Not Seeing A Category "Poetry"

[Sower]

continued; (I'm no mathematician)

          What Mathematical Paradoxes Teach Us about Paradoxes in Christianity
                                Paul Bialek

In Christian academic circles, we talk about the integration of our faith and learning. That is,
we seek to discover and develop connections between our Christian faith and our particular
discipline. This is notoriously difficult when the discipline is mathematics.
I have found that asking myself these three questions has helped me to integrate my Christian
faith with mathematics, although they could be applied to any discipline:
• How does the fact that I am a Christian affect the way I view mathematics?
• How does the fact that I am a mathematician affect the way I view Christianity?
• What do Christianity and mathematics have in common?
In my opinion, one reason it seems so difficult to integrate the Christian faith and mathematics,
especially to non-mathematicians, is because the focus is on the first question. Is mathematics any
different for Christians? Are the rules of calculus different? Is there such a thing as a Christian
chain rule? There are ways in which Christian thought has influenced mathematics. For example, at
the 1991 ACMS conference, Gene Chase spoke about how Christian theology has furthered
mathematics. I have found, however, that in terms of integrating my faith and mathematics, the
second question is more fruitful. How does my mathematical background affect the way I view
Christianity? How does it enable me to make a contribution to the kingdom of God?
An excellent example is the work of one of our main speakers, William Dembski. In his
1998 book [1], he uses a sophisticated mathematical analysis to show how incredibly unlikely it is
that we are merely products of random evolutionary chance.
Today I am going to present another example of how our mathematical background affects
the way we view Christianity, namely what mathematical paradoxes teach us about paradoxes in
Christianity.
What do we mean by a paradox? A paradox is an apparent contradiction in a field of
study. It may be a genuine contradiction or it may involve two truths which merely appear to
contradict each other.
One thing that mathematical paradoxes teach us about paradoxes in Christianity is that the
existence of paradoxes does not automatically invalidate a body of knowledge.
67
Again, some mathematical paradoxes have been genuine contradictions. A well-known
example is Russell's paradox from set theory. This paradox forced set theorists to revise the
definition of a set, which was too broad and all-inclusive.
But not all paradoxes are genuine contradictions. Some involve truths which only appear to
contradict. For example, Georg Cantor showed that the cardinality (the size) of R (the set of real
numbers) is greater than the cardinality of Q (the set of rational numbers). From this we can
conclude that the cardinality of the set of irrationals is greater than the cardinality of Q. So in some
sense, there are more irrationals than rationals. Yet between any two irrational numbers there are
infinitely many rationals. How can there be more irrationals than rationals if there are infinitely
many rationals between any two irrationals? This is paradoxical, yet mathematicians today accept
the idea that the cardinality of the set of irrationals is greater than the cardinality of the set of
rationals. This paradox is only an apparent contradiction, not a genuine contradiction.
Thus the existence of paradoxes does not necessarily mean that a system of thought has errors
or flaws. Even sound bodies of knowledge may have paradoxes.
Christianity has many paradoxes:
• There is one God, yet he is three persons. How can three persons be one God?
• God is sovereign, yet humans have free will.
• Jesus Christ was fully God and fully man.
• God is all-powerful and perfect in his love, yet there is great suffering and
injustice in the world.
• God acts in the world (he is immanent) yet he does not reside in the world (he is
transcendent).
Some skeptics questions the validity of Christianity or even the existence of God because of
these paradoxes, yet the existence of paradoxes is no guarantee that a system of thought is false.
Even bodies of knowledge which are true may have paradoxes. This insight from mathematics
should be a great comfort to us Christians, because it shows that paradoxes in Christianity do not
necessarily invalidate the Christian faith.
To be sure, the existence of paradoxes in Christianity does not guarantee that Christianity is
true. It is also possible for false bodies of knowledge to have paradoxes. But rather than conclude
that a system of thought or a model is false because it has paradoxes, we should be asking ourselves
these two questions:
68
• Is this model superior to the alternatives?
The alternative models may have their paradoxes, too. For example, atheism has its paradoxes, the
existence of higher life forms being one of them. We have to weigh the evidence for and against
each of the models. David Neuhauser discusses this in greater detail in [3].
If we encounter a paradox, another question we should ask is:
• Are we exacerbating the paradox by making unwarranted assumptions?
