The spy who flunked it: Kurt Gödel's forgotten part in the atom-bomb story | Hacker News Reader

The spy who flunked it: Kurt Gödel's forgotten part in the atom-bomb story

nature.com

139 pointsdrdee98 comments

One does not send Kurt Gödel to do this. He had to be coached during his citizenship test. “1947, he took the test to become a U.S. citizen, sponsored by Einstein and their mutual friend, the economist Oskar Morgenstern. When he appeared before the judge, after answering the first questions satisfactorily, he told the judge that he had analyzed the U.S. Constitution and that it had a drafting problem, which allowed the U.S. to legally become a dictatorship or a fascist state, as had happened in Austria. Gödel’s two friends were horrified because they saw that his citizenship was in jeopardy, and he could be deported to Europe. The judge himself stopped Gödel by uttering the pragmatic phrase “Oh God, let’s not get into this” and ending the examination before the logician could say anything more.” [1] https://english.elpais.com/science-tech/2023-07-01/kurt-gode...

I wonder if this may be related to FDR serving four terms as President. If he hadn’t died in office, and had term limits not been implemented in 1951, he may have had even more terms than that.

https://en.wikipedia.org/wiki/Franklin_D._Roosevelt

https://en.wikipedia.org/wiki/Twenty-second_Amendment_to_the...

My understanding is that the "flaw" was that it allows itself to be modified via the amendment process. Given his connection to self-referential problems, this makes sense.

Ironically the real problem with the U.S. constitution is that it’s much too hard to modify. The text is arcane and vague and often meaningless in a modern context, so courts can make up any interpretation they prefer.

Does the constitution contain a right to privacy that allows abortion? It did for 50 years, then it suddenly didn’t. What’s a “well-regulated militia”? Anything that a particular court wants it to be. What about “emoluments”? Same.

Instead of trying to squeeze new meaning out of 18th century vocabulary, a constitution should rather have a self-destruct clause. Rewrite it completely every fifty years or so. There are nations that do this kind of constitutional reset when times are good (e.g. Finland whose latest constitution is from 2000), and they seem stronger for it.

Of course this is not possible in America also because culturally the constitution is much closer to scripture than to ordinary law. The Puritan heritage is in plain sight in how a legal text is treated like Moses had brought it down from the mountain.

"There are nations that do this kind of constitutional reset when times are good (e.g. Finland whose latest constitution is from 2000), and they seem stronger for it."

Don't nations do it when times are good for the current regime, often to get rid of term limits?

Mr. Trump helpfully pointed out another flaw: a president can pardon whoever they want, including themselves. So it's apparently legal for Mr. Biden to send a team to assassinate all his political rivals and then just pardon them.

Legal thinking these days is that self-pardons are meaningless, but the fact remains that the party will march so much in lockstep that there are no consequences anyway.

Nixon apparently considered pardoning himself:

The Pardon of President Nixon: Annotated

https://daily.jstor.org/the-pardon-of-president-nixon-annota...

> Nixon had considered pardoning himself while in office[0], and sought a legal opinion considering the constitutionality of doing so (his lawyers believed that it was constitutional). However, three days before his resignation, the Justice Department issued its own opinion to the contrary: “Under the fundamental rule that no one may be a judge in his own case, the President cannot pardon himself.”[1]

[0] Pardon Me?: The Constitutional Case against Presidential Self-Pardons

https://www.jstor.org/stable/797310

[1] https://www.justice.gov/sites/default/files/olc/opinions/197...

In the context of Trump:

Explaining the presidential self-pardon debate

https://constitutioncenter.org/blog/explaining-the-president...

> On Monday, June 4, 2018, President Donald Trump said on Twitter that, “As has been stated by numerous legal scholars, I have the absolute right to PARDON myself, but why would I do that when I have done nothing wrong?”[2] His comments echoed those made by Rudy Giuliani, his attorney, on Sunday.

> Here is a brief explanation of the President’s clemency powers, some past cases, and a theoretical look at a Nixon-era question: Can a President pardon himself or herself?

[2] https://twitter.com/realDonaldTrump/status/10036162109221478...

Not so according to the Justice Department.

https://www.justice.gov/sites/default/files/olc/opinions/197...

