What an Engineer thinks of Zeno's paradoxes.
I have at some or other times thought about Zeno's paradoxes.
Here is what I have come up with lately
[1]
/
[2]
. There
are some possible answers to Zeno's paradoxes
in the Google world, but they don't sound entirely convincing
to me. So I decided to roll my own.
Most of the material for the following discussion is in the
foot-notes which cover about 2/3 of the text.
Please look there if you want to get some of the finer points.
Zeno's paradoxes touch some of the most
delicate foundations of human thinking, and cannot be dealt
with in a curt manner.
Even if I approach the matter with a certain style of
robustness, I have to make provision that there
exists some philosophical balance. Because I am also a
philosopher, not only an engineer.
For the engineer,
[3]
Zeno's
paradoxes are a prime example of a certain style of thinking
that the ancient Greek philosophers as a group excelled
in:
Extremely tricky and brilliant in the details, but entirely
impractical.
[4]
There were many accomplished practical engineers in ancient
Greece.
but beyond the engineers there were only philosophers that are
well-known today.
[5]
/
[6]
/
[7]
/
[8]
/
[9]
This ancient thinking style is also called confusionism, or
illusionism, or Hegelism.
[10]
Zeno's problem was the idea of continuum in the Universe
The underlying natural-philosophical problem of Zeno was the
idea of the continuum.
Continuum is just an idea, with nothing corresponding to it in
the Real Universe.
To add some more confusion, the mathematicians called
Continuum Numbers the Real Numbers.
Zeno's paradoxes are based on these assumptions:
1) that the Real Universe is a continuum.
2) that the Real Time is a continuum.
3) that movement takes place in the above continuum of time
and place.
Here we deal with Zeno's paradoxes on the level of engineering
and science. What all the scientific
exploration in the last 400 years or so has shown, is that
there is no such thing as a continuum.
Everything in the Real Universe comes in quanta, even time.
[11]
But let us note that these quanta are not the quanta of
"quantum theory"
.
[12]
A more common physical term to use would be the word
"particle"
, but there is no particle of time.
And there is no particle of gravity.
So:
the quantum of matter is the atom.
the quantum of light is the photon.
[13]
/
[14]
the quantum of electricity is the electron.
the quantum of time is the clock-tick.
[15]
the quantum of movement or (loco-)motion is the diameter of an
atom.
There still exist some incompatibilities:
[16]
/
[17]
A short excursion into illusionism in mathematics
Since the modern science of mathematics is based on the
ancient Greek work, we can find many examples
of illusionism in mathematics. Real numbers of the irrational
kind exist only as a concept.
[18]
A good example is Pi.
[19]
Some more examples are in the footnotes:
[20]
/
[21]
The calculus works with this illusionist assumption: That one
can go down to "infinitesimal"
,
and then go up to "infinite"
. The
reasoning is, that infinite cancels out infinitesimal,
[22]
and so, the calculus works and gives practical
results.
The engineer has the following calculations to make
Let us start with Achilles and the tortoise:
This shall be a real assignment for an engineer, and one does
best to disregard all the above
philosophical logisticisms, and especially Zeno's own
misleading prescriptions.
One must make certain assumptions and base some calculations
on these.
First, we must set a certain distance of the race course. The
curse of the modern metric system of the
French Revolution is that it correlates to nothing of the
human proportionality, nor the human ability,
which was the base of all the ancient measuring systems.
[23]
The modern metric system is a
procrustean
[24]
measure of the kind: One size
fits no-one.
This makes it very difficult to visualize an actual travelling
or traversing process,
for which we are seeking an engineering solution in a computer
simulation.
The
best
[25]
standard to use
for a practical evaluation of the Zeno paradox is the Roman army standard.
So we set the distance of the race course at the Roman army
standard:
[26]
100 strides of human legs.
[27]
The ancient measure of stride is conserved in the roman
stride:
[28]
gradus = single stride = 2½ pes = 740.880 mm = 0,740880
m = 74,0880 cm.
passus = double stride = 2 gradi = 1,48176 m = 5
feet
The most widely used ancient distance constant was the Roman
mile. This is the constant distance which
a Roman legion traversed with 1000 double strides or: mille
passuum
[29]
mille passus = mile = 1000 passus = 1,48176 km = 5000
feet.
The milia passuum is a very constant measure, and it is
recorded that all the Roman legions
traversed exactly this distance with 1000 passuus (or double
strides), when travelling on even ground.
[30]
This was so accurate, that all the geography of the Roman
empire depended on it. Some inaccuracies are
introduced by the factors of terrain and weather:
[31]
When the march goes steeply uphill, the
stride will
shorten. But to compensate, when it goes downhill, the stride
will lengthen.
The Romans tended to build straight roads, probably to ease
their distance computations.
[32]
But in ordinary terrain, roads are winding more or less, so on
a winding road, the real distance of x miles
travelled, computes to somewhat more than the absolute
distance of two points on the map.
[33]
The latter is also called the bird-flight or straight
distance.
[34]
From the Roman measure derives
the english
or american mile of today.
[35]
The Roman legion marching speed was
variable, depending on terrain and
urgency, from 3 to 4 miles per hour.
[36]
3 miles per hour computes to 4,8280 km/h.
[37]
4 miles per hour computes to 6,4373 km/h.
[38]
Back to the example:
For the example we count here in single strides
(gradi).
