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Re:Monumental advance in number theory posted to the ArXiv b... by maegul on Monday November 07, 2022 @09:20PM (#33515219)
> but it now requires so much knowledge just to get to the frontiers of human knowledge, not to speak of making a dent into uncharted territory, that results are being obtained later and later in life. When mathematicians have had time to accrue sufficient knowledge while still being sharp enough to make the intellectual leap. The sad part is that as the trend continues we may reach a point where a mathematician's intellectually productive life is not sufficient to contribute anything novel, statistically speaking.

IMO, an under appreciated dynamic across the board of human endeavours.

With increasing complexity comes the need for more time to understand and master anything.

[dead]
Re:Monumental advance in number theory posted to the ArXiv b... by deanCommie on Monday November 07, 2022 @11:39PM (#33515914)
> Applying AI to education should supercharge the smartest.

What does that mean?

Re:Monumental advance in number theory posted to the ArXiv b... by wnkrshm on Tuesday November 08, 2022 @02:47AM (#33516885)
I don't know but it also fails to disregard that the smartest people (whatever that means) can have executive function issues that are orthogonal to their mental capacity.

Additionally, the belief that intelligence alone will improve the world is misguided imo. You need empathetic, intelligent people.

Re:Monumental advance in number theory posted to the ArXiv b... by kwhitefoot on Tuesday November 08, 2022 @06:30AM (#33518066)
As we do not yet have artificial intelligence making confident statements about what its effects might be is surely rather rash.
Re:Monumental advance in number theory posted to the ArXiv b... by ConceptJunkie on Tuesday November 08, 2022 @08:40AM (#33518804)
You must be an optimist if you think what passes for AI right now could in any way help education. It's an incredibly useful tool, but not in the way you almost certainly are thinking.
Re:Monumental advance in number theory posted to the ArXiv b... by inamberclad on Monday November 07, 2022 @10:21PM (#33515559)
Agreed, it feels like the days of maverick engineers leading huge efforts is mostly over. Will there be any more Kelley Johnsons, or Gene Kranzs in our lifetimes?
Re:Monumental advance in number theory posted to the ArXiv b... by zip1234 on Monday November 07, 2022 @11:06PM (#33515772)
Tools can be used for this purpose. New tools can unlock new abilities for engineers. I’m not sure if it applies to math but I don’t have any reason why it shouldn’t.
Re:Monumental advance in number theory posted to the ArXiv b... by colordrops on Monday November 07, 2022 @11:09PM (#33515784)
Engineering is different from math. A large part of the reason why engineering innovation is lacking in a particular field is due to entrenched players, heavy bureaucracy, and government regulation, for instance with spaceflight and EVs.
Re:Monumental advance in number theory posted to the ArXiv b... by ciphol on Tuesday November 08, 2022 @01:54AM (#33516578)
There is still plenty of amazing engineering going on, for example EUV development at ASML.
Re:Monumental advance in number theory posted to the ArXiv b... by earthboundkid on Monday November 07, 2022 @11:52PM (#33515980)
The “lack of progress” in philosophy comes down to this. By the time of Socrates or Confucius, there had already been a lifetime’s worth of philosophical thought done that you needed to grapple with. Everyone since then has necessarily had to let something slip through in order to move forward at all.
Re:Monumental advance in number theory posted to the ArXiv b... by woodruffw on Tuesday November 08, 2022 @12:06AM (#33516039)
I think we sometimes understate philosophy's progress: it's certainly not measurable in decades, but the world of 2022 looks very different (in terms of philosophical priors and their consequences) than the world of Socrates or Confucius. A handful of examples that come to mind:

* (Nearly) everybody on our planet lives under a government whose fundamental structure and right to power comes from modern (meaning 17th century) political philosophy.

* Everybody posting on this board is using a machine that only existed in the mind games (and works) of math philosophers just over a century ago.

* The philosophical foundations for contemporary science are very young, practically within living memory. Popper died less than 30 years ago!

