‘Partial differentiation’ is an important mathematical technique which, although I have used it for decades, always confused me until a few years ago. When I finally had the blinding insight that de-confused me, I vowed to share that insight so that others could be spared the same trouble (or was it just me that was confused?). It took a while to get around to it, but here it is:
My daughter Eleanor make a drawing for it, of a maths monster (or partial differentiation monster, to be specific) terrorising a hapless student. The picture only displays in a small frame at the linked site, so I’m reproducing it in all its glory here.
Bondi Junction, October 2016
Here’s a piece I wrote explaining the mathematics behind the peculiar phenomenon of acoustic ‘beats’.
It’s a bit maths-y. But for those that don’t love maths quite as much as I do, it also has some interesting graphics and a few rather strange sound clips.
Bondi Junction, August 2016
Books are too long. People talk for too long. Academic papers are too long. Almost everything is too long.
Why? Partly, because to be concise is very difficult. Urban legend has it that Blaise Pascal once wrote at the end of a letter to a friend: ‘I’m sorry this letter is so long. I didn’t have time to write a short one’.
I struggle with conciseness. Part of the problem is that, when I am trying to explain something, I worry about whether what I have said is clear enough, so I keep on saying it over, in a slightly different way each time, in the vague hope that one of the attempts will make the connection.
I think a better strategy might be to make one brief attempt at an explanation and then wait for a response. If more is needed, I imagine my interlocutor will tell me. If they do, the particular nature of their response will better enable me to tailor my next statement to fill in the information that was missing in my first.
But that requires discipline, and nerves of steel. It is like being silent in an interview after giving a short reply to a question – forcing the interviewer (or interrogator) to make the next move. Few people can carry that off, and I suspect I am not one of them.
Academic papers can be particularly irritating, droning on about all the references and who has written what, so that by the time one gets to the bit about what the authors have done that’s actually new, one is exhausted and wants to retire for a tea break. It’s not clear to me whether this is a stylistic practice, imposed by the producers and reviewers of journals, or whether it reflects insecurity on the part of the authors, who may feel that they need to mention some minimum number of other papers in order to be taken seriously.
Arthur Schopenhauer railed against this sort of writing in a series of essays collected under the title ‘The Art of Literature’. He opens with an unrestrained broadside ‘There are, first of all, two kinds of authors: those who write for the subject’s sake, and those who write for writing’s sake.‘ Schopenhauer loved the first (and of course considered himself to be one of them) and loathed the second.
If someone really has something important to say, it usually doesn’t take very long. When Neville Chamberlain announced the grim news to the British people in 1939 that Britain had declared war on Germany, the message had been delivered by the end of the 67th word. I did a test reading just now and it took about 26 seconds, including pauses for effect.
Einstein’s legendary 1905 paper that presented his special theory of relativity to the world, ending decades of confusion amongst physicists, is only 24 pages, and the key part that resolves the paradoxes by which physics was previously beset is complete by the end of page 12! John Bell’s paper that turned the world of Quantum Mechanics upside down in 1964 is only six pages. Bell cited only five references. Einstein cited none.
In general communication, most people use too many words. I do too, but I am trying to correct that. I feel that, where possible, I would like to conduct a post-mortem on every sentence I utter and work out whether that sentence has added any new information. If it hasn’t, then it was probably a waste of everybody’s time.
Politicians exploit this deliberately. They are trained to, when asked a difficult question by a journalist, give a long-winded, emphatic speech about something only tangentially related, thereby avoiding the issue and (they hope) making the journalist despair of persisting with the question because of the pressure of time. Even better, if the politician sounds confident in their ‘answer’, the less analytic watchers will form the impression that the politician is competent and frank. The more analytic types just shrug their shoulders in disgust and turn the telly off.
A sentence can be very long and yet not reveal what information it contains until late in the sentence. Sometimes there is a key word that makes it all fall into place, The words before that one stack up like the numbers in a long calculation on a Reverse Polish calculator, impotent while they wait for release. Then the key word comes and it all falls into place. It attains a meaning. The wait for that word can sometimes be prolonged, like in this:
Though they all came from different social strata, sub-cultures and occupations, crammed together against their will in the prison cell from which they wondered if there would ever be any release, though none of them had known each other – or even known of each other – in their previous lives, though they squabbled and quarrelled over the tiniest of things, the one thing that bound them together despite the rivalries and petty jealousies, the perceived slights and reconciliations, the development, disintegration and reformation of cliques, was a single shared emotion, an emotion so powerful that they could feel it oozing out of one anothers’ pores, smell it on their breath and discern it in the tones of voice – the emotion of fear.
