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Saturday 8 February 2020
Wednesday 22 January 2020
Something weighing on my mind
I'm largely indifferent about whether imperial or metric should be used as the standard system of units. Admittedly it would be a lot easier if the whole world stuck to one system of measurements so that we didn't have to worry about converting units correctly, and we wouldn't have had that head-desk situation with the Mars probe, but that doesn't prove one system is 'superior'. In fairness, you could make a very similar 'argument' about different languages, different currencies, different electrical plugs et cetera that would be just as arbitrary. Personally, I think the fact that there's an argument about which system is superior says a lot more about the decimal system than anything else, but virtually no one is proposing a transition to uncial or hexadecimal. In any case, what I'm about to say is not an attack on the 'metric system' as such, just the way it is commonly abused by metric users.
When I go to objectively measure how much my figure has changed over Christmas, I use a weighing scale with two choices of scale: the imperial scale (stones & pounds) and the metric scale (kilogrammes). My weight is the force exerted on me by the Earth's gravity. I get that the weighing scale is not measuring my mass directly, but infers it by assuming I'm weighing myself on the Earth. Taking it to the Moon wouldn't be fair because it wasn't designed for that purpose. I'll even concede that my mass is probably more important to my health than how I interact with the Earth. To quote Cree in the Little Dinosaur Adventure, "There's no need to bite my head off".
No, my problem is that most people seem to think these two scales are measuring along the same dimension, leading to confused statements such as "I weigh 80 kilogrammes" or "I've put on weight", and then I'm accused of pedantry when I explain that they do not. The mentality seems to be that there are two types of people in the world: those that don't know the distinction but can nevertheless function, and those that need to understand it and do because it's fairly elementary Newtonian Mechanics.
It's possible this mindset will change when the common person regularly travels in space to other worlds, but I've already heard some people say that the kilogramme can be understood as a unit of force if it is understood as an 'Earth equivalent', i.e. a force equivalent to that exerted on a kilogramme of mass due to gravity on the Earth's surface. That's just perfect; now I have to judge which use of 'kilogramme' is being used based on the context, as if English wasn't already full of ambiguities that distract attention away from the intended message. If I was being really, really fair I'd concede that 'natural units' systems are commonly used in physics, but that actually does serve a legitimate purpose of removing pesky constants from physics equations. What purpose does an 'Earth equivalent kilogramme' serve? We already have a metric unit of force called the 'Newton', one that isn't (quite as) Earth-centric.
When I first heard about this distinction between mass & weight as a student, I conceded my mistake and immediately tried to correct my language from that point. I genuinely do not get why people try to 'push back' and defend the use of sloppy language when it is pointed out to them. Does the truth just not interest or matter to you? People who say they don't like arguments seem very good at perpetuating them. Stop misusing language and I'll stop complaining. (:
The distinction might not seem to matter on face value, but it's unnecessarily confusing to children (or even adults) taking an interest in the subject for the first time. It's a bad move to first give them inaccurate information that they have to 'unlearn'. I also don't think it's an accident that a lot of Flat Earth advocates or sympathisers believe that things just fall because they're 'heavy' or 'dense'. They're genuinely confused about these concepts, and honestly I don't entirely blame them.
As a final note, yes I am sorry for that title. Please accept this cookie as a token of apology. 🍪
When I go to objectively measure how much my figure has changed over Christmas, I use a weighing scale with two choices of scale: the imperial scale (stones & pounds) and the metric scale (kilogrammes). My weight is the force exerted on me by the Earth's gravity. I get that the weighing scale is not measuring my mass directly, but infers it by assuming I'm weighing myself on the Earth. Taking it to the Moon wouldn't be fair because it wasn't designed for that purpose. I'll even concede that my mass is probably more important to my health than how I interact with the Earth. To quote Cree in the Little Dinosaur Adventure, "There's no need to bite my head off".
No, my problem is that most people seem to think these two scales are measuring along the same dimension, leading to confused statements such as "I weigh 80 kilogrammes" or "I've put on weight", and then I'm accused of pedantry when I explain that they do not. The mentality seems to be that there are two types of people in the world: those that don't know the distinction but can nevertheless function, and those that need to understand it and do because it's fairly elementary Newtonian Mechanics.
It's possible this mindset will change when the common person regularly travels in space to other worlds, but I've already heard some people say that the kilogramme can be understood as a unit of force if it is understood as an 'Earth equivalent', i.e. a force equivalent to that exerted on a kilogramme of mass due to gravity on the Earth's surface. That's just perfect; now I have to judge which use of 'kilogramme' is being used based on the context, as if English wasn't already full of ambiguities that distract attention away from the intended message. If I was being really, really fair I'd concede that 'natural units' systems are commonly used in physics, but that actually does serve a legitimate purpose of removing pesky constants from physics equations. What purpose does an 'Earth equivalent kilogramme' serve? We already have a metric unit of force called the 'Newton', one that isn't (quite as) Earth-centric.
