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Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

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Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#11

Post by inglor_cz »

card_zero wrote: post #5
andrewflnr wrote: post #2

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…

> 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?

"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.)

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#12

Post by rtpg »

andrewflnr wrote: post #2

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…

> 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)

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#13

Post by pfdietz »

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.

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#14

Post by rm30 »

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.

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#15

Post by AnimalMuppet »

rtpg wrote: post #12
andrewflnr wrote: post #2

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…

> 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…

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.

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#16

Post by DangitBobby »

zozbot234 wrote: post #8

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.

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.

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#17

Post by zozbot234 »

DangitBobby wrote: post #16
zozbot234 wrote: post #8

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.

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 sa…

> 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.

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#18

Post by aebtebeten »

gilleain wrote: post #4
andrewflnr wrote: post #2

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…

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 pract…

> ...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.

Re: Naples' 1790s civil war was intensified by moral panic over Real Analysis (2023)

#20

Post by aebtebeten »

inglor_cz wrote: post #3
andrewflnr wrote: post #2

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…

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…

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 α-and-ω in finite descriptions".

Is that a fair take? Would it be one, even if it were ahistorical?

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