Paul P. Mealing

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Saturday, 19 July 2008

Epistemology; a discussion

Recently (1 July) I wrote a post on The Mirror Paradox, which arose from my reading of Umberto Eco’s book, Kant and the Platypus back in 2002. The post was an edited version of part of a letter I wrote to Eco; the rest of the letter was to do with epistemology, and that is the source of this post.

Some people think that because we can’t explain something, either it is wrong or it doesn’t exist. Two examples from the opposite sides of philosophy (materialism and fundamentalist religion) illustrate this point very clearly. In a previous post, The Ghost in the Machine (Apr.08), I reviewed an article in SEED magazine (Henry Markram’s Blue Brain project). In the same magazine, there is an essay by Nicholas Humphrey on the subject of consciousness. Effectively, he writes a page-length treatise arguing that consciousness must be an illusion because we have no explanation for it. This is despite the fact that he, and everyone he meets in life, experiences consciousness every day. Humphrey’s argument, in synopsis, is that it is easier to explain it as an illusion than as reality, therefore it must be an illusion. Personally, I would like to know how he distinguishes dreaming from living, or even if he can (please refer Addendum below, 4 April 2010). Another example from the polar opposite side of rational thinking is evolution. Fundamentalist Christians tend to think, because we can’t explain every single aspect of evolution, it can be challenged outright as false. This is driven, of course, by a belief that it is false by Divine proclamation, so any aspect of the theory that is proven true, of which there is evidence at all levels of biology, is pure serendipity. (Refer my Nov.07 post, Is evolution fact? Is creationism myth?)

I’m making a fundamental epistemological point that we don’t understand everything – another, excellent example is quantum mechanics (see The Laws of Nature, Mar.08), where I quote Richard Feynman, probably the world’s best known expert on quantum mechanics (he had a Nobel Prize to prove it), and arguably its best expositor, who said quite categorically in his book, QED, ‘…I don’t understand it. Nobody does.’ There is nothing that makes less sense than quantum mechanics, yet it is arguably the most successful scientific theory of all time. Historically, we’ve always believed that we almost know everything, and Feynman was no less optimistic, believing that we would one day know all physics. But, if history is any indication of the future, I choose to differ. In every avenue of scientific endeavour: biology, cosmology, quantum theory, neuroscience; there are enormous gaps in our knowledge with mysteries begging inquiry, and, no doubt, behind those mysteries, lay a whole gallery of future mysteries yet to be discovered.

None of this was in the letter I wrote to Umberto Eco, but it seems like a good starting point: we don’t know everything, we never have and we probably never will. The only thing we can say with confidence is that we will know more tomorrow than we know today, and that is true for all the areas I mentioned above. As I’ve already said in previous posts: only future generations can tell us how ignorant the current generation is.

Actually, this is not so far removed from Eco’s introduction in Kant and the Platypus, where he hypothesises on the limits of our ability to comprehend the universe, which may include metaphysical elements like God. He postulates 4 hypotheses based on matching items of knowledge (symbols) with items of physical entities (elements), which he calls, for convenience sake, 'atoms', and various combinations of these. As a corollary to this approach, he wonders if the graininess of the universe is a result of our language rather than an inherent feature of it, as all the hypotheses require segmentation rather than a continuum.

I won’t discuss Eco’s hypotheses, only mention them in passing, as I take a different approach. For a start, I would use ‘concept’ instead of ‘symbol’ or ‘atom,’ and ‘phenomena’ instead of ‘elements’. It’s not that I’m taking explicit issue with Eco’s thesis, but I choose a different path. I define science as the study of natural phenomena in all their manifestations, which is really what one is discussing when one questions the limits of our ability to comprehend the physical universe. Secondly, it is becoming more and more apparent that it is mathematics rather than language that is determining our ability to comprehend the universe – a philosophical point I’ve already discussed in 2 posts: Is mathematics evidence of a transcendental realm? (Jan.08) and The Laws of Nature (Mar. 08).

