I blame this phenomenon for why people feel mathematics is not useful. It is, it's just that reality is more complex than high school mathematics can properly prepare you for. To understand the mathematical solution, should you encounter it, does however require some foundational knowledge so if you don't know any then you can't even begin to understand it (possibly even failing to identify it is mathematical).
I mean take something practical like a mortgage. It's fairly easy to calculate the annuities using a geometric series, but that places it beyond most people's mathematical skills. Sure you could use a special calculator, or if you're adventurous look up the formula (beware though, Wikipedia is bound to lead to errors by defining the monthly rate as yearly rate / 12), and then it probably may not like doing mathematics at all but neither do you really understand what's happening.
It's astounding how some of math's first practical applications is to calculate movement of the stars. Absence of frictions and close bodies make physics much easier to approximate than anything on earth. Maybe to have students appreciate math, we should have them predict eclipse one again. God create astronomy to teach humans mathematics.
Yet you will find that mortage interests charged by the bank don't match your series, because they use an accrual schedule with a special way to measure years fractions, then calculate an equivalent notional for those dates, and remap that to your payment calendar.
Turns out there is a surprinsing amount of details here as well. And you don't have all parameters for the calculation, like often in life.
It's not necessarily wrong, but it's missing any disclaimer about which interest rates they're talking about so if you don't know what you're doing it will lead to mistakes.
Then you can start layering on the complexity by estimating repayments based on the expected variability of future interest rates, convert to Real Dollars based on inflation and expected wage growth, then compare with alternative options such as renting + investing, etc...
You are correct about the mortgage payments, I had a 1-2 cent error every few months, after a days struggle, I simply added a “Penny” column to add or subtract as necessary so that they would match!
I dunno. It seems to me like high school math is a pretty decent foundation for understanding many important aspects of the world (derivatives, integrals, etc.).
And I think that one can clearly and substantively “get” and intuitively understand and intelligently work with an amortizing loan without needing to understand whatever level of math is required to comprehend why pressing (g) (12i) on an HP 12c is not an atomically precise representation of of the answer out to the 12th decimal place.
My mortgage company charges interest based on the number of days in the month divided by 365. I have replicated their calculation like this. I'm not sure what they do in leap years - there are at least 3 distinct approaches.
Aside: Building stairs that way is a really bad idea. Don't use angle brackets – you cut stringers. And besides the first and last step, stringers are pretty straightforward to layout with a carpenter's square with some stair guides (two little posts that screw to the square). And, generally you just layout 8" rise, 12" deep stairs, so you really don't lay out anything at all.
But, really, don't build stairs like OP suggested.
I also don't like that fixtures at top and bottom are SPOF. Why not add pillars to the top and ~3rd from bottom steps, then tie with another beam, like an H overlapping a ladder?
That way, there will be far less chances that bolts spontaneously rip off and stairs go down, nor the lengths of pillars to have to be critical dimensions. The pillars can optionally be cut to precise lengths to be screwed through both steps and slopes for maximum Apple-ness, or, can be lazily cut to long-enough lengths, nailed through to steps from the "outside", and used as base for handrails as if it had been the plan.
(dc: not an engineering advise. consult a real engineer for safety. Also add cross beams in width-height plane, they help tremendously)
Yeah, having the steps only connected by some small screw brackets is borderline redneck engineering, someone's gonna get killed when two screws come loose.
Somehow as easy as it is conceptually, I still find cutting 3 stringers that line up well to be surprisingly difficult. In part because I’ve only has to do it a handful of times in my life.
Also related to the ending, I‘ve come to realise more and more that most people reason out of belief first and arguments second on a lot of things (maybe most things?) Climate change deniers are an obvious example. But also more nuanced views such as "the software I make makes the world a better place" (I belief this myself), "me being a healthcare worker is a great thing for humanity", or "I need to game at least once a week to relax." And it makes sense. Many things are extremely complex and so picking one side and going for that will make sense in many cases. It’s often necessary to be able to make decisions. However, the risk is getting too stuck in certain ideas. It’s easier to accept a long held belief and reject opposing information than to re-evaluate.
Professional deniers are propagandists, not good-faith debaters. Lack of nuance is one problem, but there's a far bigger problem with groups of people knowingly trying to poison and undermine anything that resembles reality-based consensus.
Absolutely - PR firms, lobbyists, astroturfers, troll farms - I'm in full agreement, and there's a lot of them around.
Also a problem are those who label the undecided or free thinking among us as not being good faith actors. And unfortunately it's a tough position to be in, as you get attacked from all sides for not choosing a side.
