How do you tell if you are in an inertial frame or not? According to Newton’s model of a fixed space, an inertial frame is one which is at rest or moving with constant velocity through that fixed space. We saw that determining this required us to observe the state of motion through that space of an object Q that was known to be not under the influence of any forces. If you observe Q to be at rest or moving with constant velocity, then you are in an inertial frame. If however, even in the absence of forces, the freely floating object Q appears to be accelerating in some direction with respect to you, you could conclude that this must be because you are in a frame that is an accelerating in a straight line in the opposite direction and hence you are not in an inertial frame.
But the interesting thing that Newton noted is that one could detect if one was in a non-inertial frame even without observing that external object Q if your frame was rotating, because then things in your own frame would start behaving oddly, with cups of tea sloshing around for no apparent reason. In fact, one could even measure the rate of one’s rotation by doing experiments purely within one’s own frame, even if one was in a windowless room so that one could not see outside.
Newton did just such an experiment to arrive at his conclusion that empty space seems to have this property that we cannot tell when we at rest or moving with constant velocity through it but we can tell when we are rotating. What he did was to hang a bucket of water from the ceiling and then twist the rope and let go. When the rope began to unwind causing the bucket to spin, he observed that the surface of the water was no longer flat but had a meniscus, with the water seeming to climb up the sides of the bucket so that the center of the surface was lower than the sides. Thus we can tell if the room we are in is rotating or not by hanging a bucket of water from the ceiling. Even if we do not cause the bucket to spin by twisting the rope, by observing the surface of the water, we can tell if the room is spinning by observing the shape of the surface of the water. We could do this in a closed room and do not need an external backdrop like the stars to know this.
The effects of rotational motion are well known. Rotation, when perceived by an observer who is also rotating, introduces fictitious forces known as centrifugal and Coriolis forces. The former gives the appearance of wanting to make things move away from the center in a radial direction. An ice skater spinning on their axis with their arms hanging down will feel a tug trying to take them away from the body. The bulge at the Earth’s equator is seen by us who are at rest with respect to the spinning planet as being due to the matter near the equator being ‘pulled away’ from the center. Someone who is hanging on the rope in Newton’s bucket experiment and rotating with it would see the water not as rotating but being pulled away from the center and towards the sides. An example of Coriolis forces is when air currents tend to veer away from their expected trajectories. The number of familiar everyday occurrences can be easily expanded. All these are examples of the fictitious centrifugal and Coriolis forces that come into play to explain the deviations from Newton’s laws of motion that we see when we are in a rotating frame.
Newton concluded that space has this property that we cannot tell if we are at rest or moving with constant velocity through it but that we can tell if we are spinning. We can even calculate how fast we are spinning, by doing experiments purely within the rotating frame. The curvature of the water in the bucket increases with how fast it is rotating.
But not all of Newton’s contemporaries were persuaded by his idea of a fixed space having these properties.The philosopher Bishop George Berkeley (1685-1753) for one rejected the idea of absolute space for the modern-sounding idea that it was unobservable. He felt that ascribing properties to space to explain physical effects made no sense and that all physical effects that were felt by any object were due to other physical bodies. If that were the case, then how does one explain the fact of the water surface becoming curved in the rotating bucket and other centrifugal force effects that can be seen even when done in a closed room far away from any other physical body? Berkeley said that it was the distant stars that caused those effects. The fact that we could not see the stars in a closed room did not matter since gravitational forces cannot be shielded. The fact that the stars were very far away could not be used to dismiss their influence since the stars were massive and there were so many of them that we could not conclude that their distance meant that their effects were negligible enough to be ignored. Far from being a passive backdrop like wallpaper, the distant stars exerted dynamical effects.
The difference between Newton’s and Berkeley’s views can be posed more starkly by this thought experiment. If there were no stars, indeed nothing at all in the universe other than Newton and his bucket, and he did the experiment again, would the water surface be curved? According to Newton, the answer is yes, since it is only the bucket’s rotation with respect to space that matters and space remains even after everything else is removed. So for Newton, as far as the bucket is concerned, nothing of significance has changed. But according to Berkeley, the water surface would remain flat because in the absence of the stars we could never tell if we were rotating or not. Similarly, according to Berkeley’s model, in the absence of the distant stars, an ice skater would not feel a tug on their arms to enable them to tell if they were spinning or not and the Earth would have no bulge at the equator.
