[HN Gopher] The spy who flunked it: Kurt Godel's forgotten part ...
___________________________________________________________________
The spy who flunked it: Kurt Godel's forgotten part in the atom-
bomb story
Author : drdee
Score : 117 points
Date : 2024-05-16 14:27 UTC (8 hours ago)
(HTM) web link (www.nature.com)
(TXT) w3m dump (www.nature.com)
| bgutierrez wrote:
| To me, the most interesting sentence in this essay is: "[H]e,
| Godel, had felt that a nuclear chain reaction would be possible
| only 'in a distant future'."
|
| Godel 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.
| JadeNB wrote:
| > Godel 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.
| Isamu wrote:
| The chain reaction was still speculative. Also physics is math,
| but math is not physics.
| genman wrote:
| Physics is math, but not all math is physics.
| andrewflnr wrote:
| Physics is, optimistically, math plus a whole load of
| initial conditions that are not findable a priori.
| wizzwizz4 wrote:
| Physics is _the study of_ reality under the assumption
| that it 's lawful. The real world doesn't run on physics:
| it just is.
| andrewflnr wrote:
| Yes, that's one of the less mathematically optimistic
| perspectives. :)
| mistermann wrote:
| The _physical portion of_ reality.
| wizzwizz4 wrote:
| 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.
| aatd86 wrote:
| 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.
| lanstin wrote:
| 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.
| andrewflnr wrote:
| You're entirely missing the critical point of my comment,
| which is the _initial conditions_.
| aatd86 wrote:
| I beg to differ. What are initial conditions if not some
| additional constraints?
| smj-edison wrote:
| 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.
| lanstin wrote:
| 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.
| jandrese wrote:
| Physics is just applied math.
| isk517 wrote:
| But not all math can be applied.
| 77pt77 wrote:
| Physics is not math.
|
| Physics uses math.
| aspenmayer wrote:
| Physics is not math any more than the map is the territory.
|
| https://en.wikipedia.org/wiki/Map%E2%80%93territory_relation
| baggy_trough wrote:
| Depends on your philosophy.
| aspenmayer wrote:
| 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. Godel 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.
| baggy_trough wrote:
| https://en.wikipedia.org/wiki/Mathematical_universe_hypot
| hes...
| aspenmayer wrote:
| 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.
| marcosdumay wrote:
| Oh, right. It "only" took the Manhattan Project.
| campbel wrote:
| 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
| lainga wrote:
| >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)
| whizzter wrote:
| 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)
| zitterbewegung wrote:
| One does not send Kurt Godel 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.
| Godel'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 Godel 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...
| aspenmayer wrote:
| 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...
| jmcqk6 wrote:
| 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.
| pavlov wrote:
| 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.
| readthenotes1 wrote:
| "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?
| cyberax wrote:
| 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.
| chuckadams wrote:
| 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.
| aspenmayer wrote:
| 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/opini
| ons/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/1003616210
| 9221478...
| aspenmayer wrote:
| 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
| mindslight wrote:
| What Godel 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.
| Apocryphon wrote:
| 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.
| kaba0 wrote:
| Libertarians are not known for their strong logical
| arguments.
| mindslight wrote:
| 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).
| Apocryphon wrote:
| I think that might fit into left-libertarianism and/or
| individualist libertarianism, perhaps.
| lifeisstillgood wrote:
| 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
| mindslight wrote:
| 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.
| gweinberg wrote:
| 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.
| mindslight wrote:
| 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.
| gweinberg wrote:
| 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.
| readthenotes1 wrote:
| 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...
| gweinberg wrote:
| 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.
| aspenmayer wrote:
| I mentioned FDR due to his being President at the time of
| Godel'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_Americ
| a...
| chuckadams wrote:
| The Nazis modeled their laws on Jim Crow, that's well-
| known. What's this about nerve gas tho?
| aspenmayer wrote:
| > 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-Juarez 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
| chuckadams wrote:
| Ok, "treated clothes with a deadly pesticide" is really
| quite different than "gassed migrants". I mean, we were
| still burning leaded gas then.
| aspenmayer wrote:
| 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.
| chuckadams wrote:
| 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?
| aspenmayer wrote:
| 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."
| Apocryphon wrote:
| 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.
| xhkkffbf wrote:
| A great story. Too bad the ending is a dud.
| a-dub wrote:
| that "hover-coat" skiing photo though...
| jdblair wrote:
| 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...
| DeathArrow wrote:
| Kurt Godel 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.
| ndr wrote:
| It doesn't make much sense... except for the fact that it works
| unreasonably well.
|
| https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
| croes wrote:
| He didn't prove it doesn't make sense, he proved it's
| impossible to prove everything.
| hucker wrote:
| 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?
| a-dub wrote:
| in my mind, i compressed it to: there are always root
| assumption(s), that are just that, assumptions.
| rvbissell wrote:
| >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.
| ducttapecrown wrote:
| 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.
| moelf wrote:
| 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?
| jmilloy wrote:
| 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.
| rvbissell wrote:
| Thank you. It seems absurd that 'next largest' could mean
| 'larger than the largest'. Hence my (OP's) interpretation.
| moelf wrote:
| 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
| andrewla wrote:
| 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.
| rational_indian wrote:
| Off topic: why is Aristotle held in such high regard? Isn't he
| the person who got almost everything wrong?
| jhbadger wrote:
| 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.
| CorrectingYou wrote:
| > 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.
| jhbadger wrote:
| 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
| chuckadams wrote:
| 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.
| aeneasmackenzie wrote:
| 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
| t_mann wrote:
| > 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).
| WhitneyLand wrote:
| He had debilitating social anxiety.
|
| It's not hard to imagine this played into him feeling
| uncomfortable in passing the message.
| chuckadams wrote:
| 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...
___________________________________________________________________
(page generated 2024-05-16 23:02 UTC)