I elaborate on this in a reply to your interlocutor downthread:

https://news.ycombinator.com/item?id=40383391

That doesn't really change much. The president can kill the members of the opposing party, and then just pardon the assassins. Who's then going to sue him?

Nation state level threats are kind of a final boss irl and in the cybersecurity realm, so I’m not sure what you want me to say. That’s a problem for Citizen Lab, Google Project Zero, and natsec but if someone individually is at risk of this happening, they need a lawyer and an insurance policy, and possibly private security, but tbh they have bigger problems like being on a naughty list. I honestly don’t think this something you can expect to move the needle on for the >99% of people who don’t have this problem.

What do you think a defense against this would even look like? To attempt to do so sounds almost absurd because it’s just such a remote possibility.

> What do you think a defense against this would even look like? To attempt to do so sounds almost absurd because it’s just such a remote possibility.

Stronger constitutional limits. For example, pardons should not be a thing that could be done unilaterally. Ideally, they shouldn't be a thing at all.

At the same time, the Constitution needs to be a bit more malleable. The Amendment process is notoriously hard, and the US Senate structure is uniquely unamendable.

What Gödel was talking about is apparent when you look at the style of logical contradiction he is known for. The US Constitution and its Bill of Rights only define rights in terms of narrowly specific negative prohibitions on what the nominal government can do, rather than positive statements of what the People should be able to do. This leaves the door open for extraconstitutional de facto government to exist along side the nominal government - not explicitly authorized by the Constitution, but also not prohibited or constrained by it. We see the resulting shape basically everywhere today, whereby most of the rights laid out in the Bill of Rights have been effectively neutered whether by the government sidestepping the Constitution's specific proscriptions, corporations exercising coercive power based on market share, or the potent combination of both.

Interesting take. Libertarians, who are usually the ones most preoccupied with finding tyranny in any government action (or any government at all), tend to be big fans of negative rights as opposed to positive rights. But that's probably because they're coming from a different ideological outlook.

Libertarians are not known for their strong logical arguments.

Yes, "it's just another system of control". I used to partial to big-L Libertarianism. I'm not any more, mostly due to this flaw in the framing and the resulting conclusions. It essentially blesses corporations' extraconstitutional governing, by simply defining away the coercion (plus a healthy dose of ignoring that a "corporation" is itself a government creation). Then offers up individualist fundamentalism to cope with its glaring failings.

(These days I identify as small-l "libertarian" because I still believe individual freedom is paramount. And it still puts me in the appropriate "other-weird" bucket with most people, rather than their friendly-kool-aid or other-enemy buckets).

I think that might fit into left-libertarianism and/or individualist libertarianism, perhaps.

Oh! I get it (I think)

Corporations can be set up that do most of the governing, and do it the way “they” want. I assume Facebook is the poster boy here, something something Content moderation.

I mean yes - kind of.

My take is that most corporations are more interested in making a quick buck than in doing any governing (especially the hard stuff).

Governing well is really really hard ( which is why it is so rare!). Outsourcing it even to your captured corporations just makes the whole thing harder surely?

I am heartened by Zuckerberg standing up in congress and saying “well you tell me what is and is not allowed”

Corporations either don’t want to govern (see above) or they already are the government (see Putin’s Russia) so it’s only extra constitutional if you think the constitution is worth more than the ink.

I think despite everything we have shared belief that our western constitutions matter - even Zuckerberg

I avoided concrete examples because they tend to bring people out of the woodwork who see the goals of that specific exercise of power as a good thing, and then argue against the general criticism.

I'd say that pigeonholing corporations as only interested in a "quick buck" is a fallacy. You're essentially focusing on one of their imperatives to write off concerns about another. From what I see corporations very much have an imperative to govern - in the small sense of controlling others, beyond what would otherwise be a disinterested smallest transactional scope.

To run with your example, Facebook does not merely present a disinterested linear feed of posts, allow for users setting their own prioritization criteria, nor design their systems to be content-oblivious conduits. Rather Facebook inserts itself into users' relationships in ways that "drive engagement", and proactively polices users' content - either preemptively and summarily enforcing what Facebook perceives as the law, and even differing from the law for internal ideological or public relations purposes. That's essentially small scale governance, with the ability to opt out being directly tied to one's dependence on Facebook. If you're just using it to keep in touch with family, you can do that in other ways. If it's a critical part of promoting your local business, running afoul of its regulations is likely more of a critical problem than say failing a health department inspection.

