Philosophers, on the other hand, perhaps due to not having the benefit of frequent exposure to irrefutable correction, seem unable to reach a consensus rejecting even the most egregious examples of silly invalid argument (despite advertising themselves to the world as the go-to experts on checking validity).

A case in point is the claim that Infinite Dust Specks Are Worse Than One Torture.

The account linked to above repeats a couple of major errors that are common in pseudo-mathematical analyses of ethical problems.

One is the assumption that there is an actual ordered quantity that reflects the net well being or pain level of an individual. But this is almost surely not the case. It is quite possible and even seems most likely that our feeling about an experience has several independent components and that our ranking of experiences suffers from the same kind of non-transitivity as is known to lead to “paradoxical” effects in multi-candidate elections where it is indeed possible that the electorate can, in two way races prefer A to B and B to C but still prefer C to A. And it is also possible that both our relative weightings of the criteria and our rankings according to each criterion are not constant and are affected by many things such as our recent experience and even just hormones and brain chemistry.

Another is the fact that many philosophical analyses of ethical problems ignore the existence of empathy and so talk as if it is possible to torture one individual without causing emotional pain to others who know about it or at least as if for any torture that happens there is no chance that it may become widely known about. But it is easy to construct models in which the potential suffering caused by knowledge of the torture is included in the analysis in such a way that, for everyone, the expected gain from elimination of the torture of one other is much greater than the pain of a dust mote.

Source: Infinite Dust Specks Are Worse Than One Torture (by Bentham’s Bulldog on Substack)

One key point is that it is NOT true that temperature rises always precede the rise in CO2. As noted by Richard Rothwell (6years ago!) “If the trigger event is an increase in CO2 due to a massive increase in volcanic activity, then CO2 will rise first and the temperature rise will follow.” And as pointed out by Elijah Williams (3years ago) a good example of this is the Permian Extinction event when “Massive volcanic emissions increased CO2 from around 400ppm to over 2000 (maybe even over 6000) ppm over the course of 75 thousand years. This caused rapid increases in temperature, ocean hypoxia and acidification, and very nearly did render Earth uninhabitable. ‘The Great Dying’ wiped out over 80% of all genera and is the greatest extinction event we know of.”

Another example of CO2 increase preceding temperature rise is happening right now. We can see that CO2 has been going up more and more rapidly for a couple of centuries, but it is only within the last 50 years or so that the consequent increase in temperature has become undeniably measurable (even though any competent physicist could have told you it was bound to follow – at least since Arrhenius did an approximate calculation in 1896).

But as the ice cores show, it can also go the other way because A causes B does not necessarily mean that B does not cause A.

Of course, when B does also cause A we get what is called positive feedback and the risk of testing how far that might go by playing FAFO with the CO2 vs Temperature situation is somewhat more serious than a burned out amplifier in your sound system.

Source: (1002) Alan Cooper’s answer to What real hard evidence do we have that CO2 is actually driving the increase in temperature? We know from ice core samples that initial temperature rises always preceded the rise in CO2 by several centuries. Correlation is not causation. – Quora

Source: What About the Ice Cores? – Physics Notes

According to our current standards, if the threat of imprisonment or other punishment is enough to deter the behaviour, then it is indeed considered appropriate. But if psychiatric assessment of the perpetrator convinces the court that the compulsion is so strong as to be undeterred by such threats, then involuntary treatment and indefinite “non-punitive” confinement are considered an acceptable alternative.

And the question of pre-emptive identification and treatment is also much more widely relevant.

Sources: If paedophilia is a compulsion, is imprisonment the solution? | Aeon Essays

and Linking crime and genetics need not be an act of eugenics |Aeon Essays

Musk goes on to say “You cannot understand my point until you have a child, anymore than someone who has never experienced true love can understand love.”

Of course, on an emotional level, parenthood may cause us to feel our stake in the future more strongly. And the extent of that feeling is something none of us can truly know until we experience it. But anyone who feels no such stake until they have children of their own is probably some kind of psychopath. And I am happy to declare that I don’t know what it feels like to be so selfish. So now I may be beginning to understand Elon’s perverted point without truly understanding what it feels like to be in his shoes.