100 strides * 74,0880 cm
[39]
results to rounded: 74 meters.
This is the distance of the race course that both Achilles and
the Tortoise must traverse.
Now come some more running speed/time/distance calculations:
[40]
Lets assume that Achilles doesn't really run but takes the
accelerated Roman marching speed of about:
[41]
6 km/h. See the speed
calculations.
[42]
This results to:
6 km/h = 6.000 m/h = 100 m/min = 1,6666 meters/sec
We also need to know how many human strides (of 74,0880 cm)
Achilles makes per sec.
This is about: 2,2521 human strides per sec. The time needed
to traverse 100 strides or 74,0880 m
at 6 km/h is 44.4 sec. Also the time step or clock needs to be
quantified. One second is an arbitrary
measure, a better one would be the average human heartbeat
when running leisurely.
But we don't need such precision, and we stay with the
Second.
Now let us assume that the tortoise has a starting advance of
37.04 meters or 50 human strides,
which is half the running distance. Then let us assume that
the tortoise has a speed of:
[43]
0,6 km/h = 0,1666 meters/sec = 32,8083 feet/minute
We need to take into account that a tortoise has a different
stride than a human.
Let us assume the tortoise stride is 7,40880 cm.
Now we can set up a time/distance computing program. This is
can for example be done with two
concurrent processes, one for Achilles, and one for the
tortoise. At each clock count, each process
communicates with the other which absolute distance from the
start both have travelled.
We can also make a graph, exactly determining the position of
Achilles at each clock second.
Likewise we can make a graph, exactly determining the position
of tortoise at each clock second.
To write a program for this, I leave this as an exercise to
computer science students.
I will not go into more programming details, but it is clear,
that in 1 clock second:
Achilles has traversed: 1.6666 meters
The tortoise has traversed: 0.1666 meters
To traverse the 37 meters of tortoise's starting advance,
Achilles needs 22.20 secs. But since we have a
computer simulation stepping with a fixed rate, we have to
look at clock cycle 23 secs.
This makes 38,3 m distance traversed.
[44]
At clock cycle 24 secs Achilles is at 40 m from
start.
At clock cycle 24 secs the tortoise is at 37 m + 4 m = 41
m.
At clock cycle 25 secs Achilles is at 41,6 m.
At clock cycle 25 secs the tortoise is at 37 m + 4,1 m = 41,1
m.
By this, the race is over.
[45]
To visualize, we can take both time/distance graphs, overlay
them and see immediately, when Achilles
has overtaken the tortoise. No matter what Zeno thinks of the
matter. Achilles will have overpassed the
tortoise at a very fixed point of distance and clock time.
This is given by natural logic, and can be shown
with the diagram. Because both Achilles and the tortoise
proceed by strides, not by infinitesimal
quantities.
The problem is the wrong application of mathematics, and
Zeno's impossible prescription. If you make an
impossible prescription, the outcome is also impossible. This
is the GIGO
[46]
principle of computing.
When we make an infinitesimal approximation, as Zeno does, it
is computationally very difficult to
calculate the exact split second when Achilles will have
reached the tortoise.
This correlates to the level of precision that we want to
achieve.
This is a quite well known computing problem, since computers
only simulate real numbers with floating
point arithmetic, and they can never go to infinitesimal, but
they must iterate or recurse n times.
[47]
This must stop at some level n because there is only a limited
amount of RAM or disk space
available to keep track of the process. Thus, it is quite easy
to calculate the exact clock step at which
Achilles will have surpassed the tortoise. When one applies
some basic engineering and computer
knowledge, Zeno's paradox dissolves into nothing.
To repeat the reasoning: it makes the matter more transparent
when we consider that both
partners move in strides, which means fixed distance quanta
that cannot be subdivided into
infinitesimals.
A human stride is 74,0880 cm, and the running distance is
74,0880 meters or 100 strides.
So the winner is clear. Believe Zeno or not. It is the
computer simulation.
The Arrow Paradox: Mass, Size, Speed, Time and Brownian Motion
The Arrow Paradox is a little different.
What Zeno didn't know, is that the speed of an object in the
mesocosmic
[48]
world
of our daily experience is something entirely different from
the microcosmic world.
As said above, there is no measurable movement on the
subatomic level,
[49]
everything in the mesocosmic world must move by n times the
quantum of movement,
which is the diameter of an atom.
So, it is actually quite difficult to figure exactly how an
arrow moves. For the engineer, we have the next
best explanation: An arrow moves just like a caterpillar on a
quantum level at quantum speeds.
This is also known as longitudinal oscillation.
[50]
It means: By each universal clock-tick,
the arrow
contracts for some nano- or pico- or femto- meter in some
nano- or pico- or femto- seconds,
and then it expands again. When it has expanded, it anchors
its tip at the new position and pulls up its tail
by some nano- or pico- or femto- xy- meters.
[51]
Since we cannot see nor measure at the
quantum level,
the exact nature of the movement of an arrow remains a
mystery. In this, Zeno is right.
We still don't really know what (loco-)motion is.
[52]
It is interesting to note that we can observe the transversal
oscillation of an arrow with a high speed
camera. The arrow bends up and down and sidewise in a
sine-wave pattern.