For example, we said earlier that one of the great paradoxes of Christianity is the existence of pain,
evil, and suffering. If God is all-powerlul and perlect in his love, then why does he permit these to
such a great extent? In asking this question, we need to examine our assumptions. Are we assuming
that because God is perfect in his love that one of his greatest goals is for us to lead pain-free lives,
free of what we judge to be excessive suffering? Our priorities in life may not be the same as his
priorities. This paradox is a vexing one, and I do not intend to minimize its difficulty. But carefully
ferreting out our hidden assumptions may help us to resolve or at least accept these paradoxes.
We have seen that the existence of paradoxes does not necessarily invalidate a body of
knowledge. Another principle that mathematical paradoxes teach us about paradoxes in Christianity
is that rules and intuitive principles which work in the finite realm do not necessarily work in
the realm of the infinite.
Suppose we are in the finite realm. We have a bag with ten candy bars. If we take one out,
there are now fewer candy bars in the bag. If we divvy up our candy bars into two bags, each bag has
half as many as the original bag.
Let's switch into the realm ofthe infinite. We have a bag with infinitely many candy bars in
it, say a countably infinite number. If we take out a candy bar, the bag still has just as many candy
bars as it had before. The sets have the same cardinality. If we divvy up our candy bars into two
bags, amazingly, each bag has just as many candy bars as the original bag. Again, the cardinality of
the set of candy bars in each bag is the same as the cardinality of the set of candy bars in the original
bag.
This goes against our intuition. We are not accustomed to a realm where you can take things
out and still have just as much as when you started. But the rules are different in the realm of the
infinite.
Let us apply this to Christianity. What are some attributes of God which are infinite? What
are ways in which God is infinite?
69
• Knowledge (he is omniscient)
• Power (he is omnipotent)
• Holiness
• Justice
• Goodness
• Truth
• Being (he is omnipresent)
• Duration (he is eternal)
Let us consider God's infinite duration, his eternity. How does this principle apply? Millard
Erickson writes, "God is ... infinite with relation to time. Time does not apply to him. He was
before time began. The question, How old is God? is simply inappropriate. He is no older now than
a year ago, for infinity plus one is no more than infinity" [2, p. 274]. What is true of God will
someday be true of us. As the hymn "Amazing Grace" reminds us:
When we've been there ten thousand years
Bright shining as the sun
We've no less days to sing God's praise
Than when we'd first begun.
Let us consider God's omnipotence. How can God listen to and respond to one million
people praying simultaneously? Since he is all-powerful, he can handle an infinite number of tasks
simultaneously. Tackling a million tasks simultaneously is no big deal for him.
Finally, let us consider God's omniscience. Those who call themselves process theologians
contend that to be real is to be in process, to be changing. How can God be real and not be in
process, not be changing? Process theologians resolve this paradox by concluding that God is
changing, growing, developing, learning. But this resolution is very unscriptural. The Scriptures are
very clear that although God is personal and does interact with human beings and the rest of his
creation, he does not change or grow or learn. A better resolution of the paradox is to note that what
is true in the finite realm is not necessarily true in the infinite realm. A finite intelligent being who is
not changing or growing in knowledge must be dead or comatose or sleeping. God, however, is an
infinite intelligent being, infinite in his knowledge, goodness, righteousness and love. He is not
changing or growing or learning. Yet he is not dead or sleeping. He already knows everything (1
John 3:20), so there is nothing for him to learn. He is already perfectly good, righteous, and loving;
he does not grow in those areas. It is difficult to conceive of an intelligent being who does not grow
or change or learn, because we are accustomed to dealing with the finite. But the rules are different
with the infinite.
70
So as we have seen, one way to integrate our Christian faith with mathematics is to consider
how our mathematical background affects the way we view Christianity. The example we have
considered is how mathematical paradoxes help us to better understand Christian paradoxes. We
have seen that the existence of paradoxes does not automatically invalidate a body of knowledge and
that rules and intuitive principles which work in the finite realm do not necessarily work in the realm
of the infinite. Christians in mathematics who do not integrate their discipline with their faith are not
in as much danger as, say, Christian philosophers or psychologists who do not pay attention to
integration issues. This is mainly because mathematics is less susceptible to distortion by non-
Christian ideologies. But those who do so lose a wonderful opportunity to deepen their
understanding of their faith.