Some people argue that mathematics is really just another language, but I would contend that this is a serious misconception of the very nature of mathematics. As Feynman points out in his book, The Character of Physical Law, translating mathematical ideas into plain English (or any other verbal language) is not impossible (he was a master at it) but it’s quite different to translating English into, say, French. To describe mathematics in plain language requires the realisation of concepts and the use of analogies and examples. Mathematics is inherently paradoxical, because it is conceptually abstract, yet it can be applied to the real world in diverse and infinitely numerous ways. Whereas plain language starts with descriptors of objects (nouns) which are then combined with other words (including verbs) that allow one to communicate actions, consequences, histories and intentions; you could argue that mathematics starts with numbers. But numbers are not descriptors – a number is a concept – they are like seeds that have infinite potential to describe the world in a way that is distinctly different to ordinary language.

Nevertheless, Eco has a point, concerning the limits of language, and one may rephrase his question in light of my preceding dissertation: is it our use of number that projects graininess onto the universe? This question has a distinctly Kantian flavour. One of the problems I had with Kant (when I studied him) was his own ‘Copernican revolution’ (his terminology) that we project our models of reality onto the world rather than the converse. As a standalone statement, this is a reasonable assertion, and I will return to it later, but where I disagreed, was his insistence that time and space are projections of the human mind rather than a reality that we perceive.

I truly struggled to see how this fitted in with the rest of his philosophy which I find quite cogent. In particular, his idea of the ‘thing-in-itself’, which essentially says that we may never know the real essence of something but only what we perceive it to be. (I think this is Kant's great contribution to philosophy.) He gave the example of colour, which, contrary to many people’s belief, is a purely psychological phenomenon. It is something that only happens inside our minds. Some animals can’t see in colour at all and some animals see colours that we don’t, for example, in the ultra-violet range. Some animals, that use echo-location, like bats, dolphins and whales, probably see in ultra-sound. It would be hypothetically possible for some creatures to see in radar, if they ever evolved the ability to transmit radar signals. But, more significantly, our discoveries in quantum mechanics and relativity theory, are proof that what we perceive as light and as time respectively are not necessarily what they really are, depending on what level of nature we examine. This leads to another aspect of epistemology that I will return to later – I don’t want to get too far off the track.

In fact, relativity theory tells us that time and space are inherent features of the universe, and, again, it is only through mathematics that we can decipher the enigma that is relativity, as well as quantum phenomena. But we don't need relativity theory to challenge Kant's thesis on the nature of space and time. We sense time and space through our eyes (our eyes are literally like a clock that determines how fast the world passes us by) and, again, this is different for different species. Many birds, and insects, see the world in slow motion compared to us because their eyes perceive the world in more ‘frames per second’ than we do (for us I think it’s around 24). The point is, contrary to Kant’s assertion, if our senses didn’t perceive the reality of space and time, then we would not be able to interact with the world at all. We would not even be able to walk outside our doors.

I once had an argument with a professor in linguistics, who claimed that 3 dimensional Cartesian axes are a human projection, and therefore all our mathematical interpretations, including relativity, based on Reimann geometry (which is curved), are also projections. The fact is, that we live in a 3 dimensional spatial world, and if we lived in a higher dimensional spatial world our mathematical interpretation of it would reflect that. In fact, mathematically, we can have as many-dimensional worlds as we like, as string theory demonstrates. Einstein’s genius was to appreciate that gravity made the universe Reimann rather than Cartesian, but, at the scale we observe it, it’s not noticeable, in the same way that we can survey our little blocks of land as if they are flat rather than curved, even though we know the earth’s surface is really a sphere.

After all that, I haven’t answered the question: is the perceived graininess of the universe a result of our projection or not? One of the consequences of Kant’s epiphany, concerning the thing-in-itself, is that it seems to change according to the level of nature we observe it at. The example I like to give is the human body, which is comprised of individual cells. If one examines an individual cell there is no way we could appreciate the human body of which it is a part. At an even smaller scale we can examine its DNA, which is what determines how the human body will eventually turn out. The DNA is actually like a code, only it’s more than an analogy, it really is a code; it contains all the instructions on how to construct the creature it represents. So what is the thing-in-itself? Is it the genome? Is it the fully grown adult body? Humans are the only species that we know of who have the ability to conceptualise this, and, therefore, are able to comprehend at least some of the machinations of the natural world. And this, I believe, lies at the heart of Eco’s introductory hypotheses. It’s not to do with matching symbols with elements, or combinations thereof, but matching concepts with phenomena, and, more significantly, concepts within concepts, and phenomena that emerge from other phenomena.