In some cases, the key word never comes. Perhaps the writer or speaker confuses themselves by their excessive verbiage and ends the sentence with an admission of defeat.
Books are too long as well! Novels are generally OK, as it takes time to get to know and care about the characters. But I have a strong sense that non-fiction books are often padded to reach whatever is considered a minimum page count for a book – usually at least 200. There isn’t really a strong market for writings that are halfway between essay and book length. In many cases a book really only has one idea, which could make a decent essay, but doesn’t justify a book. But essays don’t get to be put on a prominent shelf that catches your eye as you enter the bookshop, nor do they get listed on the New York Times best sellers’ list.
Nassim Taleb’s famous book ‘The Black Swan’ is like that. It really only contains one idea, which is that investors, bankers and other financiers have for decades been making crucial financial decisions based on theories in which they assume that the future will be like the past, and that all occurrences of randomness must follow the Normal Distribution (the nice friendly old ‘Bell Curve’). Decisions based on that erroneous, oversimplified assumption have repeatedly led to disasters, because events tend to be more extreme than is predicted by the Bell Curve. Taleb’s is a good insight, and definitely worth saying, but probably not worth stringing out to book length.
And then, if the book sells well, they write it again, ever so slightly differently, and pretend it’s a new book, with new ideas. Taleb did that. Self-help authors do it all the time – which raises the question ‘If your first book about how to live a better life was so incomplete that it needs to be supplemented by a second, why did I waste my time reading it?‘ I suspect Richard Dawkins may do it too. As far as I can tell he has written at least four popular explanations of evolution. I read The Blind Watchmaker and thought it was great (but too long, of course!). But I didn’t read The Selfish Gene, The Ancestors’ Tale or The Greatest Show on Earth because I couldn’t see any indicators that they would contain much substance that hadn’t already been covered in the one I had read. I imagine there is some new material in each of them, but I would guess it’s more likely to be a dozen pages’ worth rather than 200+.
Fiction authors and other creative artists do this too. Stravinsky acidly observed that Vivaldi wrote the same marvellous concerto five hundred times. Bach shamelessly reused his work (goodness knows he was paid little enough for it!) and Enid Blyton invented maybe a dozen adventure and fantasy stories, which she recycled into what seems like hundreds of similar tales (surely I’m not the only one that’s noticed the remarkable similarity between Dame Slap’s School for Bad Pixies and Mr Grim’s School for Mischievous Brownies?). And let’s not even mention Mills and Boon. But somehow I don’t mind that so much. We humans are story-telling animals, and telling the same story repeatedly, changing it just a little every time, is what we have always done. I find myself able to smile indulgently on the prolixity of Enid and Antonio and Mills (?), but alas not on that of Nassim or Richard, or Deepak Chopra.
I think I’ve ranted for long enough now about how We All (including me) need to work on being more concise with our communication. It’s time to relent a little.
Not all language is just about conveying information, so the efficiency with which the information is conveyed is not always the best test. In comforting a frightened child, information communication is not the purpose of our speech. I will restrain myself from objecting that the second half of the soothing phrase ‘There, there‘ is informationally redundant. In fact, I think I could even stretch to approving of its repetition, if its first invocation was insufficient to assuage the poor mite’s distress.
Declarations of love, expressions of support, telling jokes, goodbyes, hellos and well-wishes are all ‘speech acts’ that have important non-informational components. It seems appropriate to apply different expectations to those speech acts from those we apply to informational speech. Even there, there are limits though. Many’s the operatic love aria I’ve sat through where after a while I just feel like screaming ‘OK, you love him, we get it, can we move on with the plot now please?’ And waiting for Mimi to die in La Boheme (of consumption, what else?) in between faint protestations of her love for Rodolfo, can become a little trying on one’s patience after the first ten minutes of the death scene.
But communication of information is the purpose of much of the language we use, especially in our work lives. It is a pity that so much of it is ill-considered.
