Subject: From your granddaughter Sara about Math homework
From: Sara Kahan <...>
Date: Sat, 24 Jan 2009 12:06:35 -0500
To: Prof. W. Kahan <...>
Hi Grandpa,
I'm in a logic class, and I was assigned to figure out some things
about Zeno's paradox of Dichotomy by finding an authority (you) on
math. I was assigned to talk with an authority and get their opinion
on whether the dichotomy paradox is, in fact, viable; the paradox
being, when you go somewhere, and go halfway, and then go halfway
again, and then halfway again, and so on, can you get there?
This would basically be a Sum of a Geometric Series. Does it work?
Thanks so much Grandpa!
Love, Sara
===================================================================
Dear Sara,
This responds to your request for an opinion about Zeno's Paradox
and Dichotomy. Your request raises at least four questions:
1) To what extent can we make sense of an ostensibly arithmetic
assertion like " 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 " ?
2) Are space and time continua? Or must they be quantized? These
two possibilities constitute Zeno's Dichotomy; he ruled out
all other possibilities. "Continuum", "quantize" and some
other possibilities will be explained below.
3) How well can we understand infinite sets if we must describe
them only in a language restricted to well-formed (grammatical)
sentences, using only finitely many of them, each with only
finitely many words, each with only finitely many sounds or
letters each drawn from a finite list of phonemes or alphabet?
4) Are answers to these questions merely matters of opinion ?
This last question is the most important and the most difficult. It
comes down to this: Opinion polls give poor answers to reasonable
questions deserving reasoned answers. Opinions are cheap, almost
all worth less than they cost. Reasons costs you time, sometimes a
lot, perhaps more than you wish to spend; I prefer them anyway.
Regardless of whether an answer satisfies you completely, you must
not accept it, much less believe it, unless you understand it
well enough to justify it in your own terms. Otherwise the answer
is dubious, to be segregated in your memory and held in abeyance
until it is justified or else supplanted by a better answer. Thus
does one's memory accumulate a clutter of tentative truths not to
be confused with eternal truths, of which we may "know" fewer
than we think.
"The trouble with people is not that they don't know
but that they know so much that ain't so."
Josh Billings, 1874
The third question raises two issues both relevant to the first two
questions:
>> What is Infinity ? I've posted an elementary answer in the form
of a letter "About Infinity, for Schoolteachers" on the web at
.
>> How does language constrain our thoughts? Any language rich
enough to let us say whatever we wish to say is too rich to stop
us from uttering falsehoods and silliness. Sometimes silliness
is as obvious as "Purple clouds sleep furiously". Sometimes it
isn't; Zeno's paradox will turn out to be unobviously silly.
Despite their fecundity, our written and spoken languages have been
unable to communicate fully what is conveyed by a caress, a taste
or a smell, a song or a dance, a sculpture or a painting, a
garden or a stroll along the seashore. The stroll refutes Zeno's
paradox better than words can.
Instead of a stroll imagine a drive north on I-280 from the on-
ramp nearest San Carlos to San Francisco, about 30 miles.
If your car runs at 60 mph., the drive will take about 30 min.
Along the way your car will pass green roadside signs telling how
many miles remain to be covered. Passing these signs will not delay
your car. Neither would purple signs saying "Halfway yet to go",
"A quarter of the way to go", "An eighth of the way to go", and so
on, not even if there were infinitely many truthful purple signs
along the roadside each saying half as far to go as the previous
sign but none saying "You have just entered San Francisco". Of
course, there is too little space along the roadside to plant
infinitely many purple signs, but that is someone else's problem.
A different situation would arise if your car were obliged to pause
for a little while, even so little as a microsecond, at every
purple sign. Infinitely many such pauses are more than enough to
prevent your car from reaching San Francisco ever. But maybe not.
Let's be practical now. The land west of the San Andreas Fault is
sliding slowly north at a non-uniform rate exceeding an eighth of an
inch per year. Consequently nobody knows exactly where to find the
boundary of San Francisco. Its location must be uncertain, most
likely by more than a millionth of an inch. Your car would come
rather closer than that to San Francisco after it had passed
sufficiently many purple signs; and then most people would not
blame you for skipping past the rest of the (infinitely many)
purple signs and walking into San Francisco. And if someone more
finicky insisted that your car get within a billionth instead of a
millionth of an inch of the boundary before you cross it so rashly,
your car would have to pause at just ten more purple signs than
before. (Do you see why "just ten" ?)
