradox (a result of Hawking's theory), right? Has that actually been resolved, and until it is, isn't that essentially an incompatibility with [some stuff that I can't talk about intelligently]?
> Another example would be the recent test of whether the Newtonian gravitational potential affects the quantum phase of neutrons (it does). This doesn't test any relativistic effects of gravity, but it does provide evidence that gravity is an interaction in the quantum sense.
Sounds pretty cool, but I doubt I'm going to be able to discuss this meaningfully. Is this related to the Higgs boson? That's the force carrier for gravitation in the Standard Model, if I understand correctly.
So basically your issue here is that QFT doesn't predict that spacetime is curved? Or it doesn't make a prediction either way about whether spacetime is curved or flat? This amounts to saying "we don't yet have a good theory of quantum gravity". Yes, this is true.
> it sounds like you think that the final theory will not be built on QM/QFT/Standard Model foundations
I am skeptical about whether string theory will turn out to be the quantum gravity theory we are searching for, and string theory is the only candidate I'm aware of that is a QFT of any sort. So yes, if string theory doesn't turn out to be right, I think we will be in the position you describe.
> and will be different enough that QM and GR will remain incompatible approximations forever
Only in the same sense that, for example, Newton's laws and Maxwell's Equations are incompatible approximations. I don't see that sort of thing as a problem.
Er, yep. Thanks for taking the time to do roundabout arguments that lead to this very simple thing that we're not even disagreeing about. Sorry for not just stating this in a better way outright.
> string theory is the only candidate I'm aware of that is a QFT of any sort.
Loop Quantum Gravity is built from a QFT called "Group Field Theory" too, that's why they still have their renormalization problem. I suppose you could argue that it's "not comprehensive" enough, or that it's not a serious contender.
> Newton's laws and Maxwell's Equations are incompatible approximations.
...and they are therefore both failed paradigms, but we're not going to agree about that opinion. All I can add is to say that we're never going to find a fifth Maxwell equation that works with the other four and explains more things within that framework. That's all I mean by "failed" and it's probably a controversial perspective.
We don't know whether nature "does that"; renormalization is a feature of our models of nature, but not all features of our models of nature are features of nature itself. That's true of any model.
> I would argue that Newtonian Dynamics is in fact a failed paradigm
You can argue that by defining "failed paradigm" in your preferred way, but you cannot argue that it is a failed paradigm in the sense that Aristotelian physics is a failed paradigm. So by using the term "failed paradigm" to describe both, you are ignoring a very important difference between the two.
Similar remarks would apply to the Standard Model in the scenario where one of our candidates for a more fundamental theory has been confirmed and the SM is now just an approximation to that theory in an appropriate regime.
Bingo! Therefore our models are each wrong in some way. Paradigm shifts happen when a new model (theory, paradigm, etc) that is "less wrong" conquers an older incompatible model that is "more wrong". That incompatibility is the defining characteristic, not the usefulness or the scope of the model. Of course I consider Newtonian Dynamics to be incompatible with GR, one uses a single frame of reference and the other uses infinite frames of reference.
Yes.
> Paradigm shifts happen when a new model (theory, paradigm, etc) that is "less wrong" conquers an older incompatible model that is "more wrong".
If you had left out the word "incompatible", I would have agreed with this as a description of what happens as science progresses, though I would still not like the term "paradigm shift" as a term for it.
But you included "incompatible", and that makes a huge difference. By your definition, for example, no paradigm shift has happened in physics since Newtonian physics replaced Aristotelian physics, because "incompatible" requires that the old model not be usable as an approximation or special case of the new model. I realize you don't mean "incompatible" to mean that, since you say:
> Of course I consider Newtonian Dynamics to be incompatible with GR
And I disagree with that, for the reason I gave above (and have stated several times elsewhere in this discussion). The reason I consider that factor so important is the same reason Weinberg, in his review of Kuhn's book, took Kuhn to task for ignoring it: that ignoring it means ignoring how progress is made in science: by coming up with new models that still take advantage of all the true things we learned from the old model. As Weinberg says, if you have one of those T-shirts with Maxwell's Equations on it, you may have to worry about it going out of style, but not about its becoming false. But Maxwell's Equations are "wrong" and "incompatible" by your definition, since they're classical, not quantum, and classical electrodynamics is now replaced by quantum electrodynamics.
