Noldorin's Blog

For whatever reason, I remember quite clearly the first time I was introduced to the wonder of paradoxes. Curiously, it was during an English class in my first year of secondary school, and the rather eccentric teacher had a particular tendency to ramble on about any interesting topic (usually well outside of the syllabus). A criticism this is not, as it was many years before the seriousness of GCSEs and A-levels. I think that in looking back I took great enjoyment out of those classes, even if I did not so much realise it then. (And it wasn’t just for the fact that we didn’t spend countless hours analysing poetry or Shakespeare.) Moreover, it is plain now that he was, through a variety of ways, trying to open our young and malleable minds so that they might perhaps (idealistically) become sharp and inquisitive, and remain so through the future years of drudgery.

Before I continue too far on such a tangent myself, let me present the focus of this post, that is one of the paradoxes with which I became acquainted during one of those many unusual English classes. I have forgetten the precise details, but the following I think is a half-way accurate rendition of what was then told (though the many embellishments may differ to those of my former teacher). As you may guess from the title, I term this little problem the “Traveller’s Paradox”, though I don’t think it has any conventional name, and has undoubtedly been repeated in many varying forms.

Warning to unsuspecting readers: the following situation is presented in the fairy-tale style. I was told it this way, so that’s how it’s going to get repeated – can’t please everyone!

A wearied prince has travelled many leagues on his quest to reach the all but forgotten castle that is the target of his quest; the location of the the legendary gem that is the final chance to save his kingdom from ruin. The pale sun is gradually vanishing behind the horizon as the young man approaches a great fork in the road ahead, far beyond which he can glimpse the tall spires of the castle that is his destination. Having sought his goal by his wits and instincts alone thus far, he is now unsure of which path to take from this point. Alas, time permits him to delay no longer if he is to succeed in his quest, and he must make the choice of road before darkness falls.

Approaching the split in the wide path, he scarcely notices two giants standing motionlessly at the side of the path. In this moment the prince recalls the words of the sage whom he had oft consulted; these twin giants, identical in appearance, were the only living creatures who know the safe pafe to the fabled castle. Yet he has a dilemma, for one of them always lies while the other always tells the truth, and what is worse, no-one may ask either of them but a single question.

Making the wrong choice of path will lead inevitably to peril and the ultimate failure of his quest. Only one of the roads provides a safe route directly to his goal. Not willing to let the fate of his realms rest in the hands chance, the prince knows he must ask one of the giants the question that will tell him the safe path – but which giant, and what question?

Now, before continuing, do ponder for a while the nature of this paradox, if you are not already. I do promise  that this dilemma does in fact have a sane solution, and unlike other paradoxes, is not a fundamental contradiction of logic and reason. Read on only when you wish to see the answer, and more relevantly, the formal (mathematical) method we can use in treating such a paradox.

Mathematical (or formal) logic is arguably at the heart of mathematics itself, and is to many the foundation of all science, philosophy, and general reason. Logic is unfortunately not such an intuitive thing to we as humans (without exception) – at least, beyond the very superficial level. Certainly, what has developed into the field of formal logic (in particular modern research into higher-order logics) would make little to zero sense to anyone discovering it upon the first time. It does not take much consideration to realise that the human mind, not unalike to those of other creatures, is one tailored by the eons of evolution to the tasks of survival and continuation of the species. Indeed, we were never remotely designed to unravel the mysteries of the universe, and it is only through our higher level capacities developed through other means that we may begin to do so. (That is at least the brutal atheists approach, and I am among those who would argue the point at a higher level.)

Without assuming too much prior knowledge of fundamental mathematics, specifically formal logic, I will now introduce an approach to resolving in a (simple?) mathematical manner what initially appears to be a very counter-intuitive situation. Do not fret though, for I myself have barely touched the surface of these areas. Still, for the sake of conciseness, I am going to assume you either have an elementary knowledge of formal logic, or can look up the ideas involved, where necessary. If you haven’t yet seen it, I mentioned a great tutorial for starting out in a previous post. (The Wikipedia pages may be enough if you just want an overview though.)