> a "corporation" is itself a government creation

Not necessarily. Today corporations are required to be created and maintained according to government procedures, but that wasn't always the case.

To a small-l libertarian, a "corporation" is simply an obvious way for a group of people with a common goal to simplify the process of working towards that goal while interacting with the rest of the world, by creating a single virtual entity that can do things like be a party to contracts, participate in trades, etc., rather than requiring every single person working towards the goal to individually be a party to each such interaction.

Of course, to a small-l libertarian, such corporations are not empowered to do anything their individual members could not do, and they don't change the limitations of the non-aggression principle. Which already leaves out much that actual corporations do in today's world.

My characterization is primarily referring to the liability shield aspect, which is a creation of government policy. If the owners of a corporation were jointly and severely liable for the actions and debts of the corporation, I'd accept the explanation that it's just a way for people to come together to achieve a common goal. But that liability shield allows for a level of scaled-up investor non-involvement that creates significant qualitative differences in the resulting behavior.

The liability shield isn't necessarily a government creation either. 90% of it can be defined by contract law, it's only when interfacing with third parties not in contract with the corporation that it comes up, but that tends to be a minority of corporate litigation.

That's assuming some unlimited ability to unilaterally push written contracts, and with those types of terms. If businesses were able to do that, we would expect it to be prevalent for the liability they do have today. Businesses are certainly trying with things like mandatory arbitration, but it would seem like a tall order in general. Especially as it's dubious whether something like a clickwrap EULA should even actually be considered a contract (due to lack of meeting of minds), or whether that current state of affairs is more appropriately viewed as corruption.

That's not true at all. The Constitution only enumerates certain powers to the federal government, and quite clearly states that the government only has those powers. The Constitution has since been "interpreted" to mean that the government can do whatever it wants if it thinks it has a really good reason e.g. "the bill of rights is not a suicide pact", but that's not really a flaw with the text of the Constitution.

This only applies to the part of my argument/example about the nominal government overstepping, while not addressing the larger phenomenon of parallel governance structures being created independently of the nominal government.

Ok, but the Constitution doesn't authorize such a parallel government, and it's hard to see what any constitution could do to prevent one. Really, the only way constitutional government can have any chance of working is if a pretty fair percentage of people in power actual care about constitutional government for its own sake. Otherwise they'll just do as seems expedient.

The Constitution fails to prohibit parallel government. Obviously, doing so could not prevent wholesale replacement of the Constitution and US government, but it would prevent the current situation where we've got parallel extraconstitutional government while the Constitution is still in legal force and supposed to be defining our rights.

Parallel governance could be prohibited through a positive definition of rights applying to any actor that might infringe (similar to say, the prohibitions regarding homicide). This would allow for a straightforward balancing of equity between conflicting rights, rather than the current "axiomatic" approach where any given right tends to be found to apply fully or not at all.

If the intent of the Constitution was to enumerate the rights granted by the citizens to its government and keep those limited, then there should have been a bit more careful wording in "provide for the common Defence and general Welfare of the United States" since that can be interpreted to mean just about anything...

They were more careful. "provide for the common Defence and general Welfare of the United States" is a justification for the enumerated powers. The enumerated powers were much more specific like

To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;

To provide and maintain a Navy;

The so-called "elastic clause" is the one that reads "To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers". Which can indeed be read fairly expansively, particularly if you ignore the clear and explicit intent to keep government power circumscribed. But there isn't anything anywhere near as ridiculous as a power to promote the general welfare as such.

I mentioned FDR due to his being President at the time of Gödel’s interview. Perhaps he was aware of the possibilities of government overreach, Constitution or Amendments be dammed, such as the internment of American citizens and others in camps during World War 2, though they were neither suspected nor guilty of any crime. Many of them weren’t even ethnically or racially Japanese, which is eerily similar to German atrocities committed during the war.