[53]
This is quite amazing,
since
one ordinarily thinks that the arrow moves stiff and straight
to its target. In Real nature, there is never
anything stiff and straight like in mathematics or logics. It
all comes in oscillations or resonances.
Now some additional reasonings. An arrow is a momentum
machine. Speed or movement in air is a
function of the momentum of an object restricted by the
resistance of the surrounding medium, which is
air in this case. In water, the resistance is so much higher,
and there torpedoes are the weapon of choice.
The power of an arrow results from its relatively high mass,
as compared to its diameter. This is also the
reason for its transversal oscillation, since its mass center
moves around outside the straight line of a
resting arrow as it oscillates.
The increase in length of factor x results in an increase of
Mass as x**3. Mass and momentum
[54]
are
proportional. The diameter parameter is the prime factor of
air resistance. The less diameter, the less air
resistance. Obvious. On the downside, a decrease in length of
factor x results in a decrease of mass as
x**-3. If you decrease length by about x= (5-10) factors of
magnitude, mass decreases at x**-(5-10).
Which comes to almost nothing. (Infinitesimal).
Let us assume for simplicity that an arrow traverses 60 m in 1
sec.
[55]
This results to:
[56]
60
m/sec = 216 km/h
The Zeno Arrow Paradox constructs an infinite regress,
reducing length and traverse to infinitesimal.
The catch is the same as with the Achilles paradoxon. If the
arrow has a length of 1 m, this means its
stride is 1 m.
[57]
But this
is more difficult to visualize, because the arrow is at point x by some clock n,
and
at point x+1 m by some clock n + 1/100 sec. If we set the
clock cycle to 1/1000 sec, the arrow has then
moved by 10 cm in each cycle. We can set the clock cycle to
any 10**x fraction, the arrow always moves
by 10**-x m. So its stride still remains 1 m per 1 second
clock.
Infinitesimal is nice in mathematics, but not in Reality, and
also not on a computer. In the Real Universe,
there is no infinitesimal entity.
[58]
At some deeper microscopic nanometer
level, all resolves to quantum
mechanics. At the microscopic nanometer level, any particle
that is suspended in air, cannot move as it
will, since its motion is inhibited by a phenomenon called
Brownian Motion. It has so many collisions
with air molecules of the same mass, that it cannot move at
all. So it will not move anywhere. Thus, an
arrow reduced to the length of an air molecule cannot move at
all.
This is the dis-(ill- /sol-)ution of the Zeno Arrow paradox.
Zeno has cleverly constructed an illusion for
us, and the world has fallen prey to this for almost 2300
years now.
[1]
Zeno paradoxes:
Google:
(URL)
http://www.google.de/#hl=d......sclient=psy-ab&q=Zeno+pa,or.r_gc.r_pw.r_qf.&fp=273aa0d90065244e&biw=1124&bih=715
(URL)
http://www.learner.org/courses/mathilluminated/.../04.php
(URL)
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
(URL)
http://de.wikipedia.org/wi......illes_und_die_Schildkr%C3%B6
(URL)
http://plato.stanford.edu/entries/paradox-zeno/
(URL)
http://mathworld.wolfram.com/ZenosParadoxes.html
This gives also one of the shortest refutation of Zeno's
paradoxes. Quoted from the above:
The dichotomy paradox leads to
the following mathematical joke.
A mathematician, a physicist and
an engineer were asked to answer the following question.
A group of boys are lined up on
one wall of a dance hall,
and an equal number of girls are
lined up on the opposite wall.
Both groups are then instructed
to advance toward each other
by one quarter the distance
separating them every ten seconds
(i.e., if they are distance d
apart at time 0, they are d/2 at t=10,
d/4 at t=20, d/8 at t=30, and so
on.)
When do they meet at the center
of the dance hall?
The mathematician said they
would never actually meet because the series is infinite.
The physicist said they would
meet when time equals infinity.
The engineer said that within
one minute
they would be close enough for
all practical purposes.
[2]
(URL)
http://www.stamoulis.de/index_de.html
[3]
Since I am equally well
versed in Software Engineering and Philosophy and Etymology,
I can (if I will) take the hat of a philosopher, and like a a
philosopher, I can think in/of an imaginary world
of infinite impossibilities. That is nice, and it puts me in
the same rank with a theologian, if I should want
to. A theologian can think of God anything s/he wants to,
because God is beyond any limits. But most of
the times, I prefer my other hat: That of the (computer-)
engineer. With this hat, I impose on myself some
thinking restrictions: I then can think only in the limits of
a Real Universe of Finite Actualities.
And this is the base of the present discussion.
[4]
The reason why ancient
Greek philosophers were not too concerned with practical matters is that manual
work was considered not befitting to a Greek Gentleman.
(This reminds us vividly of present-day Greek
problems.)
Manual work was reserved for slaves. Therefore, the ancient
Greek philosophers didn't learn the practical
experiences that one can only gain by dirtying one's hands
with solid, messy, dirty, and sturdy matter.
Aristoteles was the exception, because in his profession as a
doctor, he had to deal with messy matters all
the time, especially blood and guts, sperm, saliva, and other
such un-nicieties.
Platon was the exact opposite. He was an aristocrat of the old
classes, and he had probably never touched
any messy base matter in his life. It is known that he had
practically no sexual life, and so he didn't have
to deal with the messy and dirty aspects of this part of human
existence.
For these and other reasons, Platon's philosophy was useless
and sterile for any practical matters.
