I believe...

• People and society are largely well calibrated. People who are deemed (by themselves or society) to be bad at maths, at sports, at arts, etc. are usually bad at them.
• People and society are not perfectly calibrated.
• People are sometimes under-confident in their abilities. This is often downstream of them lacking confidence.
• People are sometimes over-confident in their abilities. This is often downstream of them being too confident.

Our society does seem to inculcate in its members the idea that certain things are only for super-smart people to do, and whoever you are, you are not smart enough to do an impactful thing.

Most people would fail at passing the bar and the USMLE. This is why most people do not attempt them, and this is why our society tells them not to.

I believe it is load bearing, but in the straightforward way: it would be catastrophic if everyone tried to study things far beyond their abilities and wasted their time.

Most people have no hope of understanding complex topics.

No. Strongly disagree. "Most people don't understand X" is a thing I could accept, but "most people can't understand X" is usually false, with only rare exceptions.

You are confusing "Most people can't understand X." with "Most people have no hope of understanding X.". Only the latter matter for the psychological toll it has on people.

Hopelessness might be warranted or not, but it's there.

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Separately, I believe that quite often, their hopelessness is warranted.

Everyone hits their ceilings.

I know many mathematically talented people who struggle to express themselves in ways that are legible to others, or to move their body in a natural way. They will get better if they train, but it's pretty clear to them and everyone else that their ceiling is low.

In general, I know many people talented [at a field] with clear limitations in [some other field]. Arts, maths, oral expression, style, empathy, physical strength, body awareness, and so on.

Over time, they learn to acknowledge their talents and their limitations.

Nope.

I meant that even if the Evil People are Evil, and thus decide to not make all the bad things go away, the fact that they could make them go away is reassuring in itself.

I should have been clearer.

(I have edited it with the hope of making it clearer. Thanks.)

I believe our ancestors elevated and condemned people for reasons mostly separate from whether they statistically made better predictions. Things like "Does this sound good?", "Does this help give more power to the leader?", "Is the person uttering the various statements seemingly convinced of them?", "Can the person make it seem like the statements matched the reality post-hoc?", etc.

It might have correlated in some cases, but what I am pointing at something much more hit-or-miss than the process you are describing.

Largely agreed, except on one point.

I think there are many safe ways to improve things, even with our partial understanding. This seems much more practically relevant than how many good vs bad actions there are in my opinion.

I think this is more the start of a pleasant long discussion over beers (or tea/coffee) than of a thread that would take a lot of time to write, and that very few people would be interested in.

Feel free to DM me on Twitter (I am more responsive there) if you are ever in Europe, I think it would be fun!

But very quickly, to not leave you pending:

If one wants to represent an infinite partially ordered set in this fashion, one has to use an informal "Hasse-like sketch" which is not well-defined (as far as I know, although I would be super-curious if one could extend the formal construct of Hasse diagrams to those cases).

I agree, and I think that they work as pointer for people who already know the corresponding formalisations and their limitations.

If used in other situations, I expect they largely do not work. Concretely, I expect that readers/students/attendees will reliably come out with wrong intuitions, and that proofs with tend to include mistakes.

I have seen "intuitive" diagrams hiding mistakes so many times, both in ML and in type theory. Whether they represent proof derivations with holes, neural network architectures with holes, calculations with holes, infinite graphs with holes, etc. The infamous "x1 + x2 + ... + xn" that is not always well defined.

The informal part always leads, formalization can't be fruitful without it (although perhaps different kinds of minds which are not like minds of human mathematicians could fruitfully operate in an entirely formal universe, who knows).

Yup, I was going to say something similar.

I believe this is more an artefact of how we humans operate than of the truth of things.

For instance, I believe that it is possible to write an automated theorem prover that proves most of the theorem pre-15th century, and that it is impossible for people pre-15th century to prove most of the things that this automated theorem prover could prove.

In practice, we humans are quite biased towards positing the existence of entities behind any perceived regularity. (Spirits, phlogiston, ethers, new-age energies, etc.)

Usually, these entities do not exist, and there was no coherent deep generalisation behind the regularity.

Sometimes, the regularities are not even there, and we just hallucinated them.

I think maths is roughly similar. That there are things, but that said things are not Plato's squares and standard numbers, but just equivalence classes of symbolic manipulations. We may perceive regularities through our intuition, and sometimes have intuitions precise enough to be useful, but it's all downstream of that source of truth.

On one hand, mathematicians tend to downgrade proofs which are impossible to intuitively understand.

I think you have misunderstood my point (and thus that I wasn't clear enough).

Regardless of how they feel about non-intuitive proofs, when these proofs contradict their intuitions, the non-intuitive proofs will still be the source of truth.

They might feel bad about this stance, they might not identify with it, they might feel that this stance is sometimes detrimental. But it is the stance they enact. Belief in belief et al.

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I think you are case in point of the "fish-in-water" effect that I am trying to describe.

You mention Hasse diagrams. They are a modern purely formal construct.

They are a convenient notation that is strictly equivalent to describing the elements of a partial ordered set in matrix form.

In the past, what you would have gotten is a lot of prose that may or may not have maintained the partial ordering properties, with some diagrams that would be much less clear.

Sometimes they would imply irreflexivity, sometimes not.

Sometimes they would imply more structure like a directed partial order, sometimes not.

Without any understanding of the implications.

You may want to think of non-standard models of arithmetics, where it wasn't clear that First Order Peano arithmetic did not capture the integers. Or of Euclides' axioms.

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I think your mention of Scott domains is great!

So, as a student (not as a researcher), I actually got acquainted with DCPOs and Scott domains, specifically in the context of denotation semantics.

One of my teachers was actually quite interested in general (non-Haussdorf) manifolds.

