[HN Gopher] Naples' 1790s civil war was intensified by moral pan...
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Naples' 1790s civil war was intensified by moral panic over Real
Analysis (2023)
Author : OgsyedIE
Score : 106 points
Date : 2026-01-31 03:53 UTC (19 hours ago)
(HTM) web link (lareviewofbooks.org)
(TXT) w3m dump (lareviewofbooks.org)
| andrewflnr wrote:
| I really want to read an essay on this topic by someone I'm more
| confident actually understands what math is. Or truth, for that
| matter. The author smears the boundary between what people
| believe and what is logically entailed, and between mathematical
| techniques and the way they are applied in modelling the real
| world. They persist in phrasing their statements about how people
| conceptualize math in terms of "is" and "are", which I tend to
| assume is a stylistic choice to speak in the perspective of their
| subjects, but they're so sloppy about perception and truth and
| "reason" in the rest of the piece that I can't be sure.
| inglor_cz wrote:
| I studied math (Algebra and Number Theory) and I am also quite
| interested in history, and while I cannot write you a whole
| essay, this is what I would like to react to:
|
| "The author smears the boundary between what people believe and
| what is logically entailed"
|
| This is not the fault of the author. This is a fairly accurate
| description of the societal situation back then, and the
| article is more about societal impacts of math than math
| itself. Revolutionary, and later Napoleonic France had very
| high regard for science, to the degree that Napoleon took a
| sizeable contingent of scientists (including then-top
| mathematicians like Gaspard Monge) with him to Egypt in 1799.
| The same France also conquered half of the continent and
| upended traditional relations everywhere.
|
| This caused some political reaction in the, well, more
| reactionary parts of the world, especially given that the
| foundations of modern mathematics were yet incomplete. Many
| important algebraic and analytic theorems were only
| discovered/proven in the 19th century proper. Therefore, there
| was a certain tendency to RETVRN to the golden age of geometry,
| which also for historical reasons didn't involve any French
| people (and that was politically expedient).
|
| If I had to compare this situation to whatever is happening
| now, it would be politicization of biology/medicine after
| Covid. Another similarity is that many scientists were
| completely existentially dependent on their kings, which didn't
| give them a lot of independence, especially in bigger
| countries, where you could not simply move to a competing
| jurisdiction 20 miles away.
|
| If your sovereign is somewhat educated (which, at that time,
| was already quite normal; these aren't illiterate chieftains of
| the Carolingian era) and hates subversive French (mathematical
| or otherwise) innovations with passion, you won't be dabbling
| with them openly.
| aebtebeten wrote:
| I'd always kind of imagined the reactionary geometers were
| defending an order in which their tools were imperfect finite
| approximations that yielded insights into perfect infinite
| truths, where the original sin of the revolutionary analysts
| was in saying that "yes, and with compactness and continuity,
| many of these problems have their a-and-o in finite
| descriptions".
|
| Is that a fair take? Would it be one, even if it were
| ahistorical?
| inglor_cz wrote:
| I like that perspective, but I believe the conflict was
| more about "old vs. new". Geometry was _very_ old by that
| point, ancient, and it carried a lot of personal gravitas
| by being associated with Euclid, Archimedes, Thales etc.
| (Galenic theory of humors enjoyed similar ancient
| intellectual prestige, hence its long and bitter retreat
| from the scene at approximately the same time.) It was also
| "obviously right", in the sense of "everyone can look and
| see for themselves". Even uneducated peple can verify that
| a certain line touches both circles etc. No wonder it was
| an attractive safe haven for conservative minds.
|
| Meanwhile, analysis was not yet particularly rigorous and
| it took several decades to converge on a standard apparatus
| and notation that could at least be understood coherently
| by _other mathematicians_. (Laymen tend to struggle with it
| until today.) Add the political dimensions of being seen
| friendly to the French into the mix, well...
| andrewflnr wrote:
| > This is not the fault of the author. This is a fairly
| accurate description of the societal situation back then
|
| The author is not giving a helpful description if they slip
| into the same mistakes as the people of the time.