Hmmm. 1,742 words. I wonder if I could turn this into a book.
Bondi Junction, November 2015
“What is Truth” asked Pilate?
I’ve always been rather fond of that riposte in the New Testament. Good old Pontius sure put those scribes, priests and Pharisees – whatever they were – in their place. Is there more to it than just a great put-down though?
It sounds deep as well, although there’s always the risk that it may be faux deepness, or ‘deepity’ as Daniel Dennett would call it.
Now I’m not going to go all post-modern here, and suggest that everybody has their own version of truth. Nor am I going to suggest that Truth isn’t a useful concept. To ask Freda whether she thinks Bill is telling the truth is a meaningful and useful question. It is actually a speculation about the state of Bill’s mind. Is he telling us what his memory tells him happened, or not? No, what I’m wondering about is what is sometimes called Absolute Truth, the sort of thing exemplified by Bertrand Russell when he says ‘Edinburgh is North of London, whether anybody knows it or not.”
Sometimes Truth is held up as a sort of Holy Grail, the quest for which can fill one’s life with meaning. The Search For Truth is the sort of noble thing to which holy men and scientists alike are said to be dedicated. Indeed, it’s not that long ago that it was a key component of a little motto I made up for myself in an idle moment – “Truth, Beauty and Compassion” – a more inclusive version of St Paul’s triumvirate “Faith, Hope and Love”. I vacillated for a while over whether I should add “Reason” to my motto. I mostly omitted it, because things just seem so much neater in threes, as well as because there seemed to be too much overlap with Truth. Sometimes I included it though, because there seem to be things covered by Reason that are not covered by Truth. For example, one can use Reason to devise a strategy to win the heart of one’s true love, but this has little if anything to do with Truth. Recently though, I have sometimes felt inclined to drop Truth from the motto in favour of Reason, and I blame Kurt Godel for this, as I will explain in the next couple of paragraphs.
Godel and Truth
The logician Kurt Godel is most famous for his First Incompleteness Theorem, which destroyed the most ambitious mathematical project of the early 20th century – the Hilbert Program, which was an attempt to find a way to formally prove everything that was believed to be true about arithmetic and basic mathematics. What Godel’s theorem showed was that, in any logical system that we use to write proofs, there will exist some logical statements that can neither be proved nor disproved, yet are perfectly meaningful. He did this by creating an ingenious variation of the famous ‘Liar Paradox’, which says ‘this statement is false’.
Now, in their attempt to make Godel’s discovery more interesting to a lay audience, some texts and sources claim that the theorem says that there are statements that are ‘true but not provable’. The trouble is that the ‘true’ part of this statement doesn’t have any clear meaning. The closest we can get is to observe that Godel’s theorem shows that there are statements that are unprovable in the chosen logical system, but for which we can prove, using a different logical system, something that is very similar to the original statement. So in that case the word ‘true’ doesn’t relate to some fundamental cosmic reality, but just to what can be proved in an alternative logical system. The word ‘true’ is superfluous as well as ambiguous here, as we can express everything Godel discovered using variants of the word ‘proved’ instead of true.
The realisation that ‘True’ is a useless word in this critical context made me wonder if there is any context in which it expresses something important, that cannot be more accurately described by other words. It is that question that I wish to explore here.
Science and Truth
Let’s start with science. Is that a search for truth? The task of science is to construct theories that relate past experiences (‘observations’) to one another in an ordered way and enable us to make predictions about future experiences, such as the appearance of comets or the effect of mixing hydrochloric acid with sodium hydroxide.
Are scientific theories ever true? Not many scientists or philosophers of science would claim that in any absolute sense. Any scientific theory, however useful, is eligible to be replaced by a more sophisticated one that better explains our observations. The first simple atomic theories were replaced by the Rutherford model in which a nucleus of protons and neutrons was surrounded by electrons whizzing around the outside. That was in turn replaced by Quantum Mechanics, and then we had quarks and other subatomic particles coming into the theory in the 1960s. Now physicists speculate on whether the subatomic particles in turn are just symptoms of some deeper underlying phenomenon such as vibrating strings in eleven dimensional space. It seems natural to suppose that there is some underlying final theory that tells everything, and that the theory is ‘True’ regardless of whether we eventually discover it. But that is by no means the only logical possibility. Here are two others:
- There is an unending descending chain of theories, each more complicated than the one above it and including all its ancestors as special cases that are good approximations under restricted circumstances.
