Wednesday, 30 April 2014

Review article: The Outer Limits of Reason (2014)

The Outer Limits of Reason

The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us, by Noson S. Yanofsky, MIT Press
Hardcover £20.95 ISBN: 9780262019354
424pp August 2013




Reason, wrote John Locke, 'must be our best judge and guide in every thing.'

In so speaking, Locke epitomised a bright confidence in reason, widespread, which was born of the Enlightenment. More recently, however, such confidence has been lost as reason has revealed its own limits, one by one – not merely practical, but theoretical – limits which frequently would seem impassable. These are the limits which Noson Yanofsky addresses.

The survey is expansive, and adept. However, at heart, and despite some late and rather airy talk of humanity's special place in the universe (at least, at the level of quantum physics), the book represents the study of a phenomenon, rather than a quest for understanding. That is, one should not expect to find, on the whole, conceptual cohesion in the book. It is a deposit, a delivery, rather than a work of perception or wisdom. Nor does the book address the existential or the social limits of reason – so insistent in our time. Yanofsky's interest is – as the subtitle says – science, mathematics, and logic. With this in mind, then, here are the limits of reason that Yanofsky has found.

Linguistic perplexities

Yanofsky starts by looking at language. It would seem to be a pedantic start, with little relevance to the real world – yet it sets the tone: reason, and the language in which it is embedded, are more complicated than it seems. He writes:

There is one major difference between the world we live in and language: whereas the real world is free of contradictions, the man-made linguistic descriptions of that world can have contradictions.

He then gives some examples: 'original copies’; 'clearly confused'; 'larger half' and 'act naturally', adding that this last is his favourite. 'Even though the phrases do not really make sense, we human beings have no problem using them in common, everyday speech.' The famous philosophical example: This sentence is false,  he says is grammatically correct and might be either true or false, adding 'it is not self-referential and not equivalent to the original 'Liar' paradox'. But why is it not self-referential? 'This sentence has five words' is surely self-referential. So is 'This sentence is printed in blue'. Why not this one? It is left unclear. Instead, Yanofsky moves smoothly on, saying it would be 'nice to have a grammatically correct English sentence that is a self-referential paradox'. And lo, here it is, delivered by W. V. O. Quine:

"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.

That's an ugly piece of writing, whether grammatical or not, so let's try to rephrase it straight away – if not to be less ugly, at least to be more what might be called 'grammatical'. The sentence is claiming to consist of two parts – a statement of fact about a particular phrase, and a quotation representing the phrase. The statement of fact is this: This claim becomes false when a copy of itself is inserted, minus the first two words, in inverted commas after the first two words. The quotation is: 'becomes false when a copy of itself, minus the first two words, is inserted in inverted commas after the first two words'.

So the whole 'Quine sentence' would be:

This claim "becomes false when a copy of itself, minus the first two words, is inserted in inverted commas after the first two words" becomes false when a copy of itself, minus the first two words, is inserted in inverted commas after the first two words.

Yanofsky says Quine's sentence is grammatical, as it is properly constructed: the subject is the phrase in quote marks and the verb is 'yields'. He argues that if the sentence is true, then because it predicts that, under certain conditions (which have been carried out), it should be false, it is false. But if it false, then the sentence has met the original requirements, so it is true!

Yanofsky and Quine seem very pleased with this. But others may well think it is all nonsense. Where is the original grammatical sentence? It would be, on this basis, possible to say that any sentence with a quote at the beginning is 'grammatical'. But not all sentences are equal. For example: 'Eat more fruit', is not only good advice but meaningful and grammatical. But sentences like: " 'Socrates is mortal' is blue all over", or " 'There are no tables in paradise' ran up the tree" look to us to be nonsense.

Quine's sentence is not rough, but legitimate, English – it is meaningless English. Thus it illustrates nothing. On the other hand, Yanofsky's oxymorons or 'baby linguistic paradoxes' as he calls them, are not paradoxical at all and merely reflect the fact that words have multiple meanings. Someone can indeed give you the 'original copies' because the term means something like the first copies made, rather than later ones. Someone can be 'clearly confused', because the clearly refers to the observer's degree of certainty about the other person's state – it is clear to them that the other person is confused. Where's the paradox? Yanofsky simply asserts:

Even though the phrases do not really make sense, we human beings have no problem using them in common, everyday speech.

Yanofsky is wrong on this, but the reader might suppose it is just a little bit of fun. But what can we make of claims though like this?

Human language is not a perfect system that is free of discrepancies (in contrast to perfect systems like mathematics, science, logic and the physical universe).

Now this is also nonsense – mathematics, logic and science alike rest on contradictions – which is the point of the rest of the book! So how did this strange claim creep in… it is quite antithetical to the author's major thesis. Still the minor thesis gets quite a long run. Many pages are spent on Russell's 'barber' paradox – which actually demonstrates that mathematics rests on contradictions – but only in a bizarre attempt to dismiss it.