While one cannot will away the stars to test whether Newton’s or Berkeley’s view would prevail, there is some evidentiary support for Berkeley’s position that it is only rotation with respect to the distant stars, not space, that is relevant. Recall that we can measure the centrifugal and Coriolis forces within a given frame without reference to anything external to it by (say) measuring the curvature of the water surface in a spinning bucket. From that we can calculate the spinning rate even in a closed windowless room. If we do this for the observable phenomena caused by the Earth’s rotation about its axis, we obtain a value for the spinning rate that is very close to the value we obtain using the length of a day. But the length of a day is obtained using the rate of rotation of the Earth with respect to the stars. There is no a priori reason why these two numbers (the first measuring rotation with respect to space, the second with respect to the stars) should come out to be the same. Could this be just a coincidence? Berkeley thought not.
In general, scientists tend to be wary of ascribing events to coincidences. It is not that coincidences cannot happen. For example, the fact that the distance of the Moon from the Earth is such that during a solar eclipse the Moon exactly blocks out the Sun, despite them being of very different sizes, is believed to be a coincidence. There is no deep underlying reason why it should be so and indeed we expect that with time, since the distances will change, the nature of eclipses will change with it.
In the case of values of the two rotational rates being the same, one explanation might be that the stars are fixed in space, so that the rotation rate of the Earth with respect to space also happens to be the same as the rotation rate with respect to the stars. But there are no obvious reasons as to why all the distant stars should be fixed in space. Berkeley proposed that those two rotation rates should be the same because in both cases we are measuring rotation with respect to the stars. Space has nothing to do with it. (We will see later that looking for deeper reasons as to why certain seeming coincidences happen also leads to the important Principle of Equivalence.)
Berkeley’s ideas are now seen as being remarkably modern but in his time, they were seen as a little strange and did not gain much traction. The idea of stars, known even then to be extremely far away, having an effect on the motion of buckets and other objects on Earth seemed almost mystical and while astrologers might like the idea, his contemporary scientists were skeptical. The mathematician Leonhard Euler (1707-1783) said that the alleged influence of stars was “very strange and contrary to the dogmas of metaphysics”.
The search for some frame in which we can measure absolute velocity continues to exert a strong hold. Newton’s idea of an absolute fixed space initially won out and had a powerful effect on scientific thinking for a long time. The idea of an aether that is co-existent with this space and permeates all of space and acts as the medium for the transmission of electromagnetic waves remained the dominant paradigm until the arrival of special and general relativity, when (as we will see later) Berkeley’s views about the distant stars being the reference frame experienced a resurgence due to the work of Ernst Mach and with support from Einstein. Now we also have the Cosmic Microwave Background that permeates all of space and we can measure our velocity with respect to that.
The anisotropy of the cosmic microwave background (CMB) consists of the small temperature fluctuations in the blackbody radiation left over from the Big Bang. The average temperature of this radiation is 2.725 K as measured by the FIRAS instrument on the COBE satellite. Without any contrast enhancement the CMB sky looks like the upper left panel of the figure below. But there are small temperature fluctuations superimposed on this average. One pattern is a plus or minus 0.00335 K variation with one hot pole and one cold pole: a dipole pattern. A velocity of the observer with respect to the Universe produces a dipole pattern with dT/T = v/c by the Doppler shift. The observed dipole indicates that the Solar System is moving at 368+/-2 km/sec relative to the observable Universe in the direction galactic longitude l=263.85o and latitude b=48.25o with an uncertainty slightly smaller than 0.1o.
While we can choose any of these frames as our reference, the choice is arbitrary in the sense that there is no reason to think that any of them are privileged over the others with respect to the laws of physics.
(To be continued.)
I don’t understand how Berkeley’s argument is supposed to be appealing at all. And I don’t think the issue is just about being “wary of ascribing events to coincidences.” Indeed, you could say there’s not even a genuine coincidence to speak of, so it doesn’t even have that much going for it.
I mean, take Newton’s spinning bucket, with some water in it. That’s rotating in some particular plane and not in others. No matter which way it happens to be rotating (or at least seems to be) about some axis or another, there is some direction along which it isn’t, because space has three dimensions, not just two.