Unfortunately, philosophers down the ages had overlooked this
glaring problem of Platon's philosophy.
Philosophy as a grand human adventure is therefore eternally
confused by the god-like enshrinement of
the Great Grand Master Platon. But the ancient Greek
philosophers were only a small group
even though they were disproportionally influential on the
thought of humanity.
This is because of the peculiarities of the literature
traditon of the christian monks who propagated these
works. If there existed any technical writings by ancient
Greek engineers, the medieval monks couldn't
understand these, and they couldn't copy them, and so these
were lost to posterity.
Vitruvius is one of the few whose technical works were
preserved. But without the technical diagrams.
Plinius (Pliny) the elder was also one writer who had
technical skill.
(URL)
http://ancienthistory.about.com/.../a/PlinyPompeii.htm
(URL)
http://en.wikipedia.org/wiki/Pliny_the_Elder
His work was one of the few preserved down the ages.
Unfortunately without the drawings.
Greek civilization could not have survived, if it had had no
skilled engineers. There were many very
practical ancient Greek engineers and technicians, who
excelled in productions for the Greek war
machine. (See Alexander's conquests) Of these, not much is
recorded in the history books.
(URL)
http://www.google.de/searc......suvius+pompei&hl=de&
[5]
The best known of these was
Archimedes, then comes closely Heron of Alexandria.
Unfortunately, they didn't succeed to start the industrial
revolution 2300 years ago.
This was due to the slave system.
There was no need for technical improvement when slave labor
was abundant.
Archimedes is best known for his screw.
(URL)
(URL)
http://www.google.de/searc......rew+archimedes&hl=de&
(URL)
(URL)
http://en.wikipedia.org/wiki/Archimedes
(URL)
(URL)
http://de.wikipedia.org/wiki/Archimedes
Another good example: The engineers who built the Antikythera,
2300 years ago, were very accomplished
practical engineers. They invented something, that after these
2300 years, one would call a computer.
(URL)
http://de.wikipedia.org/wiki/Mechanismus_von_Antikythera
(URL)
http://en.wikipedia.org/wiki/Antikythera
(URL)
http://www.spektrum.de/alias/technikgeschichte/.../1065416
(URL)
http://www.spiegel.de/wissenschaft/...
But the technical people had a lesser social status because
they had to work with their hands.
[7]
Bronze casting in ancient
Greece 2300 years ago was at a technical level that was probably never
surpassed anywhere and anytime later. A possible exception
were the cannons of Urban or Orban,
but only for sheer mass, not for intricacy. (See: the conquest
of Byzantium).
(URL)
http://en.wikipedia.org/wiki/Fall_of_Constantinople
(URL)
http://en.wikipedia.org/wiki/Dardanelles_Gun
(URL)
http://en.wikipedia.org/wiki/Byzantine%E2%80%93Ottoman_Wars
The unsurpassed examples of the ancient skill are the Riace
bronzes, also called Riace Warriors.
(URL)
http://en.wikipedia.org/wiki/Riace_bronzes
They have survived the ravages of a lesser medieval humanity
because they had been
buried in the sea for 2000 years.
Most of the other ancient greek bronze statues had been molten
down for their metal.
There is one more exception from history:
The horses of Venice, stolen from Byzantion in the raid of
1204.
(URL)
http://en.wikipedia.org/wiki/Horses_of_Saint_Mark
(URL)
http://www.crimethrutime.com/
In Germany, we have the Bavaria Statue in Munich,
which seems to be the only specimen that comes close to the
ancient Greek technology.
(URL)
http://en.wikipedia.org/wiki/Bavaria_statue
The statue of Liberty in New York cannot compete at all with
this,
for technical reasons that would take another book to
explain.
(URL)
http://en.wikipedia.org/wiki/Statue_of_Liberty
I am an engineer, and I know this stuff. You can believe me or
not.
[8]
Also well known in
mathematics is Euklid for his foundation of geometry, but his theorems are
entirely
impractical. He had just re-formulated the ancient knowledge
of the Egyptian engineers so that absolutely
no-one could understand it. This could be called a
master-piece of reverse-engineering.
[9]
One more paradoxical
Gedankenexperiment: If one would accelerate some atomic matter to the speed of
light, which is impossible, but only here for us to think of:
Then this matter would have infinitesimal
diameter, and also infinitesimal mass. But, because of its
momentum, it would have infinite mass.
Both cannot go together.
And therefore I believe that Einstein didn't understand some
of the very basics of this universe.
Most of the rest of his followers didn't understand anything
more nor else.
And for this, there are so many Nobel Prices awarded. What a
waste.
[10]
This is for Hegel's
(in)famous exclamation:
"If the facts contradict the theory, then this is too bad for
the facts"
[11]
(URL)
http://en.wikipedia.org/wiki/Second
A good example is that with every advance in measuring
technology, the clock time step becomes
shorter. But it never beomes infinitesimal.
The present scientifically established clock tick is the
oscillation of a caesium atom. See:
(URL)
http://en.wikipedia.org/wiki/Atomic_clock
A good discussion is in Prof. Peter Janich's works about the
inter-connection of technology and theory.
But there still remains a time step. Believe Einstein or not.
There is no such thing as a continuum.
Especially not in time. There is always some illusionism, even
in science:
(URL)
http://www.spiegel.de/wissenschaft/...
