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.
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.
Btw, JD Vance claimed his wife is his spirit animal, so if he's a racist, he is at best the kind of racist who'd find it morally imperative that animals try and become human?
PS: have you figured out what math pros would call the satirical functor? It's alright to ask your kith kin or kollegen for help :)
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.
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.
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.
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?
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...
> 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)
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.
> 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."
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.)
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.
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.
> 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.
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.
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.
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+mathematicshttps://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.
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.
> considering the coordinate system as spoiling the purity
Isn't that one of the main arguments for "coordinate free" or "synthetic" anything? It's about a pursuit of elegance and increased generality, not necessarily of obnoxious real-world politics; so it seems especially weird to inherently conflate these things.
Depends way you look at it. In some ways, analytic geometry generalises a set of algebraic tools to geometric problems. Eg from the text
> For the analysts, this was irrelevant: algebra captured the essential relations expressed by the terms of the problem, which then served to guide the mathematician toward the solution. For the synthetics, by contrast, a solution to the original geometrical problem could only be geometrical in nature; and so, what the analysts were offering were not solutions but meaningless numbers.
> While the analysts strove for maximum generality, the synthetics argued for the specificity and locality of all mathematical methods. […] [For the synthetics] even more misleading would be to believe that there is a single universal method that can be applied to all kinds of problems.
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.
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
Fwiw, (exotericised)
Seems like it could be somewhat ticklish, and Could almost be.Btw, JD Vance claimed his wife is his spirit animal, so if he's a racist, he is at best the kind of racist who'd find it morally imperative that animals try and become human?
PS: have you figured out what math pros would call the satirical functor? It's alright to ask your kith kin or kollegen for help :)
Unless it be one of the functors from Mathematics Made Difficult (1971) I shall have to ask my kith for counsel...
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.
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.
> ...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.
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.
> 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.
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?
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...
>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
> 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)
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.
> 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 - ?
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.)
> (But it wasn't always that cynical, plenty of the rulers themselves were quite pious.)
It's probably easy to believe in a God that pre-approved and delivered my life of wealth, power, and leisure. :p
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.
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.
War often pushes people to the limit
> 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.
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.
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.
Did you send this from an LLM?
This is so far removed from today's politics where the electorate seems to have switched off their brains.
Ye gods, we're truly in the stupidest timeline.
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.
> 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.
Maybe because the title of the book this is a review of is “Reactionary Mathematics: A Genealogy of Purity”.
This does not mean that coordinate free geometry is reactionary per se, but considering the coordinate system as spoiling the purity of geometry is.
> considering the coordinate system as spoiling the purity
Isn't that one of the main arguments for "coordinate free" or "synthetic" anything? It's about a pursuit of elegance and increased generality, not necessarily of obnoxious real-world politics; so it seems especially weird to inherently conflate these things.
Depends way you look at it. In some ways, analytic geometry generalises a set of algebraic tools to geometric problems. Eg from the text
> For the analysts, this was irrelevant: algebra captured the essential relations expressed by the terms of the problem, which then served to guide the mathematician toward the solution. For the synthetics, by contrast, a solution to the original geometrical problem could only be geometrical in nature; and so, what the analysts were offering were not solutions but meaningless numbers.
> While the analysts strove for maximum generality, the synthetics argued for the specificity and locality of all mathematical methods. […] [For the synthetics] even more misleading would be to believe that there is a single universal method that can be applied to all kinds of problems.