The
girls, Carol, Linda, Jean and Jen had a good day shopping at Prime Outlets and
eating at Lee’s Crab Shack II. You went along for the quiet time and the food.
Tomorrow you and Carol meet Linda and Bill at their place for a trip to the
Columbia at Ybor City, the oldest restaurant in Tampa.
** **
The Patio Dining Room where we usually
eat.
**
The original Columbia Restaurant,
located in the historic Ybor City neighborhood in Tampa, Florida, is the oldest
continuously operated restaurant in Florida, the oldest Spanish restaurant in
the United States and one of the largest Spanish restaurants in the world with
1,700 seats in 15 dining rooms taking up 52,000 ft² over an entire city
block. Founded in 1905 in, the landmark is still owned by the
Hernandez/Gonzmart family and serves Spanish and Cuban cuisine.
From Wikipedia
** **
2004
hours. This restaurant was a tradition long before I was alive and I hope it
will continue to be long after I am gone. This is a part of what our family
loves; this is a part of what I love; this is a part of ‘Old Florida’, of
Spanish Florida.
Why is food and history so much of your
heart, boy? – Amorella
I do not know, but they are.
Post. – Amorella
2012 hours. I did take a photo
tonight. It has an Avant-garde flare I like. My title is “Light Interrupted”.
"Light Interrupted"
2101 hours. Now I want to return to Thinking
in Numbers and my notes. On page 30 Daniel Tammet says:
“One
hundred proverbs, give or take, sum up the essence of a culture; one hundred
multiplication facts compose the ten times table. Like proverbs, these
numerical truths or statements – two times two is four, or seven times six
equals forty-two – are always short, fixed and pithy. Why then do they not
stick in our heads as proverbs do?”
I
never thought of putting proverbs and the times table together but I have no
trouble understanding it; probably because I enjoyed learning the times table.
My math problems came in the fourth grade with long division and then with
fractions, not with multiplication. I still have trouble with ‘leftovers’ (but
not the food kind). For instance the photograph of the sunset I took tonight is
enjoyable to me because I fixed in a frame. It has no left over quality as I
see it (other than it is light interrupted by air and condensed water vapor and
water below and sky above). My concept of the universe and beyond is that
everything is accounted for, even the accidentals along the way. Everything
must be accounted for.
What of the things that don’t exist, boy?
What of the imaginings of ‘heartansoulanmind’? – Amorella
2119 hours. They are accounted for, at
least in the Merlyn books, in The Place of the Dead or
HeavenOrHellBothOrNeither.
You are surprised how quickly you came up
with a response that is from ‘heartansoulanmind’. – Amorella
I am. I thought it was a trick
question but quickly realized I was providing the trick in my imagination so I
dismissed it.
2123
hours. I have one more note then I can continue reading the book. It is on page
46. This is in the chapter “Classroom Intuitions” and the words compose the
final three lines of the chapter:
“Then
she came to a beautiful conclusion about fractions that I shall never forget.
She
said, ‘There is no thing that half of it is nothing.’”
2139
hours. I agree this is a
delightful statement even though I don’t understand the mathematics. I
immediately think of Zeno’s Paradoxes so I checked out Wikipedia as a reminder.
** **
Zeno's
paradoxes
From Wikipedia, the free encyclopedia
Zeno's paradoxes
are a set of philosophical problems generally thought to have been devised by
Greek philosopher Zeno of Elea (ca. 490–430 BC) to support Parmenides’s doctrine that contrary to the evidence
of one's senses, the belief in plurality and change is mistaken, and in
particular that motion is nothing but an illusion. It is usually assumed, based
on Plato’s Parmenides (128a-d), that
Zeno took on the project of creating these paradoxes because other philosophers
had created paradoxes against Parmenides's view. Thus Plato has Zeno say the
purpose of the paradoxes "is to show that their hypothesis that existences
are many, if properly followed up, leads to still more absurd results than the
hypothesis that they are one." (Parmenides 128d). Plato has
Socrates claim that Zeno and Parmenides were essentially arguing exactly the
same point (Parmenides 128a-b).