Also, you can easily observe that distant stars are not all distributed on a single plane like that. Instead, they’re scattered around more or less equally in every direction. And more specifically, beyond the stars not being on a plane, it’s not like they’re clustered together on that specific plane (nor are they clustered so as to avoid it) which would correspond in any coherent way with the phenomenon wer’re observing with the water in the bucket.
Besides, even if it did happen to be the case that, in some particular run of the experiment, the bucket was (at least apparently) spinning along the plane of the galaxy or what have you, then we could (arbitrarily) change the axis of rotation of our bucket any way we like, with no corresponding change (unsurprisingly) in the distribution of any of the distant stars…. Because they’re totally unrelated.
That’s pretty much what I mean by the lack of a coincidence: the phenomena simply do not coincide as that word suggests. They don’t even look like they might have anything to do with one another. The thought is that if the stars were somehow supposed to be responsible for that phenomenon, then shouldn’t we expect the water to be affected in all directions more or less equally, instead of the effect being oriented along a plane in the way that it is? Why wouldn’t the water grow or shrink radially outward or inward, or do some such thing like that, if the stars actually had anything to do with it?
cr @1:
You’re choosing an axis. Whichever axis you choose to rotate the bucket around, from the point of view of the bucket, the stars are rotating about that axis, coinciding with the orientation of the deformation of the surface of the water, which is symmetric about that same axis.
You may as well ask why, if we choose a particular direction to accelerate, we feel a force along the same axis. If space is isotropic, would you then say we should feel a force in all directions?
rob @ 2.
The axis of rotation is the key point here (but not being a physicist I am most likely missing the real point).
I would go further and say that from the point of view of the axis, the bucket (and the water it contains) is rotating about the axis. This axis of rotation would exist whether the distant stars are present or not, which is what I think was Newton’s point.
If detection of rotation is dependent upon a fixed reference point (the visible stars according to Berkeley) in the absence of a visible reference point the fictitious forces will still arise relative to the fixed point of rotation, such that the fictitious forces will still deform the surface of the water. If all the other mass in the universe were to magically vanish, we would still be left with a bucket of water that has mass and a fixed axis of rotation.
In the same vein, if our atmosphere were opaque such that we could not even detect that we had a nearby star, we could still determine the speed of rotation of the earth by measuring the fictitious forces evident in the Coriolis effect. We would just have to have developed a measurement of time that is NOT related to the rotation of the Earth relative to the sun (such as we use now). But, that said, if we had developed a measurement of time based on the fictitious forces it would still be based upon the rotation of the Earth relative to the sun due to the suns thermal affect on the atmosphere.
Having time and space embedded within each other would appear to be a problem considering that time can not only be measured by motion of matter through space -- changes of phase of matter can give a more accurate measurement of time.
Yeah, I understand that just fine. Those stars will certainly look like they’re rotating from the bucket’s POV. But that’s a sort of mathematical fact, and it couldn’t have been otherwise. So, yes, you can find a way to use “coincide” there if you like, but that’s not what I meant by “a coincidence” nor do other people normally use the word in that sense. At least generally, you should be able to imagine vaguely how things would’ve been, if there had not been a coincidence since some other (probably less remarkable) set of events had happened instead. I don’t think I can do that with the bucket though.
But for me, the more important point is that when I push the bucket to make it spin, it is definitely the thing which is affected — not the distant stars (or the entire rest of the world other than that bucket), because it’s abundantly clear that touching the bucket didn’t do that.
Even though it sort of appears that way from the bucket’s POV, I’m just not concerned about that at all and don’t think I need to be in order to make sense of things. And maybe it’s enough just to say that I don’t take it as a given that such appearances (empirical phenomena) can never be misleading in any way, or that they should always be treated like they must all be on an equal footing with one another or something like that. Because I know I definitely don’t believe that, and I can’t figure out how else you would buy into that argument. So, like I said, I’m just not moved by it at all.
Berkeley’s point is that in the absence of everything else in the universe, you cannot tell if you are spinning because there would e no external reference points and no fictitious centrifugal and Coriolis forces. You could not identify a plane of reference and an axis of rotation perpendicular to that plane. Those things would be unobservable.
Much of our intuition in this area is based on the idea that there is such a thing as absolute space and that we can tell our motion with respect to it. We think that we can imagine what things would be like if we remove everything else. What Berkeley argued is that that need not be the case.