When I first heard about this distinction between mass & weight as a student, I conceded my mistake and immediately tried to correct my language from that point. I genuinely do not get why people try to 'push back' and defend the use of sloppy language when it is pointed out to them. Does the truth just not interest or matter to you? People who say they don't like arguments seem very good at perpetuating them. Stop misusing language and I'll stop complaining. (:
The distinction might not seem to matter on face value, but it's unnecessarily confusing to children (or even adults) taking an interest in the subject for the first time. It's a bad move to first give them inaccurate information that they have to 'unlearn'. I also don't think it's an accident that a lot of Flat Earth advocates or sympathisers believe that things just fall because they're 'heavy' or 'dense'. They're genuinely confused about these concepts, and honestly I don't entirely blame them.
As a final note, yes I am sorry for that title. Please accept this cookie as a token of apology. 🍪
Thursday 16 January 2020
Why are Boltzmann brains a problem again?
I'm positive I'm about to display an embarrassing level of ignorance, but this objection to us living in a slowly dissolving random fluctuation in an infinite random universe never made much sense to me, and not because I think I am a brain floating in the vacuum of space with all of my memories implanted. Nevertheless, I'll be as careful with my wording as I can.
If I understand it correctly, the objection is there would be far more small fluctuations in an infinite random universe than there are large fluctuations, and so almost all brains in that universe would be aimlessly floating in a vacuum, the chance result of atoms coming together into that configuration. This means that if you do live in an infinite random universe, you would basically have to be a Boltzmann brain, which if nothing else sounds a bit bleak.
I accept that small fluctuations the size of brains are far more numerous than large universe sized fluctuations, but aren't we forgetting that brains in our 'fluctuation' didn't spontaneously form by chance? It was an iterative, non-random process, and you would need a fluctuation large enough to produce stars, planets & complex chemistry for that process to even happen. Once you had that situation, the evolution of brains is basically inevitable. That can't be said about small fluctuations, which could be just about anything (e.g. a dining chair).
It seems to me like this argument is conflating low entropy with complexity. Sure, brains are good examples of low entropy, but how is that being calculated exactly? Aren't crystals also examples of extremely low entropy? If so, that tells me the calculations of entropy ignore how each part of the brain relates to other parts of the brain, as well as the processes involved in the development of a brain or a crystal. In other words, it is far too reductionist to really capture the full complexities involved in what it 'means to be a brain'.
I admit I'm not qualified to form an opinion either way. Maybe low entropy vs complexity is a distinction without a difference here, or I'm playing far too fast & loose with these concepts. Hopefully I'll take the time to read Sean Carroll's paper on the matter when I get the chance, but as it stands my intuition is firmly unsatisfied with the 'Boltzmann brain' argument.
For the record I don't think we do live in a infinite random universe, but I would have thought that, in that scenario, almost all brains would appear within large fluctuations that are at least the size of a single galaxy, formed through a Darwinian process rather than random chance. There's nothing 'obviously wrong' with that idea anyway.
If I understand it correctly, the objection is there would be far more small fluctuations in an infinite random universe than there are large fluctuations, and so almost all brains in that universe would be aimlessly floating in a vacuum, the chance result of atoms coming together into that configuration. This means that if you do live in an infinite random universe, you would basically have to be a Boltzmann brain, which if nothing else sounds a bit bleak.
I accept that small fluctuations the size of brains are far more numerous than large universe sized fluctuations, but aren't we forgetting that brains in our 'fluctuation' didn't spontaneously form by chance? It was an iterative, non-random process, and you would need a fluctuation large enough to produce stars, planets & complex chemistry for that process to even happen. Once you had that situation, the evolution of brains is basically inevitable. That can't be said about small fluctuations, which could be just about anything (e.g. a dining chair).
It seems to me like this argument is conflating low entropy with complexity. Sure, brains are good examples of low entropy, but how is that being calculated exactly? Aren't crystals also examples of extremely low entropy? If so, that tells me the calculations of entropy ignore how each part of the brain relates to other parts of the brain, as well as the processes involved in the development of a brain or a crystal. In other words, it is far too reductionist to really capture the full complexities involved in what it 'means to be a brain'.
I admit I'm not qualified to form an opinion either way. Maybe low entropy vs complexity is a distinction without a difference here, or I'm playing far too fast & loose with these concepts. Hopefully I'll take the time to read Sean Carroll's paper on the matter when I get the chance, but as it stands my intuition is firmly unsatisfied with the 'Boltzmann brain' argument.