* Thus, Zeno's paradox is not really a paradox about motion along a *
* path through a continuum. It is instead an observation that anyone *
* who puts marks along the path in the manner Zeno implied will have *
* to place infinitely many marks there and none at the path's end, to *
* which all but finitely many marks will be crowded closer than any *
* nonzero distance, no matter how tiny, specified in advance. *
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
About Question (1)
~~~~~~~~~~~~~~~~~~
Some attempts to cure Zeno's Paradox instigated assertions like
" 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... = 1 "
about infinite series. This seems a cure worse than the disease.
How can anyone expect to finish adding infinitely many numbers?
The equation's left-hand side is a non-terminating infinite process
hidden in the ellipsis "...". The right-hand side's " 1 " is a
terminus assigned to the process. Can a terminus be assigned in a
reasonable way to every such process? If so, what terminus does
" 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... "
deserve? In the late eighteenth century, Leonhard Euler, a very
prolific and influential mathematician even after he became blind,
suggested " 1/2 " as an appropriate terminus for that series.
Consistent with his reasoning, he also suggested that
" 1 + 1 + 1 + 1 + 1 + 1 + ... = Infinity " and
" 1 + 2 + 3 + 4 + 5 + 6 + ... = Infinity " but
" 1 + 2 + 4 + 8 + 16 + 32 + ... = -1 " .
(Can you see why?) He must have had a perplexing sense of humor.
There are algebraic operations that can be performed unexceptionably
upon any finite sum, but generate conundrums when attempted upon
some examples of infinite series. Two such simple examples appear
in problems 1(a) and 1(b) posted (solved) on my web page at
.
During the nineteenth century a rationale evolved to separate those
infinite series called "convergent", to which a terminus could be
assigned in mathematically respectable ways, from other infinite
series that deserved no such terminus. The rationale employs subtle
ideas like one that figures in a paragraph between asterisks above.
The ideas are subtle enough that many a bright college student who
encounters them for the first time decides to major in some subject
other than mathematics. Some of the subtleties arise from criteria
for convergence that are not finite computations, so they leave
undecided whether almost all of the infinitude of infinite series
converge although their every term can be computed exactly. Despite
the rationale's shortcomings it was essential for the progress of
science and engineering in the nineteenth and twentieth centuries.
For example, 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 + ...
converges to Pi, the ratio of a circle's circumference to its
diameter. There are infinitely many other ways to compute Pi as
accurately as you like if you calculate long enough. There is no
way to compute Pi exactly by starting from integers and performing
only finitely many exact additions, subtractions, multiplications,
divisions, square roots, cube roots, fifth roots, etc. Though
millions of decimal digits of Pi have been computed, we can know
Pi only approximately, never exactly. And the same is true for
almost all the numbers that figure in science and engineering.
* Thus, Zeno's Paradox has contributed somewhat to our appreciation *
* of the crucial role in mathematics, science and engineering played *
* by approximation and computability. These two pose problems most *
* people, including most mathematicians, prefer not to contemplate; *
* consequently I have earned a good living solving such problems. I *
* guess sewer repair offers similar employment opportunities. *
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
About Question (2)
~~~~~~~~~~~~~~~~~~
Zeno's Paradox is best regarded as an attempt to choose one of his
Dichotomy's two possibilities: Either space and time are continua,
or they must be quantized. These possibilities need explanations.
"Quantized" is easier: If space-time is quantized, then every two
distinguishable events or points must be separated by at least one
irreducible nonzero quantum of space-time. If this quantum exists,
it has to be extremely tiny, smaller by far than the diameter of a
proton in the hydrogen atom, shorter by far than the time light
takes to pass by the proton. If this quantum exists, no paradox
exists because the road to San Francisco needs only finitely many
purple signs:- about log2((30 miles) / (the miles in a quantum)) .
This number is gargantuan, but finite; there must be a last sign;
the car's pauses will delay but not prevent the car's arrival there.