Mathematical puzzles

To reprise the paradox: Russell's illustration is of a village in which the barber must cut the hair of everyone who does not normally cut their own hair – and the paradox arises when the barber's own locks grow long and he realises that he cannot cut them without breaking the rule. Yanofsky says that such a village operating under such a rule 'does not exist' – thus the problem is dismissed. 'Since the real world cannot have contradictions, the village does not really exist.' But of course, the imaginary village does not exist, yet the problem it illustrates does. One might also ask about what it is to 'exist' anyway – with numbers and perfect triangles not existing in any 'real world' sense either. However, the charge at Russell's example, rather than at the mathematical issue he was raising, seems so misguided that it hardly merits a counter-thrust.

Another rotten solution appears with regard to something called the 'interesting numbers paradox'. This notes that many numbers have interesting properties: for example 5 is a prime number and 6 is a perfect number (the sum of its factors is equal to itself). What is the first 'uninteresting number'? 15? 26? Yet whatever number we might try for, we will run into a problem. The first 'uninteresting number' has at least one thing about it that makes in noteworthy – it is the first uninteresting number.

Again, Yanofsky is quick to set the linguistic rules and dismiss the paradox. He says that there is no way to define what an interesting number is – the property is subjective. 'We cannot make a paradox out of a subjective property'. At this point in the book, certain important things are certain. Numbers are all clear and exact and have precise definitions. 'The concept of the number 4 is exact had has a clear definition.' Atoms are real, objects are not. The ship of Theseus exists as atoms. 'Of course the ship exists as atoms.'

We are fortunate to live among other people who learned to give the same names to commonly occurring external stimuli. Each of us calls these similar stimuli 'the ship of Theseus'.

This is also off the mark, alas. The philosophical ship exists as a hypothetical example, no atoms are involved. The problem of whether this ship made of new wood is the ship, or that pile over there is the true ship, is entirely to be solved by abstract debates over definitions, nothing to do with external stimuli. Eventually, noting that the ship is hypothetical, Yanofsky decides the existence of the ship 'is an illusion'. Along, presumably, with all our other categories and ideas. Unicorns too, have no atoms. But neither do mathematical objects. Triangles and numbers first out of the door?

At the beginning of the book this seems not likely. But as the story unfolds, with the mysteries of infinite sets and a rather complicated theory involving the mathematical 'axiom of choice', the existence of numbers and mathematical entities in general is increasingly questioned.

Metaphysical perplexities

Yanofsky says that the problem is that people generally imagine that there are objects out there which we give names to. This is the illusion. 'What do exist are physical stimuli. Human beings classify and name these different stimuli as different objects.'

At last, we get to a much more interesting discussion of modern physics which begins by noting the divide between the two main theories, relativity and quantum mechanics. (More mysteries arrive later too with a return to the assumptions behind quantum mechanics. One of the oddest is that of 'superposition' – the ability of quantum particles to have at any particular time more than one position. Subatomic particles can be in many positions simultaneously.)

Relativity theory describes the rules governing gravity and large objects, while quantum mechanics deals with 'the other forces and small objects'. Despite this difference in scope, the two theories are in conflict, as relativity requires space and time to be continuous, while quantum theory requires them to be discrete – exactly the issue that Zeno presented in his famous thought experiments.

Here, Yanofsky presents Zeno's arguments in a refreshingly clear and perceptive way, arguing that the oft-repeated mathematical responses to the issues fail to address the more profound, underlying philosophical questions. The only way out of the arrow paradox, for example, is to suppose that time is actually made up of lots of little instants, through which the arrow jumps (as it does in a photographic sequence). That would get rid of Zeno's paradox, but at great cost elsewhere.

Modern physics and engineering are based on the fact that time is continuous. All the equations have a continuous-time variable usually denoted by t. And yet, as Zeno has shown us, the notion of continuous time is illogical.


In everyday life, an infinite number of points with zero width will not stretch very far, nor will an infinite number of moments of zero duration last very long. But all modern notions of calculus, which is the basis of modern mathematics, physics and engineering, rely on such counter-intuitive properties of infinity.

Number games

Galileo was one of the first to note the counter-intuitive character of infinite sets, and the fact that, for example, the infinite set of whole numbers is no larger than the set of all even numbers. This can be well illustrated by noting that every even number is a natural number, and for every natural number there must be one that is double – an even number. If every natural number has its 'even twin', then the set of all natural numbers and the set of all even numbers must be the same size. Yes, by everyday thinking, there should be twice as many natural numbers as even numbers. But not when you can count to infinity. Should we abandon this rather odd idea of mathematical infinity and go back to common sense? But it is the odd idea that has proved more useful in practice – for interpreting the universe and in scientific predictions.