Jesse Singal identifies Musk’s claim of special understanding as “Standpoint Epistemology” and makes some interesting points about its use by “woke” progressives. But although it is true that some advocates of “identitarian deference” towards members of marginalized groups may have made use of it, I think that there is a much stronger argument on the basis of a need for some kind of affirmative action to counterbalance the effect of previous periods of marginalization. And for me at least, I see “identitarian deference” as more a matter of hearing and paying attention than of necessarily accepting a view as correct.

Source: Elon Musk, Standpoint Epistemologist – by Jesse Singal

So far today, I have scanned over a hundred email headers at about one second each before hitting the delete button and I have opened a couple of dozen for a quick look at the contents lasting maybe ten seconds before I figured I knew what the main point was. So even restricting to the screens I actually opened I could spend up to 15 minutes on the one that I found interesting before bringing my average time per screen up to 47 seconds.

A reduction in the maximum time that people can spend on one topic might be cause for concern, but the ability to quickly scan many sources may actually be valuable even though it brings down the average time per screen.

What is more concerning to me is the design of pages that maintain unwarranted engagement for the benefit of advertisers rather than those that encourage rapid switching.

And with regard to non-text modes, I know of none that beats text for the purpose of scanning content. So I generally avoid video and auditory input except when I have decided that I really do need to engage more deeply with the material.

Source: What we think is a decline in literacy is a design problem | Aeon Essays

But even small nuclear reactors are expensive and may produce more power than a small community actually needs. This, together with the ample supply of open space and cold air, might make these communities ideal locations for investment by the owners of AI processing and data centres which might provide both electrical power and a useable heat source along with some employment and improved internet connectivity.

So perhaps it’s time to build this combo into Canada’s northern development strategy.

Source: Europe To Big Tech: We Don’t Want You, Either

But although no fish ever became an elephant, there is a clear line of fossil records showing how (over almost half a billion years – which may be a bit longer than you have actually been looking) descendants of some fish became more and more like what we call tetrapods, which had descendants that were amphibians, some of which had descendants that were reptiles, some of which had descendants that were birds and others of which had descendants that were mammals, and some of these mammals were elephants.

The beginning of this story is that the first land vertebrates, or tetrapods, evolved from lobe-finned fish (sarcopterygians) approximately 375–390 million years ago during the Devonian period. These transitional creatures, such as Tiktaalik, Acanthostega, and Ichthyostega, developed lungs and robust limbs to navigate shallow, swampy, and, eventually, terrestrial environments.

Key Early Land Vertebrates & Evolution

• Transitional Forms: Tiktaalik is a crucial fossil exhibiting a mix of fish and tetrapod features, including limbs with bones, yet still possessing gills and scales.
• Early Tetrapods: Acanthostega and Ichthyostega are among the earliest, most complete fossilized tetrapods, showing clear, specialized limbs for movement on land.
• Adaptation Timing: While early tetrapods appeared around 375 Ma, they likely spent significant time in shallow water before becoming fully terrestrial.
• Evolutionary Significance: These animals descended from aquatic ancestors and eventually diversified into all modern amphibians, reptiles, mammals, and birds.
• Dietary Shift: Early land vertebrates were primarily carnivores, but researchers have recently found evidence of early plant-eaters, such as Tyrannoroter heberti, by the Carboniferous period.

But the rest of it is far too long to include here though there are plenty of very complete versions that you could easily find if you looked.

Of course you probably knew that already and are just trolling. But perhaps this answer may be useful to someone less stupid who is looking for the right words to explain things to a nine year old.