Re:Monumental advance in number theory posted to the ArXiv b... by adalacelove on Tuesday November 08, 2022 @12:36AM (#33516210)
We measure philosophy success relative to finding the meaning of life. On the other hand everybody agrees that computers have advanced enormously without achieving yet AGI.
Re:Monumental advance in number theory posted to the ArXiv b... by woodruffw on Tuesday November 08, 2022 @01:27AM (#33516456)
“We” don’t do anything of the sort. The meaning of life is a Douglas Adams joke; philosophy’s scope extends slightly further.
Re:Monumental advance in number theory posted to the ArXiv b... by kbrkbr on Tuesday November 08, 2022 @03:15AM (#33517023)
The measure of success is different from school to school, if any exists at all in a given school. And there is no lack of schools.
Re:Monumental advance in number theory posted to the ArXiv b... by LudwigNagasena on Tuesday November 08, 2022 @12:37AM (#33516212)
Did those things appear because philosophers talked about them or did philosophers talk about them because they appeared?
Re:Monumental advance in number theory posted to the ArXiv b... by unsupp0rted on Tuesday November 08, 2022 @01:16AM (#33516396)
I can easily imagine an Earth with no philosophers after say Aristotle that is functionally identical to our current one.
Re:Monumental advance in number theory posted to the ArXiv b... by TuringTest on Tuesday November 08, 2022 @04:32AM (#33517479)
Minus evidence-based science, free markets, government of the people, freedom of religion and the end of slavery. Yep, exactly the same.
Re:Monumental advance in number theory posted to the ArXiv b... by LudwigNagasena on Tuesday November 08, 2022 @04:45AM (#33517557)
Might as well attribute all of that to Christianity. That claim probably holds as much water.
Re:Monumental advance in number theory posted to the ArXiv b... by andrepd on Tuesday November 08, 2022 @07:13AM (#33518275)
Attributing science and freedom of religion to Christianity? That's a bold thought.
Re:Monumental advance in number theory posted to the ArXiv b... by dublin on Tuesday November 08, 2022 @08:27AM (#33518694)
And a correct one. There is a reason these things did not develop until when and where they did...
Re:Monumental advance in number theory posted to the ArXiv b... by woodruffw on Tuesday November 08, 2022 @08:51AM (#33518897)
Pinning concepts like "science" and "freedom" to a specific era is a silly endeavor. But if you had to, neither would originate in Christian Europe: science as a concept predates Christianity by at least 800 years; the thing we call "freedom" today didn't take shape until 1600 years into Christian Europe's development.

Christianity, like just about every religion, has made invaluable contributions to philosophy. But it's ahistorical, to an embarrassing degree, to attribute such broad concepts and bodies of inquiry to it.

Re:Monumental advance in number theory posted to the ArXiv b... by ConceptJunkie on Tuesday November 08, 2022 @08:36AM (#33518772)
Wow. I guess you're not familiar with any philosophers other than Aristotle, otherwise you could never make such a statement.
Re:Monumental advance in number theory posted to the ArXiv b... by unsupp0rted on Tuesday November 08, 2022 @09:45AM (#33519497)
Please elaborate
Re:Monumental advance in number theory posted to the ArXiv b... by whatshisface on Tuesday November 08, 2022 @01:27PM (#33522420)
There should be a rule where you're not allowed to phrase a question as a false statement that insults a lot of people...
Re:Monumental advance in number theory posted to the ArXiv b... by Tainnor on Tuesday November 08, 2022 @10:53AM (#33520337)
Without philosophy, there is no Hegel, without Hegel there is no Marx, without Marx there is no Lenin, without Lenin, there is no Stalin, without Stalin there is no Cold War, without the Cold War there's probably not a war in Ukraine right now, and without a war in Ukraine I wouldn't be worried about my gas bill. I think that's pretty concrete.

(Just a single, rather obvious example. Easy enough to find others.)