(you started all this, Neighbor:)

[In the Clouds]

 

The Village Blacksmith

Henry Wadsworth Longfellow

Under a spreading chestnut tree
  The village smithy stands;
The Smith, a mighty man is he,
  With large and sinewy hands;
And the muscles of his brawny arms
  Are strong as iron bands.

His hair is crisp, and black, and long,
  His face is like the tan;
His brow is wet with honest sweat,
He earns whate'er he can
And looks the whole world in the face
  For he owes not any man.

Week in, week out, from morn till night,
  You can hear his bellows blow;
You can hear him swing his heavy sledge,
  With measured beat and slow,
Like a sexton ringing the village bell,
  When the evening sun is low.

And children coming home from school
  Look in at the open door;
They love to see the flaming furge,
  And hear the bellows roar,
And catch the burning sparks that fly
Like chaff from a threshing floor.

He goes on Sunday to the church
and sits among his boys;
He hears the parson pray and preach.
He hears his daughter's voice
singing in the village choir,
And it makes his heart rejoice.

It sounds to him like her mother's voice,
Singing in Paradise!
He needs must think of her once more,
  How in the grave she lies;
And with his hard, rough hand he wipes
  A tear out of his eyes.

Toiling,--rejoicing,--sorrowing,
  Onward through life he goes;
Each morning sees some task begin,
  Each evening sees it close;
Something attempted, something done,
Has earned a night's repose.

Thanks, thanks to thee, my worthy friend
  For the lesson thou hast taught!
Thus at the flaming forge of life
  Our fortunes must be wrought;
Thus on its sounding anvil shaped
  Each burning deed and thought!

[TheBlade]

May I :)

When I consider how my light is spent,

  Ere half my days, in this dark world and wide,
  And that one Talent which is death to hide
  Lodged with me useless, though my Soul more bent
  To serve therewith my Maker, and present
  My true account, lest he returning chide;
  “Doth God exact day-labour, light denied?”
  I fondly ask. But patience, to prevent
  That murmur, soon replies, “God doth not need
  Either man’s work or his own gifts; who best
  Bear his mild yoke, they serve him best. His state
  Is Kingly. Thousands at his bidding speed
  And post o’er Land and Ocean without rest:
  They also serve who only stand and wait.”

John Milton

[In the Clouds]

Thank you Thank you!  How lovely.

[Retrobyter]

If You Will Come, We'll Surely Kneel To Thee

By Retrobyter

Seven hundred years
Before Christ's timely birth, 
A prophet of great boldness came
To tell the men of earth

That there would be a pow'rful Man
Anointed to be King;
Of His surprising miracles 
The populace would sing.

Of healing pow'r to make blind see,
And deaf to hear His voice,
Of pow'r even to raise the dead, 
And cause men to rejoice;

Of all these things and so much more
The children would sing songs ...
But only children welcomed Him to 
Remedy their wrongs!

A song once written years before 
Isaiah was a child
Predicted that Christ would not fit into
Systems then gone wild;

"The builders of the pyr'mid said,
'The stone does not fit in!'
When they were done, it CAPPED the work,
Much to their chagrin!"

The Messiah was rejected,
Ridiculed and slain,
But to the shock of all involved 
He CAME TO LIFE AGAIN!

But, He did not then "cap the work"
Of which the song declared; 
Instead, He left the earth until
He hears the message dared,

"We care not whether You were He
Of whom the Christians claim;
We only know we need you now,
Or we will die in shame.

"We've heard of you, Messiah fair,
And of your pow'r we read; 
Please come and rescue us, dear Sire,
In this our time of need!