I think it's because we create a story about every part of us and our life. Every thing we do (sometimes we even do it after the fact).
I feel like those beliefs are what's keeping us grounded in the sense of understanding the world, of being in control. so it makes sense that we would start with them even if we are not aware of it.
And it makes sense that it's hard to let them go because without them you get a sense that you are just floating away and don't have any to "hang" your assumptions on.
You can reason a lot but there comes a point when things get bigger than you and you have to trust some other authority or just trust your gut.
There's not many people that could comfortably reason from first principles and be satisfied with where they end up. (I don't think I could do that)
> Also related to the ending, I‘ve come to realise more and more that most people reason out of belief first and arguments second on a lot of things (maybe most things?)
This is the thesis to the introduction to Jonathan Haidt's The Righteous Mind. Boiling it down, he argues that self-justification is the most fundamental human reaction. We reason after, not before, and our reasoning flows to align with our own justification.
> I‘ve come to realise more and more that most people reason out of belief first and arguments second on a lot of things (maybe most things?) Climate change deniers are an obvious example
Do you think most climate change believers reasoned themselves into that position? I think for >95% of people, belief in climate change is a purely social phenomenon. The overwhelming majority of people don't understand enough about physical phenomena like blackbody radiation to have any intelligent opinion about climate change one way or the other.
I do not think that. This similarly applies to both sides of some other contentious scientific/medical topics which can easily get someone labeled an “antivaxxer” as an ad-hominem directed without solid reasoning. One imagines what labels were given to those who did not support widespread leech therapy at its peak…
The untrue explanations are often the simpler ones. They are easier for people to "understand" even if they are wrong. And people don't like to be perceived as dumb, therefore they keep on arguing that they are right.
Example: Flat Earth. The horizon looks flat, therefore the Earth must be flat.
Actually correct take: You cannot slow down climate change without mass depopulation, and most climate change alarmism is overhyped. It will impose a cost to civilization and it will have to be dealt with by adapting to it, rather than some inane scheme to create carbon tax credits to justify more ways for the hyper-wealthy to stay wealthy.
I’m a data scientist, and it always surprises me how much work goes into fields that do not require a degree — easily just as much (or more) thought is required to solve “daily” work tasks.
I needed to redo a closet in my house where the original wireframe shelves were not installed correctly and fell out as soon as any weight was put on them. First you’ve got to patch the holes. But spackle will show flashing. Ok, so I’ll use joint compound instead. But there’s 27 brands and types, all for slightly different purposes. And there might still be flashing if you don’t use the right type of PVA primer. And you need the right sanding tools and, oh, it turns out I suck at applying joint compound so I have to call someone anyway to make it look like years of expertise went into that.
On to the new shelves. Well I don’t want the closet to look like I did it myself, so I’ll get some floating shelves to make it look nice and modern. Wait, $1,200 for a piece of wood?? Can’t I just find a tree in the backyard and cut that myself? Oh... I need specialized equipment to make that look nice too. Whatever I’ll just order the shelves.
Have the shelves, but now it turns out the tolerances on the walls in the closet weren’t that tight when it was built. It seems like the width of the closet varies quite a bit as you move away from the wall. So now there’s a gap. And the floating shelf metal bracket misses the nearest wall stud by 1/8th of an inch, which would halve its load capacity. I’ll just drill another hole in it. Oh, it’s stainless steel and that requires specialized equipment and fluid to prevent overheating and...
So yeah, I totally agree with this article in every way. It’s amazing anything works at all in society.
I enjoyed the stair story here quite a bit - I find that I always have a newfound appreciation for the “real” jobs when I find a fine craftsman at work.
We had a curved staircase fabricated for our house and it was absolutely fascinating to watch the planning and execution go into this task. The original measurements were plugged into a cad drawing where we could adjust the number of treads and the position of the top and bottom steps based on our preference. Then they fabricated the entire assembly offsite and brought it in via crane in one piece. Even building the handrail was fascinating as there was a jig built in place to bend the wood to the curve required.
The physical constraints and fractal complexity is fascinating for someone like me who is used to deterministic machines moving bits on a daily basis.
> If you’re trying to do impossible things, this effect should chill you to your bones.
> It means you could be intellectually stuck right at this very moment, with the evidence right in front of your face and you just can’t see it.
This speaks to me. It's terrifying knowing deep in your soul that something is possible but getting stuck trying to make it happen. When you try to talk to people, you seem a bit unhinged. Like some crazy person who's creating his own problems by avoiding the easy established solutions. People will straight up tell you it can't be done, they will ask you why you can't do it in the normal way. It's tempting to take their word for it and just give up. However, if you persevere a little and ask the right questions, they might just see that it is possible to do things differently. If you persevere a lot, you might figure out all the details and end up proving that it was possible after all.