Many folks today aren’t aware that actions of the US government in its treatment of migrant workers, Native Americans and others who were US citizens under the Bracero program, including detainment, torture, and even usage of Zyklon B itself were all known to German authorities and intentionally replicated in German treatment of Jews and other groups deemed undesirable during the war. The racism, discrimination, and xenophobia of America was pointed at and acknowledged by Germany, and Germans sought to emulate it.

https://en.wikipedia.org/wiki/Bracero_Program

https://en.wikipedia.org/wiki/1917_Bath_riots

https://en.wikipedia.org/wiki/Internment_of_Japanese_America...

The Nazis modeled their laws on Jim Crow, that’s well-known. What’s this about nerve gas tho?

> But I also unexpectedly uncovered other information at the National Archives that took my great-aunt's personal recollections beyond family lore or microhistory. These records point to the connection between the U.S. Customs disinfection facilities in El Paso-Juárez in the 20s and the Desinfektionskammern (disinfection chambers) in Nazi Germany. The documents show that beginning in the 1920s, U.S. officials at the Santa Fe Bridge deloused and sprayed the clothes of Mexicans crossing into the U.S. with Zyklon B. The fumigation was carried out in an area of the building that American officials called, ominously enough, "the gas chambers." I discovered an article written in a German scientific journal written in 1938, which specifically praised the El Paso method of fumigating Mexican immigrants with Zyklon B. At the start of WWII, the Nazis adopted Zyklon B as a fumigation agent at German border crossings and concentration camps. Later, when the Final Solution was put into effect, the Germans found more sinister uses for this extremely lethal pesticide. They used Zyklon B pellets in their own gas chambers not just to kill lice but to exterminate millions of human beings. But that's another story.

Excerpts: 'Ringside Seat to a Revolution'

From Ringside Seat to a Revolution, An Underground Cultural History of El Paso and Juarez: 1893-1923 by David Dorado Romo. Published 2005 by Cinco Puntos Press.

https://www.npr.org/5176017

Ok, “treated clothes with a deadly pesticide” is really quite different than “gassed migrants”. I mean, we were still burning leaded gas then.

It’s a matter of degree, not kind. I don’t think you could argue it’s a horse of a different color. That migrants literally rioted speaks to the depravity of the conditions.

That we inadvertently and indirectly influenced and inspired certain Germans to build their own kind of literal gas chambers modeled on those of the United States is not a matter of debate, is my point. That they took it “too far” is not our fault today, but it is all our shared legacy and responsibility to not dare repeat or enable.

I’ll go with “depraved indifference” rather than “genocidal intent” but when you spread it across a whole society, they ain’t all that different, are they?

I don’t disagree, and I’m not trying to paint all Americans or all Germans with the same brush. The migrants were here legally, and they deserved due process and protections of the Constitution, even as noncitizens, especially their rights to “life, liberty, and pursuit of happiness,” which could be interpreted as freedom from undue interference in their way of life and livelihood as legal migrant workers. That’s not to say that they weren’t subject to immigration/border controls, but the system as such was only just becoming formalized when these events occurred. Historically, the southern US/Mexico border was open and not really regulated much at all just years before these events transpired.

The events of World War 2 undertaken by Germany and other countries, individuals, and groups directly influenced the creation of the United Nations, leading to the drafting, approval, and adoption of the Universal Declaration of Human Rights in 1948, and the later International Bill of Human Rights in 1976. It’s thanks to the United Nations also we have a formal definition of what constitutes genocide.

We’ve come a long way, and have farther still to go, but “we shall overcome because the arc of the moral universe is long but it bends toward justice,” Martin Luther King, Jr. said, paraphrasing Theodore Parker. Yet we must also remember the words of John Stuart Mill, often paraphrased and misattributed to Edmund Burke:

“Bad men need nothing more to compass their ends, than that good men should look on and do nothing.”

That's not really the problem. Yes, FDR broke the tradition, but he was the one exception who broke the rule, and there were plenty of mechanics to prevent him from retaining office if the electorate truly didn't want him.

It's already clear who will most likely be POTUS on January 20, 2025, so it will be tested.

https://www.newyorker.com/news/daily-comment/kurt-godels-loo...

To me, the most interesting sentence in this essay is: "[H]e, Gödel, had felt that a nuclear chain reaction would be possible only 'in a distant future'."