If you don't save us now,
There won't be anyone left to save!
The nation Israel is doomed,
And we can see our grave!"

And, in the nation Israel's 
Darkest, gloomiest hour,
Will come the Prince (not yet the King)
In majesty and pow'r!

And, when they see Messiah riding
Down on kingly steed, 
And all the millions following
To aid the ones in need,

Then, shouts of victory will echo
All across the Land
With sighs of relief and gasps of awe,
As He saves with mighty hand!

And THEN will He be made a King,
Accepted now at last;
And THEN a King o'er many kings
As one by one they ask!

Ah! But for now He is the "Christ" -
"Anointed" TO BE King,
And although He is not King now,
He will be in the "Spring":

For now, the "Winter" slowly drags
With vernal guarantees,
And time goes on at God's own pace
With thoughtfulness and ease.

He fills each age with purpose,
And reason for each pause,
To get the most from ev'ry age,
And weed out all the flaws.

So, boast not Christian Gentile,
For you are graffed in,
To share Messiah's bounty
Prematurely; do not sin!

If God spared not the people
Whom He dealt with many years,
What will He do if you should speak
Against them without tears?

The Scriptures say so little
'Bout this time in which we live ...
Could it be that we've read into it
Much more than He did give?

Why must we feel that we're the ones
For whom the sun must shine?
Why do we think we're all who'll be
At the Marriage Feast to dine?

Get rid of that old foolish pride;
We're graffed in, I say!
This high and mighty attitude
Will come to humbler day!

It should be crushing blow to think
That Christ's return will be
When ISRAEL cries, "If you will come,
We'll surely kneel to Thee!"

 

[Michael37]

9 minutes ago, Retrobyter said:

If You Will Come, We'll Surely Kneel To Thee

By Retrobyter

Seven hundred years
Before Christ's timely birth, 
A prophet of great boldness came
To tell the men of earth

That there would be a pow'rful Man
Anointed to be King;
Of His surprising miracles 
The populace would sing.

Of healing pow'r to make blind see,
And deaf to hear His voice,
Of pow'r even to raise the dead, 
And cause men to rejoice;

Of all these things and so much more
The children would sing songs ...
But only children welcomed Him to 
Remedy their wrongs!

A song once written years before 
Isaiah was a child
Predicted that Christ would not fit into
Systems then gone wild;

"The builders of the pyr'mid said,
'The stone does not fit in!'
When they were done, it CAPPED the work,
Much to their chagrin!"

The Messiah was rejected,
Ridiculed and slain,
But to the shock of all involved 
He CAME TO LIFE AGAIN!

But, He did not then "cap the work"
Of which the song declared; 
Instead, He left the earth until
He hears the message dared,

"We care not whether You were He
Of whom the Christians claim;
We only know we need you now,
Or we will die in shame.

"We've heard of you, Messiah fair,
And of your pow'r we read; 
Please come and rescue us, dear Sire,
In this our time of need!

If you don't save us now,
There won't be anyone left to save!
The nation Israel is doomed,
And we can see our grave!"

And, in the nation Israel's 
Darkest, gloomiest hour,
Will come the Prince (not yet the King)
In majesty and pow'r!

And, when they see Messiah riding
Down on kingly steed, 
And all the millions following
To aid the ones in need,

Then, shouts of victory will echo
All across the Land
With sighs of relief and gasps of awe,
As He saves with mighty hand!

And THEN will He be made a King,
Accepted now at last;
And THEN a King o'er many kings
As one by one they ask!

Ah! But for now He is the "Christ" -
"Anointed" TO BE King,
And although He is not King now,
He will be in the "Spring":

For now, the "Winter" slowly drags
With vernal guarantees,
And time goes on at God's own pace
With thoughtfulness and ease.

He fills each age with purpose,
And reason for each pause,
To get the most from ev'ry age,
And weed out all the flaws.

So, boast not Christian Gentile,
For you are graffed in,
To share Messiah's bounty
Prematurely; do not sin!