Firstly, let us formulate the given problem in the language of propositional logic, which involves nothing more than the manipulation of true and false values, in essence. There is no fool-proof way of doing this translation, of course, but read on and I think you will see it all works out pretty well.

Note that I use for the truth value and for the false value here.

To begin let us define the propositional variables and functions we will be using:

By arbitrarily labelling the pair of giants, let represent the truth-telling one and the lie-telling one.

Note that $\equiv$ is not actually a symbol in propositional logic, but just syntax for defining a variable.

Now, using this notation, and carefully analysing the situation (problem) presented in the above text, we can see that the problem is to find a formula for that satisfies a proposition that represents the problem. First, observe that whichever giant you ask, the reply you get will be either or in response.

The proposition that defines the situation is then:

In words, that is: asking the particular question to either giant, you will get the same response (yes/no answer) every time, and this answer will also indicate that you should take path (if true, the “first” path; if false, the “second” path).

Unsurprisingly perhaps, there is no well-defined procedure for finding the correct formula for . (There are in fact a number of perfectly valid/equivalent solutions that are relatively simple.) Now, since is simply a function of the propositional variable , we can quite easily factor out the dependence on and still write without loss of generality as:

If one wanted to be utterly rigorous, one could then take this formula for , substitute it into the problem definition, and use the rules of inference (for whatever formal system) to manipulate it into a form that explicitly gives and . On the other hand, I’d rather not lose any readers I still have at this point, so let’s do things the slightly more intuitive way by using some simple human analysis!

The propositional operator , while defined as the bidirectional application of the logical implication operator (i.e. ), can also be seen as representing equivalence of two formulas. (This can indeed by proven formally, though again it’s not really worth the space here I feel.)

Taking the original proposition, we can see (again, with a bit of higher-level analysis) that it implies both:

and

By straightforward substitution for , and a bit of simplification, we can then deduce that:

With the understanding of as representing identity, as previously stated, we can substitute the values for and back into the previous formula to get the following.

Problem solved… right? Well no, not quite actually. If we had mechanically applied the rules of inference from the axioms and the formulaic representation of the problem, we would undoubtedly be able to stop at this point (requiring a good few more pages of derivation however). Since we have not been completely rigorous, we must now prove that this definition of is indeed a correct solution to the problem. Let us substitute the formula back into the problem definition and see.

Hence, we now know that the above solution is correct. What remains is only to translate this formula into plain English. Not so trivial, I think we would agree. The first thing to note is that the parameter is unknown to the prince who asks the question. Well, enough hints…  let’s see if anyone can figure it out first (no cheating), and I’ll update the post with an answer in a few days.

Reading Frank Herbert’s Dune was an experience that has likely influenced me in a number of ways, some quite subtly. For certain, however, it has stimulated a great deal of my own philosophical thought. The Dune universe, while on the surface highly esoteric in certain ways, has in my view astounding relevance in the present day with regards to such matters as politics, philosophy, and interest – to a degree not possessed by any other work of science fiction (and few others) I have encountered. Moreover, it is most often expressed so elegantly that one cannot help but interpret it in a profound way.

The quote I wish to share here is undoubtedly one of those that should be meaningful to any man or woman alive today. The so-called “Litany against Fear” is a passage of text that first appeared in Herbert’s original Dune novel near the start of the book (when Paul undergoes his test of “humanity” under the threat of a Gom Jabbar). Call this passage a litany, mantra, or what you will – to me it is something that has significance well beyond it words.

I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past I will turn the inner eye to see its path.
Where the fear has gone there will be nothing.
Only I will remain.

While I would not be inclined to say I use it in the same way the fictional Bene Gesserit do, it is nonetheless a surprisingly effective passage to recite in ones mind at various times. What is more, the Litany against Fear is not simply confined to combat feat; to me, it is equally effective to resist any other “overwhelming” emotion, be it anger, hatred, regret, or obsession. Perhaps “resist” is the wrong word here even; the way the Litany helps to overcome fear is somewhat less obvious and forceful, not to be easily expressed in words, I think.