Some of Zeno's nine surviving paradoxes
(preserved in Aristotle’s Physics and
Simplicius’s commentary thereon)
are essentially equivalent to one another. Aristotle offered a refutation of
some of them. Three of the strongest and most famous—that of Achilles and the
tortoise, the Dichotomy argument, and that of an arrow in flight—are presented
in detail below.
Zeno's arguments are perhaps the first
examples of a method of proof called reductio
ad absurdum also known as proof by contradiction. They are also credited as
a source of the dialectic method used by Socrates.
Some mathematicians and historians, such
as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for
which modern calculus provides a mathematical solution. Some philosophers,
however, say that Zeno's paradoxes and their variations remain relevant metaphysical
problems.
The origins of the paradoxes are somewhat
unclear. Diogenes Laertius, a fourth source for information about Zeno and his
teachings, citing Favorinus, says that Zeno's teacher Parmenides was the first
to introduce the Achilles and the tortoise paradox. But in a later passage,
Laertius attributes the origin of the paradox to Zeno, explaining that
Favorinus disagrees.
Achilles and the tortoise
Distance vs. time,
assuming the tortoise to run at Achilles' half speed.
In a race, the quickest runner can never
overtake the slowest, since the pursuer must first reach the point whence the
pursued started, so that the slower must always hold a lead.
– as recounted by Aristotle, Physics
VI:9, 239b15
In the paradox of Achilles and the
Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the
tortoise a head start of 100 metres, for example. If we suppose that each racer
starts running at some constant speed (one very fast and one very slow), then
after some finite time, Achilles will have run 100 metres, bringing him to the
tortoise's starting point. During this time, the tortoise has run a much
shorter distance, say, 10 metres. It will then take Achilles some further time
to run that distance, by which time the tortoise will have advanced farther;
and then more time still to reach this third point, while the tortoise moves
ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he
still has farther to go. Therefore, because there are an infinite number of
points Achilles must reach where the tortoise has already been, he can never
overtake the tortoise.
Dichotomy paradox
That which is in locomotion must arrive
at the half-way stage before it arrives at the goal.–
as recounted by Aristotle, Physics
VI:9, 239b10
Suppose Homer wants to catch a stationary
bus. Before he can get there, he must get halfway there. Before he can get
halfway there, he must get a quarter of the way there. Before traveling a
quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
The resulting sequence can be represented
as:
This description requires one to complete
an infinite number of tasks, which Zeno maintains is an impossibility.
This sequence also presents a second
problem in that it contains no first distance to run, for any possible (finite)
first distance could be divided in half, and hence would not be first after
all. Hence, the trip cannot even begin. The paradoxical conclusion then would
be that travel over any finite distance can neither be completed nor begun, and
so all motion must be an illusion.
This argument is called the Dichotomy because it involves repeatedly
splitting a distance into two parts. It contains some of the same elements as
the Achilles and the Tortoise paradox, but with a more apparent
conclusion of motionlessness. It is also known as the Race Course
paradox. Some, like Aristotle, regard the Dichotomy as really just another
version of Achilles and the Tortoise.
There are two versions of the dichotomy
paradox. In the other version, before Homer could reach the stationary bus, he
must reach half of the distance to it. Before reaching the last half, he must
complete the next quarter of the distance. Reaching the next quarter, he must
then cover the next eighth of the distance, then the next sixteenth, and so on.
There are thus an infinite number of steps that must first be accomplished
before he could reach the bus, with no way to establish the size of any
"last" step. Expressed this way, the dichotomy paradox is very much
analogous to that of Achilles and the tortoise.
Arrow
paradox
If everything when it occupies an equal
space is at rest, and if that which is in locomotion is always occupying such a
space at any moment, the flying arrow is therefore motionless.