Re:Monumental advance in number theory posted to the ArXiv b... by woodruffw on Tuesday November 08, 2022 @01:26AM (#33516445)
Each of the examples I gave have direct origins in philosophy. It’s not even clear what the alternative would be for Hobbes and Popper; for Frege you could call him a mathematician instead, I suppose, if you ignore all of the other philosophy he did.
Re:Monumental advance in number theory posted to the ArXiv b... by kbrkbr on Tuesday November 08, 2022 @02:46AM (#33516871)
The causality here is hard to prove, even untangle. I think the French Revolution was largely more influential for the state of governments today. I also think that science was not influenced so much by Popper, but much more by Fisher and Neyman. People tend to use what is a successful tool, and be influenced by power and theatre, not so much by fragile logical constructs without empirical value found sitting in the armchair.
Re:Monumental advance in number theory posted to the ArXiv b... by Tainnor on Tuesday November 08, 2022 @11:03AM (#33520473)
Do you think the French Revolution happened in a vacuum? Without the ideals of Enlightenment philosophy it would have amounted to nothing more than mob violence (if it had even gotten started at all).
Re:Monumental advance in number theory posted to the ArXiv b... by kbrkbr on Tuesday November 08, 2022 @02:20PM (#33523086)
Good point, I agree that it did not. Were those ideas however progress, or a tool that the Committee of Public Safety used because it fit their agenda?

Was this philosophy finding truth that guided policy, or was it making up rhetorical arguments in a power struggle on behalf of the rising middle class and its policy?

In absence of criteria of non empirical truth that are independent of the very non empirical truths they are criteria for I tend to believe the latter.

In addition to this inherent weakness philosophical arguments are not for the masses anyways. Not even philosophers usually agree what they mean or entail. How could those idea(l)s have guided the revolution as more than just superficial self justification?

Remember that the enemies of the French Revolution had their philosophers too, they just lost.

Historical events usually have very mundane reasons, class struggles, geopolitics etc. Ideas may play a part too. But I believe not in the sense that they are the horses in front of the cart.

Re:Monumental advance in number theory posted to the ArXiv b... by Tainnor on Tuesday November 08, 2022 @04:25PM (#33524515)
Well, the original point was whether philosophy had any influence on world events, not whether that influence was good or not.

To me, the idea that ideas don't play a major role in world history and politics seems odd. There's nothing more fundamentally human than trying to understand the world around you and wanting to shape it in a certain way. Of course material conditions are important; the French Revolution wouldn't have happened had the state not gone bankrupt. But it also wouldn't have happened in that way without Enlightenment ideas. Very basic principles of the revolution, some of which were kept by Napoleon and in subsequent generations and still live on today were formulated by those enlightenment thinkers. Just consider the declaration of human rights.

Re:Monumental advance in number theory posted to the ArXiv b... by kbrkbr on Wednesday November 09, 2022 @06:14AM (#33529863)
I understood the thread roughly this way: „philosophy makes progress. The evidence is that the modern world has adopted the more progressed rules of enlightement and scientific rules.“ And my reply was that this is highly questionable.

I’m saying that the enlightenment ideas happened because the uprising middle class needed justifications in their power struggle. Not that the ideas formed a middle class that then acted upon them.

I agree with you that humans need ideas to tell a story of themselves. And that story better paint them as good! But that story is usually not the driving force.

So I fail to see progress as I don’t see a criterion to measure progress (in general. Aristoteles‘ logic is a tool. Here I see how to compare it to FOL and judge which is better or worse for specific use cases).

As for human rights: do I agree with them? Yes! Can I justify them? No! Do I know where to exactly draw the lines when two rights collide? No! Is it important to justify them in a philosophical way? I doubt.

Re:Monumental advance in number theory posted to the ArXiv b... by LudwigNagasena on Tuesday November 08, 2022 @04:54AM (#33517595)
I don’t see any direct origins. Karl Popper was a small child when Einstein came up with relativity. Hobbes was just a reactionary apologist of absolute monarchy. And Frege’s philosophical contributions were world-changing as long as we can pretend that propositional calculus is philosophy.
Re:Monumental advance in number theory posted to the ArXiv b... by woodruffw on Tuesday November 08, 2022 @08:24AM (#33518669)
I don’t think this is going to be productive, but to be clear: the claim isn’t that science didn’t exist before Popper, or that Popper somehow gets credit for all of science. It’s that Popper has produced the best description of science’s epistemic underpinnings thus far, allowing science to progress without imploding under the demands of positivism. In other words: when we “do” science, the epistemic model we use is less than a century old.