Gödel was second-to-none in analytic ability and he was paranoid. What made him so certain that nuclear chain reaction was a distant possibility? If it were anyone else, I'd say they were just trying to comfort themselves by not thinking about terrible consequences. I wish I knew what made him come to the conclusion he came to.

> Gödel was second-to-none in analytic ability and he was paranoid. What made him so certain that nuclear chain reaction was a distant possibility? If it were anyone else, I'd say they were just trying to comfort themselves by not thinking about terrible consequences. I wish I knew what made him come to the conclusion he came to.

The universe does not subscribe to rigorous, certain analysis. Being an unparalleled genius in mathematics can make you too certain of the applicability of conclusions deduced from axioms about the real world; in mathematics axioms are true because you decide to work in a model of them, but the real world usually doesn't care to cooperate with the axioms that make sense to you. My suspicion would be that that's what happened here.

The chain reaction was still speculative. Also physics is math, but math is not physics.

Physics is math, but not all math is physics.

Physics is, optimistically, math plus a whole load of initial conditions that are not findable a priori.

Physics is the study of reality under the assumption that it's lawful. The real world doesn't run on physics: it just is.

Yes, that's one of the less mathematically optimistic perspectives. :)

The physical portion of reality.

No, the causal portion of reality. (Philosophers of metaphysics are still arguing whether acausal aspects of reality are, in any real sense, real, but we don't need to listen to them.) If we discover something non-physical (the usual example being ghosts), that'll just become a new branch of physics.

Physics currently talks about dark energy, the geometry of spacetime, and quantum superposition. I don't think you can get much less physical than that.

> No, the causal portion of reality.

You are welcome to make an assertion that causal forces lie only in the physical realm, but declaring something to be true does not necessarily make it true, though at scale it can certainly make it seem true.

See:

https://en.wikipedia.org/wiki/Ideology

>> An ideology is a set of beliefs or philosophies attributed to a person or group of persons, especially those held for reasons that are not purely epistemic...

https://en.wikipedia.org/wiki/Illusory_truth_effect

>> The illusory truth effect (also known as the illusion of truth effect, validity effect, truth effect, or the reiteration effect) is the tendency to believe false[1] information to be correct after repeated exposure.

[1] The "false" designation here is unnecessary and flawed imho (or better: a fine example of how ubiquitous this problem is).

> (Philosophers of metaphysics are still arguing whether acausal aspects of reality are, in any real sense, real,

What meaning do you assign to the symbol "real" in this context?

Is it flawless?

> ...but we don't need to listen to them.)

In an absolute sense, no you don't. But to achieve certain desires you may have to. Like the saying goes: "You may not be interested in metaphysics (or truth, etc), but it might be interested in you".

But it's even more interesting: not only do you not have to listen to or seek truth, you can engage in mass collective story telling on the internet, confusing the population at scale. Don't believe me? There's a search function right here on HN, the evidence is there for the viewing!

> If we discover something non-physical (the usual example being ghosts), that'll just become a new branch of physics.

a) Science does like their Motte and Bailey.

b) I doubt you can actually see the future. If you disagree, please explain, using only physics, how you can.

> Physics currently talks about dark energy, the geometry of spacetime, and quantum superposition. I don't think you can get much less physical than that.

Do you believe that aggregate reality is constrained by your cognitive abilities, or are you perhaps more so describing your opinion about "reality"?

I don't think you understand what I'm saying. If we discover causal forces "outside the physical realm" (as we did when we first developed the theory of electromagnetism), we will simply redefine "the physical realm" to include the new, previously-unknown physics. (That, or we'll expand the domain of physics while preserving the legacy terminology, like when we decided that "world" meant a planet, or an "atom" could be separated into parts.)

> What meaning do you assign to the symbol "real" in this context?

Doesn't matter, because physics doesn't care whether or not its domain is "real". We can ignore the metaphysicians because physics is the study of models, observations, and discrepancies, and would work just as well if reality were an illusion.

> I doubt you can actually see the future. If you disagree, please explain, using only physics, how you can.

Information enters my brain. My brain gradually builds a predictive model, according to some (presumably physical) process that I call "me". I can then make confident conditional statements about the future, which historically have been overwhelmingly (though not exclusively) correct where the antecedent is satisfied. By induction, I infer that I possess the general ability to see the future with high accuracy, in certain domains.