For the record I don't think we do live in a infinite random universe, but I would have thought that, in that scenario, almost all brains would appear within large fluctuations that are at least the size of a single galaxy, formed through a Darwinian process rather than random chance. There's nothing 'obviously wrong' with that idea anyway.
Sunday 18 February 2018
Friday 16 February 2018
Shortest distance between a point and a line
Problem
There is a point P with co-ordinates [xp,yp] and a line with known parameters. Find the distance between P and the nearest point on the line, or equivalently, the perpendicular distance between P and the line.
Approach
Find the co-ordinates of this point O and then use Pythagoras' Theorem to determine distance.
--------------------------------------
Point of intersection: O = [xo,yo]
Point outside line: P = [xp,yp]
y-axis intercept: Q = [xq,yq] = [0,C]
P_ = OP, Q_ = OQ
Perpendicular vectors <=> P_.Q_ = 0
Line equation (slope-intercept form): y = M*x + C
P_ = (xp - xo)*ex_ + (yp - yo)*ey_
Q_ = (xq - xo)*ex_ + (yq - yo)*ey_
= -xo*ex_ + (C - yo)*ey_
P_.Q_ = -xo*(xp - xo) + (C - yo)*(yp - yo) = 0
(C - yo)*(yp - yo) = xo*(xp - xo)
C - yo = xo*(xp - xo)/(yp - yo)
C = xo*(xp - xo)/(yp - yo) + yo
yo = M*xo + C
M*xo = yo - C
M = -(C - yo)/xo
= -(xp - xo)/(yp - yo)
= (xo - xp)/(yp - yo)
xo - xp = M*(yp - yo)
= M*yp - M*(M*xo + C)
= M*yp - M^2*xo - M*C
xo + M^2*xo = M*(yp - C) + xp
xo*(1 + M^2) =
xo = (xp + M*(yp - C))/(1 + M^2)
yo = M*(xp + M*(yp - C))/(1 + M^2)) + C
= (M*(xp + M*(yp - C)) + C*(1 + M^2))/(1 + M^2)
= (M*(xp + M*yp) - M^2*C + C + M^2*C)/(1 + M^2)
= (M*(xp + M*yp) + C)/(1 + M^2)
L² = (xo - xp)² + (yo - yp)²
xo - xp = (xp + M*(yp - C))/(1 + M^2) - xp
= (xp + M*(yp - C) - xp*(1 + M^2))/(1 + M^2)
= (M*(yp - C) - M²*xp)/(1 + M²)
= M*(yp - C - M*xp)/(1 + M²)
yo - yp = (M*(xp + M*yp) + C)/(1 + M^2) - yp
= (M*(xp + M*yp) + C - yp*(1 + M²))/(1 + M^2)
= (M*xp + C - yp)/(1 + M²)
L² = (M*(yp - C - M*xp)/(1 + M²))² + ((M*xp + C - yp)/(1 + M²))²
= M²*(yp - C - M*xp)²/(1 + M²)² + (M*xp + C - yp)²/(1 + M²)²
= (M²*(yp - C - M*xp)² + (M*xp + C - yp)²)/(1 + M²)²
= (M²*(yp - C - M*xp)² + (-1)²(yp - C - M*xp))²/(1 + M²)²
= (M²*(yp - C - M*xp)² + (yp - C - M*xp)²)/(1 + M²)²
= (M² + 1)*(yp - C - M*xp)²/(1 + M²)²
= (yp - C - M*xp)²/(1 + M²)
=> L = |yp - M*xp - C|/√(1 + M²)
---------------------------------------------
Distance equation using line equation in standard form
A*x + B*y + C' = 0
B*y = -A*x - C'
y = -(A/B)*x - C'/B
Equate coefficients -> M = -A/B, C = -C'/B
L = |yp - M*xp - C|/√(1 + M²)
= |yp - (-A/B)*xp - (-C'/B)|/√(1 + (-A/B)²)
√(1 + (-A/B)²) = √(1 + (-1)²*A²/B²)
= √((B² + A²)/B²)
= √(B² + A²)/B
=> L = |yp + (A/B)*xp + C'/B|/(√(A² + B²)/B)
= |(B*yp + A*xp + C')/B|/(√(A² + B²)/B)
= |A*xp + B*yp + C'|/√(A² + B²)
----------------------------------------------
Sanity Checks
xp = 0, yp = 1, C = 0, M = 0
L = |1 - 0 - (0)*(0)|/√(1 + (0)²)
= |1|/√(1)
= 1
xp = 1, yp = 1, C = 0, M = -1
L = |1 - 0 - (-1)(1)|/√(1 + (-1)²)
= |1 + 1|/√(1 + 1)
= 2/√(2)
= √(2)
There is a point P with co-ordinates [xp,yp] and a line with known parameters. Find the distance between P and the nearest point on the line, or equivalently, the perpendicular distance between P and the line.