For example, if we sit on a chair that can rotate and shut our eyes and have someone spin the chair, we can tell we are rotating because the fluid in our ears moves in such way that we can detect our motion. But if there were no liquids in our body that can slosh around, we would have no way of knowing if we were spinning or not.
Reminds me of Cuttlefish’s post back when…
https://freethoughtblogs.com/cuttlefish/2013/07/20/oh-nothing-really/
As far as I understand, because there is no way of confirming or otherwise either standpoint without direct experimentation (no experiment can be devised as a fascimile of the removal of all other matter from the universe) we have only our predictions to go on and how well they affirm, or otherwise, observation.
I myself understand the need (or desire) of an absolute reference point against which all motion can be measured. I think this is a human atribute, evolved with our senses. Discarding that is difficult.
I just don’t see how, when all other matter is conceptually removed, the bucket of water ceases to be a point of reference in its own right. If it is the only matter in the universe is it not the only reference point?
Just because there are no external references, the “normal” inertial forces in a bucket of water cease to exist?
But as I said, I am not a physicist, so I take your word on it, Mano.
In fact, I have long wanted to ask these questions of someone with your expertise. Finding you asking similar questions is a little….. disconcerting.
Mano @ 5
This is the bit that I don’t get. So a bucket of water would not “slosh” about when shoved if it were the only thing in the universe?
John @ 6.
That caused me to go all a-chuckle. Thanks.
tuatara @#8,
Yes, that is pretty much what Berkeley’s view is.
Thanks Mano.
Personally, due to Berkeleys ideas being built on his belief that the forces of nature exist in the mind of the observer only (having no independant existence outside the ability to perceive them) I find his ideas on the subject quite easy to discard.
Perhaps it is telling that Brian Greene does not even mention him in his excellent book “The Fabric of the Cosmos” in which the subject is quite thoroughly explored.
Mach, on the other hand, is an entirely different proposition.
tuatara
//Personally, due to Berkeleys ideas being built on his belief that the forces of nature exist in the mind of the observer only (having no independant existence outside the ability to perceive them) I find his ideas on the subject quite easy to discard.//
Why should any forces literally exist any more than centrifugal force? Maybe they’re just useful fictions. And certainly one can hold this view whist still believing in a material world (Berkeley was an immaterialist).
Responding to tuatara
This seems wrong. All you would have to do is accelerate to a constant velocity that is a high fraction of the speed of light relative to the co-moving frame, aka relative to the nearby and faraway stars. That should break the spherical symmetric of the nearby and faraway stars. Then, repeat the experiment with water in a bucket, and see if there’s a preferred direction to the experiment. I’m gonna bet that there isn’t a preferred direction, aka the stars don’t matter at all for the outcome of the experiment.
@Rob
Am I right?
@Mano
Are you being a contrarian right now? Or are you still genuinely confused? It’s so weird seeing you doubt the ability to use experiments to tell if you’re in an accelerating vs inertia frame, and also whether you’re in a rotating vs inertia frame. Or maybe I’m just reading too far into what you’re writing.
Rob, would a velocity near the speed of light relative to the comoving frame aka relative to the CMB break all spherical symmetries with respect to the stars? In particular, would length contraction break the spherical symmetry of apparent distribution of mass from nearby and faraway stars around my observer? I think so… That should introduce a preferred direction in Berkeley’s model, right? I think so… And I’m betting experiments would show that there is no such preferred direction.
Gerrard @13: Mano is giving us Berkeley’s views. In this series, he’s following his own path of exposition. I counsel patience!
Understood! Lol. Sorry.
Gerrard @#13,
I am not being a contrarian. I am trying to work my way gingerly through rather subtle ideas and trying to keep an open mind while doing so. It is so easy to let one’s strong feelings about something dominate one’s thinking, even in the absence of evidentiary support.
Far be it from me to discount the use of experimental tests to try and distinguish between theories! What I am saying is that we cannot do a direct test of Berkeley’s theory because we cannot get rid of the rest of the universe. We have an indirect test in the fact that the spinning rate of the Earth is the same whether we use the fictitious forces or the stars. If we discount coincidences, that seems to support Berkeley.
@Mano
Did you read the rest of my post #13? Isn’t that a direct test of Berkeley’s ideas? I think we could test it.