– as recounted by Aristotle, Physics VI:9, 239b5
In the arrow paradox (also known as the
fletcher’s paradox), Zeno
states that for motion to occur, an object must change the position which it
occupies. He gives an example of an arrow in flight. He states that in any one
(durationless) instant of time, the arrow is neither moving to where it is, nor
to where it is not. It cannot move to where it is not, because no time elapses
for it to move there; it cannot move to where it is, because it is already
there. In other words, at every instant of time there is no motion occurring.
If everything is motionless at every instant, and time is entirely composed of
instants, then motion is impossible.
Whereas the first two paradoxes divide
space, this paradox starts by dividing time—and not into segments, but into
points.
Three other paradoxes as given by
Aristotle
Paradox of Place:
"… if everything
that exists has a place, place too will have a place, and so on ad infinitum."
Paradox of the Grain of Millet:
"… there is no
part of the millet that does not make a sound: for there is no reason why any
such part should not in any length of time fail to move the air that the whole
bushel moves in falling. In fact it does not of itself move even such a
quantity of the air as it would move if this part were by itself: for no part
even exists otherwise than potentially."
The Moving Rows (or Stadium):
"… concerning the
two rows of bodies, each row being composed of an equal number of bodies of
equal size, passing each other on a race-course as they proceed with equal
velocity in opposite directions, the one row originally occupying the space
between the goal and the middle point of the course and the other that between
the middle point and the starting-post. This...involves the conclusion that
half a given time is equal to double that time."
Proposed solutions
According to Simplicius, Diogenes the
Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in
order to demonstrate the falsity of Zeno's conclusions. To fully solve any of
the paradoxes, however, one needs to show what is wrong with the argument, not
just the conclusions. Through history, several solutions have been proposed,
among the earliest recorded being those of Aristotle and Archimedes.
Aristotle (384 BC−322 BC) remarked that
as the distance decreases, the time needed to cover those distances also
decreases, so that the time needed also becomes increasingly small. Aristotle
also distinguished "things infinite in respect of divisibility" (such
as a unit of space that can be mentally divided into ever smaller units while
remaining spatially the same) from things (or distances) that are infinite in
extension ("with respect to their extremities").
Before 212 BC, Archimedes had developed a
method to derive a finite answer for the sum of infinitely many terms that get
progressively smaller. Modern calculus achieves the same result, using more
rigorous methods. These methods allow the construction of solutions based on
the conditions stipulated by Zeno, i.e. the amount of time taken at each step
is geometrically decreasing.
Aristotle's objection to the arrow
paradox was that "Time is not composed of indivisible nows any more than
any other magnitude is composed of indivisibles.” Saint Thomas Aquinas,
commenting on Aristotle's objection, wrote "Instants are not parts of
time, for time is not made up of instants any more than a magnitude is made of
points, as we have already proved. Hence it does not follow that a thing is not
in motion in a given time, just because it is not in motion in any instant of
that time.” Bertrand Russell offered what is known as the "at-at theory of
motion". It agrees that there can be no motion "during" a
durationless instant, and contends that all that is required for motion is that
the arrow be at one point at one time, at another point another time, and at
appropriate points between those two points for intervening times. In this view
motion is a function of position with respect to time. Nick Huggett argues that
Zeno is begging the question when he says that objects that occupy the same
space as they do at rest must be at rest.
Peter Lynds has argued that all of Zeno's
motion paradoxes are resolved by the conclusion that instants in time and
instantaneous magnitudes do not physically exist. Lynds
argues that an object in relative motion cannot have an instantaneous or
determined relative position (for if it did, it could not be in motion), and so
cannot have its motion fractionally dissected as if it does, as is assumed by
the paradoxes.
Another proposed solution is to question
one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy),
which is that between any two different points in space (or time), there is
always another point. Without this assumption there are only a finite number of
distances between two points, hence there is no infinite sequence of movements,
and the paradox is resolved. The ideas of Planck length and Planck time in
modern physics place a limit on the measurement of time and space, if not on
time and space themselves. According to Hermann Weyl, the assumption that space
is made of finite and discrete units is subject to a further problem, given by
the "tile argument" or "distance function problem".
