This is a great take because I suspect that the airline industry is exactly the reason why all these companies are now in the "death by a thousand cuts" stage of their strategies. Literally no one prefers the airlines today and the experience of flying to the same degree (and especially not more) than they did a decade ago or more. Everything about the experience, from the boarding to the seating to the food, is objectively worse than it was before and yet people still need to fly.
Is the fact that these systems cannot prove their own consistency actually a feature of the incompleteness theorem? I thought it effectively boiled down to "you can keep either consistency or completeness, not both". It's been a while since my metamathematics course ...
It depends on exactly what you mean with those terms.
When you say "you can keep consistency or completeness but not both" you are essentially stating Godel's first.
Morally speaking, Godel's first is "no system is complete", but there are two exceptions: if the system isn't powerful enough to formulate the Godel sentence then the proof doesn't work, and any inconsistent system is trivially complete because ex falso quodlibet sequitur, i.e. any wff is true.
The part about a system not being able to prove its own consistency is Godel's second theorem.
But the theorem only does what it says on the tin: the system not being able to prove its own consistency doesn't mean that it being inconsistent! [1]
This is the case for ZFC, for example. We can't prove ZFC's consistency within ZFC: that would violate Godel's second, and we know that either ZFC is inconsistent (that is, you can derive an absurd from the axioms) or ZFC is incomplete (that is, there exists a well-formed formula of ZFC that cannot be proved or disproved within ZFC).
We don't know which one it is.
[1] This implication and similar ones are often made in pop-culture presentations of the foundation problem. I generally vigorously object to them because they're not only imprecise -- that's understandable -- but they leave a layman with a completely wrong impression.
> the system not being able to prove its own consistency doesn't mean that it being inconsistent!
Another funny thing that can happen is that a system proves its own inconsistency, despite being consistent. The short summary is to never trust a system talking about its own consistency.
Not that you were, but I don't quite understand why people get so caught up on this fact. There are objective facts about the nature of reality, and we are all (or at least competent practitioners in the field) are thoroughly convinced that we have identified a subset of these facts.
These presumed facts have helped us do things like go to the moon and build skyscrapers, but then someone comes along with the old "but how do you actually know" argument of a college freshman, and then we get into a conversation about the potential social relativism of math.
All the while, people will see a half-assed psychology study with a questionable procedure, weak at best, erroneous at worst statistics and therefore tenuous at best conclusions, and this study is taken to be "true" and might legitimately impact notable institutions. Yet when we're talking about extremely complicated topics that exist on the edge of the horizon of human intuition, no matter how obvious the impact some people just refuse to accept things as objective simply because they fail to intuitively understand them.
Foundational fields like mathematics and physics are as objective as we can get. If you don't accept that, your belief about what is objectively true ends at cogito ergo sum and that's that. This has always been such a pointless conversation in my mind.
> There are objective facts about the nature of reality
This is a pretty bold claim and you would have to do a bit of work to make it more convincing. Besides, it's not really how science works. Different theories wax and wane over time all the time. What we're really doing with science is coming up with ever better models that give us greater predictive power.
You could argue that at the macro scale we're asymptotically approaching some kind of objective truth with the whole endeavor of science, but you can't simply tunnel across that gap and make the leap to say that we know there is an objective truth.
The best that we can do is probably say something along the lines of "these are the best methods of getting closer to the truth that we have available - if anyone claims to have better methods, they are very likely wrong", but you still need to have the humility to accept that even the best models that we have to date are not infallible. They do not give us the objective truth, nor do they promise to, but they are the most effective tools that we have available to us at the current time. This is critically not the same as claiming that they give us the objective truth.
You could say that for all intents and purposes/everyday usage, "sure, these are objective facts about reality" - but I don't actually think that helps anyone and it serves to foment mistrust towards science and scientists. I think the best chance we have at preserving the status of science and scientists in our society starts by being honest about what it actually is giving us - which is quite a lot, but let's not oversell it for the sake of convience or whatever.
As Heissenberg said, "What we are seeing is not nature, but nature exposed to our mode of questioning."
And the mode -- we invented it as it is because of a whim of history, because it is a game, and we like the game, and it's useful for us. But as far as facts go, Nietzsche summed it up the most concisely: "there are no facts, only interpretations."
If by "objective truth" we mean the qualities of nature that exist irrespective of any individual's perception, then I think the continued reliance of our scientific knowledge in producing effective and consistent results are at least some measure of that.