- There are multiple theories that can explain the full range of possible experiences, all as valid as one another. The competing corpuscular and wave theories of light are crude examples of this. In the end, they both proved inadequate to explain all the experimental results obtained, but for a while they were both successful in explaining different aspects of the behaviour of light. There is no apparent reason why there could not be two or more different theories, each of which can by itself explain everything we experience.
Sometimes the Common Sense view of the world is that of course there is an underlying Truth – that quarks and electrons really exist and we just need to find out more about them. While I am passionately in favour of finding out more about quarks, I regard that as a desirable refinement of our theories, not a discovery of Truth. If we are to think in terms of common sense, I cannot help but observe that common sense tells me the table in front of me is a hard, solid, opaque object, not the strange collection of pinpricks and force-fields in empty space that quantum mechanics suggests. That is not to say I don’t ‘believe’ quantum mechanics. Quite the contrary. Rather it is to say that I don’t want to give common sense any privileged role in such a deliberation. And also, that the notion of Truth is compromised. Which is more true, my perception of the table as hard and solid, or the idea of it as an empty space with the odd tiny quark and electromagnetic force field? The practical response is that the question is meaningless. Both representations of the table have their uses, and are complementary. Table as solid object is useful if I want somewhere to put my kettle. Table as quarks is useful if I want to understand why it doesn’t collapse under the weight of the kettle (or how heavy a kettle I can put on it).
Perhaps a belief in Truth equates, in this context at least, to a Realist view of the world – that the world really is made of a specific sort of thing or things, and we just need to find out what it is. It is a belief in Reality, with a capital R.
But hold on, any moderately sane person believes in Reality don’t they? Well, it depends what one means by reality. I am sure I have been guilty of saying ‘so and so lives in a dream world – they have only a passing acquaintance with reality’. My defence is that I do not mean by this that so-and-so doesn’t believe in quarks. Rather, I mean that they do not reason about their experiences in such a way as to give a strong likelihood of their expectations of future experiences being fulfilled. Perhaps so-and-so has gone sun-bathing on the beach in the middle of a summer’s day, under the illusion that a coating of sunscreen will be enough to protect her for two hours in full sun. The ‘reality’ from which she is disconnected is that that evening she will have the experience of red, horribly painful skin, and two days after that she will have the experience of the skin peeling off in sheets. So-and-so’s delusion has nothing to do with her metaphysical opinion of quarks. It is about her inability to use reason to control the nature of her future experiences.
So it seems that when I use the word ‘reality’ in everyday intercourse, I am referring to what experiences we can expect in the future, and what we might be able to do to control those experiences.
Note that Reason, or Rationality, has stepped in here to fill the vacuum left by our jettisoning the notion of Truth.
Truth in other contexts
A belief in Truth is not limited to just quarks. At various times, people have proposed the existence of other forms of truth in the form of objects that exist independently of what people know. Moral rules, aesthetic truths and mathematics are key examples of this. A Moral Realist will claim that moral rules exist, and the purpose of moral education and ethical inquiry is to discover them. A mathematical realist will say the same thing about Pythagoras’s Theorem, or the number six. An aesthetic realist may be a rarer beast, but if I could find one, I would expect them to assert that Mozart’s Requiem is beautiful and Milli Vanilli is not.
Yet there is no more need to believe in such Truths than there is with quarks. I can develop a system of moral rules for my own use, or discover them in my own mind by reflection on my feelings. If I prefer something less subjective, I can define a moral truth to be any feeling about how to treat others that is shared by a large proportion of the human race.
With mathematics I can, if I wish, regard any mathematical concept as an invented idea that helps me arrange things in my mind and reason about what might happen next. I can take the view that everybody invents the mathematics that they use for themselves. So every toddler invents the number six, rather than discovering a pre-existing abstract object. Or perhaps they find a pre-existing pattern for the number six somewhere in their brain, placed there by evolution because of its usefulness to survival.
The more I reflect on this, the less value I see in maintaining a concept of Absolute Truth in the way one thinks about the world.