PS: It’s highly impractical, but not impossible, to do that test. We have designs for ships that could accelerate to substantial portions of the speed of light right now. They’re just hideously expensive and require expanding our nuclear arsenal by many orders of magnitude.
https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propulsion)
tuatara @#11,
Berkeley does not say that forces exist only in the mind of the observer. He says that only other material bodies can influence material bodies. Space is irrelevant.
I have not read Greene’s book. The fact that Greene does not mention him does not mean anything by itself. Authors have to choose what to include and what to omit in their books. He may have felt that this topic was too esoteric or subtle for the audience he was writing for,. Does Greene say anything about the radiation paradoxes being discussed here? If not, does that mean there is no issue here? Or that he fet it was too esoteric?
PS:
Anyone want to hazard a guess as to how accurate we could make the “bucket test” with modern equipment and methods? And how that would translate into the minimum required lorentz factor and thus the minimum required speed relative to the CMB? I assume 0.1 c would be good enough. I wonder if much lesser speeds would be good enough. Related: What is our current speed relative to the CMB anyway?
PPS:
And google says our current speed relative, aka Earth’s current relative speed, to the CMB is 368 km/sec, aka about 0.001 c. I think that means a length contraction ratio of like 1 : 1.0000007534. That’s, what, 7 decimal digits before a discrepancy should be observable? How precise can we make an apparatus to test whether we’re in a rotating frame or not? Would putting it on the space station help? That seems like it probably is within the bounds of modern engineering precision and accuracy.
Mano, #5:
But isn’t that just wrong? I think you could tell, without reference to any other external objects, that the surface of the water in spinning vs. non-spinning buckets is different, since that changes whether or not it forms a meniscus.
A somewhat different example: if there were two objects rotating about one another and tied together with a rope, you could tell (even if that’s the only stuff in the world) that there’s tension on that rope. Possibly, it may even be enough to break the rope.
And unless I’m missing something, this isn’t just a matter of perspective or whatever, a point which is maybe a little trickier to make with the bucket example. If for instance the rope breaks in some frame or other, then there are no frames in which it doesn’t break. Observers would all agree on that sort of thing. So, you can’t choose to break the rope or not break the rope by changing frames somehow. (Or maybe not literally changing frames at any particular time, but just by deciding out of convenience to pretend that one is “inertial,” even though it isn’t really, and having to invoke “fictitious forces” to make up for that.)
In any case, I don’t think you need to check out whatever might be going on with “distant stars” or what have you, in order to detect such phenomena. I mean, all you would need to look at is the rope (e.g.), in order to see that it either stays in one piece or it doesn’t. No?
Yes, I get it that it’s possible to say (at least semi-coherently) that the rest of the world is rotating and your little system is not. So, sure, you could try to explain things that way. In that case, the funny thing is that the rope breaking would seem to be an indication that your system is just quietly sitting there “at rest” and doing basically nothing … certainly an odd way to think about it. No changing velocities within the system, until eventually the rope breaks (under some kind of stress, presumably) and the two objects it had connected begin to fly apart. Somewhat mysteriously, this purportedly happens due to “the rotation of the rest of the universe” and not what seems like the more obvious choice of the system itself rotating…. I think that’s just a very silly approach to take, even if we could technically do it without contradicting ourselves, because explanations are for making things more comprehensible for ourselves and not less.
Gerrard @14: Yes. You’d still have axial symmetry around the direction of motion (relative to a local comoving observer), but light in front would be blue-shifted, light behind would be red-shifted, and (I think!) star density would be concentrating at 90 degrees to your motion.
The way I’ve always heard about this dilemma is in terms of General Relativity. Under that paradigm, any frame of reference is equally valid, including this case of the rotating bucket.
In that frame, we do see the effect of rotation (the curved water surface) and that leads to only two possible conclusions: the effect is caused because we’re rotating w/r/t space itself (implying that a universal reference frame exists), or we’re rotating w/r/t the distant stars/galaxies. Since the first is ruled out both theoretically (by GR) and experimentally (Michelson-Morley et al), we’re left with the conclusion that the rotational effect is somehow produced by the distant matter in the universe.
Unless, of course, GR is incorrect. But its predictions have been confirmed over and over again, so good luck overturning Einstein!
DonDueed. I think you’re mistaken. In GR, there is no special speed, but there is a special rotational state. Rotating frames are not inertia frames in GR.