If God spared not the people
Whom He dealt with many years,
What will He do if you should speak
Against them without tears?

The Scriptures say so little
'Bout this time in which we live ...
Could it be that we've read into it
Much more than He did give?

Why must we feel that we're the ones
For whom the sun must shine?
Why do we think we're all who'll be
At the Marriage Feast to dine?

Get rid of that old foolish pride;
We're graffed in, I say!
This high and mighty attitude
Will come to humbler day!

It should be crushing blow to think
That Christ's return will be
When ISRAEL cries, "If you will come,
We'll surely kneel to Thee!"

 

An epic example.

[Retrobyter]

1 hour ago, Michael37 said:

An epic example.

Shalom, Michael.

How do you mean?

[Michael37]

4 minutes ago, Retrobyter said:

Shalom, Michael.

How do you mean?

It's an academic term for a style of poetry, lengthy poetry of triumph and glory.

[Retrobyter]

2 minutes ago, Michael37 said:

It's an academic term for a style of poetry, lengthy poetry of triumph and glory.

Shalom, Michael.

Thank you. I was hoping that was an academic reference. I've also set it to music as a ballad.

[Retrobyter]

On 2/18/2024 at 1:37 AM, Sower said:

continued; (I'm no mathematician)

          What Mathematical Paradoxes Teach Us about Paradoxes in Christianity
                                Paul Bialek