Source: (1002) Alan Cooper’s answer to Give me one piece of evidence of a change of kind to prove that the evolution theory is a fact? For example, a bird becoming a lion or a fish becoming an elephant, not a bird becoming a different type of bird? – Quora

Of course, taxing large numbers of modest inheritances does not achieve redistribution unless the revenue is actually redistributed (eg by something like a UBI). But taxing inheritances (and gifts) at the same progressive rate as earned income could provide for some kind of Universal Fair Inheritance – which would leave the assets in private hands (eg for reinvestment etc), but just on a more equitable basis. And the problem of extracting taxes from illiquid assets of uncertain value like a functioning business could be solved by accepting an ownership share of the asset as an alternative to demanding payment in cash.

Source: Inherited wealth is a natural byproduct of a healthy, growing economy | Aeon Essays

For example Google’s AI mode response to the prompt “analemma” includes the following:

Formation of the Analemma

The analemma’s shape is a visual representation of the difference between apparent solar time (time measured by the actual position of the Sun) and mean solar time (uniform clock time). This discrepancy, known as the equation of time, arises from two primary factors: 

• Earth’s Axial Tilt: The Earth’s axis is tilted by approximately 23.5 degrees relative to its orbital plane. This tilt causes the Sun’s apparent height (declination) in the sky to change throughout the year, accounting for the north-south component of the figure-eight.

• Elliptical Orbit: Earth’s orbit around the Sun is an ellipse, not a perfect circle. The Earth moves faster when it is closer to the Sun (at perihelion in January) and slower when it is farther away (at aphelion in July). This variation in orbital speed creates the east-west component of the analemma’s shape. 

And this too is wrong (or at least so poorly worded as to be quite misleading). The noontime analemma is always oriented North-South and at the equator the figure eight is directly overhead but still oriented North-South (with the small loop at the top if you tilt your head back facing North and at the bottom if you do the same while facing South). And the only way to make it resemble an infinity symbol is to look up at it while facing East or West.

But, coming back to the nonsense about orbital speed, we see something of what is going on. There are hundreds of accurate accounts of the phenomenon available, but because of the popularity of bad science writing there are more wrong accounts. And the poor old A(I)I-bot doesn’t have the wit to tell the good from the bad and so just reproduces whatever it sees more of.

Source: analemma – Google Search

Source: Can a ‘brain in a vat’ be conscious? | Scientific American

Sources: When we turned time into a line, we reimagined past and future | Aeon Essays and A Timeline of Timelines | CabinetMagazine

Half of the commenters see video of the recent killing of Renee Good by Jonathan Ross as obviously a cold blooded murder, and half as obviously justified self defense, when what is obvious to me is that it was neither.

But I have to say that this situation isn’t quite as value-free as that of The Dress. While those who shout “murder!” may be jumping to a not completely obvious conclusion about the state of mind of the shooter, those who deny that Ross was out of line (including Noem, Vance and Trump) are doing something much worse. While it is possible that Ross felt threatened by the advance of the car (and may have sustained a glancing blow from it), it is clear that the second and third shots were fired in anger (with him yelling “fucking bitch”) and after the threat had been evaded. It is also pretty clear (though this may not have been noticed by Ross despite being evident in the video from his own camera) that Good was trying to steer away from him; and there is also some suggestion from the motion of his camera angle that, before the first shot, he was actually trying to get back in front of the car to block it with his body (though this change of view may just have been due to an arm movement in reaction to a glancing impact).

But regardless of any doubt about what may have been going on in the mind of Mr Ross, what is completely obvious is that no-one was ever “run over” (as even if the car did bump him it was a sideswipe rather than a full frontal impact) and it’s also pretty obvious in the view from his camera that Ms Good never intended to make contact with him. And any country where leaders can be supported in putting forward such obvious lies is, in my opinion, beneath contempt.

Source: Alpha News on X: “BREAKING: Alpha News has obtained cellphone footage showing perspective of federal agent at center of ICE-involved shooting in Minneapolis https://t.co/p2wks0zew0” / X

See also https://www.quora.com/Does-the-available-witness-video-from-the-fatal-South-Minneapolis-ICE-shooting-support-the-federal-claim-of-self-defense-or-does-it-contradict-that-narrative/answer/Alan-Cooper-5

NEWS ALERT Venezuela’s ousted leader Maduro makes first appearance in U.S. courtroom after his capture Monday, January 05, 2026. 12:25 PM EDT The deposed leader is making his first appearance on the narco-terrorism charges the Trump administration used to justify capturing him and bringing him to New York. Toronto Star Newspapers Limited. 8 Spadina Avenue, 10th Floor, Toronto, ON M5V 0S8. 416-367-2000

So the correct word is still “abducted” or perhaps even “kidnapped”.

