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nb_christseeker

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and now for some quotes......... yay!

I browsed www.quotemeonit.com by author and took the ones I liked. Feel free to peruse the collection. Peace!




Aesop

Beware that you do not
lose the substance by
grasping at the shadow.

A crust eaten in peace is better than
a banquet partaken in anxiety.

Men often applaud an imitation
and hiss the real thing.

A doubtful friend is worse than a
certain enemy. Let a man be
one thing or the other, and we
then know how to meet him.

Wealth unused might as well not exist.

The injuries we do and those
we suffer are seldom weighed
in the same scales.





Arthur Ashe

When bright young minds can't
afford college, America pays the price.

If I were to say, "God, why me?"
about the bad things, then I should
have said, "God, why me?" about the
good things that happened in my life.





Isaac Asimov

Violence is the last refuge
of the incompetent.




Margaret Atwood

In the spring, at the end of the
day, you should smell like dirt.

Does feminist mean large unpleasant
person who'll shout at you or someone
who believes women are human beings.
To me it's the latter, so I sign up.

If the national mental illness of the
United States is megalomania, that of
Canada is paranoid schizophrenia.

For years I wanted to be
older, and now I am.

An eye for an eye only
leads to more blindness.






Rowan Atkinson

I have a plan so cunning you could
stick a tail on it and call it a weasel.

For telling a good and incisive religious joke,
you should be praised. For telling a bad
one, you should be ridiculed and reviled. The
idea that you could be prosecuted
for the telling of either is quite fantastic.




W H Auden

A real book is not one that we
read, but one that reads us.

A daydream is a meal at which images
are eaten. Some of us are gourmets,
some gourmands, and a good many take
their images precooked out of a can and
swallow them down whole, absent-
mindedly and with little relish.

The center that I cannot find is
known to my unconscious mind.

All sins tend to be addictive, and the
terminal point of addiction is damnation.

Man is a history-making creature who can
neither repeat his past nor leave it behind.

We are here on earth to do good to others.
What the others are here for, I don't know.

When one has great gifts, what answer to
the meaning of existence should one require
beyond the right to exercise them?






James Augustine

It is as painful perhaps
to be awakened from a
vision as to be born.






Jane Austen

In nine cases out of ten,
a woman had better
show more affection
than she feels.

One does not love a place the less
for having suffered in it
unless it has all been suffering,
nothing but suffering.

For what do we live, but to make
sport for our neighbors,
and laugh at them in our turn?

There is safety in reserve,
but no attraction. One cannot
love a reserved person.









Mary Austin

Man learned to resort to the
dance when he felt helpless
or fragmentary, when he felt
dislocated in his universe.

That curious social warp which
obligates us most to impeach the
validity of a woman's opinion at
the points where it is most
supported by experience.

No man naturally can imagine
any more compelling business
for a woman than being
interested in him.





John Ashcroft

Islam is a religion in which Allah demands you
send your son to die for him; Christianity is the faith
in which God sent his son to die for you.





Hank Aaron

I don't feel right unless I have a sport to play
or at least a way to work up a sweat.

You can only milk a cow so long, and
then you're left holding the pail.

It took me seventeen years to get three
thousand hits in baseball. I did it in one
afternoon on the golf course.

I never smile when I have a bat in my hands.
That's when you've got to be serious.
When I get out on the field, nothing's a joke
to me. I don't feel like I should walk
around with a smile on my face.









Edward Abbey

Anarchism is founded on the
observation that since few men
are wise enough to rule
themselves, even fewer are
wise enough to rule others.

Society is like a stew.
If you don't keep it stirred up,
you get a lot of scum on top.

Growth for the sake of growth is
the ideology of the cancer cell.

If the end does not justify
the means - what can?

There is science, logic, reason; there
is thought verified by experience.
And then there is California.

One man alone can be pretty
dumb sometimes, but for real
bona fide stupidity, there ain't
nothin' can beat teamwork.

May your trails be crooked, winding,
lonesome, dangerous, leading to the most
amazing view. May your mountains
rise into and above the clouds.

The only thing worse than a
knee-jerk liberal is a knee-
pad conservative.

Our culture runs on coffee
and gasoline, the first often
tasting like the second.






Lyman Abbott

Patience is passion tamed.

It is easy to condemn,
it is better to pity.