> The best that we can do is probably say something along the lines of "these are the best methods of getting closer to the truth that we have available - if anyone claims to have better methods, they are very likely wrong", but you still need to have the humility to accept that even the best models that we have to date are not infallible.
This last sentence slightly conflates the scientific method with the models they produce. I am not claiming that the models are "true", I am claiming that the scientific method is the only reasonable means of gaining a reliable understanding of the objective nature of reality, assuming it exists; and that you cannot pick and choose what you believe in based on your intuition.
Quantum principles have been proven in experiments that have as tight a margin of error as measuring the width of the United States to one human hair, producing shockingly consistent and effective models that were absolutely critical to the development of modern technology. Yet some people somehow refuse to accept these models as an "accurate" reflection of reality, whereas they'll take, at face value, psych/sociological/economic studies that are frankly nothing short of pathetic in comparison.
In regards to science, I am saying that there is a hierarchy of belief. You can draw the line wherever you like in terms of what you think is "true", but you cannot reorder this hierarchy and believe these sorts of psych studies while at the same time questioning the physical models that power the technology that is used to publish them.
And this isn't speaking about math, which is a particularly special case given that it is not scientific but still produces shockingly effective results.
I think I would agree with pretty much all of the above. There is a sort of hierarchy of belief, and quantum mechanics is probably close to the top of that. However, even so, it is still not infallible. It is possible for us to discover a regime where it breaks down and we need a new theory to supplant it.
This is basically what I'm arguing - no matter how accurately our theories line up with observation, we can never be sure that we have reached "the final theory" AKA the truth. I think this is where a lot of misunderstanding and mistrust for science originates. It will never deliver to us the truth - if it did, how would we ever know?
It is a method of getting closer and closer to what we believe is the truth. But there is still a gap there, however small it might be in the case of quantum mechanics. The scientific method by it's very construction is unable to bridge that gap.
Still, to date there does not exist a more effective method we know of as a species at getting closer to what we believe to be the truth. I think the above is a maybe subtle distinction there that is worth pointing out and educating people on. Just sort of making that distinction between the process and the results. That it is the best process we have, but even so, it cannot cross that gap and definitively say "this is the truth". That that is a gap we have to choose whether to cross ourselves with a leap of faith (or sometimes a very tiny hop of faith in the case of quantum mechanics). I think that might help people cement their faith in the process even if they dont necessarily place their faith in the results (in the case of questionable psych studies for example).
If objective reality exists, which is still a pretty big if last time I checked, not only do we not know what it fully is, we don't even know what any part of it is. The best that we can do is get better and better at modeling it in ways that are useful to us (which is what science is doing for us).
"Science" (scientists) studies the physical realm almost exclusively, and where it does venture somewhat into the metaphysical, it brings a lot of baggage that works excellently in the physical realm, but is often detrimental in the metaphysical. Plenty of scientists, I'd bet money even most, find[1] metaphysics to be ~silly...except of course when they are whining about it (politics, economics, society, etc..."the real world").
[1] The context, or set and setting, makes a big difference in how they will behave. But the range of behavior is actually pretty simple, far simpler than much of the programming problems we breeze through on a daily basis, without even thinking twice about it. I'd say the problem isn't so much that it's hard, it's more so that it is highly counterintuitive, a lot like Einstein's relativity when first encountered (or even after understood).
> This is a pretty bold claim and you would have to do a bit of work to make it more convincing. Besides, it's not really how science works. Different theories wax and wane over time all the time. What we're really doing with science is coming up with ever better models that give us greater predictive power.
Yes, but is mathematics like that? Is it even science?
I wouldn't reject objective facts, but I also wouldn't believe they exist any more than I would believe Santa Claus exists, unless someone can successfully argue for their existence. AFAIK this has yet to be done by philosophers (though there have been many attempts).
E: I should mention that it's not just a binary yes or no here, there is a 3rd option of "I don't know" and I would rabidly defend the "I don't know" camp until someone can convincingly argue one way or the other. All of this has nothing to do with the actual usefulness of science which is unquestionable in my opinion.
This is strictly talking about science "overreaching" into the philosophical realm if you will, where even from the start, methodologically it doesn't have the right tools to answer these questions. You don't prove scientific theories "true", you just accumulate more and more supporting evidence. It never hits a magical moment where the neon lights turn on and a sign says "Your theory is now true! Congratulations!". And even if it did, it would be fleeting anyway because there are no sacred cows here - your theory can just as easily get supplanted by a better theory in light of more evidence.
Yes it's a bold claim philosophically. How would you justify it?