I think this is precisely one of the realms where formalism shines.

I expect it would have been impossible for the field to take off without formalism.

There are so many broken intuitions in non-metric geometry! If we were left at the intuitions as being the ground truth, the field could not have succeeded. (Even in beginner metric geometry, it is easy to get screwed when one is not clearly introduced to specific examples where the definitions of closed/bounded/compact differ.)

There are just too many different types of spaces/topologies/manifolds, with different axioms behind them. The "true objects" here have little to do with standard numbers, intuitive euclidean spaces et al.

To the extent one has "true objects" in minds, these object are of a purely formal nature.

They have the shape of "these different formal systems are isomorphic in some sense" (you can encode the rules of one in the other, and get the same constructions as a result), and "I have some heuristics about these formal equivalence classes" (in the sense of an A-star heuristic).

The telescope that lets mathematicians check whether something is there or not is formalism, not their intuition.

Again, this is the practice that mathematicians have. As you said, when you start to prove, you check against the formal low-level. So de-facto, this is where you ground truth is.

I have met many mathematicians who were calibrated about this. They developed an internal calibro-meter that tells them how strong an intuitions is and in which rule systems they apply. And they transcend the feeling of the object existing, they just take intuitions as useful heuristics.

I think this is mostly a fish-in-the-water effect. The Platonic realm has little to do with Plato's realm anymore. The Platonic realm is now that of syntax.

In the past, the epistemology of Platonic mathematics was that of the intuition ("I feel it") or observation ("I see that literally all triangles obey the Pythagorean theorem").

But now, however Platonist mathematicians may describe themselves, they take syntax is the de-facto source of truth.

If you have an intuition for a proof of a theorem or the existence of an object, and when you try to formalise it (even in a proof sketch, not necessarily Coq), you consistently fail, it's strong evidence that the intuition was groundless. This is because we trust syntax much more than intuition or observation nowadays, which is what it means to live in the formalist era.

If an AI was to generate mechanistic counter-proofs of theorems that were "proven" in prose, regardless of how illegible the counter-proofs was, mathematicians would defer to it and not their intuition or the perception of the platonic object.

This is in stark opposition to the pre-formal times, where people routinely created paradoxes (think Zeno more than Russel) and faulty reasonings, even in the realm of maths.

I see this as just correct calibration: if tomorrow, someone developed a Platonistic telescope, that let us see the mathematical abstractions for what they are in the Platonic realm of ideas, in a way that we would trust more than formal proofs when both disagreed, then we would move on from the formalist era.

But the formalist era has this air of definitiveness, in that no mathematician expects to ever discover such a Platonistic telescope. In practice, we all defer to the syntactic rules, and intuition are just heuristics that sometimes let us go faster or bring us some nice intuitions, when they don't lead us astray.

This is a big part of the insight behind separation of powers.

Yup!

I do not plan to defend all the claims that I make, and neither do people outside of mechanised proofs.

Beyond this, I think you have misread the article! I believe that your examples are in fact not examples of unsubstantiated claims, but of you misidentifying what are the actual claims!

For easy reference, I quoted you in bold, and myself in italics.

You say that when there is more wealth concentration, that leads to less freedom "empirically" but you don't present any empirical evidence

I have used "empirically" only twice, namely "Capitalism empirically leads to wealth concentration." and "Plutocracy is empirically not conducive to liberty and democracy."

I expect these two to be pretty uncontroversial. The latter should be obvious. The first one is why there is so much redistributive taxes and progressive tax systems everywhere.

I am not saying all these taxes are good, I am saying they result from the wealth concentration that happens when there are more investment opportunities for people with more money, r > g, higher RoI on capital than on labor past a threshold, etc.

But more importantly, I did not say that "when there is more wealth concentration, that leads to less freedom empirically". This is why I am making a case in the first place! Else, I would have just said that!

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You say that when one side has bargaining power over another, that's bad per se, but it's not explained why

I do not say this! I specifically oppose mere power from bargaining power, explaining that power is much stronger. And the article starts with an explanation for why power over another is bad.

When someone has power over us, they can compel us. Power is the ultimate form of bargaining power: it morphs compromises and trade relationships into compulsion and coercion. This is the logic behind big stick ideology and gunboat diplomacy.

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You give Elon Musk buying Twitter as an example of the negative influence of billionaires, but in fact the Twitter Files reveal an apparently more serious threat to freedom from the state, whose power is actually counterbalanced by wealthy individuals

"negative influence of billionaires"?? I know billionaires who do good things! That's not the type of things I would write!

Elon Musk buying Twitter is an example of how wealth concentration (billionaires) can lead to power concentration (economic power -> media & political power) through the fungibility of money! This is one of the core theses.

Let me actually quote the article:

From the perspective of the Enlightenment Philosophy, Elon Musk is a great example of the type of power concentration we ought to avoid in a single person.

He became a billionaire, with outsized economic power.

Then he used that economic power to buy Twitter, acquire a massive amount of media power, and use it to steer the discourse where he wanted.

Finally, he leveraged all of that to get himself into a close alliance with Trump and get massive amount of political power through DOGE.

[...]

This is not about the personality of billionaires. Some are mean, some are nice. Some are evil, some are good. But that's beside the point. The point is that separation of power is a core principle of the Enlightenment Philosophy. Defeating kings was not about them being bad people, it was about splitting the levers of power so that no one may wield them them unilaterally.

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I wanted to go through the exercise of answering comments a couple of times.

There were several comments like yours, and I was wondering if I was going crazy, because these didn't read like responses to what I had actually written.

I don't have a confident understanding of what's happening. My best guess is "People get triggered online and answer what they've been used to seeing, as opposed to what's actually written".