Peter Abelard

It took me fifteen years to discover that I had
no talent for writing, but I couldn't give it up
because by that time I was too famous.




Ralph Abernathy

Bring on your tear gas,
bring on your grenades,
your new supplies of Mace
your state troopers and even
your national guards. But
let the record show we ain't
going to be turned around.

I'm sick and tired of black and
white people of good intent
giving aspirin to a society that
is dying of a cancerous disease.







Peter Abrahams

To get where you want to go
you can't only do what you like





Bella Abzug

Women have been trained to
speak softly and carry a lipstick.
Those days are over.






Jane Ace Sherwood

Home wasn't built in a day.

I'm a ragged individualist.

Time wounds all heels.








Dean Acheson

A memorandum is written
not to inform the reader
but to protect the writer.

The first requirement of a
statesman is that he be dull.
This is not always easy to achieve.
(think Howard Dean)

Time spent in the advertising
business seems to create a
permanent deformity like the
Chinese habit of foot-binding.





Diane Ackerman

I don't want to get to the end of my
life and find that I have just lived
the length of it. I want to have
lived the width of it as well.






John Dalberg Acton

Look at a day when you are supremely
satisfied at the end. It's not a day when you
lounge around doing nothing; it's when you've
had everything to do, and you've done it.

The long term versus the short term
argument is one used by losers.

If some great catastrophe is not announced
every morning, we feel a certain void.
Nothing in the paper today, we sigh.

There are two things which cannot be
attacked in front: ignorance and narrow
mindedness. They can only be shaken by
the simple development of the contrary
qualities. They will not bear discussion.

The danger is not that a particular
class is unfit to govern. Every class
is unfit to govern.





Abigail Adams

A little of what you call frippery is
very necessary towards looking
like the rest of the world.

Learning is not attained by chance.
It must be sought for with ardor
and attended to with diligence.

I am more and more convinced that
man is a dangerous creature and
that power, whether vested in many
or a few, is ever grasping, and
like the grave, cries, "Give, give.''






Ansel Adams

There are always two people in every picture:
the photographer and the viewer.

You don't take a photograph, you make it.






Douglas Adams

He hoped and prayed that there wasn't
an afterlife. Then he realized there was a
contradiction involved here and merely
hoped that there wasn't an afterlife.

A common mistake that people
make when trying to design something
completely foolproof is to underestimate
the ingenuity of complete fools.

The major difference between a thing
that might go wrong and a thing that
cannot possibly go wrong is that when
a thing that cannot possibly go wrong
goes wrong it usually turns out to
be impossible to get at or repair.

He felt that his whole life was some
kind of dream and he sometimes
wondered whose it was and
whether they were enjoying it.

Space is big. You just won't believe
how vastly, hugely, mind- bogglingly
big it is. I mean, you may think it's a long
way down the road to the drug store,
but that's just peanuts to space.

Human beings, who are almost
unique in having the ability to learn
from the experience of others, are
also remarkable for their
apparent disinclination to do so.

I love deadlines. I especially
like the whooshing sound they
make as they go flying by.

It is no coincidence that in no
known language does the phrase
'As pretty as an Airport' appear.

To give real service you must add
something which cannot be bought or
measured with money, and that
is sincerity and integrity.

I really didn't foresee the Internet.
But then, neither did the computer industry.
Not that that tells us very much, of course -
the computer industry didn't even foresee
that the century was going to end.

You live and learn.
At any rate, you live.









Scott Adams

There's a fine line between
participation and mockery.

Be careful that what you write
does not offend anybody or cause
problems within the company.
The safest approach is to
remove all useful information.

There's nothing more dangerous
than a resourceful idiot.

You can never underestimate the
stupidity of the general public.





Jane Addams

The excellent becomes the permanent.

Old-fashioned ways which
no longer apply to changed
conditions are a snare in
which the feet of
women have always
become readily entangled.

Civilization is a method of
living and an attitude of equal
respect for all people.






Joseph Addison

The friendships of the world are oft
confederacies in vice,
or leagues of pleasures.

If we may believe our logicians, man is
distinguished from all other creatures by
the faculty of laughter. He has a heart
capable of mirth, and
naturally disposed to it.

It seems only imperfection complains of what
is imperfect. The more perfect we are
the more gentle and quiet we become
towards the defects of others.





Konrad Adenauer

History is the sum total
of things that could have
been avoided.