Okay. But this is only the tip of the problem. If (counter-intuitively) it turns out that there as many even numbers as there odd and even numbers put together – how many 'real' numbers will there be? Is this set any greater? With real numbers there is no one to one correspondence to the set of natural numbers. In fact, for every natural number there is an infinite number of real numbers. Worse, this infinity is larger than the usual one! Sets like this are called uncountably infinite and are indeed 'vastly larger' than sets that are countably infinite (like natural numbers). You can 'subtract' an infinite set from an uncountably infinite set and still have an uncountable infinity of numbers. Modern mathematics creates different levels of infinity.

The last function of reason is to recognise that there is an infinity of things which are beyond it. It is but feeble if it does not see so far as to know this.

      - Blaise Pascal


Kurt Gödel, the mathematician who formally confirmed the limits of logic and mathematics, actually thought that the human mind could transcend logic, for example by understanding certain statements that mere computers cannot ever prove no matter how many calculations they perform in the process. Yanofsky rightly contrasts this with the famous claim (well, it is within philosophy anyway) of the French mathematician, Laplace, his confident prediction in the 18th century that:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.'

      - Pierre Simon Laplace


However (and most certainly since the work of Edward Lorenz, who was both a mathematician and a meteorologist, in the 1960s) it has been clear that many things in the world around us are not predictable at all. Not in practice, and not in theory either. These are chaotic systems, where prediction of effects would require infinite precision about the starting points – the causes, precision that is physically and logically impossible. Yanofsky remembers too that Hindu priests thousands of years ago had already realised that the exact length of the lunar months was impossible to ever calculate – because the moon is affected by both the Earth and the Sun, and in turn affects the movements of the Earth.

In terms of weather, as Edward Lorenz memorably put it, the mere flap of a butterfly's wing in one country could 'cause' a hurricane a week later somewhere else, as a cascade of tiny effects change outcomes at higher and higher levels. And it is not just weather that is 'chaotic'. Economists recognise that stock movement fluctuations are equally dependent on small initial movements and feedback effects, and biologists recognise the interplay of forces that can affect phenomena such as the rise and fall of populations or the spreading of diseases.

Yanofsky writes:

The truth is that science was never really about predicting. Geologists do not really have to predict earthquakes; they have to understand the process of earthquakes. Meteorologists don't have to predict when lightening will strike. Biologists do not have to predict future species. What is important in science and what makes science significant is explanation and understanding.

As Alan Turing first pointed out (and this was before Crick and Watson the conventionally credited discovers of human genetic codes and DNA) human cells perform a very complex job in working out what kind of cell they need to become. In an organism, all cells have the same DNA- it is their location in the organism that decides the form they eventually take. But as one cell's decision affects its neighbour, their decisions also affect it – the process is replete with feedback effects.

Chaos theory describes processes that are deterministic but not predictable. However, quantum mechanics describes processes that are not only beyond prediction, but not even deterministic. 'We cannot determine what a single object in a quantum system will do in the short term. This takes us one more step outside the bounds of reason.'

Subatomic physics offers another challenge to conventional science: within it, the experimenter cannot avoid being part of the experiment and influencing the outcome. The world takes on the shape we see because we are looking at it – it is not that humans find out the properties of things by looking at them. When properties of an object do not exist before they are measured, it is the death of 'what philosophers call 'naïve realism', says Yanofsky – adding:

Before any measurements, the properties are in superposition. When X is measured, the X property collapses to a single value while the Y value remains in superposition. If the Y property is then measured, then it too collapses. The point is that if the measurements were done in a different order, then the values could collapse into different values.

This leads on, naturally enough, to Schrödinger's Cat. This is the celebrated thought experiment of Erwin Schrödinger in which a cat is locked in a box with some radioactive material that may or may not decay. If it does, it kills the cat, and if not, not.

Quantum Entanglements

The notion of superposition means that until Schrödinger observes the process, the radioactive material is in 'superposition', and has both released and not released the atomic particle, and has both killed and not killed the cat. But what, asked Eugene Wigner, what if instead of Schrödinger opening the box and forcing the universe to make up its mind by either emitting the particle or not emitting it, he gets a friend to do it, and then the friend reports to him the fate of his cat afterwards? Wigner's point was that surely the friend's perception would be enough. Any conscious being would seem to be enough. (All of this assumes that cats are not conscious, an assumption many cat owners wold puzzle about.)

Yanofsky writes of the experiment:

Quantum mechanics places simple materialism in jeopardy by highlighting a new entity in the universe called consciousness. This consciousness is not made of physical objects and yet it affects how the universe works. Consciousness causes a superposition to collapse to a position. No longer are there only physical objects and spaces between them. Scientist and materialists must incorporate consciousness into their worldview.