Many people talk about the recursive ability of the human brain, which is to hold multiple relationships within one’s mind, like my friend’s mother’s lover has a cat with an injured foot. I understand that 5 is the norm, after which we tend to lose the thread. In which case, I ask: how can we follow a story, or even an argument, like the one I’m writing now? In another post (Imagination, Mar.08) I suggest that maybe it was storytelling that originally developed this aspect of our intellectual ability. We tend to think of words as being the ‘atoms’ of a story, but, as a writer of fiction, I know better, as I will explain shortly. Individual words do have a meaning of their own, but, as Wittgenstein pointed out, it is only in the context of a sentence that the true meaning is apparent. In fact, it is the sentence, or phrase, that has meaning rather than the individual words, as I’m demonstrating right now. But it really requires a string of sentences, and a lengthy one at that, to create an argument or a story. The shortest component of a story is actually a scene, and a scene is usually delineated by a break in time or location at its beginning and its end. But, of course, we don’t keep all the scenes in our memory for the course of the story, which may unfold over a period of days, so how do we do it?

Well, there is a thread (often times more than one) which usually involves a character, and we live the thread in the moment just like we do with our lives. It’s like when we are in contact with that thread we have the entire thread in our mind yet we are only interested in its current location in time and space. The thread allows us to pull out memories of it, make associations, into the past and future. This is the really extraordinary attribute of the human brain. I’ve no doubt that other animals have threads as well, but I doubt they have the same ability as we do. It is our ability to make associations that determines almost everything intellectually about us, including our ability to memorise and learn. It is only when we integrate new knowledge into existing knowledge that we actually learn it and understand it. To give an example, again, from Wittgenstein, if you come across a new word, you can only comprehend it when it is explained in terms of words you already know. In a story, we are continually integrating new information into existing information, yet we don’t see it as learning; we see it as entertainment. How clever is that?

I argue that recursiveness in the human brain is virtually limitless because, like the cells and the human body, we can conceptualise concepts within concepts ad infinitum, as we do in mathematics. For example, calculus requires the manipulation of infinite elements yet we put them all into one function, so we don’t have to even think of an infinite number of elements, which, of course, would be impossible.

I’ve made the point in other posts, that the reason we comprehend the universe to the extent that we do is because we have this ability to perceive concepts within concepts and the universe is made up of elements within elements, where the individual element often has nothing in common with the larger element of which it is a part, so graininess is not the issue. I don’t believe this is a projection; I believe that this is an inherent attribute of the entire universe, and the only reason we can comprehend it, in the esoteric way we do, is because we are lucky enough to have the innate ability to perform the same trick mentally (see The Universe's Interpreters, Sep.07).

I’ve almost exhausted this subject, but I want to say something about schemas. I mentioned, earlier in this essay, Kant’s assertion that we project our ideas, or models, onto the universe in order to comprehend it. I discuss this as well in The Laws of Nature, but in a different context. Eco also talked about schemas, and while he said it was different to the psychological term, I will attempt to use it in the same sense as it is used in psychology. A schema is a template, is the best description I can give, whereby we apply it to new experiences and new knowledge. We even have a schema for the self, which we employ, subconsciously, when we assess someone we meet.

I argue that the brain is a contextual instrument in that it axiomatically looks for a context when it encounters something new, or will even create one where one doesn’t readily exist. By this I mean we always try and understand something on the basis of what we already know. To give an example, taken from Eco’s book, when Europeans first saw a platypus they attempted to categorise it as a mammal or a reptile (it lays eggs). But, if I was a European, or from the northern hemisphere, I would probably think it was a type of otter or beaver, assuming I was familiar with otters and beavers, because it is air-breathing yet it spends most of its time in river water or underground. Another example: assume you had never seen a man on a horse, but mythically you had seen pictures of centaurs, so the first time you saw a mounted man you might assume it was all one animal.

