44 | New Scientist | 8 January 2022
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on this argument to apparently show that,
no matter how far you have travelled towards
point B, you will always still have to walk
halfway from your current position to
point B and this takes at least some amount
of time. The conclusion is that all journeys
should take an infinite amount of time – yet
clearly that isn’t true.
When we encounter a paradox, our
instinct is that something has gone wrong.
Contradictions like this shouldn’t exist
and we want to solve or explain away the
inconsistencies. In the case of Zeno’s paradox,
mathematicians have since discovered that it
is possible to divide up a distance into an
infinite number of portions – but also to add
up that infinite series and get a result that is
finite. For most of us, that seems counter-
intuitive, but it is a truth showing that reality
doesn’t always conform to our preconceptions.
In my work, I have identified a number of
general strategies that logicians use to tackle
paradoxes. For instance, the aforementioned
solution to Zeno’s paradox is an example
of what I call the “odd guy out” strategy,
where we spot that one of the premises of
the paradox is dodgy. Sometimes, paradoxes
are extremely difficult to explain away, and
in these cases logicians can employ what
I call the “detour”, where you admit that
the paradox can’t be solved on its own terms,
but propose that some deep assumption
about reality needs revising.
This kind of thinking has been helpful on
many occasions, not least through the famous
Schrödinger’s cat paradox, which has helped
us interrogate the true meaning of quantum
theory for decades. But using logic alone to
analyse paradoxes ignores an important
element of what makes them so engrossing
in the first place. A lot of the language used to
discuss paradoxes uses terms like “seem”,
“apparently” or “appears to show”, which
subtly demonstrates that human
understanding is at the core of how paradoxes
work. Paradoxes don’t exist in a vacuum, they
are puzzles that take shape in our minds.
This is what prompted me to begin
developing a new way of analysing paradoxes
a few years ago. My method puts us, the readers
of the paradox, at the heart of things. It hinges
on an old idea called subjective probability,
often used in maths but not applied to
paradoxes before. Subjective probability is the
degree to which a rational person will believe
something. Take a statement like “two plusThe power
of paradoxes
Grasping the role of human intuition in
mind-bending logic puzzles can help us all think
more clearly, says philosopher Margaret Cuonzo
A
WOMAN once approached me with
a curious problem concerning her
husband. Like most people who
choose to get married, she had promised to
love her spouse to the exclusion of all others.
But there was a problem: according to her,
the man she married simply wasn’t the same
person any more. He had the same name
and career, the same memories and skills.
But over many years, an accumulation
of small changes had, she felt, made her
husband a completely different person.
This woman had approached me not
because I’m an expert in matters of the
heart, but because I had just given a talk
about paradoxes. These puzzles have
entertained and perplexed us for millennia.
They force us to grapple with some of the
deepest matters of logic and meaning.
What does it mean for something to be
“the same”, for instance?
I couldn’t offer the woman any simple
answers. I reminded her that she had probably
changed quite a bit since her youth too. And
I pointed out that sometimes our intuitions
about concepts like identity can be unhelpful.In fact, the point goes well beyond
relationships. Chewing over paradoxes
can show us places where our intuitions
need tweaking, and this applies everywhere
from the foundations of mathematics to
social media and our efforts to live more
sustainable lives. Paradoxes have helped
thinkers resculpt our understanding of key
concepts and attain fresh scientific insights
time and again. Now, a new way of thinking
through paradoxes is emerging, one that
holds promise because it puts our mushy
human intuition front and centre.
One reasonable way to define a paradox is
as “a set of mutually inconsistent claims, each
of which appears to be true”. One of the oldest
and most famous of these puzzles is Zeno’s
dichotomy paradox, developed by Zeno of Elea,
a thinker who lived in Greece in the 5th century
BC. Imagine a person walking from point A to
point B. To reach point B, they first have to walk
half the distance, and this takes some finite
amount of time. When they get halfway, they
still have to walk halfway between where they
are now and point B, and this also takes a finite
amount of time, albeit a little less. Zeno carried>