According to this, the length of the hypotenuse of a right angled triangle in
discretized space is always equal to the length of one of the two sides, in
contradiction to geometry. Jean Paul Van Bendegem has argued that the Tile
Argument can be resolved, and that discretization can therefore remove the
paradox.
Hans Reichenbach has proposed that the
paradox may arise from considering space and time as separate entities. In a theory
like general relativity, which presumes a single space-time continuum, the
paradox may be blocked.
The paradoxes in modern times
Infinite processes remained theoretically
troublesome in mathematics until the late 19th century. The epsilon-delta version
of Weierstrass and Cauchy developed a rigorous formulation of the logic and
calculus involved. These works resolved the mathematics involving infinite
processes.
While mathematics can be used to
calculate where and when the moving Achilles will overtake the Tortoise of
Zeno's paradox, philosophers such as Brown and Moorcroft claim that mathematics
does not address the central point in Zeno's argument, and that solving the
mathematical issues does not solve every issue the paradoxes raise.
Zeno's arguments are often misrepresented
in the popular literature. That is, Zeno is often said to have argued that the
sum of an infinite number of terms must itself be infinite–with the result that
not only the time, but also the distance to be travelled, become infinite.
However, none of the original ancient sources has Zeno discussing the sum of
any infinite series. Simplicius has Zeno saying "it is impossible to
traverse an infinite number of things in a finite time". This presents
Zeno's problem not with finding the sum, but rather with finishing
a task with an infinite number of steps: how can one ever get from A to B, if
an infinite number of (non-instantaneous) events can be identified that need to
precede the arrival at B, and one cannot reach even the beginning of a
"last event"?
Today there is still a debate on the
question of whether or not Zeno's paradoxes have been resolved. In The
History of Mathematics, Burton writes, "Although Zeno's argument
confounded his contemporaries, a satisfactory explanation incorporates a
now-familiar idea, the notion of a 'convergent infinite series.'".
Bertrand Russell offered a
"solution" to the paradoxes based on modern physics, but Brown concludes
"Given the history of 'final resolutions', from Aristotle onwards, it's
probably foolhardy to think we've reached the end. It may be that Zeno's
arguments on motion, because of their simplicity and universality, will always
serve as a kind of ‘Rorschach image; onto which people can project their most
fundamental phenomenological concerns (if they have any)."
Quantum Zeno effect
Main article: Quantum
Zeno effect:
In 1977,physicists E. C. G. Sudarshan and
B. Misra studying quantum mechanics discovered that the dynamical evolution
(motion) of a quantum system can be hindered (or even inhibited) through
observation of the system. This effect is usually called the "quantum Zeno
effect" as it is strongly reminiscent of Zeno's arrow paradox. This effect
was first theorized in 1958.
Zeno behaviour
In the field of verification and design
of timed and hybrid systems, the system behaviour is called Zeno if it
includes an infinite number of discrete steps in a finite amount of time. Some
formal verification techniques exclude these behaviours from analysis, if they
are not equivalent to non-Zeno behaviour.
In systems design these behaviours will
also often be excluded from system models, since they cannot be implemented
with a digital controller.
A simple example of a system showing Zeno
behaviour is a bouncing ball coming to rest. The physics of a bouncing ball,
ignoring factors other than rebound, can be mathematically analyzed to predict
an infinite number of bounces.
Selected and edited from Wikipedia –
Zeno’s Paradoxes
** **
You find delight in the darkness of the
joke, of the paradox and math has little to do with it. For you this shows a
foolishness in too much reason, that ‘trickery’ is afoot and you neatly conjure
up Milton’s character of Satan in Paradise Lost. And, you think, reason
alone does not a human make. Tomorrow, when time permits, continue your
reading. This sort of exercise clears the mind boy, gives the mind a vacation,
so to speak. Post. - Amorella
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