My point is that we apply schemas to everything we meet and perceive, often subconsciously, and when we become more familiar with the new experience, phenomenon or knowledge, we adjust our schema or create a new one, which we then apply to the next new experience, phenomenon or whatever.

There is a logical connection here, to what I suggested earlier, that we only understand new knowledge when we integrate it into existing knowledge. A schema is a consequence of existing experiences and knowledge, so cognitively it's the same process. The corollary to this is that when we encounter something completely alien, we need a new schema altogether (not unlike Kuhn's paradigm-shift).

I read recently in New Scientist (31May 2008) that someone (Karl Friston) had come up with a Bayesian interpretation of the brain (using Bayesian probability), at all levels, including neurons (they strengthen connections based on reinforced signals). The brain makes predictions, then adjusts its predictions based on what it senses in a reiterative process. He gives the everyday example of seeing something out of the corner of your eye, then turning your head to improve your prediction.

Schemas, their interaction with the world and our modification of them accordingly, is such a reiterative process, only on a different scale. Previously, I've talked about the dialectic in science between theory and observation, or theory and experimentation, which is another example of the same process, all be it's at another level altogether and is performed in a more disciplinary manner.

This is where I should write a conclusion, but I think I already have.

Having completed this essay, it has little resemblance to my letter to Umberto Eco in 2002, in either content or style, but some ideas and some arguments are the same.


Addendum (4 April 2010): I may have misrepresented Nicholas Humphrey - please read the addendum to my post Consciousness explained (3 April 2010).

Monday, 7 July 2008

Layers of Being

You’ve probably noticed that a recurring theme of some of my essays is the virtue of self-honesty. I guess that’s why I am attracted to existentialism – in fact, I think it’s fair to say I was attracted to it long before I knew what it was.

I am going to discuss 3 layers of being, based on my own experience and observations, and I am sure some will argue that there are more, while others may argue that they don’t exist at all, but, basically, this is possibly the most personal of my essays thus far on this blog, so it’s not very scientific.

What do I mean by layers of being? I’ve already said that it’s important in philosophy to define one’s premises and concepts. I think a good starting point is another one of my recurring themes: the inner and outer world. Some people, especially some philosophers, would prefer not to make any distinction, but I find it unavoidable. I’m a writer of fiction, and it was whilst writing fiction that I first appreciated the significance of the inner and outer world. Fiction, in a Paul Mealing defined nutshell, entails a character’s journey. Once you take that approach, it generates its own corollary: the character is changed and altered by the events in the plot that he or she encounters. To extend the metaphor, the plot becomes a vehicle for the character’s own inner journey. I was aware of this from my very first attempts to write fiction. Of course, it’s exactly the same in life, only we don’t use the terms, plot and character, in real life.

So I already have 2 layers: the inner and outer world. Before I introduce the third, I need to elaborate on these 2, as they are the most obvious and also, they are experienced by everyone, even if you would prefer to conflate them. The most obvious interface or interaction between these 2 layers is found in relationships. It is through relationships that we practice integrity or deception, generosity or rejection, engagement or apathy. There are other terms: love, jealousy, anger, hate, envy, revenge, charity, empathy, compassion. All these terms only acquire meaning within the context of relationships, but, of course, it’s unavoidable that they also reflect something deeper within the individual.

But there is one simple rule or criterion, which, I believe, puts all relationships into perspective, and that’s expectation. In any relationship: family, work, love, sport, even legal; there are expectations. It is when an individual’s expectation is in agreement with the group’s that there is harmony. When this expectation is either above or below the group’s, or the other’s, there will be conflict. By above or below, I mean we either expect more or less of our own role compared to what others might think. And one can see that honesty plays a key role here – if we deceive someone into an expectation that can’t be met, either by them or by us, then we have already started on a bad footing.