I achieved a small victory just like this a few days ago and I'm really proud of it, I'm actually preparing to Show HN.
This is an unusually rare, worthwhile piece. It's rare for somebody to be both very grounded, with a personal history of hands-on practical competence, yet simultaneously abstracting his experience to a high philosophic level.
A nitpick is this statement: "The massive difference in weight between a rocket full of fuel and an empty one means that a reusable rocket can’t hover if it can’t throttle down to a very small fraction of its original thrust, which in turn means it must plan its trajectory very precisely to achieve 0 velocity at exactly the moment it reaches the ground."
I don't think this is how SpaceX does it. When you have closed loop feedback control using velocity and distance measurements from radar and vectoring thrusters, it's no longer an impossibly difficult ballistic problem. I suppose that highlights the necessity of not making assumptions about solution methods.
The original statement is correct: the rocket cannot throttle down to a small fraction, to an equivalent thrust to the mass of the rocket. SpaceX rockets indeed do not "hover" for any significant length of time. The trajectory is planned to achieve 0 velocity at just above ground level. The feedback control comes in because it's impossible to know the exact performance and atmospheric characteristics, so if you tried to do it by dead reckoning it would be off vertically by tens of meters, so the trajectory is continuously adjusted.
Readit News
I mean take something practical like a mortgage. It's fairly easy to calculate the annuities using a geometric series, but that places it beyond most people's mathematical skills. Sure you could use a special calculator, or if you're adventurous look up the formula (beware though, Wikipedia is bound to lead to errors by defining the monthly rate as yearly rate / 12), and then it probably may not like doing mathematics at all but neither do you really understand what's happening.
Saw this quote on an art blog showing SEMs of diatoms [2]
[1] https://butdoesitfloat.com/Diatoms [2] https://en.wikipedia.org/wiki/Diatom
edit link
Turns out there is a surprinsing amount of details here as well. And you don't have all parameters for the calculation, like often in life.
Lead to errors relative to what? The mathematical idealisation or the actual practice?
Are you referring to this page?
https://en.wikipedia.org/wiki/Mortgage_calculator
"Since the quoted yearly percentage rate is not a compounded rate, the monthly percentage rate is simply the yearly percentage rate divided by 12."
Do you think that the explanation above is wrong?
In practice interest payments are calculated in many "wrong" ways, but that's what it is:
https://en.wikipedia.org/wiki/Day_count_convention
It's not necessarily wrong, but it's missing any disclaimer about which interest rates they're talking about so if you don't know what you're doing it will lead to mistakes.
https://en.m.wikipedia.org/wiki/All_models_are_wrong
And I think that one can clearly and substantively “get” and intuitively understand and intelligently work with an amortizing loan without needing to understand whatever level of math is required to comprehend why pressing (g) (12i) on an HP 12c is not an atomically precise representation of of the answer out to the 12th decimal place.
Looks like all the calculators do that as well. What's the right way?
$$ \left( 1 + \frac{i_a}{n_a} \right)^{n_a} = \left( 1 + \frac{i_b}{n_b} \right)^{n_b} = \left( 1 + \frac{i_{\text{annual}} }{1} \right)^{1} $$
https://www.investopedia.com/terms/e/effectiveinterest.asp
https://en.m.wikipedia.org/wiki/Day_count_convention
Deleted Comment
But, really, don't build stairs like OP suggested.
That way, there will be far less chances that bolts spontaneously rip off and stairs go down, nor the lengths of pillars to have to be critical dimensions. The pillars can optionally be cut to precise lengths to be screwed through both steps and slopes for maximum Apple-ness, or, can be lazily cut to long-enough lengths, nailed through to steps from the "outside", and used as base for handrails as if it had been the plan.
(dc: not an engineering advise. consult a real engineer for safety. Also add cross beams in width-height plane, they help tremendously)
What OP was really making were misused ladders.
https://www.blocklayer.com/stairs/straighteng, click "Show Notching Template", then "Diagrams to PDF"
Also a problem are those who label the undecided or free thinking among us as not being good faith actors. And unfortunately it's a tough position to be in, as you get attacked from all sides for not choosing a side.
I think it's because we create a story about every part of us and our life. Every thing we do (sometimes we even do it after the fact).
I feel like those beliefs are what's keeping us grounded in the sense of understanding the world, of being in control. so it makes sense that we would start with them even if we are not aware of it.