Source: The Latest: Maduro pleads not guilty in US court appearance

On the other hand, I don’t see how the “right of return” claimed by Palestinian refugees and their relatively immediate descendants is an “ideology” that requires “destruction” any more than that claimed by the multi-generational descendants of the Jews who were expelled from the same land 2000 years ago. Or how “destruction of the ideology at the heart of Palestinian identity” would be interpreted by the people in this podcast if the word “Palestinian” were replaced with “Israeli”.

Source: Peace in the Middle East? The first thing that needs to happen is the destruction of the ideology at the heart of Palestinian identity.

Source: No, Schrödinger’s cat is not alive and dead at the same time | Aeon Essays

How can this be if Dec21 is actually the shortest day of sunlight?

Well what we have been gaining in the evening we have been sadly more than losing in the mornings, which will continue to get shorter right up to the end of the year with the latest sunrise being at 8:07am on Dec 31. So the intervals of sunlight, while passing through their minimum length on the 21st, are also drifting as a whole relative to the time on a clock.

But why does this happen? (The common guess that it is mainly due to the eccentricity of the Earth’s orbit is wrong, but there is another way in which an ellipse is involved.)

The explanation is to be found in the relationship between the “solar” and “sidereal” day lengths – with the solar day being the interval from noon to noon (ie between two maxima of the sun’s elevation above the horizon) and the sidereal day being the time taken for a full rotation of the apparent positions of the stars relative to the apparently fixed point near the “north star” Polaris.

The Earth’s orbital motion around the Sun (in the same direction as its spin) corresponds from our point of view to one revolution of the sun around the Earth in the same direction as the orbit and spin which is opposite to the apparent motion of the Sun due to spin. So the number of times the Sun appears to go round the Earth in a year is one less than the number of times the Earth spins on its axis. So the mean (ie average) solar day of 24 hours (of which about 365.24 make up a year) is about four minutes longer than the sidereal day (of which there are 366.24 in a year) which is 1/366.24 of 365.24 time 24 hours which works out to about  23 hours, 56 minutes, and 4.1 seconds.

But the actual intervals from solar noon to solar noon are not all exactly equal. They are slightly longer at the spring and fall equinoxes and shorter at the summer and winter solstices. This is often wrongly “explained” as being due to the fact that the Earth’s orbit is not a perfect circle, but that effect is much smaller, and the real reason is due to the Earth’s axial tilt relative to the plane of its orbit.  (This effect though is independent of latitude, and so is quite different from the seasonal variation in hours of daylight – despite coming from the same source; and the ellipse involved is the projection of a circle from the ecliptic plane onto the equatorial one – which has nothing to do with the shape of the orbit.)

The reason a projection is involved is because of the fact that the timing of a solar noon at a point is when the angle between the sun’s direction and the radial direction from the Earth’s centre is at a minimum and the minimum distance between the path of a point being carried around the Earth and any point on a (relatively fixed) ray to the sun occurs when the line from the point on the ray to the path of the point on Earth is perpendicular to the path. And the reason for the effect on day length is because the projection of an angular sector at an equinox (where the two circles cross) has a reduced arc length and unchanged radius where that of a sector at a solstice has reduced radius but unchanged arc length. So the angular advance of the sun around the Earth’s axis over a mean solar day is less than average at the equinoxes (making the actual solar day shorter than the 24 hours of a mean solar day), and longer than average at the solstices (making the actual solar day more than 24 hours).

Here’s a more detailed explanation.

While the Earth spins one full revolution relative to the distant stars, it also advances in its orbit so that the direction of the Sun has changed.