Paradoxically, this leads to the third layer of being, and the one I started off with: deception to oneself. Our relationships with others have a direct internal reflection and vice versa. To take an example, if we hate someone it corrodes our own soul, leaves us bitter in a way we can’t fathom. Likewise, jealousy alienates the person we love. These are contradictory causes and effects, yet we have all experienced them. Surely, you say, this is not dependent on a third layer, this is merely a further extrapolation of the inner and outer world.

What then is self-deception? I’m talking about neurosis where one has a distorted view of oneself. The dissociation that can occur between individuals and others can also occur within oneself. I know this because I have experienced it. When I read of people who have gone off the rails, I can sometimes see myself, as I know how easy it is to have a distorted view of oneself and feel like one has lost their core, or what we sometimes call our identity. Of course, this self distortion directly affects our relationships with others – it has an impact on the outer world – the two are not independent.

And this is why I place so much emphasis on self-honesty, because, without it, one can’t be honest in one’s relationships with others. But, I believe, it is also this third, deeper layer of being that provides the spiritual dimension that some people claim. In other words, it comes from a self-examination and a level of self-honesty that most of us fail to achieve. It doesn’t require a belief in God, but, ideally, it should lead to a sense of egolessness. What Buddhists most likely call the no-self, though I’m no expert in Buddhism.

One must also define what one means by ego. Again, I think Buddhism provides a key - to do with attachment, though I’m not opposed to attachments per se. There are healthy and unhealthy attachments, all to do with choices, but I’m getting off the track. Buddhism deals with attachment to life in general (samsara) and I would say that ego is an essential aspect, arguably, the very consequence of this. Again, ego can be healthy or unhealthy, so the egolessness I refer to is an ideal, whereby one becomes ‘unattached’ even to oneself, albeit sounds like a complete contradiction.

Have I personally reached this state? No. Maybe when I die. After all, one doesn’t have to be a Buddhist to believe that death is the final letting go of ego (I think a Jewish philosopher once said that, but I can’t remember who it was). Of course, I’m yet to prove it.

Tuesday, 1 July 2008

The Mirror Paradox

The mirror paradox is best stated as a question: why is a mirror image seen as left to right reversed but not top to bottom? I’m not sure why this is always seen as a philosophical conundrum, when it involves science, and, to a lesser extent, psychology. Having said that, I’ve rarely seen it explained correctly, so maybe it is a philosophical conundrum after all. Certainly, Stephen Law, in his excellent book, Philosophy, believes it’s ‘a puzzle science cannot solve’. But I beg to differ: it’s a real optical phenomenon entailed by the 3 dimensional spatial world in which we live.

In the Oxford Companion to the Mind (1987), Richard L. Gregory (Professor of Neuropsychology at University of Bristol, UK) in my view, talked all around the solution, without actually delivering it. He certainly understood that there is no rotation in the mirror (see below). However, he seemed to think (like Stephen Law and Umberto Eco) that there is no reversal at all, though he once obliquely referred to an 'inversion', so maybe he knew without knowing that he knew.

In 2002, whilst working in America, I read Umberto Eco’s Kant and the Platypus, which was as stimulating a book as I’ve ever read on epistemology, though I think it would be a difficult read for anyone who wasn’t familiar with Kant, at least at a rudimentary level. When I had completed it, I wrote a long letter to Umberto, who acknowledged receipt of my letter but never followed up with a response. I learned later that he had a fight with throat cancer, so I take it as no slight that he didn’t respond further.

In one of his chapters, he gives a lengthy discourse on the mirror paradox, and that was one of my points of contention. He argued that a mirror ‘reverses nothing’, but a second reflection did reverse left to right, which restores the image to what we normally see. I pointed out to him that this was illogical, nevertheless there is a specific case where he is correct. I will return to this specific example at the end of my discourse. I need to say that I have great respect for both Stephen Law and Umberto Eco, as both of these men are far more knowledgeable than me in their respective fields.