Speaking of spinning buckets, a quick nod aside to the people at U Arizona who spun a giant vat of molten glass so the meniscus, when it cooled, made an 8.4 meter mirror for the Magellan telescope!
https://www.sciencefriday.com/segments/giant-magellan-telescope/
They should have named the resulting mirror “Bishop Berkeley” because it will be looking at the same distant stars that, according to him, shaped it.
DonDueed @24:
Not sure what you mean by “equally valid”. Any frame in any paradigm is valid in the sense that you can describe the motion of bodies in terms of that frame, even though that description might be horribly complicated.
In GR, inertial frames are defined by geodesics (the generalization from flat spacetime of straight lines). These concepts are defined locally (at or in the neighbourhood of a given spacetime point). So rotation of a bucket can be described by local deviations from geodesic motion.
That doesn’t mean there is a universal reference frame; any spacetime point is intersected by an infinite number of geodesics corresponding to different local relative velocities and orientations.
Neither does it have anything to do with distant stars/galaxies. It’s all down to local curvature.
Correction: a geodesic doesn’t define an orientation. A geodesic corresponds to an infinite number of frames with different orientations (i.e choice of x, y and z axes).
You can’t tell empirically whether, in absolute terms, your velocity is 0 or whether it’s 42 (pick whatever units you like). Not just according to Einstein, either. Newton approves, because he didn’t think you could do that either.
You can, however, tell when velocities change over time, which of course is all an acceleration is…. Kind of the whole idea behind F=ma. So you can’t tell whether you went from 0 to 42, say, but you can definitely tell that it didn’t remain the same.
To take my example from #22 again, we observe a change in the system when the rope breaks. You can attribute that sort of change in the system to the system’s own state of motion changing over time (AKA, stuff is accelerating somehow). You had some set of events that happened over a period of time, when the system noticeably changes and which you want to understand, so for an explanation of it, you’ll need to appeal to something changing over time…. More or less because you don’t think that sort of thing just happens spontaneously and/or inexplicably.
Alternatively, it may in some cases be that stuff outside your system is actually causing the observed changes in it, so you could attribute it to stuff like that instead, which is sometimes (but not always or necessarily) a more appropriate/sensible choice. Maybe the whole world is indeed rotating around the Earth, while the Earth itself is not, and that is why it bulges around its equator…. But nobody thinks that helps us understand Earth’s shape any better than supposing the Earth is actually the thing that’s rotating. If you asked them, probably most people would think that only makes things less understandable. And even if some do think that’s the better option, it’s not like that’s forced upon us somehow (because of physics or math or whatever).
Anyway, notice how you only had to appeal to things outside of the system because you were assuming that the system itself was inertial. With its state of motion not changing over time (by hypothesis), there would be no changes like that in it which you could appeal to, in order to explain the changes over time that you do in fact see in the form of various phenomena that seem to require explanation (such as why the rope breaks, for instance). You’d still have to point at something changing over time, like the whole universe rotating, in order to cook up an explanation that at least on paper might look adequate (although in that case it’s still just plain wacky if you ask me). So, you’re kind of giving yourself this little puzzle, and it can be resolved by looking at other external objects.
However, if you start out by supposing that your system is rotating, not that it’s inertial — a perfectly acceptable thing to do, since we can see all sorts of rotating things in the world — then within the system itself, you’ve already got what you need, in order to make sense of the phenomena in question. Then, you can just ignore whatever is going on outside of it, as long as none of that is causing any other relevant effects that might also need to be taken into account. (Or that stuff may just not be very noticeable or significant, so you don’t really need to worry about it, at least for some practical purposes.)
#27 JORE
Lol! We totally should.
Bishop Berkeley also told Leibniz and Newton you can’t divide by zero, a major calculus problem that we didn’t really solve until the theories behind limits were developed. They were all ‘but it works’. Dude was surprisingly ahead of his time in some ways.
Mano @19
Of course you are right. Not mentioning something can arise from perfectly innocent reasons. Green does not touch on the radiation paradoxes that you are exploring. It’s subjects are (or appear to be) more specifically space and time.
I look forward to seeing where you lead us to.
Mano. My prior exposure to the ideas of Berkeley were all filtered through the biases of others. Having now been encouraged to investigate him further by what I have read here, I offer my apologies for my ignorance of him and subsequent unkind characterisation.