Consider the binary expansion of Pi. Let F(n) be the statement that, for any m greater than or equal to n, there are more 1s than 0s in the first m bits of that binary expansion. Let S be the statement that there is some number n for which F(n) is true. There is no way of proving or disproving S by trial and error, because we cannot try out what happens with all the infinite number of combinations of values of m and n. Perhaps there’s a clever way of proving it or disproving it that doesn’t require trying an infinite number of combinations, just as we can prove that the sum of two odd numbers is always even, without trying it on all possible pairs of odd numbers. But nobody knows of one, and it seems entirely plausible that there is no such proof.
Yet most people would say that S is either true or false, but we just can’t know which. This seems to be a problem, because there are no words to explain what we mean by true or false here. Let us explore some more.
The Liar Paradox
The Liar Paradox is a statement that generates a contradiction, however you interpret it. One of its more sophisticated versions is ‘This sentence is not true’. If we suppose it’s true then what it says must be true, so it is not true, which contradicts our supposition that it’s true. If we suppose it’s false then what it says cannot be true, so it can’t be true that the sentence is not true, so the sentence must be true, which contradicts our supposition that it’s false. The third possibility is that the sentence is meaningless. But if it’s meaningless then it’s not true, which is exactly what the statement says is the case, so the statement is true, so it’s not meaningless, which contradicts our supposition that it is meaningless.
There are a number of ways to resolve this paradox. Most of them involve questioning what we mean by truth in this context. One possible meaning is provable – we say that something is true if and only if it can be proved. This has shades of Godel’s Theorem. If we take this approach then the paradox disappears because ‘This statement is not provable’ is not paradoxical if we require our proofs to be strictly logical (although one has to have at least a partial understanding of Godel’s theorem to see why that is so).
Another possible meaning is that a sentence is true if it accurately reflects our perception of reality (or just our perception, if we want to avoid the equally troublesome word ‘reality’), false if the opposite of the statement reflects that perception, and meaningless otherwise. This is a popular interpretation of ‘Truth’, which is called the ‘Correspondence Theory of Truth’. With that meaning, it is ‘true’ to say a soccer ball is round because, when we look at it, it looks round. With this meaning, we can resolve the paradox by observing that the sentence is meaningless because it doesn’t say anything about our perception of reality. All it does is talk about itself.
In both these resolutions, we have only been able to make sense out of the sentence by getting rid of the vague, undefined word ‘true’ and replacing it with a clear, pragmatic definition.
This realisation strengthens my feeling that the word True is a troublemaker, that sows confusion wherever it is used. Perhaps we really should dispense with it, except in causal, imprecise, everyday slang. But if so, what should we replace it with?
What can we replace Truth with?
The words ‘true’ and ‘truth’ are so much a part of everyday language that, if we were to discard them, replacement would be required on a large scale. For someone that subscribes to the Correspondence Theory, mentioned above, that theory provides a satisfactory way of defining the terms, and hence justifying their retention. The price that must be paid for that though, is believing that there is such a thing as Ultimate Reality – a unique description of the way things are, to which any other descriptions are only approximations. Furthermore, that description must exist regardless of whether anybody knows it.
Most people seem to hold that belief, and for them the dilemma ends there. But, for the reasons given above, I am skeptical, and certainly see no justification for such a belief in terms of other, more fundamental and intuitive concepts. All we can say with confidence is that we have experiences, and that there are models available – called scientific theories – to connect these experiences together in a patterned way and make predictions about what we may experience in future.
For someone that suffers this same ‘skeptical disability’ as me, the Correspondence Theory is not an option. So what, if anything, can we use instead?
My proposal is to replace it by ‘consistency with experience’. We will call a statement ‘true’ if it is consistent with both actual and potential future experience. By ‘consistent’ we mean there is no possible rational deduction from actual or potential experience that contradicts any part of the statement. So if June tells us that she has not eaten the last biscuit, that is not true if, upon inspecting the biscuit barrel, we would see that there are no biscuits left in there and, if we used an endoscope to inspect her stomach contents, we would find traces of that type of biscuit therein. It would also fail to be true if June’s statement was inconsistent with her own experience – that is, if she has had the experience of taking and eating the last biscuit. It would even fail to be true if June were a sleepwalker who has been known in the past to sleepwalk to the kitchen, take a snack and return to her bed without waking, and she woke this morning with a rash of a kind that she has only ever had before after eating that type of biscuit. In this latter case, June has not experienced eating the biscuit, but she can rationally deduce from her experience of the rash that she has probably eaten it in her sleep.