The claim that propositional calculus is in the domain of philosophy and mathematics is not controversial in either domain. Nor is it unusual: the history of philosophy is the history of discharging subjects once they develop a field of their own.

Re:Monumental advance in number theory posted to the ArXiv b... by LudwigNagasena on Tuesday November 08, 2022 @10:01AM (#33519701)
> It’s that Popper has produced the best description of science’s epistemic underpinnings thus far, allowing science to progress without imploding under the demands of positivism

I am not sure what that claim means precisely. Popper saved science from… a certain group of philosophers?

> when we “do” science, the epistemic model we use is less than a century old

The epistemic model is whatever the scientist in question uses. One can be a Popperian, another can claim they are guided by God, and someone else would just say “shut up and calculate.”

Popper didn’t like Bohr and his Copenhagen interpretation from the philosophical standpoint yet he didn’t deny that Bohr was a good physicist.

> The claim that propositional calculus is in the domain of philosophy and mathematics is not controversial in either domain. Nor is it unusual: the history of philosophy is the history of discharging subjects once they develop a field of their own.

The context of this discussion is that philosophy progresses over time. You cannot say that it progresses just because we put a label “philosophy” on every new thing that isn’t formalized yet. In that trivial sense, of course, philosophy had great progress; especially in its subfield of natural philosophy also known nowadays as science. But did Frege really stand on the shoulders of Hegel, Spinoza and Kant and improved on their ideas?

Re:Monumental advance in number theory posted to the ArXiv b... by FabHK on Tuesday November 08, 2022 @10:32AM (#33520064)
> Popper has produced the best description of science’s epistemic underpinnings thus far

Ok, sure.

> allowing science to progress without imploding under the demands of positivism

How is giving a description of something necessary for its continued progress? Were people not having sex before evolutionary psychologists and biologists elucidated how and why?

Re:Monumental advance in number theory posted to the ArXiv b... by Tainnor on Tuesday November 08, 2022 @10:59AM (#33520405)
You might be interested in the fact that Einstein himself claimed that he might not have stumbled upon relativity if it hadn't been for David Hume:

https://aeon.co/essays/what-albert-einstein-owes-to-david-hu...

Re:Monumental advance in number theory posted to the ArXiv b... by pmontra on Tuesday November 08, 2022 @07:01AM (#33518205)
I second your other two points but I think this one is inaccurate:

> * (Nearly) everybody on our planet lives under a government whose fundamental structure and right to power comes from modern (meaning 17th century) political philosophy.

A large number of countries are ruled by dictators and their close supporters. Sometimes they position themselves within the boundaries of a more or less modern ideology but ultimately care only about being the one in command and use that ideology as lipstick. This is a straight continuation of prehistory, not something stemming from 17th century.

Re:Monumental advance in number theory posted to the ArXiv b... by woodruffw on Tuesday November 08, 2022 @08:15AM (#33518603)
I meant more the structure of and justification for the state and sovereign, not liberal ideology. You’re absolutely right that a large number of people don’t live under the latter.
Re:Monumental advance in number theory posted to the ArXiv b... by jll29 on Tuesday November 08, 2022 @03:57PM (#33524191)
What is most annoying is: many of the dictators were installed by "democracies" so that these can continue their unsustainable lifestyles, at the expense of extreme suffering elsewhere.

...and I disagree with the parent's parent's point on progress in philosophy: all three examples are not progress of philosophy but at best influence or application of it, and in the last case did not involve trained/professional philosophers (e.g. Turing was a mathematician/chemist-turned-cryptoanalyst/computer scientist).

