Logic Café
Sir Frederick
http://www.newscientist.com/channel/fundamentals/mg19425991.400-impossible-things-for-breakfast-at-the-logic-caf.html;jsessionid=OGIAENJMHEOM
Impossible things for breakfast, at the Logic Café
a.. 14 April 2007
b.. From New Scientist Print Edition.
c.. Robert Matthews
CHRIS ISHAM has a problem with truth. And he suspects his fellow physicists
do too. It is not their honesty he doubts, but their
approach to understanding the nature of the universe, the laws that govern
it and reality itself. Together with a small band of
allies, Isham is wrestling with questions that lie at the very core of
physics. Indeed they run even deeper, to such basic concepts
as logic, existence and truth. What do they mean? Are they immutable? What
lies beyond them?
After years of effort, Isham and his colleagues at Imperial College London
and elsewhere believe they can glimpse the answers to
these profound questions. They didn't set out to rethink such weighty
issues. When they started nearly a decade ago, the researchers
hoped to arrive at a quantum theory of the universe, an ambitious enough
task in itself. Yet in the process they might have bagged
something bigger.
For if their results stand up, Isham and his colleagues appear to have found
a new way of making sense of reality using concepts
even more fundamental than mathematics and logic. Not only could their
insights be good news for quantum theory, they could lead to
a whole new way of constructing theories of reality.
Since its emergence around a century ago, quantum theory has become one of
the cornerstones of modern science. It underpins
everything from the behaviour of quarks and semiconductors to the power of
medical scanners. And it has passed virtually every test
thrown at it, its predictions agreeing with experiment to many decimal
places.
With a track record like that, quantum theory might seem ideal for casting
light on the ultimate questions about the universe, such
as why it exists at all. Not so. In fact, it runs into very big trouble very
quickly, because quantum theory has a problem with
truth.
With hindsight, perhaps this shouldn't be so surprising. Right from the
start, quantum theory has had a reputation for giving odd
answers to even seemingly simple questions. In the everyday world,
everything has nice, clear-cut properties: people are either dead
or alive, electrons either spin up or down. Yet according to quantum theory,
what we're seeing is just one manifestation of a whole
panoply of possibilities, all mixed together.
How all those possibilities turn into just the one reality we see has caused
endless debate among theorists. Their efforts have
produced various interpretations of quantum theory, the most famous of which
is the Copenhagen interpretation, named in recognition
of its inventor, the Danish quantum pioneer Niels Bohr. According to this
view, it is the act of observing that triggers the panoply
of possibilities to collapse down to the single reality we experience.
Quite how this collapse process works isn't exactly clear. What is plain is
that it raises profound questions about the whole notion
of truth in quantum theory. For it implies that it is impossible to know the
truth of any statement about, say, an electron until it
has been observed. Unless that happens, it doesn't really make sense to talk
of the electron - or anything else for that matter - as
being real.
Things get much worse when quantum theory is applied to the entire universe.
If the universe is as real as we believe, then it must
have been cast into that state by an observer able to view it all. Yet since
the universe includes everything, there can be no
external observer.
Theorists have come up with all kinds of alternative interpretations to
avoid the problem, which others have in turn torn apart.
Small wonder, perhaps, that most workaday physicists are happy to leave them
to it. Alas that's just not an option for quantum
cosmologists, who have to find some way of turning the cosmic cornucopia
into the one real universe we actually inhabit.
But who says they have to? Perhaps we're reading quantum theory all wrong,
and there is no need to force the universe or anything
else into rude reality. Rethinking quantum theory is an appealing thought,
not least because it would pull the quantum world into
line with common sense. Yet there is a problem with this vision. It is ruled
out by an elegant result published in 1967 by
mathematicians Simon Kochen and Ernst Specker.
Kochen and Specker's theorem puts some pretty severe constraints on anyone
hoping to rid quantum theory of its weirdness. Put
simply, the theorem shows that it is pointless expecting to get simple true
and false answers from quantum theory. Every statement
about a quantum system must either depend on a host of assumptions, or
refuse to obey the standard rules of logic - and possibly
both.
For quantum cosmologists, Kochen and Specker's theorem is particularly bad
news. It rules out all hope of squaring quantum theory
with the common-sense view that the universe is real and has simple,
clear-cut properties. Or at least it does for those who believe
the laws of logic are set in stone. What if they aren't? Could the problem
lie not in quantum theory, but in our notion of truth?