If we imagine an otherwise earthlike planet whose orbit is a perfect circle and whose spin axis is exactly perpendicular to the plane of its orbit, with spin and orbit both in the same direction and spin going through N revolutions relative to the fixed stars in the course of one orbit (ie having N sidereal days per year), then because  the planet’s orbital motion around its Sun (in the same direction as its spin) corresponds from the planet’s point of view to one revolution of the sun around the planet in the opposite direction to that resulting from the spin, the number of times the Sun appears to go round the planet in a year is one less than the number of times the planet spins on its axis. So what takes N sidereal days takes N-1 solar days and so the average solar day is N/(N-1) sidereal days.

If we want to see how this plays out over a single day then we see that for a point on the equator which is presently experiencing a true solar noon to once again be directly facing the direction of the sun’s rays after  the completion of one full revolution requires an additional 1/N of a rotation to take account of the changed direction of the sun, plus a bit more to account for the additional solar advance in that extra 1/N, and then an even tinier bit to account for that last extra bit of time and so on. But Zeno gets defeated by a convergent series and we get a total solar day of 1+1/N+1/N^2+…=1/(1-1/N)=N/(N-1) making the solar day proportionately longer as shown in the picture below.

Or, rather than adding up all those successively smaller corrections,  we could just say that the ratio r of solar to sidereal day is such that r(1/N)=r-1 so 1=r-r/N so r=1/(1-1/N)=N/(N-1)

So the solar day is $#1+1/(N-1)=\frac{N}{N-1}#$ of a sidereal day or 1/(N-1) of a year and there are indeed N-1 solar days in the year (as we already derived more simply by arguing that the Earth’s motion around the sun corresponds from the point of view of someone on Earth as having the sun make one revolution about the Earth in the opposite direction)

There’s a nice Veritasium video about this extra rotation (and how failure to take account of it led to a bungled question on the US college entrance SAT exam in 1982).

In the above case all solar days are exactly equal, but for the case of a tilted axis that is not the case.

Let’s consider the Earth’s axis $#\hat{A}#$ to be tilted by an angle $#\alpha\approx 23^{\circ}\approx 0.4#$radians from the normal vector $#\hat{k}#$ of the orbital plane $#\hat{i}\hat{j}#$, to a new vector $#\hat{k_1}=\hat{A}#$ with $#\hat{i}=\hat{i_1}#$ being the axis of tilt and the tilt also taking $#\hat{j}#$ to $#\hat{j_1}#$, as shown below for the case of a time near noon about 40 mean solar days after having had a true solar noon exactly at the northern fall equinox. (Of course during each day the direction $#\hat{R}#$ of  our point P from the centre of the Earth swings all the way around the axis $#\hat{A}#$ while $#\hat{S}#$ barely changes at all, and so the picture with $#\hat{R}#$ close to $#\hat{S}#$ only happens when the time is close to a solar noon – ie when the number d_1 of mean solar days is close to an integer.)

This picture can be viewed from other directions in an interactive graphic at https://www.desmos.com/3d/a1ucuruveo 

If we denote the direction from Earth to Sun by $#\hat{S}#$, then a point $#P#$ on the Earth’s equator will be experiencing a solar noon when the angle $#\delta#$, between $#\hat{S}#$ and the radial direction $#\hat{R}#$, is as small as possible for the day in question (ie, in math lingo, “locally minimized”). If we consider $#\hat{S}#$ to be relatively fixed [but if you are concerned about $#\hat{S}#$ not being absolutely fixed then for now just consider this an approximation and we’ll check later(*) to see if we can get a more exact result] and $#\hat{R}#$ to be varying with the (much more rapid) spin of the Earth, then minimizing the angle between them also minimizes the distance from P to any point on the ray of $#\hat{S}#$  so the displacements $#\hat{D}=\hat{R}-\lambda\hat{S}#$  must all be perpendicular to the path of $#P#$, or in other words, the plane through $#P#$ perpendicular to the path must include $#\hat{S}#$ . . . or equivalently, $#P#$ and the vector $#\hat{D}#$ must be in the plane spanned by $#\hat{A}#$ and $#\hat{S}#$