Most people explain the mirror reflection in terms of rotation, as it appears that the mirror rotates the image around, and this is particularly compelling for mirror-reflected writing, as I explicate below. But this merely raises another question, effectively transcribing the paradox, not solving it. Why does it rotate the image about the vertical axis, not the horizontal axis? Stephen Law gives an analogy: if you walk through a door that opens on the right side, why does it open on the left side when you come back the other way? The answer is because you turn yourself around. Law argues that if we lived in zero gravity, whereby you could turn yourself upside down to open the door, it would still open the same way, so the implication is that it’s gravity that creates the emphasis on the vertical axis. In fact, Stephen Law speculates that if we lived in a weightless environment then perhaps the ‘mirror puzzle would not even be a puzzle’. But I believe he'd be wrong: it's the left-right symmetry of the human body that creates the emphasis on horizontal over vertical reflection.

All these explanations and descriptions seem to overlook the fact that there is no rotation in the mirror at all – in fact, it’s the lack of rotation that gives us mirror reflection. I believe that most of these explanations actually appreciate this fact; they just fail to explain it. But I have been keeping you in suspense – the answer to this puzzle is deceptively simple: the mirror doesn’t reverse left to right, or top to bottom, it reverses back to front. We live in 3 dimensions, not 2, and a mirror reverses everything in the dimension perpendicular to its plane. So the rotation is a genuine illusion (it doesn’t happen), but the reversal is a true optical phenomenon.

Below is an edited version of my exposition that I sent to Umberto Eco.

Normally, if we want to see something back to front we have to turn it around. Generally we do this by turning the object through its vertical axis but we can also turn it through its horizontal axis. If we turn it through its vertical axis, as happens when someone turns to face us, their left side appears on our right and their right side appears on our left. This is unavoidable. But they could also turn to face us by standing on their hands, in which case they would appear upside down but their left side would still be on our left and their right side on our right. Then if they stood by doing a half cart wheel they would resume their normal stance but left to right would be reversed. The mirror quite literally reverses the image back to front without rotating it through any axis at all. So we don’t see the image upside down but likewise we don’t see the left side on the right or the right side on the left. This is the illusion pure and simple. The illusion, when we face a mirror, is that it appears to rotate us around a vertical axis, when in fact it doesn’t, it turns us back to front. If we look at something between us and the mirror, we see the front of it facing us, and the back of it facing us in the mirror. This is the key to the illusion. When we look at ourselves in a mirror we expect to see ourselves as others see us, but we can only do this when we have 2 mirrors, which appears to really rotate everything about the vertical axis (as Eco contended), but, in actuality, restores front to back to front again. But I’m fast-forwarding - I will elaborate on double reflections later.

In other respects we are not fooled by the mirror’s conservation of left and right. If we see in the mirror someone standing behind us and to our left, we automatically look over our left shoulder, not our right. Where we are fooled is when we reach for something on a table between us and a mirror, as in the case of an object on a dressing table or a bathroom bench top, while watching the object in the mirror. If we reach for an object at the back of the table, we appear in the mirror to be reaching forward towards us, rather than away. Likewise, if we drag an object on the table towards us, we appear in the mirror to be pushing it backwards not forwards. If you doubt this, try shaving or combing your hair with your left hand instead of your right (or your right hand if you're left-handed). We’ve trained our preferred hand through years of practice.

When we look in the rear view mirror at a car parked behind us while standing at a traffic light, we see that the driver is sitting on the same side of the car as we are and we are not confused. Because we know the car is behind us, the same as in the previous example, when we knew that the person standing behind us in the mirror was on our left or right side just as the mirror dictated. If the car was traveling towards us, we would expect to see the driver on the opposite side to us because the car has been turned around it’s vertical axis. If we turned around to look at the driver behind us at the traffic light, we would still see that he or she is on the same side of the car as we are, because both cars are facing the same direction, even though we have turned around to look backwards. Therefore, when we look at the driver in the rear view mirror we can see that left and right have been conserved. So why is it that when we look at the number plate we have to read it backwards, as if it's been rotated?