In Christian academic circles, we talk about the integration of our faith and learning. That is,
we seek to discover and develop connections between our Christian faith and our particular
discipline. This is notoriously difficult when the discipline is mathematics.
I have found that asking myself these three questions has helped me to integrate my Christian
faith with mathematics, although they could be applied to any discipline:
• How does the fact that I am a Christian affect the way I view mathematics?
• How does the fact that I am a mathematician affect the way I view Christianity?
• What do Christianity and mathematics have in common?
In my opinion, one reason it seems so difficult to integrate the Christian faith and mathematics,
especially to non-mathematicians, is because the focus is on the first question. Is mathematics any
different for Christians? Are the rules of calculus different? Is there such a thing as a Christian
chain rule? There are ways in which Christian thought has influenced mathematics. For example, at
the 1991 ACMS conference, Gene Chase spoke about how Christian theology has furthered
mathematics. I have found, however, that in terms of integrating my faith and mathematics, the
second question is more fruitful. How does my mathematical background affect the way I view
Christianity? How does it enable me to make a contribution to the kingdom of God?
An excellent example is the work of one of our main speakers, William Dembski. In his
1998 book [1], he uses a sophisticated mathematical analysis to show how incredibly unlikely it is
that we are merely products of random evolutionary chance.
Today I am going to present another example of how our mathematical background affects
the way we view Christianity, namely what mathematical paradoxes teach us about paradoxes in
Christianity.
What do we mean by a paradox? A paradox is an apparent contradiction in a field of
study. It may be a genuine contradiction or it may involve two truths which merely appear to
contradict each other.
One thing that mathematical paradoxes teach us about paradoxes in Christianity is that the
existence of paradoxes does not automatically invalidate a body of knowledge.
67
Again, some mathematical paradoxes have been genuine contradictions. A well-known
example is Russell's paradox from set theory. This paradox forced set theorists to revise the
definition of a set, which was too broad and all-inclusive.
But not all paradoxes are genuine contradictions. Some involve truths which only appear to
contradict. For example, Georg Cantor showed that the cardinality (the size) of R (the set of real
numbers) is greater than the cardinality of Q (the set of rational numbers). From this we can
conclude that the cardinality of the set of irrationals is greater than the cardinality of Q. So in some
sense, there are more irrationals than rationals. Yet between any two irrational numbers there are
infinitely many rationals. How can there be more irrationals than rationals if there are infinitely
many rationals between any two irrationals? This is paradoxical, yet mathematicians today accept
the idea that the cardinality of the set of irrationals is greater than the cardinality of the set of
rationals. This paradox is only an apparent contradiction, not a genuine contradiction.
Thus the existence of paradoxes does not necessarily mean that a system of thought has errors
or flaws. Even sound bodies of knowledge may have paradoxes.
Christianity has many paradoxes:
• There is one God, yet he is three persons. How can three persons be one God?
• God is sovereign, yet humans have free will.
• Jesus Christ was fully God and fully man.
• God is all-powerful and perfect in his love, yet there is great suffering and
injustice in the world.
• God acts in the world (he is immanent) yet he does not reside in the world (he is
transcendent).
Some skeptics questions the validity of Christianity or even the existence of God because of
these paradoxes, yet the existence of paradoxes is no guarantee that a system of thought is false.
Even bodies of knowledge which are true may have paradoxes. This insight from mathematics
should be a great comfort to us Christians, because it shows that paradoxes in Christianity do not
necessarily invalidate the Christian faith.
To be sure, the existence of paradoxes in Christianity does not guarantee that Christianity is
true. It is also possible for false bodies of knowledge to have paradoxes. But rather than conclude
that a system of thought or a model is false because it has paradoxes, we should be asking ourselves
these two questions:
68
• Is this model superior to the alternatives?
The alternative models may have their paradoxes, too. For example, atheism has its paradoxes, the
existence of higher life forms being one of them. We have to weigh the evidence for and against
each of the models. David Neuhauser discusses this in greater detail in [3].
If we encounter a paradox, another question we should ask is:
• Are we exacerbating the paradox by making unwarranted assumptions?
For example, we said earlier that one of the great paradoxes of Christianity is the existence of pain,
evil, and suffering. If God is all-powerlul and perlect in his love, then why does he permit these to
such a great extent? In asking this question, we need to examine our assumptions. Are we assuming
that because God is perfect in his love that one of his greatest goals is for us to lead pain-free lives,
free of what we judge to be excessive suffering? Our priorities in life may not be the same as his
priorities. This paradox is a vexing one, and I do not intend to minimize its difficulty. But carefully
ferreting out our hidden assumptions may help us to resolve or at least accept these paradoxes.
We have seen that the existence of paradoxes does not necessarily invalidate a body of
knowledge. Another principle that mathematical paradoxes teach us about paradoxes in Christianity
is that rules and intuitive principles which work in the finite realm do not necessarily work in
the realm of the infinite.
Suppose we are in the finite realm. We have a bag with ten candy bars. If we take one out,
there are now fewer candy bars in the bag. If we divvy up our candy bars into two bags, each bag has
half as many as the original bag.
Let's switch into the realm ofthe infinite. We have a bag with infinitely many candy bars in
it, say a countably infinite number. If we take out a candy bar, the bag still has just as many candy
bars as it had before. The sets have the same cardinality. If we divvy up our candy bars into two
bags, amazingly, each bag has just as many candy bars as the original bag. Again, the cardinality of
the set of candy bars in each bag is the same as the cardinality of the set of candy bars in the original
bag.
This goes against our intuition. We are not accustomed to a realm where you can take things
out and still have just as much as when you started. But the rules are different in the realm of the
infinite.
Let us apply this to Christianity. What are some attributes of God which are infinite? What
are ways in which God is infinite?
69
• Knowledge (he is omniscient)
• Power (he is omnipotent)
• Holiness
• Justice
• Goodness
• Truth
• Being (he is omnipresent)
• Duration (he is eternal)
Let us consider God's infinite duration, his eternity. How does this principle apply? Millard
Erickson writes, "God is ... infinite with relation to time. Time does not apply to him. He was
before time began. The question, How old is God? is simply inappropriate. He is no older now than
a year ago, for infinity plus one is no more than infinity" [2, p. 274]. What is true of God will
someday be true of us. As the hymn "Amazing Grace" reminds us:
When we've been there ten thousand years
Bright shining as the sun
We've no less days to sing God's praise
Than when we'd first begun.
Let us consider God's omnipotence. How can God listen to and respond to one million
people praying simultaneously? Since he is all-powerful, he can handle an infinite number of tasks
simultaneously. Tackling a million tasks simultaneously is no big deal for him.
Finally, let us consider God's omniscience. Those who call themselves process theologians
contend that to be real is to be in process, to be changing. How can God be real and not be in
process, not be changing? Process theologians resolve this paradox by concluding that God is
changing, growing, developing, learning. But this resolution is very unscriptural. The Scriptures are
very clear that although God is personal and does interact with human beings and the rest of his
creation, he does not change or grow or learn. A better resolution of the paradox is to note that what
is true in the finite realm is not necessarily true in the infinite realm. A finite intelligent being who is
not changing or growing in knowledge must be dead or comatose or sleeping. God, however, is an
infinite intelligent being, infinite in his knowledge, goodness, righteousness and love. He is not
changing or growing or learning. Yet he is not dead or sleeping. He already knows everything (1
John 3:20), so there is nothing for him to learn. He is already perfectly good, righteous, and loving;
he does not grow in those areas. It is difficult to conceive of an intelligent being who does not grow
or change or learn, because we are accustomed to dealing with the finite. But the rules are different
with the infinite.
70
So as we have seen, one way to integrate our Christian faith with mathematics is to consider
how our mathematical background affects the way we view Christianity. The example we have
considered is how mathematical paradoxes help us to better understand Christian paradoxes. We
have seen that the existence of paradoxes does not automatically invalidate a body of knowledge and
that rules and intuitive principles which work in the finite realm do not necessarily work in the realm
of the infinite. Christians in mathematics who do not integrate their discipline with their faith are not
in as much danger as, say, Christian philosophers or psychologists who do not pay attention to
integration issues. This is mainly because mathematics is less susceptible to distortion by non-
Christian ideologies. But those who do so lose a wonderful opportunity to deepen their
understanding of their faith.