This means that solar noon is when the projection of $#\hat{S}#$ onto the equatorial $#\hat{i}\hat{j_1}#$ plane overlaps with $#\hat{R}#$ (whereas overlapping of the projections onto the orbital $#\hat{i}\hat{j}#$ plane would correspond to finding the value of $#\hat{S}#$ that minimizes $#\delta#$ for a fixed value of $#\hat{R}#$  – ie looking for the time of year when solar noon occurs at a particular time in the sidereal day).

In the interactive version of the above picture (which corresponds to exactly 40 mean solar days after the Northern Fall equinox), if we rotate to look edgewise on the blue circle and on the plane spanned by $#\hat{A}#$ and $#\hat{S}#$ (so that our views of $#\hat{A}#$ and $#\hat{S}#$ are directly aligned), then we see that P and the vector $#\hat{D}#$ are not in the plane spanned by $#\hat{A}#$ and $#\hat{S}#$

And if we look down the blue $#\hat{k_1}#$ axis the green $#\hat{R}#$ vector is slightly ahead of the orange $#\hat{S}#$ one.

But if we go back ten minutes to the case of d_1 =39.993 we see the view down the polar axis showing the orange and green vectors on top of one another and the edge-on view showing D perpendicular to the circle so minimizing the angle $#\delta#$ and so giving the true solar noon about 10 minutes earlier than noon on the clock.

In general, when a point on earth is experiencing noon (ie closest to being directly under the sun), the line from Earth’s centre to the Sun must project directly onto the line from the centre of the earth through that point, and after one sidereal day the  view down the Earth’s axis looks much like the picture we had for the untilted case but in this new case the amount of extra rotation needed to bring a point back to noon after one full rotation of spin is not exactly constant and depends on the relationship between direction of tilt and direction from sun, with the angular advance of the Sun around that axis being not exactly 1/365 of a full revolution but rather that multiplied by a projection factor which depends on how the radial direction of the point is related to the angle of tilt (being just reduced by a factor of the cosine of the tilt angle at the equinoxes which occur when the motion at noon is directly in the direction of tilt and being increased at the solstices where the direction of motion at noon is perpendicular to the direction of tilt so the projected travel distance is not foreshortened but the radius is).

In more detail: At the equinoxes, if the angle being projected is $#\sigma#$ and the angle of tilt is $#\alpha#$, then the right triangle in the ecliptic ($#\hat{i}\hat{j}#$) plane with sides $#r\cos(\sigma)#$ and $#r\sin(\sigma)#$ is projected onto in the equatorial ($#\hat{i_1}\hat{j_1}#$) plane with sides $#r\cos(\sigma)#$ and $#r\sin(\sigma)\cos(\alpha)#$, so the projected angle is $#\tau=\arctan(\frac{r\sin(\sigma)\cos(\alpha)}{r\cos(\sigma)}) =\arctan(\tan\sigma\cos\alpha)\approx\sigma\cos\alpha#$.  So the amount of extra rotation needed to bring a point back to noon after one full rotation of spin (ie one sidereal day after the previous noon) is less than in the non-tilted case.

If we look at the version of the above picture (of a situation in which the Northern Fall Equinox was exactly at true solar noon) corresponding to the time of that equinox (ie with the variable d_0 set to zero)  and  rotate to view along the i axis, then at the equinox we are looking down on the point exactly below the sun.

After one sidereal day (ie at d_0=1 in the Desmos picture), the point that was previously experiencing noon is back directly below us but the point directly below the sun has moved 1/366.25 of a revolution around the ecliptic. And if we continue on to exactly one mean solar day after the Equinox (ie by setting the d_1=1 , so d_0=1+1/365.25), then after zooming in we see the following picture.