Any regular readers, please excuse this brief diversion into emotional and philosophical. Juxtaposed with all my technical and scientific posts, it may not fit in very well, but alas, this blog has never had much cohesion and is rather a brain dump for me. And while I may be unique in this opinion, I actually find that discovering the ongoing variety of thoughts in someone’s mind is far more interesting than consuming formalised articles.

In case anyone is interested in reading the essay I wrote for a college assignment, entitlted “The Future of Space Exploration” (no need to explain the topic I think), I’ve made the full article available to read here. Be warned, however, it is around 3,500 words of mainly rambling – though the content isn’t too technical, I’d like to believe. As always, I would be curious to hear whatever opinions readers have on the subject and my writings.

These  famous words of the French philosopher Voltaire are probably better understood as one of the many examples of his keen wit, rather than a profound statement; yet they do contain a certain poignancy. The self-referential nature of this quotation makes it so wonderfully ironic and perhaps even paradoxical. As with so many quotes that are worth knowing, there are often several levels of meanings: some obvious, others concaled; some sincere, others facetious. I would in fact beg to disagree with the title of this post (though I, as perhaps anyone) cannot say whether Voltaire made this comment literally or not. To me, it is the very nature of most, if not all, witty quotes, to in some way express a deeper truth – in some respect I think it is part of the definition. After all, truth is beauty, they say.

Enough of this meta-discussion, though. The purpose of this post is really just to share the small selection of the more memorable quotes I’ve picked up over the years (or at least the ones I’ve managed to jot down before they slipped away). Indeed, rather than launching into a commentary here of all the quotes, I thought it would be better simply to list them as a (hopefully dynamic) collection, leaving it as an “exercise for the reader” to gather what interpretation they will from the words. It is my opinion that they are both better understood and appreciated in such a way – at best it gives the reader undesirable preconceptions, at worst it demeans the thing.

Well, here they are then: my collection of quotations, aphorisms, adages, or whatever you want to call them. I’ve written them in what I believe to be their most commonly accepted form, and attributed them to their most widely acknowledged sources. Nonetheless, I have little doubt that some of them have been distorted, and may even be apophrycal, yet this never made a difference to me and I don’t see why it ever should from any but a historical perspective. Good sayings are often improved over time, sometimes so much that they cannot be attributed to any more than folk wisdom. Whatever they are, they certainly give pause to ponder.

Note: The juxtaposition between the more profound and the more humurous quotes below may seem slightly awkward, but I feel it would be too artificial to separate them out in a clear-cut fashion, so read them as you will.

Socrates (469 – 399 BC) – Ancient Greek philospher

I cannot teach anybody anything, I can only make them think.

Education is the kindling of a flame, not the filling of a vessel.

I know that I am intelligent, because I know that I know nothing.

The unexamined life is not worth living.

Plato (428 – 348 BC) -  Ancient Greek philospher

Be kind, for everyone you meet is fighting a hard battle.

For a man to conquer himself is the first and noblest of all victories.

Courage is knowing what not to fear.

Excess of liberty, whether it lies in state or individuals, seems only to pass into excess of slavery.

Wise men talk because they have something to say; fools, because they have to say something.

Voltaire (1694 – 1778 AD) – French Enlightenment philosopher

Judge a person by their questions, rather than their answers.

He who thinks himself wise, O heavens! is a great fool.

Prejudice is opinion without judgement.

The multitude of books is making us ignorant.

I disapprove of what you say, but I will defend to the death your right to say it.

Common sense is not so common.

This agglomeration which was called and which still calls itself the Holy Roman Empire was neither holy, nor Roman, nor an empire.

Oscar Wilde (1854 – 1900 AD) – Irish poet and playright

Man can believe the impossible, but can never believe the improbable.

What is a cynic? A man who knows the price of everything and the value of nothing.

A man who does not think for himself does not think at all.

Nothing that is worth knowing can be taught.

Albert Einstein (1879 – 1955 AD) – German theoretical physicist

There are two ways to live: you can live as if nothing is a miracle; you can live as if everything is a miracle.

Science without religion is lame, religion without science is blind.

Logic will get you from A to B. Imagination will take you everywhere.

The important thing is not to stop questioning. Curiosity has its own reason for existing.

Everything should be as simple as it is, but not simpler.