Though information theory is usually considered a branch of mathematics, the (incredibly eldritch) field of thermodynamics allows theorems of information theory to be translated into theorems of physics, and (in some cases) vice versa.

> Do you believe that aggregate reality is constrained by your cognitive abilities,

No, but again, that's irrelevant.

> I don't think you understand what I'm saying.

You are telling stories. Stories are the foundation of the culture you have been raised in, they trump everything.

I know you don't understand what I'm saying.

Banach spaces looking like lightweight Euclidian spaces (in terms of constraints), and reality being experienced as data and applications, I'm going to try and say that:

Physics is a subset of Mathematics but which subset we don't exactly know.

In a sense, physics is the self-referential discovery of a subset of mathematics.

Makes sense since "existence" is impredicative.

I find infinite dimensional spaces even of zero curvature to be a bit creepy. Especially with a vector space structure, that just makes it worse.

You're entirely missing the critical point of my comment, which is the initial conditions.

I beg to differ. What are initial conditions if not some additional constraints?

Mathematics is perhaps a subset of logic? I'm not very familiar with this space, but I think it's pretty cool that so much of math can be represented with very simple rules building up, especially with things like the Lean proof language.

My understanding is that QFT is not axiomatized - it works but there's a bit of hand-waving at the level of actual proofs. There are people working on this, but currently physics is physics and maths is maths.

QFT also becomes increasingly unwieldy for large systems. E.g., try calculating bond energies of complex molecules. You can do it for hydrogen and "hydrogen like" molecules, but once you get beyond a few interactions, the differential equation does not have closed form solutions. This is not a total loss as you can resort to numerical solutions in at least some cases.

Science is mostly about producing models that "work" whether this involves reductionism to more fundamental principles or not. At worst however, the model which describes the larger system should not conflict with the description of the smaller one.

Verseon is doing raw quantum mechanics computations about various potential drugs and various important receptors. https://www.verseon.com/

Physics is just applied math.

But not all math can be applied.

Physics is not math.

Physics uses math.

Physics is not math any more than the map is the territory.

https://en.wikipedia.org/wiki/Map%E2%80%93territory_relation

Depends on your philosophy.

There are an uncountable number of atoms in the universe. There are nondeterministic occurrences in many natural processes. Even if the universe is a simulation, that doesn’t mean that it’s able to be represented or formalized from within the simulation.

Math is possibly one of the best tools of the scientific method to make logical sense of the universe, but that doesn’t mean that we can compute any given thing due to the scales involved. Gödel was likely familiar with the unknowable nature of the universe given certain hard limits of calculation, computation, and measurement.

Physics is the tool we use to move the world, yet we may never be able to build a lever or fulcrum long enough to actually do so - physics and math only give us a place to stand and means to orient ourselves. Everything else is engineering, I’d say.

> Give me a place to stand and with a lever I will move the whole world.

https://en.wikiquote.org/wiki/Archimedes

I would like to hear your thoughts, however. I think you’re right in a sense, that physics and math are possibly inseparable syntactically, similar to the identity property. If a thing is what it does, physics is math as much as it is logic, or vice versa.

Like math, logic is what we understand it to be, regardless of correspondence to reality. Physics rather seeks to represent and model more than the abstract truths of math and logic, and physics has an element of necessary utility and correspondence to hypothetical or actually existing realities, possibilities, and observable phenomena, whereas math and logic are not burdened by testability, but are rather proven or disproven via internal consistency and formalisms.

Like many tools, their proper usage comes down to holding them correctly, both in hand and in mind.

https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...

Awesome reference! Thanks for that.

To further cite:

On Math, Matter and Mind

https://arxiv.org/abs/physics/0510188

If I’m following your train of thought, I read it like this:

All squares are rectangles, but not all rectangles are squares. All physics is math, but not all math is physics.

As it applies to my understanding of your point and to my understanding of the MUH, Tegmark argues that the universe is necessarily pigeonholed by physics, and must be constrained by and embodies physics, which seems logical and consistent to me.

My point was perhaps a bit more pedantic regarding the distinctions between A: actually existing external reality independent of our understanding of it, and B: our representation of the universe that physics seeks to formally define.