A thick skin is
a gift from God.

All parts of the human
body get tired eventually -
except the tongue.

The art of politics consists in
knowing precisely when it is
necessary to hit an opponent
slightly below the belt.

In view of the fact that God
limited the intelligence of man,
it seems unfair that He did
not also limit his stupidity.

Only the stupidest
calves choose their
own butcher.

We all live under the same
sky, but we don't all have
the same horizon.







Alfred Adler

It is easier to fight for one's principles
than to live up to them.

The test of one's behavior pattern;
relationship to society,
relationship to one's work,
relationship to sex.

The greater the feeling of inferiority
that has been experienced, the more
powerful is the urge to conquest and
the more violent the emotional agitation.

Exaggerated sensitiveness is an
expression of the feeling of inferiority.

We must interpret a bad temper
as a sign of inferiority.

Man knows much more
than he understands.






Freda Adler

The passionate controversies of one
era are viewed as sterile preoccupations
by another, for knowledge alters
what we seek as well as what we find.

It is not only by the questions we
have answered that progress may
be measured, but also by those
we are still asking.

Major social movements. . . eventually
fade into the landscape not because they
have diminished but because they
have become a permanent part of our
perceptions and experience.

The type of fig leaf which each culture
employs to cover its social taboos offers
a twofold description of its morality. It
reveals that certain unacknowledged
behavior exists and it suggests the
form that such behavior takes.

Stripped of ethical rationalizations
and philosophical pretensions,
a crime is anything that a group in
power chooses to prohibit.

There is another side to
chivalry. If it dispenses leniency,
it may with equal justification
invoke control.

Rape is the only crime in
which the victim becomes
the accused.








James Agee

As a whole part of "psychological education"
it needs to be remembered that a neurosis
can be valuable; also that "adjustment" to
a sick and insane environment is of itself
not "health" but sickness and insanity.

God doesn't believe in the easy way.

You must be in tune with the times and
prepared to break with tradition.








Bette Midler

I don't know much,
but I know I love you,
and that may be
all I need to know.







Spirow Agnew

An intellectual is a man who doesn't
know how to park a bike.

Yippies, Hippies, Yahoos, Black Panthers,
lions and tigers alike -- I would swap
the whole damn zoo for the kind of
young Americans I saw in Vietnam.






Troy Aikman

Losing doesn't eat at me
the way it used to. I just get
ready for the next play, the
next game, the next season.

When you get in a game like
this it comes down to a few plays...
They made the most of their
opportunities and made some
big plays, and we failed to
do that. We had our chances
and we let them get away.





Edward Albee

I have a fine sense of the ridiculous,
but no sense of humor.

Good writers define reality; bad ones
merely restate it. A good writer turns
fact into truth; a bad writer will, more
often than not, accomplish the opposite.

If Attila the Hun were alive
today, he'd be a drama critic.







Louisa May Alcott

Let my name stand among those
who are willing to bear ridicule and
reproach for the truth's sake, and
so earn some right to rejoice
when the victory is won.

It takes very little fire to make a great
deal of smoke nowadays,
and notoriety is not real glory.

A faithful friend is a strong
defense; And he that hath found
him hath found a treasure.

Far away there in the sunshine
are my highest aspirations. I
may not reach them, but I
can look up and see their beauty,
believe in them, and try
to follow where they lead.

People want to be amused, not preached at,
you know. Morals don't sell nowadays.

Talent isn't genius and no amount
of energy can make it so. I
want to be great, or nothing. I
won't be a commonplace dauber, so
I don't intend to try any more.

Fame is a pearl many dive for
and only a few bring up. Even
when they do, it is not perfect,
and they sigh for more, and lose better
things in struggling for them.

"Stay" is a charming word
in a friend's vocabulary.

Have regular hours for work
and play; make each day both
useful and pleasant, and prove
that you understand the worth
of time by employing it well. Then
youth will be delightful, old age
will bring few regrets, and life will
become a beautiful success.









Alan Alda

Here's my Golden Rule for a tarnished
age: Be fair with others, but keep
after them until they're fair with you.

Be as smart as you can, but remember
that it is always better
to be wise than to be smart.






Shana Alexander

Trying to squash a rumor is
like trying to unring a bell.

The sad truth is that excellence
makes people nervous.





William R Alger

We give advice by the bucket,
but take it by the grain.

