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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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