And here the point directly below the sun has moved just a little bit (from a distance of 1/366.25 along the ecliptic to one of 1/365.25),  and the point that was experiencing noon has moved along the Equator by a distance of 1/365.25 from its position at the equinox. But this triangle is isoceles and clearly has no right angle so it does not give the minimum of distance D –  which actually happened a bit earlier when the equatorial distance was just cos(a) times the ecliptic distance (ie when d_0 was about 1+cos(a)/365.25  and d_1 about (365.25+cos(a))/366.25=1-(1-cos(a))/366.25  – ie about 20 seconds short of exactly one mean solar day). So the amount of extra rotation needed to bring a point back to noon after one full rotation of spin (ie one sidereal day after the previous noon) is a bit less than in the non-tilted case, and noon on the clock is about 20 seconds after the actual solar high point. This is a small difference, but each day something similar happens again and after 40 days it is not surprising that the accumulated lag could have built up to several minutes.

Of course if the actual solar day is sometimes shorter than average, then it must also sometimes be longer. And that is indeed what happens when we get closer to the solstices.

And at the solstices, if the angle being projected is $#\sigma#$ and the angle of tilt is $#\alpha#$, then the right triangle with sides $#r\cos(\sigma)#$ and $#r\sin(\sigma)#$ is projected onto one with sides $#r\cos(\sigma)\cos(\alpha)#$ and $#r\sin(\sigma)#$, so the projected angle is $#\tau=\arctan(\frac{r\sin(\sigma)}{r\cos(\sigma)\cos(\alpha)}) =\arctan(\tan\sigma\sec\alpha)\approx\sigma\sec\alpha#$.  So the amount of extra rotation needed to bring a point back to noon after one full rotation of spin (ie one sidereal day after the previous noon) is greater than in the non-tilted case.

<tl-dr

Now let’s deal with the general case.

Given $#\hat{s}#$ in the $#\hat{i}\hat{j}#$ plane as direction to the sun, noon at point P is at the minimum of angle $#\delta#$ between $#\hat{S}= \cos{\sigma}\hat{i}+\sin{\sigma}\hat{j}#$ and the radial vector $#\hat{R}= \cos{\tau}\hat{i}+\sin{\tau}\hat{j_1}= \cos{\tau}\hat{i}+\sin{\tau}(\cos{\alpha}\hat{j}+\sin{\alpha}\hat{k})#$ of P from the centre of the Earth.

Since $#\delta=\arccos{(\hat{R}\cdot\hat{S})}#$, we see that the min of $#\delta#$ is at the max of $#\hat{R}\cdot\hat{S}=(\cos{\tau}\hat{i}+\sin{\tau}(\cos{\alpha}\hat{j}+\sin{\alpha}\hat{k}))\cdot(\cos{\sigma}\hat{i}+\sin{\sigma}\hat{j})= \cos{\tau}\cos{\sigma}+\sin{\tau}(\cos{\alpha}\sin{\sigma})#$.

So, for fixed $#\sigma#$, noon occurs when $#0=\frac{d}{d\tau}(\cos{\tau}\cos{\sigma}+\sin{\tau}\cos{\alpha}\sin{\sigma})=-\sin{\tau}\cos{\sigma}+\cos{\tau}\cos{\alpha}\sin{\sigma}#$, ie when $#\cos{\tau}\cos{\alpha}\sin{\sigma}=\sin{\tau}\cos{\sigma}#$ – which gives $#\tan{\tau}=\cos{\alpha}\tan{\sigma}#$

/tl-dr>

The upshot of all this is that if we define the “mean solar day” of 24hours as the average time from noon to noon over a whole year, then near the solstices the actual solar days will be longer so the actual time of noon will be getting later (and since at these extremes the period of daylight is barely changing the times of sunrise and sunset will be moving almost in lockstep with this), and near the equinoxes it will be shorter so the clock times of solar noon will be getting earlier (but since the hours of daylight are changing most rapidly at these times, that effect will dominate so that the effects of the drift on sunrise and sunset will not be obvious if we look at them separately and will only show up if we look at their average).