| bgilroy26 wrote:
| >Therefore, there was a certain tendency to RETVRN to the
| golden age of geometry
|
| Echoed in the LaRouchians of the 2000s. I don't know what
| they're up to now
| gilleain wrote:
| Oh! I really liked the essay - the idea that French 'analysis'
| was seen as a dangerous modern invention and contrasted with
| 'synthetic' geometric understanding of the world had political
| implications is fascinating. There could be parallels with the
| present day use of computer modelling (and now AI) being seen
| as a risky way to organise and run societies.
|
| I agree that there is a lot of vague language around the
| practice of mathematics as a social and philosophical construct
| ('analysts' vs 'synthetics') but I'm not sure how that
| indicates the author does not understand what truth is. My
| understanding of the history of mathematics and science is that
| these areas of knowledge were much more intertwined with
| philosophy and religion than they are considered to be today.
|
| So Newton saw no issue with working on the calculus at the same
| time as being an alchemist and a non-trinitarian. Understanding
| the world was often a religious activity - by understanding
| Nature, you understood God's creation - and in Naples it seems
| that understanding analysis was tied to certain political and
| nationalist ideas.
| aebtebeten wrote:
| > _...these areas of knowledge were much more intertwined
| with philosophy and religion..._
|
| Indeed, consider Laplace to Napoleon "I had no need of that
| hypothesis", ca 1799.
| card_zero wrote:
| > statements about how people conceptualize math in terms of
| "is" and "are"
|
| What do you mean? I searched the page for "are", it doesn't
| appear much at all, I'm ruling that one out. So do you mean for
| instance this statement - ? "This zealous quest
| for universal problem-solving algorithms is precisely what made
| the synthetics uneasy."
|
| What's wrong with that?
| inglor_cz wrote:
| "What's wrong with that?"
|
| Political context. Rationalism was associated with atheism,
| which, for the first time in European history, started making
| visible inroads into the intellectual class. If you can solve
| all your problems using your reason, do you really need a
| God? And plenty of French philosophers hinted that the answer
| could be "no".
|
| It wasn't just a religious question. Atheism or suspicion of
| thereof was seen as politically subversive, in the age when
| most ruling feudal dynasties still relied on God's grace as
| the ultimate fount of their power - at least in their eyes of
| the subjects. (But it wasn't always that cynical, plenty of
| the rulers themselves were quite pious.)
| Blackthorn wrote:
| Very good reminder when it comes to thinking about that
| period of time. The famous apocryphal conversation between
| Napoleon and Laplace comes to mind here.
| rtpg wrote:
| > The author smears the boundary between what people believe
| and what is logically entailed, and between mathematical
| techniques and the way they are applied in modelling the real
| world.
|
| I think the clue here is the section mentioning Cauchy and
| rigor.
|
| Without a certain flavor of rigor, "proofs" given by people,
| _especially in analysis_, can feel unsatisfying and can
| outright be incorrect, even if the thing they are trying to
| prove is true!
|
| Imagine a proof of the intermediate value theorem like: well if
| you try to go from point A to point B you _have_ to pass
| through C in between eventually or else you'll never get to B.
|
| This might be a sketch of a proof, vaguely. And it's not like
| the IVT is _wrong_, right? But a non-rigorous proof is not
| convincing. A non-rigorous proof might leave out details that
| would otherwise guarantee that a proof isn't left up to
| interpretation.
|
| If your proof hand waves away some cases that feel trivial to
| you, to others that might look like a hole in your proof! Or
| you might think it's trivial, and actually it's not trivial..
| but you haven't done it.
|
| Anyways this is, I think, the core here. A new style of
| mathematics with new foundations... that haven't quite been
| smoothed out yet. The conclusions being reached are all kinda
| mostly right, but the reasons the conclusions are correct have
| not been actually properly set up. So skeptics can drive a
| truck through that contradiction.
|
| Knowledge is about knowing the right thing for the right
| reasons... and in its infancy I could see a universe where a
| lot of mathematicians are running around using its tooling
| without having the right foundations for it.
|
| We are lucky to live downstream of all this hard work. In the
| moment things were messier (see also calculus' initial growing
| pains)
| AnimalMuppet wrote:
| Francis Schaeffer defines epistemology as "what we know, and
| how we know, and how we know we know". Sounds like they were
| missing that third part. And because they were, they could
| not be quite sure of the other two.
| arduanika wrote:
| War often pushes people to the limit
| bawolff wrote:
| > The Neapolitans did not reject modern analysis simply because
| they considered it French.