And it makes sense that it's hard to let them go because without them you get a sense that you are just floating away and don't have any to "hang" your assumptions on.
You can reason a lot but there comes a point when things get bigger than you and you have to trust some other authority or just trust your gut.
There's not many people that could comfortably reason from first principles and be satisfied with where they end up. (I don't think I could do that)
This is the thesis to the introduction to Jonathan Haidt's The Righteous Mind. Boiling it down, he argues that self-justification is the most fundamental human reaction. We reason after, not before, and our reasoning flows to align with our own justification.
(Luther would be proud.)
Do you think most climate change believers reasoned themselves into that position? I think for >95% of people, belief in climate change is a purely social phenomenon. The overwhelming majority of people don't understand enough about physical phenomena like blackbody radiation to have any intelligent opinion about climate change one way or the other.
Deleted Comment
Example: Flat Earth. The horizon looks flat, therefore the Earth must be flat.
"support rather than illumination"
I needed to redo a closet in my house where the original wireframe shelves were not installed correctly and fell out as soon as any weight was put on them. First you’ve got to patch the holes. But spackle will show flashing. Ok, so I’ll use joint compound instead. But there’s 27 brands and types, all for slightly different purposes. And there might still be flashing if you don’t use the right type of PVA primer. And you need the right sanding tools and, oh, it turns out I suck at applying joint compound so I have to call someone anyway to make it look like years of expertise went into that.
On to the new shelves. Well I don’t want the closet to look like I did it myself, so I’ll get some floating shelves to make it look nice and modern. Wait, $1,200 for a piece of wood?? Can’t I just find a tree in the backyard and cut that myself? Oh... I need specialized equipment to make that look nice too. Whatever I’ll just order the shelves.
Have the shelves, but now it turns out the tolerances on the walls in the closet weren’t that tight when it was built. It seems like the width of the closet varies quite a bit as you move away from the wall. So now there’s a gap. And the floating shelf metal bracket misses the nearest wall stud by 1/8th of an inch, which would halve its load capacity. I’ll just drill another hole in it. Oh, it’s stainless steel and that requires specialized equipment and fluid to prevent overheating and...
So yeah, I totally agree with this article in every way. It’s amazing anything works at all in society.
We had a curved staircase fabricated for our house and it was absolutely fascinating to watch the planning and execution go into this task. The original measurements were plugged into a cad drawing where we could adjust the number of treads and the position of the top and bottom steps based on our preference. Then they fabricated the entire assembly offsite and brought it in via crane in one piece. Even building the handrail was fascinating as there was a jig built in place to bend the wood to the curve required.
The physical constraints and fractal complexity is fascinating for someone like me who is used to deterministic machines moving bits on a daily basis.
https://news.ycombinator.com/item?id=29429385
https://news.ycombinator.com/item?id=22020495
https://news.ycombinator.com/item?id=16184255
Reality has a surprising amount of detail (2017) - https://news.ycombinator.com/item?id=38304840 - Nov 2023 (1 comment)
Reality has a surprising amount of detail - https://news.ycombinator.com/item?id=36309597 - June 2023 (1 comment)
Reality has a surprising amount of detail (2017) - https://news.ycombinator.com/item?id=29429385 - Dec 2021 (118 comments)
Reality has a surprising amount of detail (2017) - https://news.ycombinator.com/item?id=28006256 - July 2021 (1 comment)
Reality has a surprising amount of detail (2017) - https://news.ycombinator.com/item?id=22020495 - Jan 2020 (115 comments)
Reality has a surprising amount of detail (2017) - https://news.ycombinator.com/item?id=16184255 - Jan 2018 (294 comments)
> It means you could be intellectually stuck right at this very moment, with the evidence right in front of your face and you just can’t see it.
This speaks to me. It's terrifying knowing deep in your soul that something is possible but getting stuck trying to make it happen. When you try to talk to people, you seem a bit unhinged. Like some crazy person who's creating his own problems by avoiding the easy established solutions. People will straight up tell you it can't be done, they will ask you why you can't do it in the normal way. It's tempting to take their word for it and just give up. However, if you persevere a little and ask the right questions, they might just see that it is possible to do things differently. If you persevere a lot, you might figure out all the details and end up proving that it was possible after all.
I achieved a small victory just like this a few days ago and I'm really proud of it, I'm actually preparing to Show HN.
I don't think this is how SpaceX does it. When you have closed loop feedback control using velocity and distance measurements from radar and vectoring thrusters, it's no longer an impossibly difficult ballistic problem. I suppose that highlights the necessity of not making assumptions about solution methods.