No, flat earth "theory", if you can call it that, has close to zero supporting evidence and AFAIK has no actual predictive power. Stick with consensus science if you want actually useful theories, but that is very different from claiming they are giving you objective truth.
Let me ask you this, when a theory that was previously accepted as consensus science loses support in light of new evidence and gets supplanted by a new and better theory, does that mean that the objective truth changed?
It’s frustrating when nearly every discussion of hard math in an open forum devolves (in part or whole) into endless, pointless, off-topic epistemological navel gazing.
This fact is interesting because many people grew up with an idea of mathematics as having discovered fundamental truth, and some of us, when we get deep into the field, realize that Math is made up of human ideas, exactly the same as a very complex boardgame we play in our heads, and that the belief that these boardgames represent something fundamental about reality is it's own leap of faith.
How is it a leap of faith when mathematical models that describe reality demonstrate continued reliance in making predictions? Does this not demonstrate that there exist mathematical relationships between certain physical quantities?
> All the while, people will see a half-assed psychology study with a questionable procedure, weak at best, erroneous at worst statistics and therefore tenuous at best conclusions, and this study is taken to be "true"
...
> Yet when we're talking about extremely complicated topics that exist on the edge of the horizon of human intuition, no matter how obvious the impact some people just refuse to accept things as objective simply because they fail to intuitively understand them.
I think the intersection of these two groups is the null set.
> Foundational fields like mathematics and physics are as objective as we can get.
Objective? Yes. But objective does not equate to "true"[1]. One requires data, and the other lives only in the mind. It is not at all problematic to ponder over whether mathematics is "true" - most mathematicians have an opinion one way or another, and they are not unanimous in their opinions.
[1] True w.r.t reality, not true in the logic sense.
Quite the opposite. The majority of people I will believe some random bogus psych or nutrition study the news, while denying quantum mechanics results.
I have no idea what "true" means if it does not mean "a quality of a statement as being an accurate description of objective reality". While mathematics itself may be a human invention, it clearly describes a fundamental truth as evidenced by its continued reliance in producing effective models that describe the measurable world, which must be taken as a reflection of objective reality. Again, you can feel free to deny this, but all you are left with is cogito ergo sum.
> While mathematics itself may be a human invention, it clearly describes a fundamental truth as evidenced by its continued reliance in producing effective models that describe the measurable world, which must be taken as a reflection of objective reality.
It often describes the fundamental truth, because much of its origins was for that purpose, not because all mathematics is fundamentally true.
There's plenty of mathematics that is not modeled by the measurable world - almost by design. Just pick a different set of axioms and you'll get mathematical truths that may conflict with reality.
According to many mathematicians, uncountable infinities don't exist, and they focus on doing math without relying on them. There goes the real number line. From a scientist's perspective, either they exist or they don't. Both cannot be true. From a mathematician's perspective, both are truths.
Similarly, either the Axiom of Choice is true or it isn't. Yet the bulk of mathematics is fine with or without it.
> There are objective facts about the nature of reality
Such as?
> "but how do you actually know" argument of a college freshman
Epistemology has been studied by some of the greatest thinkers since the ancient greeks ( probably even before ) and not just college freshmen.
> no matter how obvious the impact some people just refuse to accept things as objective simply because they fail to intuitively understand them.
If you have to intuitively understand them, it isn't very objective is it?
> Foundational fields like mathematics and physics are as objective as we can get.
What in math are objective facts about the nature of reality? Where in nature is the number 1? Also do you realize that many mathematicians don't even accept the 'reality' of real numbers.
I think as you think more deeply about these topics, you will change your tune.
> If you have to intuitively understand them, it isn't very objective is it?
They were talking about things that you don't have to (and indeed can't) intuitively understand.
> What in math are objective facts about the nature of reality? Where in nature is the number 1?
In all sorts of places, not least counting things. Mathematics is about equivalences; the point of saying 2 + 2 = 4 isn't to make some funny marks on paper or pray to the platonic void, it's to say that if you have something that's 2-like and combine it with something that's 2-like in a way that's +-like, then the result will be =-like to something 4-like, in the same sense of "like".
> They were talking about things that you don't have to (and indeed can't) intuitively understand.
He was talking about objective things that people fail to to intuitively understand.
> In all sorts of places, not least counting things.
So where? Just one. Give me where 1 exists so I can check it out.
> Mathematics is about equivalences;
No. Mathematics is about theorem generation. Going from axioms to theorems via proofs. Though some go from theorems to find axioms.