That is the question that Isham and his colleagues have dared to ask, with
intriguing results.
Abandoning the standard laws of logic in order to make the universe real
seems like a hefty price to pay. Yet some theorists have
believed it is worth it, says Steven French, a philosopher and quantum
physicist at the University of Leeds in the UK. That's one
reason why mathematicians over the years have developed other systems of
logic. "Along with standard true/false logic there are
so-called non-classical logics out there which include true, false and
indeterminate values," says French. "People looked at them as
a way of dealing with problems in quantum theory, but it died out in the
1970s and 1980s because it wasn't really illuminating very
much."
A big sticking-point lay in finding the alternatives to "AND", "OR" and
"NOT", the logical operators of the standard or Boolean
algebra that are routinely used by everyone from philosophers to computer
programmers to make logical deductions. While this
familiar form of logic works well enough in everyday situations, it fails to
describe the behaviour of quantum systems.
Isham illustrates this using the example of ordering breakfast in a cafe.
Imagine looking through the menu and finding that eggs,
bacon and sausage are on offer. It states the choice as "eggs AND bacon, OR
eggs AND sausage", but the chef could equally offer the
same breakfast choice in a shorthand version: "eggs AND (bacon OR sausage)".
That's because the operator AND possesses a
mathematical property called distributivity, which links eggs with whatever
is inside the brackets. Distributivity is vital for
making common-sense deductions. Lose it - as you do in quantum theory - and
you can expect some unusual results.
Take that cafe menu, for instance. If you ask for eggs AND bacon in the
quantum world, you could get nothing at all says Isham. Ask
for eggs, and bacon or sausage, and you'll get eggs plus some weird quantum
mix of bacon and sausage. Clearly, relying on quantum
logic to reason your way to a decent breakfast is likely to lead to
disappointment.
There is a serious point lurking behind all this. Systems of logic lacking
distributivity are very hard to reason with and quantum
logic is one of them. Worse still, Kochen and Specker's theorem rules out
any hope of tinkering with quantum logic to force it to
give us simple true/false answers to statements about physical systems. Yet
without those simple answers, it doesn't make sense to
say a physical system has certain properties and is thus "real". Why does
quantum logic have to be so frustrating?
Isham and Jeremy Butterfield at the University of Oxford decided to dig
deeper into the problem. They dug so deep, in fact, that
they found themselves under the foundations of standard mathematics and
staring at something far more fundamental. That something is
a concept called a topos, and it could be the basis of a whole new way of
constructing theories of reality.
The idea of a concept even more general than mathematics and logic may seem
mind-bending, yet mathematicians have happily
contemplated such things for years. They have long known that the whole of
standard mathematics and logic can be constructed from
entities called sets. A set is just a collection of objects - anything from
the infinite set of prime numbers to the set of all
mammals or even the set of all universes. Crucially, sets obey the laws of
standard logic and Boolean algebra.
Mathematicians have since discovered that sets themselves are merely the
most familiar example of the even more general concept of a
topos. The precise definition of a topos is highly technical, but all topoi
share one key feature: each gives rise to its very own
variety of logic. Suddenly an astonishing possibility opens up: we can break
away from the familiar set-based variety of logic and
describe the world via other topoi.
Isham and his colleagues saw topoi might offer a way to break the shackles
of Kochen and Specker's theorem. The trick was to find
topoi whose associated logic would reconcile quantum theory with the notion
of a real universe. That meant searching for new
definitions of the logical operators AND, OR and NOT. Others have tried this
before, and it is far from trivial, says Isham. "In
practice, the procedures have been rather hit and miss."
To pin them down precisely, he and his colleagues turned to the bigger
mathematical palette offered by topos theory. Now they could
see Boolean algebra for what it is: merely the most familiar of many
possible types of algebra, each of which could act as the basis
of entire new forms of logic.
Armed with these, Isham and his colleagues have identified the topoi for
quantum theory. Not surprisingly, they are very different
from anything we're familiar with, and of course come with their very own
form of logic. That logic does at least have one familiar
feature: it is distributive. At a stroke, this removes one of the most
perplexing aspects of quantum theory. It allows us once more
to make common-sense deductions about quantum systems. Finally, the universe
can be real without having to fret about "outside"
observers.