Writing not only provides the best illustration of the illusion, it also provides the best means to understand it. If you hold up a page of a book with writing on both sides while facing a mirror, the side facing you is readable, but the side facing the mirror is mirror-reversed. However, if the page was transparent, then the writing on the other side would also appear mirror reversed exactly as it does in the mirror. Take a sheet of plastic or cellophane, or anything clear that can be written on. If you hold up this transparent sheet so that the writing is mirror-reversed to you then it will also appear mirror-reversed in the mirror. Likewise if you hold it up so that the writing is readable to you then it will appear readable in the mirror. So where did the illusion of rotation go?

The illusion has gone but the reversal hasn't. Because when you hold up the sheet so you can read it, you are looking at the front of the sheet whereas the image you see in the mirror is the back of the sheet. Left to right is not reversed but front to back is. The front you see in the mirror is actually the back to you. If you were to place yourself between the sheet and the mirror, without changing its orientation, you would see the writing mirror-reversed. The mirror mirror-reverses the back of the sheet. And, of course, you would have to turn yourself around to read it, which only emphasises the illusion that the mirror rotates the image, but actually it doesn’t. The reason writing always appears reversed left to right, is because we always turn it left to right to face the mirror. We do the rotation, not the mirror.

This brings me to the third image created by a second mirror. If you set a book upright on a dressing table (or a table with a mirror behind it) with the front cover facing you, then the back cover will be mirror-reversed in the mirror. If you then took a small mirror (say a shaving mirror) and place it between yourself and the book, but facing back into the main mirror (or background mirror) you can create a third image of the cover in the main mirror. This is very easy to do by small adjustment of the angle of the foreground mirror. Naturally enough (but only because we know in advance) we can read the front cover in the third image exactly as it appears to us on the table. If we didn’t already know this, I believe it would be a complete surprise. The important point is that the image is not rotated at all, it is simply reversed back to front twice, using an intermediary mirror that is facing away from us.

In fact the foreground mirror behaves in exactly the same way as the transparent sheet I referred to in the previous example. If you could see through the foreground mirror so it’s image could be read from the back (in other words if it was a transparent screen with the book cover projected onto it) we would be able to read it exactly as we can in its reflection in the background mirror. The point is though, that the foreground mirror reverses the image, not from left to right but back to front. The foreground mirror only has the writing in the right order because it is facing away from us. If you were to place yourself between it and the background mirror (and turn yourself around) you would see the writing is mirror-reversed as you would with the transparent sheet. So the background mirror mirror-reverses the foreground mirror.

But, as I alluded to earlier, there is a specific situation, and a common one, where a second mirror does translate the image directly from left to right, which upholds the illusion of rotation. We often find ourselves in a bar, or a bathroom, with 2 vertical mirrors joined at right angles like 2 walled mirrors. In this case the image you would see is a double reflection no matter which mirror you looked in. In fact, if there were 2 extended wall mirrors, then there would be 4 images of you, including the prime image. If you were to press your finger into the corner, you would see 4 symmetrical images of it, one of which would be you. Another unique feature of this third image is that it would always remain in the corner of the room as you moved about, whereas the other 2 images would follow you around. This also means, of course, that everyone in the room would see themselves in the corner (assuming they had a clear line of sight).

The (apparent) non-reversed image results from a secondary reflection coming off a primary reflection that you cannot see, because the two reflections simultaneously 'swap' over on the adjacent mirrors. This, in fact, does resemble a rotation about the vertical axis, simply because the mirrors are joined on a vertical axis. And this is what led Umberto Eco to argue that the first mirror image is not reversed left to right but the second image is. He is correct, in this specific case, but only because we create a virtual vertical axis of rotation by the very careful alignment of the 2 mirrors.

So now I have turned a simple answer into more than 2,000 words, and either have confused you completely or explained a common phenomenon thoroughly. I hope the latter.

Footnote: I had a brief discussion with Stephen Law on this topic. We agree to disagree over my belief that science does solve this puzzle. You can visit his post on this subject (and our dialogue) at the following: http://stephenlaw.blogspot.com/2007/04/mirror-puzzle-solution.html And explore the rest of his excellent site.

Addendum: If you think this exhausts the subject of mirrors, you should read Richard Feynman's quantum mechanical explanation of reflection in his truly fantastical book, QED, The Strange Theory of Light and Matter.