Once we accept our limits, we go beyond them.

Small is the number of people who see with their eyes and think with their minds.

Try not to become a man of success but rather to become a man of value.

Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.

Common sense is the collection of prejudices acquired by age eighteen.

Mark Twain (1835 – 1910 AD) – American author

All generalizations are false, including this one.

Do not put off till tomorrow what can be put off till day-after-tomorrow just as well.

Good friends, good books and a sleepy conscience: this is the ideal life.

The man who does not read good books has no advantage over the man who cannot read them.

Never let formal education get in the way of your learning.

It is better to keep your mouth shut and appear stupid than to open it and remove all doubt.

Karl Weierstrass (1815 – 1897) – German mathematician

When I wrote this, only God and I understood what I was doing. Now, God only knows.

Oliver Heaviside (1850 – 1925) – English mathematician and physicist

Why should I refuse a good dinner simply because I do not understand the digestive processes involved?

As I said, I hope to update this list over time as I pick up more – do however please feel free to suggest any particularly worthwhile along the same themes expressed here.

This post essentially follows on from the Notes on Kant post by David, which having prompted rather a lot of comments and one or two conversations, led to a few interesting conclusions on the subject of morality. Here, however, I mainly intend to express my own views and conclusions on the nature of morals (though David seems to be of much accordance, at least in his end point). I’ll leave it to anyone who wishes to comment to counter my points.

To start, I should mention that my own philosophy on morals seems to be largely in accordance with rule utilitarianism (or at a minor variant of it). What follows is pretty much the set of ideas that guided me to the eventual conclusion regarding optimality. Specifically, I argue that an action is moral if it is beneficial to either oneself or society (or both) and not detrimental to the well-being and continuation of the society as a whole. The intentions of the individual performing the action must also satisfy these conditions if the action is to be deemed moral, else the action must be morally neutral at best. Importantly, this specification implies that choices made with self-interest in mind can be moral so long as the communal benefit is non-negative. This becomes quite obvious given the assumption that the well-being of individuals in general contributes to the well-being of society (at least in an indirect way). Note that the arbiter in all these cases must be hypothetical as well as purely objective (nature itself, if you will), meaning that even though a certain action may be considered immoral as a consensus of society, it may nevertheless be neutral or even moral in actuality. Saying this, in a well-functioning and successful society, there would seem to be a general requirement that the judgement of moral worth of actions is reasonably accurate in a high proportion of situations.

Considering these points and their commonalities is primarily what led me to believe that when you boil everything down, morals are nothing but an approximation to optimality of society. Now as soon one mentions optimality, the question of a measure automatically follows. Of course, most people probably have some vague notion of what an “optimal society” is, but since the aim here is to be as formal and specific as possible, I really need to define a cost function, at least in loose terms. At this point, I would imagine that the opinions of most people would tend to diverge rapidly. Some would reason that the cost function is purely dependant on well-being/happiness/pleasure (whether more for the individual or society separates the hedonists from the utilitarians), while others would state the straightforward biological (yet to many cold and unpleasing) view that optimality is but a measure of the size of the population and thus the continued ability to self-replicate. Finally, the more religious among us might contend that optimality is simply the perfectly obedient following of teachings passed down to us by God. In essence, this cost function is nothing other than the “meaning of life” (in the widest possible sense) – something that may never be defined, and certainly not a discussion I’m going to include in this post! Whatever view people wish to take, I believe that the basic statement that “morals are an approximation for optimality” holds well in all cases. Like the nature of this optimality, the mechanism by which the concept of morality has been instilled in us (evolution, creationism, Spinoza’s God, or whatever) is also open to debate, but nonetheless is not able invalidate this theory in itself, at least in the way I see it. Yet all these concepts of optimality surely do have similarities. Another important feature of the cost function is that parameters should be not only the current state of humanity and the world, but also the states at points in the future (perhaps stretching infinitely far ahead in time). In the end, I think I can say that I do personally feel reasonably content with this definition of morals (albeit most likely an incomplete one). In my mind one cannot proceed any further in a formal definition without invoking reasons akin to the “meaning of life” such as those just mentioned – all very contentious or speculative and therefore not terribly helpful as bases for any fundamental theories, in my view.