|
| And yet after reading the article, it sounds like that is exactly
| what happened. They took some minor philosophical dispute in math
| and blew it up for cultural reasons to stick it to the invader.
| It doesn't sound like it ever really was about the math for most
| people in that context.
| 2b3a51 wrote:
| I think it depends on what one regards as a minor dispute. The
| Newton/Leibniz calculus dispute a generation or so earlier was
| pretty major, with Newton defending his deductive geometrical
| method of fluxions against Leibniz's more algebraic concepts.
| Leibniz was also much into his universal calculus. I was
| wondering what this Fergola would have thought about Newton and
| his geometrical method (fluxions)!
|
| The Naples state at that time was around 5 million people. You
| had the landowners (I imagine) looking around at the
| 'enclosures' of common land in Britain and other parts of
| Europe and thinking about rents. You had the engineers and
| Jacobins thinking about new roads and canals and all. The ones
| who lost out appear to have been the peasants as they lost the
| feudal protections and access to common lands. And so it goes.
| zozbot234 wrote:
| Hot take: the author sneaks in a premise that synthetic
| mathematics is per se "reactionary", but this is itself pure
| reactionary copium for not getting it:
| https://ncatlab.org/nlab/show/synthetic+mathematics
| https://en.wikipedia.org/wiki/Synthetic_mathematics . There's
| nothing wrong with wishing to pursue a "coordinate-free" approach
| to any mathematical field: the old geometers were quite right
| about this.
| DangitBobby wrote:
| I don't understand how you got that at all. The impression I
| got was the synthetic mathematicians were wary of applying the
| new "analysis" methods because lack of rigor made them question
| how safe and valid it was to use.
|
| > Using another set of metaphors: Analysts followed the fast
| flights of their feverish and uncontrolled imagination, while
| synthetics kept their feet on the ground. Their procedures were
| slow but safe.
| zozbot234 wrote:
| > I don't understand how you got that at all.
|
| The word "reactionary" appears multiple times on that page,
| in association with either synthetic mathematics itself or
| the politics that's said to be closely connected to it. I'd
| say that's by far the most unambiguous part of my argument
| about this text.
| pfdietz wrote:
| The critique of calculus as lacking in rigor goes much further
| back than that. Bishop Berkeley famously argued calculus was no
| more dependable than theology. It was only with Cauchy and the
| formalization of analysis in the 19th century that this issue
| would be put to bed.
|
| https://en.wikipedia.org/wiki/The_Analyst
|
| I wonder if the issues that this essay claims came up in Italy
| persisted in any way. I ask that, because there was later
| (1885-1935) an infamous breakdown in Italian mathematics (the
| "Italian School of Algebraic Geometry") due to foundational
| issues.
|
| https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
|
| History doesn't repeat but it sometimes rhymes.
| aebtebeten wrote:
| From a contemporary viewpoint, Berkeley seems to have been a
| bit of an edgelord in his defense of faith:
| https://en.wikipedia.org/wiki/Appeal_to_the_stone#Origin
|
| (or am I doing him an injustice and he was running with the
| whole Cartesian Demon thing to see just how far he could take
| it?)
|
| Mumford (a grand-advisee of Castelnuovo's) on the infamous
| school: https://ftp.mcs.anl.gov/pub/qed/archive/209
| rm30 wrote:
| If we review the history we can notice that there was always an
| influence from politics/religion to science, literature, arts,
| philosophy and the use of them by politics, maybe to justify some
| decision and state of facts.
|
| It helps to empower control over population and fits perfectly in
| the social and historical context: the emperor blessed by God,
| the evolution theory, the epic poems, theory of race, the
| industrial revolution, and modern times don't escape these
| patterns too, we just suppose to be neutral.
| amelius wrote:
| This is so far removed from today's politics where the electorate
| seems to have switched off their brains.
| kazinator wrote:
| Since the dawn of electronics, mathematics has entangled with
| politics indirectly, via technological change. The article
| describes a de facto technological change (emergence of Real
| Analysis), a kind of technology with many practical applications
| with economic implications.
| boothby wrote:
| Ye gods, we're truly in the stupidest timeline.
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(page generated 2026-01-31 23:01 UTC)