So your definition basically agrees with Kuhn that no progress is ever made in science; all we can do is lurch from one paradigm to another, each incompatible with the last and throwing away all the knowledge we gained from the last. And in the transition from Aristotelian physics to Newtonian physics, that is what happened. Whatever true things Aristotelian physics had taught people, nobody has been taught since we got rid of it. The same is not true of Newtonian physics. And it will not be true of GR, QFT, or the Standard Model in some future time when we have found a new fundamental theory to which those are all approximations.
I should make clear, given what I've said about what I think actually count as paradigm shifts, that I agree with you that when a paradigm shift happens, it doesn't happen through pure rationality with no politics involved. Certainly the shift from Aristotelian to Newtonian physics did not.
This is why I qualified my statements by saying "once the field becomes a science". The story of plate tectonics finally being accepted (and, as you say, not by any sort of purely rational process) is the story of that part of geology becoming a science. The previous view was not a scientifically established view; it was just the unsupported opinion of the people who happened to be bigwigs in the field.
The reason I am using such a strict definition is that looser ones make the term "science" a useless label that can be put on anything one likes, or withheld from anything one doesn't like. To make the term useful and meaningful, it has to focus on something more objective than individual preferences. The something more objective that I have been implicitly using is that hypotheses have to be tested using evidence and experimental data, and discarded if they are falsified; otherwise, even if the person making the hypothesis is a respected "scientist", the hypothesis doesn't get to be counted as "science".
> we're dangerously close to a no-true-scotsman problem
The way to avoid that is to have a reasonably objective criterion for applying the term. That's what I've tried to provide above.
Sure it does. Newtonian mechanics, relativity, and QM/QFT are all real sciences. So are astronomy, chemistry, and biology, at least.
> that really means we have never changed to a new paradigm?
No. The old paradigm was Aristotelian physics (at least it was in physics). That paradigm, as I've already said, does not survive at all today, not even as a special case or an approximation.
If you want examples from other sciences, here are some:
Astronomy: old paradigm - astrology and other schemes that knew how to keep track of where stars and planets were in the sky, but did not construct or test any models to explain why the appearances were as they were; new paradigm - Ptolemaic astronomy, which was the first model that actually purported to explain why the appearances were as they were and to test itself against observations, further progress has all been on that same basic pattern, although the underlying models that have survived being tested against data have changed a lot.
Chemistry: old paradigm - alchemy, phlogiston, caloric, pre-Lavoisier chemistry; new paradigm: Lavoisier and post-Lavoisier chemistry, which started with a proper understanding of combustion and went on to the atomic theory, kinetic theory of gases, periodic table of elements, and now is applying QM to analyzing why atoms and molecules have the chemical properties they have.
Biology: old paradigm - purely phenomenological and taxonomic classification of organisms, without trying to explain any patterns or test any hypotheses against data; new paradigm - attempts to explain inheritance and the development of basic evolutionary theory.
I'm a modernist in certain things, but not in all things. I'm not much of a fan of modern politics, for example.
Ah, ok. I'm certainly not a post-modernist or post-post-modernist or meta-modernist or any of those things.
alas, I cannot back this up with any data
but note that backing my assertion up with references is not sufficient to make my statment scientific.
but I can add some resasoning to my claim: modern science seems to be performed with a goal in mind (often funding of some sort). it is no longer a free-form pursuit. So then, does science have to be a free-form aimless pursuit? (maybe?)
This is way too pessimistic without being more specific than just "modern science". Modern science is accurate enough to produce technology that allows your phone to tell you where you are to within a meter or so, to communicate with a space probe that has exited the solar system (not to mention launching the probe in the first place and having it go where it was intended to go, visiting multiple planets and moons over several decades), to produce the global networking system that we are having this conversation on, and so on for many, many more examples.
