I thought we were well past trying to understand mathematics. After all, John von Neumann long ago said "In mathematics we don't understand things. We just get used to them."
Many ideas in math are extremely simple at heart. Some very precise definitions, maybe a clever theorem. The hard part is often: Why is this result important? How does this result generalize things I already knew? What are some concrete examples of this idea? Why are the definitions they way they are, and not something slightly different?
To use an example from functional programming, I could say:
- "A monad is basically a generalization of a parameterized container type that supports flatMap and newFromSingleValue."
- "A monad is a generalized list comprehension."
- Or, famously, "A monad is just a monoid in the category of endofunctors, what's the problem?"
The basic idea, once you get it, is trivial. But the context, the familiarity, the basic examples, and the relationships to other ideas take a while to sink in. And once they do, you ask "That's it?"
So the process of understanding monads usually isn't some sudden flash of insight, because there's barely anything there. It's more a situation where you work with the idea long enough and you see it in a few contexts, and all the connections become familiar.
(I have a long-term project to understand one of the basic things in category theory, "adjoint functors." I can read the definition just fine. But I need to find more examples that relate to things I already care about, and I need to learn why that particular abstraction is a particularly useful one. Someday, I presume I'll look at it and think, "Oh, yeah. That thing. It's why interesting things X, Y and Z are all the same thing under the hood." Everything else in category theory has been useful up until this point, so maybe this will be useful, too?)
It's probably a neurological artefact. When the brain just spent enough time looking at a pattern it can suddenly become obvious. You can go from blind to enlightened without the usual conscious logical effort. It's very odd.
Just because someone said it doesn't mean we all agree with it, fortunately.
You know the meme with the normal distribution where the far right and the far left reach the same conclusion for different reasons, and the ones in the middle have a completely different opinion?
So on the far right you have people on von Neumann who says "In mathematics we don't understand things". On the far left you have people like you who say "me no mats". Then in the middle you have people like me, who say "maths is interesting, let me do something I enjoy".
von Neumann liked saying things that he knew would have an effect like "so deep" and "he's so smart". Like when asked how he knew the answer, claiming that he did the sum in his head when undoutedly he knew the closed-form expression.
I have tingling suspicion that you might have missed the joke.
To date I have not met anyone who thought he summed the terms of the infinite series in geometric series term by term. That would take infinite time. Of course he used the expression for the sum of a geometric series.
The joke is that he missed a clever solution that does not require setting up the series, recognising it's in geometric progression and then using the closed form.
The clever solution just finds the time needed for the trains to collide, then multiply that with the birds speed. No series needed.
Yeah, I wonder how exactly he meant that. I doubt that Von Neumann believed in random plug-and-chug, which is what I'd probably mean if I said I had given up on understanding something. Possibly von N was being very careful and cautious about what "understanding" means.
For example there's a story that von Neumann told Shannon to call his information metric entropy, telling S "nobody really understands entropy anyway." But if you've engaged with Shannon to the point of telling him that quantity seems to be the entropy, you really do understand something about entropy.
So maybe v N's worry was about really undertanding math concepts fully and extremely clearly. Going way beyond the point where I'd say "oh I get it!"
Brave was never good: crypto-crap, based on Chromium, and was modifying web pages from the start without your consent. I never understood why people use it.
People use it because it is essentially Chrome with uBlock-Origin built in (I think the developer of uBlock Origin is employed by Brave) and it removes the stupid cookie modals that are on every website. Between running a pi-hole and Brave, I rarely see an advert on a website.
Turning off the "crypto-crap" can be done quite easily (you literally right click on the BAT icon and it is gone) and the new tab ads are removed again with a couple of clicks. I've found it also runs much better than Firefox on older hardware.
The first and last time I tried Brave, it was injecting links (with a pretty golden picture) in each post of reddit (and I'm not talking about changing the referrals). To turn that off I had to look deep into the settings.
Depends, but with this news you will probably not be downgrading too much.
Brave really does have a bunch of very nice features, I particularly enjoyed using them on my phone to download videos from youtube for online listening. Built-in adblocking is very enjoyable too.
Do note that there had been several smaller controversies, including one that 'Honey' got recently into hot water for, which was replacing affiliate links with their own. There is currently an on-going lawsuit with Honey for this.
In honesty, look at the controversies page on wikipedia and decide for yourself, I don't think there is a good or a bad choice here.
In my opinion, if you care about the open web, then you should not be using a Blink (Chromium) based browser like Brave. The less control Google has, the better for the web.
The internet used to be controlled in large by Microsoft. Then it wasn't. It does not have to continue to be controlled by Google in the future. Not using Chromium based browsers is a first step.
Brave is my favorite so far. You can run an HTTP monitor like Charles Proxy or Fiddler in your OS if you think your browser is snooping on you. I do Brave + Ghostery and works great.
My sister lives in Anaheim Hills. A couple years ago a brush fire burned right up to the property line behind her house. Fortunately her house was undamaged. Last year the insurance company declined to renew her policy. I think you made a wise choice to avoid that area.
I guess so. I hope your sister is doing alright. Nobody wanted to insure properties in Orange or Anaheim Hills for me either, only option was to get calfair policy. We ended up buying in Aliso Viejo.
Actually, that’s probably the key insight. A democratic, successful Ukraine (not a guaranteed thing at any point) would be an existential threat to the “Russian World” narrative from Moscow, and upend the regime. Even a partially successful Ukraine with working if imperfect pluralism, and regular transitions of power would probably be an profound threat, proving that other models could work.
Interesting article. I wonder if in part 2 they will cover the work done at TRW as part of the VHSIC Phase 2 program. They succeeded with a WSI chip called the CPUAX that used some of the same redundancy techniques together with a built in self test and configuration system.
I honestly completely disagree. We had kids young and honestly it was so much easier in our 20s than it is now with a baby in our 30s. Despite having more money, babies are just harder as you get older. It's the moderate things. With our first, I was so active and nimble. Now I have slight back pain and stuff but it makes me noticeably less active with my third.
By the time my oldest is an adult, I will be in my early 40s. That's still young enough to do whatever you want to do. I encourage people to have kids young. Honestly, we should normalize having kids in college (free daycare, etc). This would be great for young women especially. Yes, it might stretch out how long it takes them to get a degree. But if dad completes it in four years and starts working, by the time dad starts working, the child is school aged and now doesn't need daycare. Then mom can complete her degree while dad works full time, and graduate with a six year old and job prospects. Kind of jealous of all my mormon friends at BYU who did this lol. By the time their kids are grown they'll be 40 year olds with professional careers, ready to do whatever they want, with the bonus of money.
On the other hand, the first-time parents I know in their mid 30s are struggling. It's harder. You're more tired. Also, by the time you're done you're getting ready to retire.
Most of my friends had kids young. They were busy working and hardly saw them grow up. A few were fortunate that the mothers could and wanted to stay home. The others all had their kids raised by daycare workers. When they found out what was happening in schools all they could do was scream about it. When covid hit and the schools shut those parents were buried and their kids suffered setbacks in their education. That's the standard routine. No thanks.
Then they're doing it wrong. The idea is that you grind when you're young to set yourself up for more free time to spend with your kids when you're older. You can afford for one parent to stay home, both if you've really nailed it
Does a bike shop owner make comparable money to an EE in a large company? How about medical insurance for themselves and the family? As a bike shop owner, what do you do when a customer brings clearly a stolen bike?
It's nice to theorize about an idyllic life if you only think about the idyllic surface of it, from inside of a less idyllic but better-off life.