(you started all this, Neighbor:)

Shalom, @Sower.

I like to think of a mathematical paradox as a misinterpretation of the facts, usually associated with the dimension in which it is viewed.

We've all seen images like this:

And asked the question, "Is the ball in front of the box or behind it?" The truth is that this is NOT a three-dimensional box as portrayed, but is a HEXAGON in two-dimensional space, with two sets of three, intersecting lines within it! Those lines and the "ball" (a shaded-in circle) are on the same sheet of paper (or the flat screen of a computer monitor). Only when the interpreter IMAGINES this to be a "three-dimensional box" does the question become a paradox.

The same is true for images like this:

Again, this is actually 2-D not 3-D! It's truly just a trick of the artist upon a two-dimensional surface.

The Hangman's Paradox is similar:

The paradox has been described as follows: ["Unexpected Hanging Paradox". Wolfram]

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left – and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday noon, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning, he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner's door at noon on Wednesday – which, despite all the above, was an utter surprise to him. Everything the judge said came true.

Other versions of the paradox replace the death sentence with a surprise fire drill, examination, pop quiz, A/B test launch, a lion behind a door, or a marriage proposal. [Chow, T. Y. (1998). "The surprise examination or unexpected hanging paradox". The American Mathematical Monthly. 105(1): 41–51.]

The problem here is that he cannot use a "chain rule" for these propositions. All he can reason is that it will either happen on Friday (which he "correctly" rules out) or on NON-FRIDAY, which is the lump of the other days in the week! He cannot logically assume that "the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday noon, the hanging must occur on Thursday, making a Thursday hanging not a surprise either." While Thursday is a part of the NON-FRIDAY set, Friday has not truly been eliminated from the problem! Indeed, THINKING that "Friday has been eliminated" makes Friday a VIABLE SOLUTION! He WON'T be expecting it!

The same is true for John 7:27:

John 7:25-27 (KJV)

25 Then said some of them of Jerusalem, "Is not this he, whom they seek to kill? 26 But, lo, he speaketh boldly, and they say nothing unto him. Do the rulers know indeed that this is the very Christ? 27 Howbeit we know this man whence he is: but when Christ cometh, no man knoweth whence he is."

However, in ASSUMING they knew from whence He came, they DECEIVED themselves with a similar paradox! It's IRONIC!

Edited March 5 by Retrobyter
to fix a wrong-word problem