There are particular areas that are called part of "modern science" but are just barely sciences (or really not even sciences yet). But there are also areas of modern science that emphatically are sciences, and not "barely" either.
> modern science seems to be performed with a goal in mind (often funding of some sort)
Modern science now mostly does, but that hasn't always been the case. The research that originally led to relativity and quantum mechanics was not. It was done out of intellectual and scientific curiosity. If your question is whether doing science for those reasons is better than doing it to pursue funding, I would say yes.
I would argue that this is better called "engineering" rather than science. In this sense engineering is a design discipline which applies science-made theories to create technology and tools.
And so engenireeng is not strictly creative (it does not create novel ideas) but refines and pushes the practice (of the theories) to more sophisticated levels.
In any case engineering does create novel technology which then can push science forward by providing better and more dieverse empirical data for science to continue.
You can't do engineering to that level of accuracy if you don't have a solid base of science to build on.
It was nothing like the shift from Aristotelean physics to Newton. The vast majority of what geologists had previously believed remained valid, and there was no change in the related sciences of physics and chemistry. And indeed it was findings in the previous geological science that lead to it.
Imagine, if you will, that plate tectonics had included that the Earth is flat and the abolition of the first law of thermodynamics and you begin to understand what a true paradigm revolution would be.
"I reformulate and derive Aristotle’s physics in modern terms."
Not that this is not a useful thing to do, but it means that what the author is calling "Aristotle's physics" is not the same thing as what Aristotle himself would have called "Aristotle's physics". The author is taking a modern viewpoint--develop a model and test its predictions against observation and experiment--that Aristotle simply did not take. That change in viewpoint is a major part of the paradigm shift from Aristotelian to modern physics that I have described elsewhere in this thread.
It's also worth noting that Aristotle's explanation of why his physics worked the way it did is completely different from that of the author of this paper. It's not so much that the author treats Aristotle's physics as an approximation whereas Aristotle did not. The author is taking a modern, mechanistic viewpoint in analyzing Aristotle's physics: hey, it turns out that you can derive it as an approximation to Newtonian physics valid under certain conditions! But Aristotle himself would have been shocked by such a claim. He viewed his physics as part of a teleological view of the universe: objects behaved as they did because of their telos. There is no analogue to his concept of "telos" anywhere in modern physics, nor in the author's version of Aristotle's physics.
Hint: Every scientific theory is built on top of a formal system.
Sure, science cannot explain everything. There was never any chance it was going to. Incompleteness has nothing to do with it.
- Stephen Hawking
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!
> 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.
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.
> 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.
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?
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 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.
Popper would say no. If we reason that ZFC should be consistent via induction, we can predict that no pixel will ever show up. But this does not seem to be a "risky prediction" at all. Unenlightened by our theory, we would have no basis to expect a different result. Maybe the pixel shows up in an hour, or maybe it shows up long after we're dead. Maybe it never shows up, why not?
> But [coupling science to decidability] 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.
Popper is surely rolling in his grave at this. Any falsifiable theory is "unilaterally decidable" to use his own words. This sounds much more like Kuhn.
I'm not going to try to elaborate on what the consequences of that would be for any given theory of science, I just want to show that whatever the difference is between science and pseudoscience, in the small scale they can look quite the same.
Tangentially, the incompleteness theorems are not scientific knowledge according to Popper. They are metamathematical propositions that suppose the consistency of mathematics as one of their hypotheses. If they could be falsified, we would have discovered an inconsistency in mathematics, and so we have nevertheless validated the theorems. You literally can't falsify them.
For more, see https://en.m.wikipedia.org/wiki/Demarcation_problem
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This is not a "deep pity" but proof that modern society has progressed beyond the sophistry of the humanities.
-- Albert Einstein
"Philosophy of science is about as useful to scientists as ornithology is to birds"