Re:Monumental advance in number theory posted to the ArXiv b... by LeanderK on Tuesday November 08, 2022 @05:15AM (#33517713)
which lack of progress? The few encounters with logic (in a philosophical sense, although it feels more "mathy") I had recently were all more recent works, certainly more recent than a large part of math I use!
Re:Monumental advance in number theory posted to the ArXiv b... by dang on Tuesday November 08, 2022 @05:00PM (#33524901)
We detached this subthread from https://news.ycombinator.com/item?id=33515219.
Re:Monumental advance in number theory posted to the ArXiv b... by kjellsbells on Tuesday November 08, 2022 @12:11AM (#33516065)
That's a very big claim, perhaps true only if human endeavor only ever builds linearly. But I dont think that is true. Certainly not in the arts or music, e.g. the Beatles did not need to ingest the entire corpus of Mozart or even Scott Joplin to be highly productive. Although, it certainly helped that they were extremely open minded to all types of music.

A second example is that of recent advances in virtualization, where the last decade of advances in things like cgroups, namespaces and containers were all done by people who I assert had no training in IBM MVS/zOS and therefore weren't building on what went before (imho, to their detriment).

Re:Monumental advance in number theory posted to the ArXiv b... by npunt on Tuesday November 08, 2022 @12:30AM (#33516164)
The Beatles themselves didn't need to ingest the entire corpus of past musicians to be productive & inventive, but they did stand on the shoulders of giants musically (vs being born in some prior century), and they were in the right time & place to be part of a growing scene whose smarts exceeded that of any of it's individual participants. This is the nature of 'scenius' and of golden ages - a lot of ideas are already teed up in the collective consciousness.
Re:Monumental advance in number theory posted to the ArXiv b... by FabHK on Tuesday November 08, 2022 @10:24AM (#33519977)
> only if human endeavor only ever builds linearly.

A lot of mathematics is cumulative, though.

And even if you can go further by going thinner (specialising more), maybe breakthroughs require lateral thinking and connections that are predicated on not being too specialised, but having breadth also. If that is the case, sooner or later we might be in trouble.

Re:Monumental advance in number theory posted to the ArXiv b... by hoseja on Tuesday November 08, 2022 @03:31AM (#33517121)
It's also a major failure in didactics. It feels like very little of the new knowledge since twentieth century has been truly digested for easy teaching. Why isn't general relativity taught in elementary school? It should be possible.
Re:Monumental advance in number theory posted to the ArXiv b... by maegul on Tuesday November 08, 2022 @03:54AM (#33517270)
Interesting! As other comments ts have pointed out, didactics as an important aspect of how humanity responds to such a dynamic.

Otherwise, I’d say our two thoughts are connected. With increasing difficulty in understanding new progress, there could be an inertial tendency to over emphasise the importance of old knowledge because it’s comforting/easier/pragmatic for teachers and parents.

Re:Monumental advance in number theory posted to the ArXiv b... by germandiago on Tuesday November 08, 2022 @03:57AM (#33517280)
FWIW I notice how remarkably bad I was taught math at school.

When I started to go to university I really noticed how bad it was. At university the jump forward was really noticeable.

For example, at school they would show you a couple of simple explanations about derivative math or integrals, briefly and start with all the formulas.

At university I used to have a teacher that started with: history of mathematics, why they were invented, made a point about its primarily practical origins.

To explain things, he could most of the time come with real-life instances of application and there were much more often intuitive or geometric interpretations of the techniques used much more often even before starting the explanation itself to have an intuitive idea and visualization of what you were achieving.

After that, I noticed that to learn math, the first thing is to develop an intuitive, non-mathy idea of what you are doing and later formalize it.

At school and high school they just taught it as almost-memorize tables, apply formulas.

Talking about Spain, btw.