Approach
Find the co-ordinates of this point O and then use Pythagoras' Theorem to determine distance.
--------------------------------------
Point of intersection: O = [xo,yo]
Point outside line: P = [xp,yp]
y-axis intercept: Q = [xq,yq] = [0,C]
P_ = OP, Q_ = OQ
Perpendicular vectors <=> P_.Q_ = 0
Line equation (slope-intercept form): y = M*x + C
P_ = (xp - xo)*ex_ + (yp - yo)*ey_
Q_ = (xq - xo)*ex_ + (yq - yo)*ey_
= -xo*ex_ + (C - yo)*ey_
P_.Q_ = -xo*(xp - xo) + (C - yo)*(yp - yo) = 0
(C - yo)*(yp - yo) = xo*(xp - xo)
C - yo = xo*(xp - xo)/(yp - yo)
C = xo*(xp - xo)/(yp - yo) + yo
yo = M*xo + C
M*xo = yo - C
M = -(C - yo)/xo
= -(xp - xo)/(yp - yo)
= (xo - xp)/(yp - yo)
xo - xp = M*(yp - yo)
= M*yp - M*(M*xo + C)
= M*yp - M^2*xo - M*C
xo + M^2*xo = M*(yp - C) + xp
xo*(1 + M^2) =
xo = (xp + M*(yp - C))/(1 + M^2)
yo = M*(xp + M*(yp - C))/(1 + M^2)) + C
= (M*(xp + M*(yp - C)) + C*(1 + M^2))/(1 + M^2)
= (M*(xp + M*yp) - M^2*C + C + M^2*C)/(1 + M^2)
= (M*(xp + M*yp) + C)/(1 + M^2)
L² = (xo - xp)² + (yo - yp)²
xo - xp = (xp + M*(yp - C))/(1 + M^2) - xp
= (xp + M*(yp - C) - xp*(1 + M^2))/(1 + M^2)
= (M*(yp - C) - M²*xp)/(1 + M²)
= M*(yp - C - M*xp)/(1 + M²)
yo - yp = (M*(xp + M*yp) + C)/(1 + M^2) - yp
= (M*(xp + M*yp) + C - yp*(1 + M²))/(1 + M^2)
= (M*xp + C - yp)/(1 + M²)
L² = (M*(yp - C - M*xp)/(1 + M²))² + ((M*xp + C - yp)/(1 + M²))²
= M²*(yp - C - M*xp)²/(1 + M²)² + (M*xp + C - yp)²/(1 + M²)²
= (M²*(yp - C - M*xp)² + (M*xp + C - yp)²)/(1 + M²)²
= (M²*(yp - C - M*xp)² + (-1)²(yp - C - M*xp))²/(1 + M²)²
= (M²*(yp - C - M*xp)² + (yp - C - M*xp)²)/(1 + M²)²
= (M² + 1)*(yp - C - M*xp)²/(1 + M²)²
= (yp - C - M*xp)²/(1 + M²)
=> L = |yp - M*xp - C|/√(1 + M²)
---------------------------------------------
Distance equation using line equation in standard form
A*x + B*y + C' = 0
B*y = -A*x - C'
y = -(A/B)*x - C'/B
Equate coefficients -> M = -A/B, C = -C'/B
L = |yp - M*xp - C|/√(1 + M²)
= |yp - (-A/B)*xp - (-C'/B)|/√(1 + (-A/B)²)
√(1 + (-A/B)²) = √(1 + (-1)²*A²/B²)
= √((B² + A²)/B²)
= √(B² + A²)/B
=> L = |yp + (A/B)*xp + C'/B|/(√(A² + B²)/B)
= |(B*yp + A*xp + C')/B|/(√(A² + B²)/B)
= |A*xp + B*yp + C'|/√(A² + B²)
----------------------------------------------
Sanity Checks
xp = 0, yp = 1, C = 0, M = 0
L = |1 - 0 - (0)*(0)|/√(1 + (0)²)
= |1|/√(1)
= 1
xp = 1, yp = 1, C = 0, M = -1
L = |1 - 0 - (-1)(1)|/√(1 + (-1)²)
= |1 + 1|/√(1 + 1)
= 2/√(2)
= √(2)
Sunday 24 January 2016
Tuesday 17 November 2015
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