False eloquence is exaggeration;
true eloquence is emphasis.

He who has no wish to be happier
is the happiest of men.

Public opinion is a second conscience.

The wealth of a soul is measured by how
much it can feel; its poverty by how little.

What is the highest secret to victory
and peace? To will what God wills,
and strike a league with destiny.

Words of love, are works of love.









George Allen

Forget the past -- the
future will give you plenty
to worry about.

If you want to catch more
fish, use more hooks.

Winning is the science of
being totally prepared.

Iron rusts from disuse, stagnant
water loses its purity and in cold
weather becomes frozen; even
so does inaction sap the vigors
of the mind.

Every day you waste is one
you can never make up.

Leisure time is that five
or six hours when you
sleep at night.

Try not to do too many things at once.
Know what you want, the number
one thing today and tomorrow.
Persevere and get it done.

Success is what you
do with your ability. It's how
you use your talent.

The achiever is the only individual
who is truly alive. I see no difference
in a chair and the man who sits
in the chair, unless he's
accomplishing something.

Work hard, stay positive,
and get up early. It's the
best part of the day.






Steve Allen

Humor is a social lubricant
that helps us get over some
of the bad spots.

Radio is the theater of the
mind; television is the theater
of the mindless.

I used to be a heavy gambler.
But now I just make mental bets.
That's how I lost my mind.

One of the nice things about problems
is that a good many of them do not exist
except in our imaginations.

Men will take almost any kind of
criticism except the observation that
they have no sense of humor.

Totalitarianism is patriotism
institutionalized.

The hair is real; it's the
head that's fake.





Woody Allen

A relationship, I think, is like a shark,
you know? It has to constantly move
forward or it dies. And I think what
we got on our hands is a dead shark.

His lack of education is more than
compensated for by his keenly
developed moral bankruptcy.

80% of success is showing up.

I am thankful for laughter,
except when milk comes
out of my nose.

More than any time in history mankind
faces a crossroads. One path leads to
despair and utter hopelessness, the other
to total extinction. Let us pray that we
have the wisdom to choose correctly.

I think being funny is not
anyone's first choice.

My one regret in life is that
I'm not someone else.







Gordon Willard Allport

People who are aware of, and
ashamed of, their prejudices are well
on the road to eliminating them.

Love--incomparably the greatest
psychotherapeutic agent--is something
that professional psychiatry cannot of
itself create, focus, nor release.





Robert Altman

If you don't have a leg to stand on,
you can't put your foot down.




Cleveland Amory

I can't take a well-tanned
person seriously.

As anyone who has ever been around
a cat for any length of time well knows,
cats have enormous patience with the
limitations of the human kind.







Tori Amos

I think that happiness is when you can let
yourself feel every emotion you want at any
time instead of being a lying little ****.





Margaret Anderson

In real love you want the other
person's good. In romantic love
you want the other person.

Intellectuals are too sentimental for me.






Marian Anderson

Leadership should be born out of
the understanding of the needs of
those who would be affected by it.

Prayer begins where
our capacity ends.







Maxwell Anderson

There are some men who lift the age they inhabit,
till all men walk on higher ground in that lifetime.

Truth's like a fire, and will burn
through and be seen.






Mario Andretti

If everything's under control,
you're going too slow

Circumstances may cause
interruptions and delays, but
never lose sight of your goal.
Prepare yourself in every way
you can by increasing your
knowledge and adding to your
experience, so that you can
make the most of opportunity
when it occurs.







Maya Angelou

Therefore we pledge to bind
ourselves to one another, to embrace
our lowliest, to keep company with
our loneliest, to educate our illiterate,
to feed our starving, to clothe our
ragged, to do all good things,
knowing that we are more than
keepers of our brothers and sisters.
We are our brothers and sisters.

There's a world of difference
between truth and facts.
Facts can obscure the truth.

The quality of
strength lined with tenderness
is an unbeatable combination,
as are intelligence and necessity
when unblunted by formal education.

If one is lucky, a solitary
fantasy can totally transform
one million realities.

Since time is the one immaterial
object which we cannot influence--
neither speed up nor slow down,
add to nor diminish-- it is
an imponderably valuable gift.

Life loves to be taken by
the lapel and told:
"I'm with you kid. Let's go."

Perhaps travel cannot prevent
bigotry, but by demonstrating
that all peoples cry, laugh, eat,
worry, and die, it can introduce
the idea that if we try and
understand each other, we
may even become friends.