[12]
Actually, this is called
"quantum mechanics". But this is another discussion and would lead too far from
the present subject.
[13]
Here comes in quantum
theory, because light can also be considered
and behaves as (continuous) wave.
(URL)
http://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality
(URL)
http://en.wikipedia.org/wiki/Light
(URL)
http://www.spectroscopyonline.com/...
(URL)
http://de.wikipedia.org/wiki/Meter
Meter m 100 Grundmaß
Zentimeter cm 10-2 10 mm Veraltet (um 1900):
Centimeter
Millimeter mm 10-3 1.000 µm 10 Millimeter sind 1
Zentimeter. Entspricht einem
Tausendstel Meter.
Mikrometer µm 10-6 0,001 mm Veraltete Bezeichnung:
Mikron, im technischen
Sprachgebrauch auch als µ (sprich mü)
bekannt.
Nanometer nm 10-9 1000 pm Entspricht einem Milliardstel
Meter (einem Millionstel
Millimeter).
Ångström Å 10-10 100 pm gebräuchlich
in der Atomphysik und in der Kristallographie
Pikometer pm 10-12 0,001 nm
Entspricht einem Billionstel Meter (einem Milliardstel
Millimeter).
[14]
To show where the
conceptions of motion are going wrong, we quote from the above
article.
In the Real Universe out there, NOTHING moves in a straight
line. It all moves in ellipsoid spiralloids.
It should be obvious that matter
— anything that has mass and takes up space — is ultimately
particulate. In fact,
matter defines what it is to be
a particle. Particles come in pieces that can be broken into smaller pieces, and
smaller pieces can be fused
together to make larger pieces. No two pieces can occupy the same exact point in
space, a concept that applies at
the atomic and molecular level for matter that is a solution, a liquid, or a
gas. Matter
in motion tends to move in a
straight line unless some force acts on it, and matter in motion tends to follow
Newton's laws of motion (which
can be reformulated into Lagrange's or Hamilton's laws of motion). Moving matter
can be blocked by the presence
of other matter, much like an umbrella blocks the paths of raindrops falling on
it.
Moving matter also can bounce
off of surfaces, as demonstrated by many a basketball player.
[15]
Actually, time is
another illusionist thing. This is because time is a measure of changes that can
be
observed, versus observed quasi-universal recurrences and
quasi-universal constants. The day and the year
are such quasi-universal recurrences, and the star
constellations of the night sky are quasi-universal
constants, because they haven't changed perceptibly in many
millennia, too slow for human observation.
But, if nothing changes, no difference can be observed, and
there might as well have been no time
involved. Since in the Real Universe, things are changing all
the time, humans got the impression, that
there is something like time, and they simply invented the
concept of time. Because modern
instrumentation can detect differences at an ever finer scale,
therefore the clock-tick becomes ever finer.
But it is never infinitesimal.
(URL)
http://en.wikipedia.org/wiki/Clock
(URL)
http://en.wikipedia.org/wiki/Detection_limit
[16]
Movement or
(loco-)motion can only be observed / measured down to the level of atoms. So the
lowest
delimiter of movement of matter is the diameter of an atom.
There is no such thing as infinitesimal
movement or (loco-)motion.
[17]
Of course, there is
movement at the subatomic level, of all kinds of xyz-ons, like phot-ons or
electr-ons
or neutr-ons or neutrino's. But it cannot be measured, and the
only practical use is in particle accelerators
(like Cern), but not in everyday households. By relativity,
the faster they move, the less diameter they
have, and at the speed of light, they have infinitesimal
diameter. Therefore, what works for photons,
doesn't work for atomic matter. This is a different ballgame,
as one would say colloquially.
It is impossible to accelerate atomic matter to the speed of
light.
(URL)
http://en.wikipedia.org/wiki/Real_number
Here we are focussing on the irrational numbers among the Real
Numbers. There are no irrational
numbers found in Reality (meaning in the Real Universe out
there). The simple reason why an irrational
number is unreal (or illusionistic), is: An irrational number
has an infinite number of non-repeating digits
behind the decimal point, or comma (german vs. english
notation). This means, if we wanted a Real
representation of this number, like as a computer value, or on
a piece of paper, we would have the
problem that the universe does not hold enough atoms to
represent it. This means, only finite
approximations to irrational Real Numbers can exist. The
practical limit of approximation is the largest
number of bytes by which a computer stores its floating point
numbers.
[19]
Let us take Pi, the best
known irrational Real Number.
(URL)
http://www.wolfram.com/mathematica/...
(URL)
http://www.google.de/searc......+circle&hl=de&prmd=i
Pi is defined as the relation of the circumference of a
perfect circle to its radius.
(URL)
http://en.wikipedia.org/wiki/Pi
This is mathematically nice, but unfortunately impossible in
practice. The first problem is that there exist
no perfect circles in the Universe. The ancient Greek
philosophers preferred to think in/of perfect circles
for their aesthetic reasons. But this caused serious problems,
for example in cosmology. In the "Real
Universe out there", everything spins around something,
because of Gravitation. And the spinning
movements are not in circles, but in ellipsoids and other
curvoids, mostly spiralloids.
For example: Planets travel around their home suns in
ellipsoid spiralloids. Also their home suns move in
their galaxies in some spiralloid fashion, and their galaxy
centers move also from somewhere to
somewhere. So far, science has established that everything
moves from somewhere to something.