> it's to say that if you have something that's 2-like and combine it with something that's 2-like in a way that's +-like, then the result will be =-like to something 4-like, in the same sense of "like".
What you are describing is just arithmetic. That's the interpretation or model for one particular set of axioms, theorems, etc. What's objective about 2-like? Feels abstract.
Funny I asked a simple question and yet you rambled on about nonsense you don't even understand.
I'm not specifying them, I'm referencing their existence. You can take them not to exist, but in that case good luck explaining the measurable consistency that every human on earth observes. In fact, if you truly believed this, there would be no point in even speaking leaving a comment because then you could not believe in the ability of language to communicate ideas or even in the existence of language itself.
> Epistemology has been studied by some of the greatest thinkers since the ancient greeks ( probably even before ) and not just college freshmen.
Yes but it is somehow seems to always be the college freshman that raise this question for fields that have contributed the most to humanity like mathematics and physics, all the while not feeling the need to question any number of academic fields built on an absolutely pathetic scientific process in comparison. It is simply because of their failure of intuition and understanding of mathematics and physics that they raise these questions - I'm not saying that these questions are not part of a worthwhile conversation in general.
> If you have to intuitively understand them, it isn't very objective is it?
I'm not understanding your argument here. The failure of a human mind to understand something does not mean that this thing is not objective. Nobody on Earth understood why lightning occurs for most of human history. This does not mean that the existence of lightning and the reasons for its occurrence are not objective qualities of nature.
> What in math are objective facts about the nature of reality?
The existence of concepts that map to reality in producing models that yield consistent, effective, and measurable results.
If there were no conscious beings, the Earth would continue its orbit around the sun and this order is what is captured by the human-invented language of mathematics. Saying that "mathematics" isn't objective may be true depending on how you define mathematics, but there is no denying that there are objective relationships which exist beyond human experience. If you deny this, you are left with cogito ergo sum which was my original argument.
We communicate with words, and people as a whole are used to being lied to and gaslit regularly especially by those in power. It's true that mathematics and the hard sciences have mechanisms for understanding that are on a different scale than, say, ethics and morality. However, it takes time for people -- especially those currently engaged in questioning the nature of their reality[1] -- to accept that in this specific instance lying and gaslighting are a lot harder[2].
The people who eventually accept and internalize the distinction around things that can be objectively shown to be true are those who by in the large have done some of the work to understand these things themselves. Godel's Incompleteness Theorem is beautiful but it takes work to understand and if it didn't, it wouldn't be much of a meaningful breakthrough. Nobody is proving that 3+5=8 and then 4+5=9.
So what the average person sees is a high level language they can't speak with people being absolutely positive that this thing is special and true and incontrovertible. That raises red flags when you're dealing with folks talking about normal everyday stuff, doesn't it? It's a lot harder to say "but I don't understand" and a lot easier to say "but what if you're wrong" socially.
[1] As all college first years do, right?
[2] Let's face it, lying to people is never impossible, it's just harder to be successful when you can be fact checked.
After the article, my opinion is going to be that people do this because they are politically privileged by a social relativistic argument. The professor in the interview is advantaged by the narrative that proofs are socially constructed.
Indeed, mathematics and any other STEM field is deeply political. People are politics. But to confuse that--that STEM as practiced is socially constructed and political--and to make a sloppy ontological conclusion about what STEM/math is, is a deeply neoliberalizing argument. So I'm inclined to make the argument that it is actually the neoliberal intellectuals who try to spin this as fundamentally only a human enterprise, because it privileges their social standing as the "correct" practitioners of this compact: capitalism requires that mathematicians be able to "sell" each other the truth of their proofs.
Indeed, the article mentions the Mochizuki controversy but I don't think they understood the social problem. See, it doesn't matter if Mochizuki is right or wrong, or if people understood his 500 pages or not. It is that in principle it could be shown that he is right or wrong. And that's unlike, say, the Bible, which is also a social compact. Reduction of STEM to social construct throws the baby out with the bathwater.
One use case is in the training of Diffusion Models. In the original formulation, the likelihood-maximizing objective is recast in terms of KL divergences. This is done because the KL divergence between Gaussians has a closed form, and the transition distributions in Diffusion Models are taken to be Gaussian, which makes the problem tenable.
Fantastic article - I would love to see an analysis comparing larger quantized models to smaller unquantized models. e.g. is a 14b quantized model better than a 7b unquantized model?
When did we start talking about the airline industry?
But seriously, the reason it works for the airline industry is because there is literally no other alternative, unlike in this case.