Jean Anouilh

Beauty is one of the rare things that
do not lead to doubt of God.
 
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Tinker Grey

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StrugglingSceptic said:
Not at all. I should think most mathematicians regard the arguments as extremely important historically, and most undergraduates are presented with them on analysis and set theory courses.

The arguments are generally explicit constructions of the set of real numbers in terms of rational numbers. That is, you build new objects from the rational numbers which you declare to be the real numbers, after showing how to define arithmetic for them and showing that they satisfy all the expected properties. Then, so long as someone believes that rationals exist, they are forced to conclude that reals also exist, because the reals can be constructed from them.

The first such construction was given by Richard Dedekind, and if you google for "dedekind cuts" you should be able to get an idea of what it involves (it can be quite technical, but a quick browse hasn't turned up anything more palatable).

The construction requires taking a real number to be a pair of infinite sets. Another construction (due to Cantor) requires taking a real number to be a class of infinite sequences. So both of these constructions require taking an infinite set to be a legitimate mathematical object, and for some philosophers and a small minority of mathematicians this is not acceptable (it does not appear to have been acceptable to Kronecker either).
Many thanks. I will look this up sometime.
 
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FreezBee

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StrugglingSceptic said:
Greek geometry talks of lines being measured by other, shorter lines, which divide them exactly. They discovered that the side of a square and its diagonal cannot be measured in this way by any other shorter line -- the two are incommensurable.

Incommensurability is usually described in terms of irrational numbers, but this is an anachronism, since the Greeks didn't have such a concept. And this brings me back to my original point: the Greeks got by just fine constructing circles and squares, without a concept of irrationals.
Again, I'll have to stir up my memory. As I recall it, PI is defined as the ratio between the circumference and the diameter (twice the radius) of a circle and as such a rational number. Now, the Greeks (again according to my memory - please correct me, if I'm wrong) only considered whole numbers existing, what we call rational numbers were considered to be ratios between whole numbers. That is, 3/4 was not 0.75, but the ratio between 3 and 4, that is, it was 3 out of 4, as if you cut a pie into 4 pieces and sold three of them, then you would have sold 3/4 of the pie.

As you write, it was assumed that any line could be divided into a whole number of shorter lines, and that for any two lines, there would be some common such shorter line, that would divide both lines evenly. And therefore that the ratio between the lengths of the two lines would be, what we call a rational number. Am I right this far?

As I recall they thought that for PI as well, attempting to figure out what whole numbers it was the ratio of.

It is true that the Greeks usually used geometric constructions with compass and ruler (the last one only to draw straight line segments, not for measuring), and therefore limited themselves to problems solvable by these two devices.

However, again according to my memory, the Pythagoreans considered numbers to be divine and thought that there had to be some unit that evenly divided into all the others, therefore thir shock to discover that there wasn't. Is this correct?


cheers

- FreezBee
 
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IiYiIiIzage

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I would first have to know how much power "god" has. However if he were to do that he would have to either change "how much" 2 was or "how much" 5 was. For instance he could change the value of 2 to 2.5 to make the equation correct or change the value of 5 to 4. On the other hand he could make the human minds much more ignorant and they wouldnt know the difference. But with that question we can only assume that their is a god and that he has the power to change the value of numbers.
 
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MoonlessNight

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psychedelicist said:
In euclidean math, no thing or person could make 2+2=5, for the same reason one couldn't create a square circle.

However, euclidean math is not some kind of absolute math, it is more relatable to a language. Anyone can make up a language any time they want. The usefulness or coherence of this language might not be so great, but it would still be 'valid'. Like language, anyone can make up a system of math, and it will still be just as 'valid', if not much less useful, than euclid's version.

Once this system is created, however, one cannot do something outside the boundaries of that system. It would be like a computer acting in a way it was not programmed, it simply isn't possible. That's not to say that you or god could not come up with a different math system in which 2+2=5, but you would have to redefine the meanings of 2, 5, +, and = to do so.

But as to whether anyone, god or otherwise, could make 2+2=5 within the boundaries of euclidean math, no. Outside those boundaries, anyone could make anything equal whatever they wanted.
What do you mean by Euclidean math? You certainly aren't talking about Euclidean geometry. Are you referring to Euclid's method of stating axioms and definitions and then proving theorems based on those axioms? But I hesitate to call anything without axioms (and especially without definitions) math, so I'm not sure what you are referring to. And you really seem to love that term.
 