So that’s my suggested replacement for Truth – consistency with experience. And since the key ingredient for assessing this consistency is Reason, I will adopt that as the third leg of my little motto: Compassion, Beauty and Reason!
Note that we haven’t really got rid of the word Truth. All we’ve done is redefine it. Or rather, defined it because, unless one subscribes to the Correspondence Theory, it was never properly defined in the first place.
This definition has some interesting implications.
Remember the above example of the statement S about the number pi? With this definition we have to conclude that, unless a proof one way or the other exists, S is neither true nor false, because there is no potential experience we could have that would confirm or deny it. In the mathematical context, true means the same thing as ‘provable’.
Further, even if there is a proof or disproof of S that we just haven’t found yet, Godel’s Theorem tells us there will be other statements that can be neither proven nor disproven. So we are now committed to saying that those things are neither true nor false.
On the other hand, we are empowered to say very clearly that provable statements like 2+2=4 are true, because we can easily go through the experience of proving them. Strictly, what we need to say is that ‘In Peano Arithmetic, 2+2=4’, because one could easily invent another system of arithmetic in which 2+2 equalled something else. So the statement is actually about the consistency of what we are saying with the rules of the system – in this case the axioms of Peano Arithmetic.
Here’s another one: ‘The universe will expand forever’. The current state of cosmological knowledge is that we do not know whether that will happen or whether the expansion will eventually slow and then reverse, ending in a Big Crunch. The Big Crunch will happen if the Cosmological Constant is positive, but not if it’s zero or negative. If the constant is positive or negative, we may one day be able to demonstrate that. But if it’s zero, we’ll never be able to demonstrate that, because there’s always the chance thet it’s a positive or negative value too small for us to detect. So if the cosmological constant is zero then the universe will expand forever but we can never know that that’s what it’s going to do. Under our new interpretation of ‘true’, we would have to say that this cosmological statement may be neither true nor false, as there can be no experience that can confirm or deny it. In fact so is the statement ‘The Cosmological Constant is zero’.
If this seems disconcerting, we can comfort ourselves with the idea that, for all the common , everyday uses of the word True, the definition gives us exactly the interpretation that we want:
- “are you telling the Truth” means ‘is what you are telling me now consistent with your own experience of what you are talking about?’
- “I feel cold” is true if the person saying it is currently having the experience of feeling cold.
So we can preserve a role for the word ‘True’ in our everyday language. What I think we can’t do without having to make unfounded assumptions about the existence and nature of Ultimate Reality, is elevate the concept to some universal principle that guides our understanding of the universe. “Reason” does a much better, and more practical, job.
Andrew Kirk, Bondi Junction, February 2012
It occurred to me a few days after finishing the above essay that, if somebody nevertheless wishes to retain a notion of Absolute Truth in their worldview, perhaps for aesthetic reasons, or just because it fits with long-accustomed habit, one way to do so is to incorporate an omniscient conscious entity in the worldview. That way, Absolute Truth can simply be defined as what it is that this entity knows. This is entirely consistent with the above suggestion of defining truth in terms of experience because, as the entity is omniscient, it experiences every object or event.
This entity may sound a bit like what some people call God. But I should note that other attributes ascribed to God by the most popular Middle-Eastern religions of Christianity and Islam, such as omnipotence and being a creator, law maker and law enforcer, are not necessary in order to take this route to believing in Absolute Truth. Nor does this approach necessitate that there be only one omniscient entity.
If the entity is infinite then it could even know the truth status of the mathematical proposition S above. Knowing the truth status of S requires knowing an infinite amount of information but perhaps that would not be a problem for an infinite entity.
This approach has echoes of the way that George Berkeley completed his Idealist theory of existence. Berkeley’s theory says that only ideas exist, not matter, and he addressed the problem of whether objects exist when nobody is looking at or thinking of them by saying that they still exist as ideas in God’s mind.
Personally, I like the idea of a universe without Absolute Truth and have no need or wish to hypothesise such an entity. But it may be of comfort to those who do like the notion of Absolute Truth to know that there is a rational basis on which they can do so.