Thus far, so much the usual stuff. However, Yanofsky extends the debate slightly further by arguing that quantum entanglement - where the state (the 'spin') of one particle describes the state of another – spells the end of reductionism – which he calls a 'fundamental superposition of all science. Today, the conventional view of modern physics, the so-called Heisenberg-Bohr hypothesis, is that there 'is really no underlying physical universe. Values [as in electron states] do not exist as a conscious observer measures the property. The value is not here before hand, rather the measure causes the value to come into being.'

As Yanofsky puts it, for three thousand years, the main goal of science has been to provide deterministic – cause and effect – rules for all phenomena. Yet now in the subatomic realm, science has started to argue for the reverse approach, the laws of quantum mechanics are non-deterministic, it describes a universe where space and time are discrete rather than continuous as in general relativity. Yet why should what is true for particles have no implications for thing s made up out of particles?

In everyday life, we normally think of the Earth as our reference point, and that reassuring notion has survived many theoretical insights into the nature of the universe. But given that the Earth spins on its axis at about 1000 miles per hour, and furthermore rotates around the sun at 67 000 miles per hour, let alone that the solar system itself is thought to whirl around our galaxy at half a million miles per hour – where is the fixed reference point? 'A stationary observer on Earth is far from stationary. There are no absolute observers, no absolute measurements and no absolute space and time. All is relative', says Yanofsky. How do we make sense of claims such as that 'every object that is moving has more mass than when it is stationary'?

Entanglement shows that there are no closed systems. Every part of a system can be entangled with other parts outside of the system. All different systems are interconnected and the whole universe is one system. One cannot understand a system without looking at the whole universe. That is, 'the whole is more than just the sum of its parts.

Einstein dismissed entanglement, calling it 'spooky action at a distance', and Yanofsky says that, more generally, physicists have made a mistake in trying to make mathematics the final arbiter of truth. 'They want their theories to be as mathematical as possible. A theory is not really acceptable to physicists until they see nice equations. Whereas in earlier times math was considered a language or a tool to help with physics, nowadays mathematics is the final arbiter of a theory. Physicists have placed their faith in the symbols and equations of mathematics. If the math works, then the physics must be correct. Galileo's lead has been followed too literally, it seems. Galileo, that is, who wrote:

Philosophy is written in that great book which continually lies open before us (I mean the Universe). But one cannot understand this book until one has learned to understand the language to and to know the letters in which it is written. It is written in the language of mathematics, and the letters are triangles, circles and other geometric figures. Without these means it is impossible for mankind to understand a single word, without these means there is only vain stumbling in a dark labyrinth.

- Galileo


The culmination of this approach is the pursuit of string theory. This tries to be the long-sought Theory of Everything, uniting all the known forces in the physical universe in one explanatory framework. Yet it remains a theory beyond empirical investigation: 'there is not a shred of empirical evidence that it is true'. Similar problems arrive with the new idea of not one but infinitely many universes – the multiverse. As Yanofsky says, Neil Manson, a philosopher, has called the concept of the multiverse 'the last shout of the desperate atheist'. The parallel universes are by definition unobservable, unverifiable, as 'unscientific' as any deity could be. They require instead a 'leap of faith'.

When Copernicus upturned the Solar System to put the sun at the centre, the mathematics did not work, all the planets' predicted movements were wrong. The geocentric system of Ptolemy was better. It was only when Kepler turned Copernicus's perfect circles into ellipses that the planets began to follow the mathematics.

Again, Euclid's ten axioms, upon which modern mathematics was constructed, contained within them what the French mathematician Jean-Baptiste d'Almbert called a scandal: 'the scandal of geometry'.

The fifth axiom, about parallel lines never meeting, was hard to demonstrate. It evaded deduction even by taking all the other nine axioms as true as a starting point. Only in the nineteenth century did Johann Gauss finally show that the problem with the axiom was that it could be either true or false! Two geometries in fact were needed – two contradictory ones. They were split asunder and called Euclidean and non-Euclidean geometry.

When Einstein was attempting to put his ideas into mathematical language, he found it impossible to describe the curves that space makes to influence the way matter moves – which is at the heart of his notion of gravity. That is, until someone pointed him towards non-Euclidean geometry. 'To this interpretation of geometry, I attach great importance, for should I have not been acquainted with it, I never would have been able to develop the theory of relativity.'

It's a simple point, but a good one, and  often overlooked. Yanofsky's book is full of such useful pointers. There are plenty of non-fiction books that start well, posing brilliant and witty challenges, and then dribble out into repetition and platitudes. Publishers, of course, like such books as they reason (probably correctly) that as long as the book is purchased it does not matter how bad the main body of it is. But here is a book that starts badly, with a floundering look at philosophical paradoxes, and then slowly but surely finds its feet and becomes a stimulating and confident account of new thinking in science and mathematics. One might say, Noson Yanofsky warms to his theme. So, too, does the reader.

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