Re:Monumental advance in number theory posted to the ArXiv b... by robertlagrant on Tuesday November 08, 2022 @05:23AM (#33517753)
When I came across the (obvious, in retrospect) visual explanation for why (a + b)(a + b) == a^2 + 2ab + b^2 I was blown away by how simple it was, and by why on earth I had to just memorise that formula in school.
Re:Monumental advance in number theory posted to the ArXiv b... by germandiago on Tuesday November 08, 2022 @05:32AM (#33517807)
This is exactly the kind of intuitions I am talking about. Much easier to explain like that.
Re:Monumental advance in number theory posted to the ArXiv b... by n4r9 on Tuesday November 08, 2022 @10:21AM (#33519936)
By "visual explanation" do you mean imagining a square whose sides are length (a + b), and breaking it up into smaller squares and rectangles?

Or do you mean mulitplying out the terms:

  (a + b)(a + b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2

?
Re:Monumental advance in number theory posted to the ArXiv b... by robertlagrant on Tuesday November 08, 2022 @10:25AM (#33519986)
The former, yes.
Re:Monumental advance in number theory posted to the ArXiv b... by n4r9 on Wednesday November 09, 2022 @09:59AM (#33532484)
It's interesting that some people find that way easier to remember. For me, multiplying out the terms seems faster, probably only because I've practised so many times in my life that I can picture the algebra in my head. It also seems more general, as it's a technique which you have to know anyway. I guess it comes down to being more geometrically minded vs algebraically minded. School should try to cater to both!
Re:Monumental advance in number theory posted to the ArXiv b... by robertlagrant on Thursday November 10, 2022 @03:20PM (#33552672)
I wouldn't use it to calculate, but I would use it to explain why the calculation is what it is.
Re:Monumental advance in number theory posted to the ArXiv b... by rcarr on Tuesday November 08, 2022 @09:23AM (#33519234)
This! Identical situation in the UK from my experience. Why maths is taught completely separated from its history and its purpose is baffling. Well actually it’s not: it’s because it serves the teachers. You hammer the formulas and patterns into the kids so they can pass the exams. Your school gets good grades and the principal is happy because he can now market his school as successful and the government inspectors are happy because the school is hitting its metrics. Meanwhile, the kids haven’t actually learned anything. As soon as the exam is passed they forget the formulas and go on with their lives.

We should be explaining the story of maths and how it benefitted society. We should be asking kids about their interests and then showing them how mathematical tools can be used in those areas. We need to show kids how maths is tied to real life rather than just presenting them with a boring formulas to memorise.

Re:Monumental advance in number theory posted to the ArXiv b... by RugnirViking on Tuesday November 08, 2022 @09:59AM (#33519683)
> because it serves the teachers

I don't think this is the case. I think it's because the teachers themselves don't have a good grasp on "the story of maths and how it benefitted society".

There is a silly meme about asking high school maths teachers "how will we use this in life", and imo it's not because there isn't a good response, but rather that it requires a good understanding of the ways math is actually used. Few high school teachers have actually themselves used the math they teach for anything other than academic exercise. Someone trained in control theory or using physics equations can make things that appear almost magical using maths, and if they are talented, they can find a way to explain it to laypeople. However, people with that combination of talents are desired by just about everybody, from universities to companies, and high schools simply have no way to compete (not least because teaching high schoolers is a soul-crushing job for bureaucratic reasons)

Re:Monumental advance in number theory posted to the ArXiv b... by rcarr on Tuesday November 08, 2022 @01:53PM (#33522738)
Yeah this is a very good point. The amount of people who basically go from school to university to school teacher is alarmingly high and bad for society in my opinion. Grown adults who have never known anything but the school system teaching the next generation. It’s ridiculous. I do still think the original point about hitting metrics is a major contributing factor though along with low pay and underfunding.
Re:Monumental advance in number theory posted to the ArXiv b... by vecter on Tuesday November 08, 2022 @08:12AM (#33518590)
I see no reason to believe that general relativity could be taught in elementary school. It requires advanced undergraduate or graduate mathematics, not to mention that the physics itself is quite difficult (to put it mildly).

To put the blame on didactically seems to miss the more important factor that humans just aren’t that intelligent, save the one in a million genius who might have the intellectual capacity to learn something so difficult at such a young age.