Another reality
But there is a price to pay, and it is precisely what the Kochen-Specker
theorem warned of: the demise of simple truth and falsity.
For all its drawbacks, Boolean algebra does at least allow every statement
about our universe to be either true or false. Yet this
turns out to be the exception among all the different types of algebras -
including the one underpinning quantum theory. The logic
associated with quantum topoi encompasses true, false and many shades of
grey in between.
Does that mean we must accept a universe that is real, but about which any
question will receive myriad answers, all of them true?
According to Isham and his colleagues, the answer - appropriately enough -
is both yes and no. If we are content to view reality
through the window of classical physics, then we can enjoy straightforward
true/false answers to our questions - as long as we avoid
the realm of atoms. But if we insist on making statements about atoms, we
must use the logic of quantum topoi and accept the
existence of a whole host of realities, all as valid as each other.
And that might just be the start; after all, there are more topoi than just
the standard and quantum ones. In a series of papers
unveiled last month, Isham proposes an even more mind-bending idea: there
may be myriad ways of viewing reality, each based on its
own topos. Together with Andreas Doering of Imperial, he has shown that
every physical system - from an electron to the whole
universe - has a unique mathematical identity that dictates how it will
appear when viewed through the prism of a particular topos.
Seen via the topos of set theory, an atom takes on its classical appearance
with nice, well-defined properties. Viewed through the
topos associated with quantum theory, it becomes altogether fuzzier and
strange.
We needn't stop there. Why not opt for another topos? It could lead to a
view of reality even more astonishing and successful than
quantum theory. "What we're hoping is that topos theory becomes the basis
for a whole new way of constructing theories", says Isham.
"Topos theory could lead to a view of reality more astonishing and
successful than quantum theory"
It is an exhilarating possibility, and one that could hardly be better
timed. Theoretical physicists feel growing disquiet about the
lack of progress on the truly fundamental questions. Attempts to understand
the ultimate origin of the universe have spawned a host
of ideas, but no consensus as to which is right. Meanwhile the search for a
"theory of everything" that would unify all the forces
and particles of nature has run into innumerable problems.
Not surprisingly, this has led to mounting suspicions that current theories
of fundamental physics are missing something big. Could
topos theory open the way? "There's no doubt that we need something
radical", says Max Tegmark, a theorist at the Massachusetts
Institute of Technology. "Whether this is it is another question. In the end
the real test is: does it get us anywhere?"
Isham agrees, but stresses that he and his colleagues have only just begun
to scratch the surface of topos theory. He hopes
researchers will see his latest papers as a framework for going beyond
quantum theory, perhaps to something even more profound.
So will topos theory trigger as big a change in our perceptions of reality
as quantum theory did a century ago? That depends at
least in part on how other theorists react to these first papers. Isham is
under no illusions about that: "We are trying to change
the way we construct theories of what reality is like," he says. "And that's
always going to be problematic."
Robert Matthews is visiting reader in science at Aston University in
Birmingham, UK
From issue 2599 of New Scientist magazine, 14 April 2007, page 30-33
footprints in the snow
on its own topos. Together with Andreas Doering of Imperial, he has
shown that every physical system - from an electron to the whole
universe - has a unique mathematical identity that dictates how it
will appear when viewed through the prism of a particular
Kant marches on. How a cognitive system apprehends the world depends
upon the template(s) it is using to organize information. Humans
inherent the one that describes the world as an experience in the
classical, Newtonian vein. But we can still contrive our own abstract
templates for representing existence in other ways.
Anthony G. Rubino
Consider:
Will (e.g., the Creative Force) as the primary topus with topoi being
secondary individual acts of will of persons, or instantiations of the
primary topus which appear to individual wills as if they had a will of
their own rather than as the instantiations of the primary topus that
they are.
The similarities of the topoi of individual persons with the topoi not
of their own making could then be understood as the basis of knowledge
through analogy, metaphor, or models.
Tony, philosopher
http://www.geocities.com/trisector/
So many misconceptions, so little time.
jusholm
"Sir Frederick" <mmcn...@fuzzysys.com> wrote in message
news:kje123pn19trcvqkm...@4ax.com...
Fascinating!