After that, finding a place to park the car in San Francisco may
take longer than it took to get there.
If this quantum exists, motion takes place as a sequence of tiny
jumps too tiny and brief for us to perceive them individually; all
motion appears to be continuous so far as we humans can see. This
is very nearly how hands move in a lady's small quartz-electro-
mechanical watch with hour, minute and second hands to tell time.
What is a Continuum? At first it seems easy to imagine: It is a
region whose every two points can be joined by a Continuous curve
inside the region. "Continuous" means the curve is like a string
that can be stretched straight so that it will resemble an interval
on the line of real numbers, say from 0 to L = (line's length) .
Infinitely many points reside on the line (and on the curve); and
they are Dense there: Between any two of these points are more,
infinitely many more. For example, all the rational numbers (the
ratios of integers) between 0 and L are dense there. So are
the Real numbers; these are ALL the numbers obtainable each from
a convergent infinite series of rational numbers. The real numbers
constitute all the numbers between -Infinity and +Infinity, each
representable by a sign, "-" or "+" or blank, followed by a string
of arbitrarily (perhaps infinitely) many decimal digits 0 to 9
with one decimal point among them. This representation of numbers
first came into use more than a millennium after Zeno's time.
Though he struggled to grasp an infinitude's paradoxical properties,
Zeno could not possibly have known that the continuum's infinitude
of numbers (points) in the interval between 0 and L is actually
infinitely bigger than the infinitude therein of rational numbers
plus any other numbers he knew existed.
Strange! How can one infinitude be infinitely bigger than another?
The question's two surprising answers emerged during the nineteenth
century. First the continuum was found to include an infinitude of
so-called "Transcendental" numbers, each obtainable from at least
one convergent infinite series; their existence had been entirely
unsuspected previously. And then, at the end of the century, the
mathematical world was shocked by Georg Cantor's demonstration
that the rational numbers together with all the other numbers whose
existence anyone had contemplated before the nineteenth century,
though those numbers are dense, constitute a merely Countable
infinitude no bigger than the infinitude of all integers (which is
not dense!); however our continuum's real numbers comprise an
Uncountable infinitude infinitely bigger and denser than all the
non-transcendental numbers Zeno or anyone else believed existed.
The distinction between Countable and Uncountable infinitudes is
articulated a little better in my letter "About Infinity, ..." and
its reading list. That letter describes another separation of all
the real numbers into the countable infinitude of Computable real
numbers and the infinitely bigger uncountable infinitude of real but
Uncomputable numbers. This distinction will matter in a moment. A
constructive demonstration that, despite their different densities,
the integers and the rational numbers are each exactly as numerous
as the other solves Problem 2 of my web page posting at
.
Many mathematicians, still a minority, believe that Cantor made
a mistake when he applied human language and Aristotelian logic to
infinite sets. The mistake amounts to the assumption of Dichotomy,
namely that every ostensibly meaningful sentence about an infinitude
must be either True or False, assumed because we have failed to
imagine anything else. These mathematicians doubt that uncomputable
numbers exist; maybe the universe evolves by executing a program on
some vast but finite computer. I am not sure that they are wrong.
Now we know Zeno's dichotomy was defective from the outset. The
continuum as he conceived it has too small an infinitude of points.
The continuum as most of us conceive it nowadays may have infinitely
too many points. Worse, his and our preoccupation with infinitudes
distracted him and us from strange and fundamental flaws in widely
held beliefs about space, time, and the objects that move in them.
Zeno believed these were separate and independent. Everyone else
believed the same for millennia before and afterwards. In the late
seventeenth century Isaac Newton's laws of motion and gravity gave
these beliefs a mathematical form by which the motions of planets,
fluids and diverse machanisms became predictable with unprecedented
accuracy. His laws ushered in the Age of Reason upon which our
Western Civilization is based and by which it distinguishes itself
from the theocratic regimes prevailing then and prevalent still in
too many jurisdictions. For all their unprecedented accuracy and
influence, his laws are approximations inaccurate at speeds higher
than a small fraction of the speed of light, at distances so small
as atomic or so big as galactic, or for masses so small as atomic
or so big as stellar.