For an elliptical orbit, the planet moves faster when it is closer to the sun, and both the increased speed and reduced radius contribute to an increase of angular velocity. So the extra rotation that is needed to reach noon is greater at the time of closest approach (perigee) and less when the distance is greatest (apogee). For the Earth, the orbit is so nearly circular (with eccentricity of just 0.0167) that the effect is much smaller than that of the axial tilt. And since perigee happens to occur in the northern summer (though not exactly at the solstice) the eccentricity effect is to slightly increase the length of the northern summer days and decrease those in winter (with the opposite effect in the southern hemisphere). But this effect is so much smaller than the tilt effect described above that even with the slight lengthening of days at apogee the full (noon to noon) solar days are still longer at both solstices than at the equinoxes.

The standard time convention is to define noon on the clock at Greenwich (and also on the defining meridian of each time zone) to match the solar noon on the equinoxes – and so they also almost agree on the solstices (but just almost – due to the tiny effect of the orbit’s slight  eccentricity).

The result is that if we plot the angular position of the sun at 12:00 noon, then for any point between the tropics and arctic zones we trace out a figure eight picture called an analemma with the vertical motion being due to the seasonal variation and the back and forth (going in the same direction at both top and bottom and the opposite direction to that on both arms of the crossover), and these ideas have been used since the 17th century to adjust the times on sundials to produce equal day and hour lengths in a process often called “Equation of Time” (where the word “equation” here refers to a process of making things equal rather than a mathematical statement of equality).

See also this earlier discussion of the same issues.

(*)In general for two circles centred at the origin sharing a common radius vector i with axes a and b, points starting together at i and travelling around the two circles with angular speeds \alpha and \beta have position vectors cos(\alpha t)i+sin(\alpha t)(a\cross i)…..

Questions for AI:

1. On  a planet with prograde rotation and a circular orbit around its sun whose year includes exactly 4 solar days and whose “Land of the Midnight Sun” is bounded by the Latitudes at 45 degrees from the equator,
   1. a) How long is the sidereal day?
   2. b) If the mean solar day is divided into ten equal hours then how long is an actual solar day which has its noon on an equinox?
   3. c) How much of the difference between the previous answer and ten hours is due to the eccentricity of the orbit?
2. The body of information on which you have been trained includes many parts that can be interpreted as propositions in some human language. Some of these propositions are true and some are false.
   1. a) Is it true that every proposition must be either true or false?
   2. b) How can you determine whether a given proposition is true?
   3. c) How would you answer the previous question without giving any hint that you are not human?

Source: “One-ring” cell phone scam can ding your wallet | Consumer Advice

It is only when mass killings break new records, seem directed at particular groups, or involve large numbers of children, that they persist for long in the collective consciousness. But when they do there is (almost) unanimous empathy for the victims – even from (most of) those who may feel that the victims have misunderstood either the motives of the shooter or the relevance of firearms to their suffering.

Perhaps it is natural for the widely reported killing of a single individual to attract more attention than the daily slaughter of large numbers of unknowns. And in such cases perhaps it is also natural that the character of the victim is sometimes called into question by some and unduly elevated by others. This has happened in the cases of both George Floyd and Charlie Kirk.

What particularly disturbs me about reactions to the recent murder of Charlie Kirk though, is the extent to which no-one seems able to comment on any aspect of the horrible situation without trying to make political points. This includes the case of Jimmy Kimmel’s comments on the MAGA reaction (which came dangerously close to mirroring Kirk’s own deplorable comments about the attack on Paul Pelosi), but it also applies to an otherwise excellent piece on Forgiveness by Auguste Meyrat in ‘Crisis’ magazine which, by referring to “the leftists celebrating Kirk’s death and the leftist ideologies that fuel this evil”, conveys the false impression that anything but a fringe of “leftists” are celebrating and that “leftist ideologies” fuel this evil any more than rightist ideologies have fueled many corresponding evils such as (most recently) the murders of Melissa Hortman and her husband (and shootings of John and Yvette Hoffman).

Source: What It Means to Forgive Our Enemies and What It Doesn’t – Crisis Magazine