It is without doubt important to stress that morals have their own imperfections and limitations, like analytical reasoning, and depend on the individuals (or society thereof) in which they have formed. Yet depending on how you look at it, morals have  evolved or been designed specifically for the purpose of optimal society. Although morality may be less adaptable than intelligence (at least over the timespans ranging from days to maybe lifetimes), it assuredly has a more “tailored” purpose, and therefore has its place alongside, and arguably ahead of, analytical reasoning.

If I were to now summarise how I believe optimal behaviour should be guided, I would say that it’s necessary to be somewhat careful not to propose something too uncompromisable. In reality, it’s almost always the case that reasons are more intricate and subtle than immediately apparent. In stating an emphasis on paying due attention to intrinsic morals (loosely, which can be recognised as principles and codes that typically “feel right” and are “seen to be right” by consensus of society), and contrarily wariness in ignoring these morals in favour of some sort of pure reasoning. “Reasoning”, after all, when performed by humans, cannot help but be intruded by egotistical motives, among other notable imperfections. Do we not after all have a fear of so-called purely “rational” or “logical” machines not hesitating to perform tasks that are undeniably immoral in the eyes of man (if not only founded in science fiction and our imaginations)?

As a quick final note, I ought to mention that nowhere in my musings have I required morals to be static in nature. Equally, there would not seem to be any issue with them being unchangeable. At this point I’m further tempted to divide morals into two categories: intrinsic and social. Again, this is a matter on which I’m only going to lightly touch. The latter is the one of particular interest in that it could imply a varying cost function for optimality. It also suggests a mutual feedback cycle between the will of society and contemporary moral values (with analytical reasoning somewhere in the process, potentially acting both positively or negatively).

Right, so this post has wound on long enough by now, and is only getting increasingly vague and leaving more loose ends. Still, I hope that I’ve at least partially conveyed my theories and general impressions on the subject. I’m not sure how everything appears to others who haven’t followed the continuous discussion on the wider topic (largely originating with David’s post). I’d certainly be keen to hear what anyone else thinks on the subject and the ideas presented here. I would not at all be surprised to receive opinions that this relation of morals to optimality seems distasteful or even incomplete to many people. Indeed, I am not sure that I am wholly satisfied with this explanation as the “root” of morals. (How can I, having already cynically accepted the fallibility of human rationalisation?) Maybe it is as a student of physics that I realise our theories of the nature, physical or human, are always but approximations to a more profound reality.

This post is the result of a discussion between David (a friend of mine and engineering student) and myself. It ought to be read alongside his post on the same topic, which takes a quite different perspective on many of the matters. As he states, it is quite unlikely that (even between us) we will cover all the points made by the both of us, though I will certainly make an effort to do so.

I now forget how the debate arose, but its main theme ended up as follows: Are modern computers conscious/self-conscious to any degree and what is it exactly that makes them so, or indeed differentiates them from humans in this respect? It began as a rather scientific/technological discussion but turned out to involve a good deal of metaphysics (in which neither of us can claim to be well versed, though we certainly learnt much in the process).

To start I should note that where David refers to intelligence, I more often that not mean consciousness. In my opinion, intelligence of certain kinds is something already possessed by computers to varying degrees; their ability to perform calculations and analysis of some forms of data far surpasses that of humans whereas they are not nearly so adept at holistic analysis or creative thinking for example.