Then, the earth is not a perfect sphere, and it turns around
at about:
1,670 km per hour at the equator, which makes 0.46 km per
sec.
(URL)
http://geography.about.com/od/learnabouttheearth/...
(URL)
http://geography.about.com/library/faq/blqzearthspin.htm
(URL)
http://en.wikipedia.org/wiki/Earth
So, when you start drawing a circle on a presumably solid
earth ground, this will not be a perfect circle
when you have finished. Assuming that it took you 1 sec for
the drawing, the end of the circle is at
0.46 km distance from the start, if you were at the equator.
By the time you end drawing your circle,
your planet has moved so much in a spiralloid manner. First by
rotating around itself, then by travelling
around the home sun.
See:
(URL)
http://curious.astro.cornell.edu/question.php?number=356
This is at about 100,000 km/h, making it 27,777...
km/sec.
And then, our own sun has travelled from somewhere in the
galaxis to somewhere else.
Therefore, Pi, although mathematically nice, does not exist in
this Real Universe.
(URL)
http://en.wikipedia.org/wiki/Circle
(URL)
http://en.wikipedia.org/wiki/Circumference
(URL)
http://www.mathopenref.com/radius.html
(URL)
http://www.mathgoodies.com/lessons/vol2/circumference.html
[20]
That an irrational Real
Number has no correspondent in the Real Universe, is a matter of clumsy
naming only, and one can still work with the concept. But this
style of mis-naming is one reason why so
many normal people find it hard to work with higher
mathematics.
Another good example of illusionism in mathematics is the name
of "imaginary numbers". Irrational Real
numbers are as imaginary as imaginary numbers. That naming
scheme doesn't distinguish them in any
meaningful sense. Of course this doesn't inhibit a trained
mathematician to work with them successfully.
The problem is again that common-sense lay people tend to get
confused.
[21]
And there are many more
counter-intuitive terms in the sciences, like in physics. The term "energy" is
entirely misleading ( for common-sense lay people), because
en-ergia in ancient Greek means "the
potential for work". But this is called "free energy" in
physics.
The term "entropy" poses a similar problem, because en-tropia
in ancient Greek means "the potential for
change". In physics, it is defined as exactly the opposite.
The wording: "unavailable for doing useful
work" makes it sound like it comes from a lawyer. See this
quote:
(URL)
http://en.wikipedia.org/wiki/Entropy
"Entropy is a thermodynamic property that is the measure of a
system’s thermal energy per unit temperature that is unavailable for doing
useful work."
(URL)
http://en.wiktionary.org/wiki/entropy
[22]
"Illusionist" means here
that one makes an impossible assumption, but somehow gets away with
it.
(URL)
http://www.learner.org/courses/mathilluminated/.../04.php
This is a quote from the above:
The idea of a limiting value to
an infinite process is at the heart of calculus.
So, infinity became a tool that
could be used, as long as one didn't look too closely at exactly how it worked.
Mathematicians came to accept
that one could indeed have a finite limit to an infinite sum. This concept made
it
possible to arrive at a finite
magnitude by summing an infinite number of infinitely small pieces. Such pieces,
which
became known as infinitesimals,
have, in some disturbingly vague sense, arbitrarily small but non-zero
magnitudes.
The great Newton, one of the
fathers of the calculus, the revolutionary new theory of the 1600s that
described
motion both in the heavens and
on Earth, at first based his ideas on these troublesome infinitesimals. It
wasn't until
the 1800s that Augustine Cauchy
turned matters around and developed a sound base for the subject by speaking of
limits. This had a profound
effect on the concept of "number," for Cauchy also found a consistent way to
give
meaning to irrational
quantities, essentially defining them as limits of sequences of rational
quantities. For example,
let's return to the mysterious
quantity 0.101001000100001000.... Cauchy would define this as the limit of the
sequence of rationals, 0.1,
0.101, 0.101001, 0.1010010001, .... This shift of perspective represented a
marrying, of
sorts, of the potential and the
actual infinite, and it brought some logic to the concepts of the infinity of
irrationals
and the infinite sums that arise
in calculus.
As calculus began to assume a
larger and larger role in both math and science, the need to understand infinity
became greater. This quest for
understanding ultimately required a shift in thinking, away from looking at
whole
numbers and magnitudes, toward
thinking about sets. In the next section, we will see some of the fundamental
ideas in this new way of
thinking.
[23]
(URL)
http://www.metric-conversions.org/
Interestingly, even though people come in many different
sizes, their bodily proportions are much more
uniform. Proportionality is one of the keys for understanding
nature. The golden Section and the
Fibonacci Series play an essential role there.
[24]
(URL)
http://en.wikipedia.org/wiki/Procrustes
(URL)
http://www.zeit.de/2012/30/Puritanismus
(URL)
http://www.zeit.de/lebensart/partnerschaft/...
[25]
"Best" is a matter of
preference. This is entirely based on experience.
What works best, is best. This is one of the prime rules of
engineering. It has nothing to do with beauty,
nor elegance, nor originality. Mostly, engineering style is
neither beautiful, nor elegant, nor original.
It just works, does what needs to be done. It is only what
works time-trusted and proven. And
mathematical beauty and originality are entirely irrelevant.