What Zeno and Newton and everyone else used to take for granted
turns out to be wrong. Space, time, and the objects that move in
them actually influence each other in ways first explained about a
century ago by Albert Einstein. He combined space and time into
one entity, space-time, in which, as speed increases towards the
speed of light, space and time become more nearly indistinguishable
in a way that has been confirmed by experiments upon clocks. They
run slower when moved at high speeds. This is difficult to grasp
with the aid of mathematics and impossible without. More so is that
space-time is dimpled, not flat. Its curvature near a dimple comes
from the mass of an object there and accounts for the attraction of
gravity. Mass is so densely concentrated at a black hole that the
dimple there is a sharp cusp whose physical behavior is currently
controversial among physicists and astronomers; see the article on
"Naked Singularities" in the issue of Scientific American dated
Feb. 2009.
The evolution of human language over more than forty millennia has
yet to catch up with a dimpled space-time, much less with twentieth
century Quantum Theory that predicts enough about how atoms behave
to sustain technological developments ranging from nuclear weapons
to the nanotechnology of pharmaceuticals and electronics. Quantum
theory assigns to each atom a discrete assemblage of states between
which instantaneous transitions can occur only with the emission or
absorption of at least one quantum of energy. When atoms collide,
or sometimes spontaneously, they may fuse or split and emit a lot
of energy in the form of fast-moving subatomic particles. Quantum
mechanics provides rules whereby the possibilities may be enumerated
if not predicted in every instance. Uncertainty is intrinsic.
Perhaps quantum jumps will be understood better in a space of many
dimensions in which our four-dimensionl space-time is embedded not
smoothly but microscopically wrinkled like the furrowed fissured
flaky bark of some trees, or like the skin of a raisin shrunk from
the smooth round skin of a grape. Then an atom's quantum jump could
resemble a flea's jump across a furrow rather than a slow crawl down
into a tiny fissure and up out of it. Zeno never thought of this.
So far as I know, nobody knows whether space-time is quantized or a
continuum, smooth or wrinkled; and nobody has conceived experiments
that would tell us which. Maybe someone in your generation will.
"Experiments" brings up the most significant difference between our
era's approach to knowledge and Zeno's. In his time and for almost
two millennia afterwards, almost everyone took for granted that all
knowledge was garnered from the pronouncements of ancient authority,
from patient passive observations, and from philosophical musings.
Perhaps Archimedes, two centuries after Zeno, knew better. We
shall never know for sure because too many of Archimedes' insights
have been lost to religious zeal that either destroyed copies of his
writings or turned them into palimpsests.
Early in the seventeenth century Francis Bacon and Galileo Galilei
promulgated two more ways to garner knowledge. One was experiments,
especially critical experiments designed to corroborate or refute a
proposed explanation. The second was instrumentation contrived to
reveal and measure things otherwise invisible; telescopes and later
microscopes and thermometers are examples. Nowadays a theory is not
deemed fully scientific unless it could conceivably be refuted by
the outcome of an experiment or observation. An outstanding example
is the Michelson-Morley experiment that found the speed of light
to be independent of its direction of propagation relative to the
Earth's motion, thus refuting the existence of the "Luminiferous
Ether" postulated late in the nineteenth century to explain the
propagation of light and other electromagnetic waves. Einstein was
inspired by this experiment to reformulate our concepts of space and
time, thus explaining Newton's gravity and the conversion of mass
into energy that sustains nuclear power-plants and nuclear weapons.
A theory that cannot conceivably be refuted by experiment is at best
mathematics, at worst pseudo-science like "Intelligent Design"
advocated as an alternative to Darwinian Evolution via Natural
Selection; see the issue of Scientific American dated Jan. 2009.
* Zeno contrived his Paradox as a thought-experiment to resolve his *
* Dichotomy. His Paradox has turned out irrelevant, his Dichotomy *
* too simple. Still, he deserves a lot of credit for the attempt. *
* *
* What we can learn best from the history of Zeno's Paradox is this: *
* Nature is more varied, stranger and more interesting than will be *
* conceived by our imaginations unaided by exploration and experiment. *
With warm affection,
your grandpa.