I think it’s reasonable to assume that the universe is internally consistent and rational; that is to say that it’s bounded by so-called laws of physics, but we may not be able to reason about it because we currently lack the tools to make sense of it; that is to say we don’t know how to represent it mathematically.

I think we likely agree - the universe computes itself and proves itself by its essential nature and very existence. I don’t mean to speak for you, but only to find points of agreement.

Oh, right. It "only" took the Manhattan Project.

The only reason that nuclear energy/weapons are possible at all is because there exists isotopes that release more neutrons than they absorb during fission (the chain reaction). Nuclear fission itself wasn't fully understood until 1938.

I think this is the hard part of analyzing systems. If there exists mechanisms outside of your current understanding, there is no way to predict what is and isn't possible

"It always seems impossible until it's done." - Nelson Mandela

>If there exists mechanisms outside of your current understanding, there is no way to predict what is and isn't possible

See Lord Kelvin on heavier-than-air flight (he was fixated IIRC on the theoretical maximum power-to-weight ratio of a steam engine)

It seems like less well researched hunches on nuclear processes often led programs astray with bad calculations.

The Germans focused on heavy water for their reactors instead of just using graphite, because they miscalculated neutron absorption allegedly because of measuring graphite pieces rich in boron. The entire program was subsequently after the heavy water sabotages in Norways seemingly taking too long and was depriorititzed (Did the world luck out on small happenstances like this and Lenard?).

The Swedish program likewise initially overestimated the required amount of fissile materials needed by an order of magnitude and it wasn't clear until years later, could an accurate figure have made the program progress quicker before public opinion turned? (a lot of complex and expensive designs were worked out to produce at the higher rate)

>He obsessively pursued the ‘continuum hypothesis’, which states roughly that the infinitude of real numbers is next largest to that of natural numbers.

That sounds backwards? There are infinite real numbers between just 0 and 1, alone.

EDIT: my respondents so far seem to be stating the same thing I was. I interpreted the phrase "next largest" from TFA to mean that the infinitude of real numbers was a smaller than that of natural numbers. (So: not the largest, but rather the "next largest"). That, to me, is what sounded backwards.

This speaks of the mathematical notion of cardinality, which is the number of elements in a set. For infinite sets, it turns out there are multiple sizes. The natural numbers have the smallest infinite size, and it is the same size of set as the integers and perhaps surprisingly, the rationals. But the real numbers have more elements. A quick proof is called Cantor's diagonalization argument, I recommend looking it up!

In fact, I recently learned there is one cardinality for each possible way to order an infinite set. This is slightly confusing because there are many ordinals of each cardinality, but there are just that many cardinals I've decided.

I missed your suggestion about Cantor's diagonalization earlier today, but I thank you for bringing it to my attention. That is quite clever.

two sets can both be infinite and yet one is "larger" than the other (c.f. https://en.wikipedia.org/wiki/Aleph_number).

You can establish two infinite sets are as large as one another by finding a bijection between them. These two sets would have the same "cardinality"

We know the real numbers has larger cardinality than natural numbers, but we don't know if there's anything in between -- can you construct an infinite set that has natural numbers < X < real numbers in terms of cardinality?

OP understands that, the quote says that the "the infinitude of real numbers is next largest to that of natural numbers", which makes it sound like it is next largest as in second largest as in less large as in lower cardinality. So it seems backwards. Maybe the quote means "next largest" as in larger.

Thank you. It seems absurd that 'next largest' could mean 'larger than the largest'. Hence my (OP's) interpretation.

I see. yeah I can see why it's confusing now -- we use "next-to-leading order" to mean a smaller effect than the leading order, but here, "next largest" means the one larger than natural numbers. Language is hard

Tangential to your point about the poor phrasing is that having "infinite ... numbers between just 0 and 1" is not evidence of a higher infinity.

Specifically, there are infinite rational numbers between 0 and 1, but rationals have the same cardinality as the natural numbers.

That's an interesting statement about rationals. My intuition would be to consider them a subset of the reals, and for natural numbers to be a subset of the rationals. How is my intuition failing me, in this case?

EDIT: found the answer: with Cantor's diagonalization, you can count all the rationals -- effectively mapping each one to a natural number. Since this mapping is demonstrably possible, they have the same cardinality.

that "hover-coat" skiing photo though...

for sure! I was curious if I could find anything else about it, and there's a patent!