This is the Roman style of thinking, versus the
ancient Greek style. The Romans were always oriented to
practical matters, and eventually, because of
this, they conquered the Greeks because of this thinking
style. Modern engineers are a consequential
follower group in the spirit of Roman engineers.
(URL)
http://www.roman-empire.net/army/army.html
(URL)
http://www.roman-empire.net/army/training.html
(URL)
http://www.roman-empire.net/army/soldier-practice.html
(URL)
http://en.wikipedia.org/wiki/Military_step
(URL)
http://library.thinkquest.org/26602/onthemarch.htm
[26]
The Greeks had their
specific olympic standard measure of the stadion.
(URL)
http://en.wikipedia.org/wiki/Stadion_%28unit_of_length%29
(URL)
http://www.google.de/searc......eek+stadium&hl=de&pr
But such trifles are not important here. Here the task is to
construct a realistic engineering solution to the
problem.
[27]
(URL)
http://www.ehow.com/how_23......measure-ones-step-stride.htm
[28]
(URL)
http://de.wikipedia.org/wi......e_Ma%C3%9Fe_und_Gewichte_%28
[29]
The ancient Roman milia
passuum.
(URL)
http://www.google.de/#hl=d......gs_nf=1&cp=13&gs_id=,or.r_gc.r_pw.r_qf.&fp=273aa0d90065244e&biw=1124&bih=715
(URL)
http://de.wikipedia.org/wiki/Meile
(URL)
http://www.dl.ket.org/latinlit/mores/measures/measure.htm
[30]
The success of Arminius
against Varus owes not only to the tactics, but also to the fact that the Roman
legionaries couldn't maintain their usual step frequency
because of morass ground. So they literally fell
out of step, and could not fight efficiently.
(URL)
http://de.wikipedia.org/wiki/Arminius
[31]
When it snows, the
stride will shorten to almost nothing. This is called: The Great Slide. No
wonder
that the Romans hated snow. All these insecurities and even
more, were recorded in the timeless manual
of warfare:
(URL)
http://de.wikipedia.org/wiki/Carl_von_Clausewitz
(URL)
http://gutenberg.spiegel.de/buch/4072/1
(URL)
http://www.clausewitz.com/readings/VomKriege1832/TOC.htm
There is another, also timeless manual of warfare: Sun Tsu/
Sun Tze/ Sun Tzi:
(URL)
http://en.wikipedia.org/wiki/Sun_Tzu
(URL)
http://de.wikipedia.org/wiki/Sunzi
(URL)
http://en.wikipedia.org/wiki/The_Art_of_War
(URL)
http://de.wikipedia.org/wi......_Kunst_des_Krieges_%28Sunzi%
(URL)
http://www.aphorismen.de/d......_aphorismen.php?search=9&
[32]
(URL)
http://www.google.de/searc......man+roads&hl=de&prmd
(URL)
http://en.wikipedia.org/wiki/Roman_roads
[33]
Some Greek islands have
extremely winding roads. There, a bird-fly distance of 40 km can easily
translate to 80 km road distance. To make matters worse: On an
extremely winding road, the car travelling
speed will also be reduced to, say: 30 km/h in high gear. This
gives an extremely increased fuel
consumption of about 200 %. Also, there are no gasoline
stations on the way, and so you might be stuck
in the middle of nowhere because you mis-calculated the time
and gasoline for the trip. And this damned
island is only 40 km across. I have experienced it myself.
This was hard-won experience.
[34]
Not even this is
straight, because the earth is a more or less flattened sphere, and: Because of
the wind
shift, the bird must compensate for the aberration. This will
lengthen the trip a more or less trifle bit.
[36]
"Horses and mules can
travel at about the same speed as a man, 3 to 4 miles per hours."
(URL)
http://www.google.de/searc......e&noj=1&site=webhp&a
(URL)
http://garyb.0catch.com/march2_basics/march_basics.html
[37]
(URL)
http://www.machinehead-software.co.uk/...
1.0000 mph =
1.6093 kmh =
0.4470 meters/sec =
87.9999 feet/minute
[38]
(URL)
http://www.machinehead-software.co.uk/...
4.0000 mph
6.4373 kmh
1.7881 meters/sec
351.9999 feet/minute
[39]
In these calculations,
it is important to note that english / american usage for numerical
"." is equivalent to ","
[40]
Running speed
calculation:
(URL)
http://www.coolrunning.com/engine/4/4_1/96.shtml
(URL)
http://www.machinehead-software.co.uk/...
[41]
If needed, a legion
could maintain this speed for 7 hours a day (42 km), 7 days a week (294 km), and
with 7 weeks marching (2058 km), they would have covered the
whole Roman Empire from one end to
the other. This was the strategic calculation rule by which
the Romans moved their troops.
This was also not much different from Napoleonic
times.
(URL)
http://www.roman-empire.net/army/army.html
(URL)
http://www.roman-empire.net/army/training.html
(URL)
http://www.roman-empire.net/army/soldier-practice.html
(URL)
http://www.therthdimension.org/AncientRome/...
An ordinary day's march for the Roman army consisted of 15-18
miles done in 7 of our hours
(or 5 of the Roman summer hours).
(URL)
http://www.romanarmytalk.com/rat/...