Before I get to the core of the discussion, it is important to firstly (try to) define a few terms. There is no general consensus on the exact meaning of consciousness but the introduction of the Wikipedia article offers a good idea of what I refer to when using the word. Self-consciousness (or self-awareness more accurately) is a much easier to define concept, if still not a concrete one: if anything can actively identify itself in a mirror (whether it be a physical or conceptual one), then it can be deemed self-aware. Several animals other than humans have been labelled as such on the basis of this test, such as chimpanzees, dolphins, and elephants. Now the question is whether computers can currently demonstrate this. An example given by David was a computer recognising its existence within a network by pinging itself via a remote device (if I remember correctly). His argument is that if the computer receives a successful reply, then it can clearly determine that it exists (the remote device would act as the mirror in this example) and is therefore self-aware. I dispute this argument primarily by asking whether the computer actively/explicitly realises that it exists. Firstly consider that it would be easy enough to fool the computer into believing that it does not exist on the network by returning a fake reply (or none at all). Also, in effect the programmer is telling the computer that it exists if it receives a successful reply, which fails to meet my criteria for self-awareness. In a way, the programmer is imparting his own realisation of the computer’s existence into it. Humans on the other hand can actively come to the conclusion that they exist (even without sensory information). They need not be told that they exist, but rather only to think about it. The famous statement by Rene Descartes, “Cogito, ergo sum” (“I think, therefore I am”) can be seen as proof of this. The same argument applies to the mirror test for self awareness in animals, although the difference there is that observers have to make the decision (albeit with very high probability) that the animal has shown signs of self-awareness. David refuted this explanation, suggesting that a person raised without any contact with others would not have the ability to come to the conclusion of their own existence. However the situation in fact then becomes similar to that of other intelligent self-aware animals which have not been trained in any meaningful way. I do concede that it is theoretically impossible to be sure of self-awareness in anything other than yourself on the basis of “Cogito, ergo sum”, though the fact that humans and animals have not been explicitly/consciously programmed gives a good indication that self-awareness arose independently.

This whole argument leads onto the (wholly philosophical and non-empirical) issue of from where consciousness is derived. It is believed (or has at least been proposed) by some that all biological organisms have a certain level of consciousness (though not necessarily self-awareness). For example, the cells that compose an organism could be seen to have a certain level of consciousness (by the definition given earlier), while the whole organism could be seen to have a greater one. Similarly, the Gaia hypothesis (especially that presented by Isaac Asimov in his Foundation series) proposes that the Earth has a supreme level of consciousness, which is greater than the sum of its component consciousnesses (including humans and other organisms). It goes as far as to suggest inanimate matter has a minute amount of consciousness, though I suspect this was a unique idea for the sake of fiction. This theory can be summarised by the statement “the whole is greater that the sum of its parts”, which comes up in various places but I feel is perhaps most appropriate here. As I warned, the topic has now diverged completely from empirical science, since no-one currently knows a way to measure consciousness quantitatively (or even define it in a concrete way). Continuing nonetheless; a computer may be said to derive its consciousness from either its programmers, internally, or from a combination of both. Humans may be considered to derive their consciousness internally (the neural networks of the brain are created from inanimate matter via biological growth and are developed with learning). Whether an entity derives its consciousness from a few other highly conscious entities (such as the programmers) or a multitude of entities with very low consciousnesses (such as cells and micro-organisms) could perhaps define what is to be considered independently conscious (though there is clearly a grayscale here). We did not discuss this particular area too far as it was becoming horrifically abstract, though I think we both agreed that it was an interesting idea.

The final point made by David in his post is regarding the increase in the complexity (again another loosely defined concept) of an entity (system) in order to completely understand itself. His point is that the complexity will eventually converge to a finite value as a system grows indefinitely in order to understand itself. (See his post for a proper explanation.) A solely hypothetical question, but nonetheless intriguing. This view seems intuitively wrong to me, but specifically it would seem necessary that the system would have to re-comprehend its entire self as it increases its complexity (and therefore level of consciousness), since fully understanding the original system and the additional parts of it would not imply an understanding of the overall system (if you subscribe to the view that “the whole is greater that the sum of its parts”).

I don’t think I can comment very well on my general philosophical views as David has (though take what has been offered already). Looking briefly at some of the terminology however, I seem to largely subscribe to the philosophies of Holism and Emergentism, which appear to contradict with his views, as I might expect. (Why else would I be writing a post on the same topics?) Still, I subscribe very much to empiricism, with the small caveat that our knowledge of metaphysics is too small and basic to yet apply it to that too. (As a student of physics, I would be worried if I didn’t!)

Now that I’ve finally made this post (after much goading to fulfill my promise), and David has likewise made his own, I’m hoping that this debate is ended for the time being, but that these posts stand well as records of our philosophical views, to which we may return at some time.