Sept. 3, 1940. H. THIRRING BRAKING DEVICE FUR SKIING Filed m. 4, 1957 INVENTOR ATTOR NEYS Patented Sept. 3, 1940 oFr cE BRAKING DEVICE FOR lKHNG Hans Thirring, Vienna, Austria Application August 1937, Serial No. 157,278 in Austria April 1, 193'? 5 Claims.

https://patents.google.com/patent/US2213754A/en?oq=US1572783...

He had debilitating social anxiety.

It’s not hard to imagine this played into him feeling uncomfortable in passing the message.

We’re talking about frigging A-bombs, I think the phrase “sack up” is apropos here. Oh well, we lucked out in this case at least…

Off topic: why is Aristotle held in such high regard? Isn't he the person who got almost everything wrong?

Not really. He certainly got some things wrong like heavier things falling faster than lighter ones, but he wrote on an enormous variety of topics, from physics, to biology, to literature, to ethics. His knowledge was broader than it was deep. His works summarized and synthesized most of what was known in his time, and was useful for centuries afterwards. The problem with Aristotle isn't so much him as the fact that the Catholic Church decided in the Middle Ages that (even though he lived before Christianity) that he was right about everything and that people who felt otherwise were heretics.

> The Church is... le BAD

Aristotelian Thomism was not enforced, nor was it agreed upon by theologicians at the time within the Church. Aquinas would only gain more support to his ideas later on, and after his death. But this was never enforced Church teaching that is infallible, it was simply one way (out of many) to do theology and arrive at certain conclusions. No one has ever been labeled a heretic by disagreeing with Aristotelianism.

Tommaso Campanella certainly was and was jailed by the Inquisition. Although like Giordano Bruno, it is hard to know exactly which of his many nonstandard beliefs the Church disliked more. Campanella wrote a book attacking Aristotle ("Philosophy demonstrated by the senses"), but he also was into astrology and magic and was also a sort of proto-Marxist with his "City of the Sun" utopia.

https://en.wikipedia.org/wiki/Tommaso_Campanella

The Inquisition was more about persecuting people who published dissenting works and finding any justification after the fact. Galileo was just repeating Copernicus’s theory, but he used the Vatican’s own press to do it, which made him a target.

You must sit in the comfy chair until lunch time with only a cup of coffee at 11.

Aristotle said that denser things fall faster than less dense things and even gave a correct expression for their speed given the limits of his mathematics and notation.

https://arxiv.org/abs/1312.4057

> He certainly got some things wrong like heavier things falling faster than lighter ones

That's wrong only if you stop at Galileo. If you go a bit further, Newton's laws tell you that the gravitational force between earth and a higher-mass object is stronger than with a lower-mass object (specifically, F = G m_1 m_2 / r^2). So the heavier object will hit the ground faster, not by virtue of accelerating faster, but by pulling earth towards it a bit more than the lighter object.

Arguably, that's not what Aristotle meant, though, and it's completely negligible for all practical purposes. But then again, once you factor in air drag and especially buoyancy, it's really not as simple as saying that all objects "fall" at the same rate. Unless you want to argue for contrived definitions where you say that perfectly stable floating objects are "falling". It is in fact relatively easy to perform experiments where objects fall at close to zero or even negative speeds (upwards).

Aristotle was the smartest guy around. Yes he got almost everything wrong but it took many centuries to find better theories.

Carlo Rovelli: "Aristotle's Physics: a Physicist's Look"

https://arxiv.org/abs/1312.4057

A great story. Too bad the ending is a dud.

Kurt Gödel kind of proved that it doesn't make much sense to build math based on axioms. This is what I consider the most important result of his work.

It doesn't make much sense... except for the fact that it works unreasonably well.

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

He didn't prove it doesn't make sense, he proved it's impossible to prove everything.

This is not.. really the right takeaway from his work. That not every true statement within a system is provable from its axioms, or that the consistency of a system is not provable within that system, does not mean that axioms are not useful. Indeed, what math is not grounded in the end by axioms?

in my mind, i compressed it to: there are always root assumption(s), that are just that, assumptions.

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