Marching speed depends, of course, on the terrain covered, the
baggage carried and the length of the
march overall, but I believe it's often stated that the Roman
legion was expected to cover 20 miles on a
good day - Caesar's forced marches I think did 25. In reality,
I would think 15-18 more likely for a
standard legion with all its baggage during normal operations.
The baggage train is the real marker for the
speed of the legion as a whole, and Caesar often rushed his
troops on ahead of the train (when he was
marching on the Nervii, for example) to pick up the
pace.
As for individual speeds, they could be very swift indeed -
Suetonius (Caesar, 57) has Caesar travelling
100 miles in one day during a journey from Rome to western
Gaul. Plutarch writes that the freedman
Icelus carried the news of Nero's death from Rome to Galba in
Spain in only seven days, which would
have required a speed of around 10 miles an hour: certainly
possible with regular relays of horses, and
these were available for rapid carriage of important news (or
people). Without the rush, journeys would be
much longer - 20 miles a day on horseback is quite tiring
enough!
This was in summer, of course - winter could make journeys
infinitely protracted, or even impossible. Vitellius' legions took two months to
march from the Rhine to northern Italy in the early spring of 69AD.
1.6666 meters/sec
328.0839 feet/minute
0.3728 mph
0.1666 meters/sec
32.8083 feet/minute
[44]
There is a cognitive
trap to easily fall into, since in a second, Achilles makes 2,2521 strides,
which is
unreal. Achilles can make only 1 stride per clock time, and
the clock cycle should be set to one stride. So
the internal clock would be 0.44 sec's, so that Achilles can
make exactly one stride per clock cycle.
[45]
At clock cycle 26 secs
Achilles is at 43,3 m.
[46]
Garbage in, Garbage
out.
[47]
The problem is also
known as the floating point rounding error, when we take a floating point number
function, and recurse it many times, feeding the result into
the input. After a certain number of back-feeds,
the calculation will consist of rounding errors only. This
pattern can be plotted as a fractal, and from there
results the strange self-similarity which is a distinct
repeating pattern of float number rounding errors.
(URL)
http://books.google.de/boo......6AzR86m780wC&pg=PA89&
(URL)
http://books.google.de/boo......PS0LAAAAIAAJ&pg=PA199&am
[49]
The electrons in the
atom shell, for example, don't move at all. They form a cloud of probability
positions.
(URL)
http://en.wikipedia.org/wiki/Longitudinal_wave
(URL)
http://demonstrations.wolfram.com/...
(URL)
http://www.google.de/searc......ngitudinal+oscillation&h
[51]
One can also rationalize
that as space-time vibrations, since by relativity theory, any moving object
contracts a little, and expands when it is still. By this, the
caterpillar-like motion of an arrow can also be
visualized. But the worlds of relativity and of the quantum
level can not be computationally translated.
They are incommensurable.
(URL)
http://www.thefreedictionary.com/incommensurable
(URL)
http://simple.wikipedia.org/wiki/Incommensurability
(URL)
http://plato.stanford.edu/entries/incommensurability/
(URL)
http://www.learner.org/courses/mathilluminated/.../03.php
[52]
(URL)
http://wiki.answers.com/Q/What_is_motion
(URL)
http://en.wikipedia.org/wiki/Motion_%28physics%29
(URL)
http://www.wisegeek.com/what-is-motion-physics.htm
[53]
In the literature, this
is also known as:
horizontal flexural vibration and vertical flexural vibration.
(URL)
http://www.vsgstapelfeld.de/training/...
(URL)
http://de.wikipedia.org/wiki/Pfeil_%28Geschoss%29
spine:
Die Anpassung des „spine“ an Bogen und
Schütze ist besonders bei den traditionellen Schützen wichtig,
da die Sehne sich beim Schuss genau auf den Bogen zubewegt,
der Pfeil sich aber um den Bogen
herumwinden muss. Ein falscher „spine“ bedeuten
einen unruhigen Flug oder das (unerwünschte)
Anschlagen des Pfeilschafts an den Bogen. Idealerweise sollten
die Schwingungen des Pfeilschaftes nach
einigen 10 Metern Flug gedämpft sein.
(URL)
http://en.wikipedia.org/wiki/Momentum
(URL)
http://en.wikipedia.org/wiki/Velocity
[55]
Google will give you
exact numbers for everything. Search for: arrow speed calculator
and you will get some interesting data:
(URL)
http://www.vsgstapelfeld.de/training/...
(URL)
http://www.backcountrybowhunting.com/articles/calc/
(URL)
http://www.stickemarchery.com/stickemcart/...
(URL)
http://www.kyudo-sum.de/ballistik/.../uebersicht.html
(URL)
http://de.wikipedia.org/wiki/Pfeil_%28Geschoss%29
(URL)
http://www.kyudo-sum.de/ballistik/...
(URL)
http://www.campus-lingen.hs-osnabrueck.de/...
(URL)
http://www.bogensportverlag.com/.../index.html
The actual speed of an arrow is about 200 - 300 feet per sec.
216.0000 kmh
60.0000 meters/sec
11811.0234 feet/minute
[57]
In reality, the length
is about 75 cm which is quite exactly the Achilles stride measure.
[58]
(URL)
http://de.wikipedia.org/wiki/Meter
(URL)
http://books.google.de/boo......Rhi7odTe2BEC&printsec=fr
(URL)
http://books.google.de/boo......YMv7jPGUUk0C&printsec=fr