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MoonlessNight

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As I recall it, PI is defined as the ratio between the circumference and the diameter (twice the radius) of a circle and as such a rational number.

Pi is a ratio, but rational numbers are ratios between two integers (whole numbers). Everything can be a ratio, just divide it by one. Pi being irrational means that you can never find two integers a and b such that pi exactly equals a/b. (Pi is in fact trancedental which for the purposes of this discussion we can consider to mean "really irrational", that is in a sense it is ever further removed from rational numbers than things like the square root of 2.)

Now, the Greeks (again according to my memory - please correct me, if I'm wrong) only considered whole numbers existing, what we call rational numbers were considered to be ratios between whole numbers. That is, 3/4 was not 0.75, but the ratio between 3 and 4, that is, it was 3 out of 4, as if you cut a pie into 4 pieces and sold three of them, then you would have sold 3/4 of the pie.

The Greeks never used or saw decimals, so the question of whether they would think 3/4=.75 is impossible to answer. If they knew that 75/100 = .75 and that the decimal is essentially a short hand for the fraction, they would agree that 3/4=.75. But you are right in saying that the greeks thought that everything ultimately reduced down to whole numbers. That is not to say that they thought every number was a whole number, as you have noted things between numbers (non-integer rational numbers) are represented as the ratio between numbers.

As you write, it was assumed that any line could be divided into a whole number of shorter lines, and that for any two lines, there would be some common such shorter line, that would divide both lines evenly. And therefore that the ratio between the lengths of the two lines would be, what we call a rational number. Am I right this far?

As I recall they thought that for PI as well, attempting to figure out what whole numbers it was the ratio of.

It is true that the Greeks usually used geometric constructions with compass and ruler (the last one only to draw straight line segments, not for measuring), and therefore limited themselves to problems solvable by these two devices.

Yep, but since irrational numbers do not divide anything, this process was doomed to failure with things like pi (as where many other important greek problems).

However, again according to my memory, the Pythagoreans considered numbers to be divine and thought that there had to be some unit that evenly divided into all the others, therefore thir shock to discover that there wasn't. Is this correct?

Well, one divides into everything and nothing really divides into fractions if you wanted to be technical. But if you mean "the greeks thought that for every number, there was some number such that the product of the two numbers was a whole number." you would be correct (this is the same as saying that every number is the ratio of two whole numbers).

And is true that the Pythagoreans were very committed to this belief in particular. There are stories that a pythagorean who proved that root 2 was irrational was immediately killed, but I don't think there's any historical basis to it.
 
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Citizen of Earth

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caitsey said:
Does God have the power to make 2 + 2 = 5?

No, only INGSOC has the power to do that. ;)

Just kidding.


To answer your question, we accept 2 + 2 = 4 because we've arbitrarily defined "2 + 2" to mean the identical thing as "4". In other words, there is no difference between saying the phrase, "2 plus 2" and by saying, "4".

2 + 2 could easily equal 0, or 5, or 10, or all three at the same time if that's how we define the terms before the fact.

The question is not one of fact, but of arbitrary definitional convention, which humans create and can change at will.

Now, if you are using the terms in the way you would during a regular math test, then no, not even God could, because it would be a logical impossibility by definition.
 
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MoonlessNight

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GreenDragon said:
Beat me to it :p

You don't have to go to God to make 2+2=5, I can do it myself. All I have to do is persuade everyone else to believe 2+2=5 and it will.
Just like how if I persuade other people that the sky is purple, the color of the sky magically changes.
 
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StrugglingSceptic

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FreezBee said:
It is true that the Greeks usually used geometric constructions with compass and ruler (the last one only to draw straight line segments, not for measuring), and therefore limited themselves to problems solvable by these two devices.

- FreezBee
It's been a while, but I'll just deal with this point since there has been some more response.

The Greeks were not limited to compass-ruler constructions. Euclid's Elements was limited to compass-ruler constructions. The Greeks often made use of more exotic curves, such as the trisectrix and the quadratix, and Archimedes' made much use of spirals. Such curves could solve problems which were not solvable by compass-ruler construction (non-plane problems). There was also significant study made of the conic sections, principally by Apollonius.
 
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