The document explores the concept of paradoxes, particularly mathematical paradoxes, which are statements that contradict themselves or seem illogical but can still hold truth. It covers various types of paradoxes, examples from literature, and introduces notable paradoxes like the grandfather paradox and those related to time travel. Additionally, it discusses self-referential statements and circular reasoning as core aspects of understanding paradoxes in both recreational mathematics and everyday language.

1. Mathematical Paradoxes
(Recreational Mathematics)
K.THIYAGU
Assistant Professor
Department of Education
Central University of Kerala, Kasaragod

2. A paradox is a statement that goes against
our intuition but may be true,
or a statement that is self-contradictory.
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3. No Keyboard Detected. Press F1
Paradox
Self-Contradictory Statement
or Concept
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4. A paradox is a statement or group of
statements that leads to a contradiction or a
situation which defies intuition. The term is
also used for an apparent contradiction that
actually expresses a non-dual truth.
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5. The Green
Button is True
The Red
Button is False
Paradoxes
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6. A Paradox is a form of the figure of
speech which when heard or read can
seem to be completely
self-contradicting.
The Phrase may be contrary to
what is expected in the situation
and can lead to a conclusion which
is not logically acceptable
conclusion.
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7. A Paradox is a paradox, one that
baffles and confuses, and is a
statement that seemingly
contradicts itself.
Examples:
I Must be cruel to be kind. (William Shakespeare) Hamlet
All animals are equal, but some are more equal than others.
(George Orwell) Animal Farm
The statement may appear to be silly but in
reality, contains a truth which seems
unreasonable.
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8. Paradox of Yore:
Which came first,
The egg or the hen?
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9. A mathematical paradox is any statement (or a set of statements) that
seems to contradict itself (or each other) while simultaneously seeming
completely logical.
Paradox (at least mathematical paradox) is only a wrong statement that
seems right because of lack of essential logic or information or application
of logic to a situation where it is not applicable.
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10. Example
• War is Peace
• Freedom is Slavery
• Ignorance is Strength
• My Weakness of my Strength
In which a statement appears
to contradict itself
Paradoxes
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11. Paradox
• "I know one thing," Socrates famously said.
"That I know nothing.“
• It's a crucial insight from one of the founders
of Western philosophy: You should question
everything you think you know.
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12. Paradox in Everyday Language
He is a wise fool.
He is Nothing.
He is Nobody
The truth is honey, which is bitter
Paradox in Everyday Language
I can resist all things apart from temptation
Nobody goes to that bar, it is too busy
Let God create a stone which he cannot lift
It is weird to not be weird
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17. Patrick Hughes outlines three laws of the paradox:
Self reference :
An example is
"This statement is false",
a form of the Liar paradox.
The statement is referring to
itself.
Contradiction:
The statement cannot be false
and true at the same time.
Vicious circularity or infinite
regress :
"This statement is false" --
If the statement is true,
then the statement is false.
"The statement below is false."
"The statement above is true."
Logical
Paradox
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18. Types of Paradox
Veridical Paradox
A veridical paradox produces a result that
appears absurd but is demonstrated to be true
nevertheless.
Example: Frederic’s birthday paradox.
Falsidical Paradox
A falsidical paradox establishes a result that not
only appears false but actually is false due to a
fallacy in the supposed demonstration.
(e.g. that 1 = 2, Horse paradox)
Antinomy
A paradox which is in neither class may be an
antinomy, which reaches a self-contradictory
result by properly applying accepted ways of
reasoning.
Example: Grelling–Nelson paradox.
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19. Example
Consider a situation in which a
father and his son are driving
down the road.
The car collides with a tree and
the father is killed. The boy is
rushed to the nearest hospital
where he is prepared for
emergency surgery.
On entering the surgery suite,
the surgeon says,
"I can't operate on this boy.
He's my son."
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20. If the surgeon is the
boy's father,
the statement
cannot be true.
The paradox is
resolved
if it is revealed that the
surgeon is a woman,
the boy's mother.
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21. Paradox of Time Travel
• In a TV programme "Into The Universe With Stephen
Hawking" on Discovery Channel, Stephen Hawking
described the following paradoxical condition:
• "Suppose you have somehow built a time machine. Now
you open up a kit containing parts of a revolver. You
assemble all the parts and then load the revolver with
bullets. Now you go into past, 5 minutes back (the time
when you have actually not set up your revolver) via the
time machine and you shoot yourself (the man that
belongs to your past who was kind of going to assemble
the revolver) using the same revolver.“
• Here comes the twist. First of all, if the revolver has
actually not been assembled, then with what have you
killed yourself, and the second part is, "is it really
possible to kill yourself in the past, if a time machine
would have existed?"
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22. • This is in fact known as grandfather's paradox.
By definition, it is described as follows:
• "Suppose a time traveller goes back in time
and kills his grandfather before his
grandfather met his grandmother. As a result,
the time traveller is never born. But, if he was
never born, then he is unable to travel through
time and kill his grandfather, which means the
traveller would then be born after all, and thus
we arrive at a contradiction and are ‘stuck’ in
this ‘loop’.
• It was first described by the science fiction
writer Nathaniel Schachner in his short story
Ancestral Voices and by René Barjavel in his
1943 book Le Voyageur Imprudent (Future
Times Three).
Paradox of Time Travel
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23. Prove
2 = 1
a2 = ab
a2 - b2= ab - b2
(a+b) (a-b)= b(a-b)
a + b= b
2a= a
2 = 1.
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24. Howlers
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25. Prove 1 = 0
X = 0
Therefore x(x-1) = 0
Therefore x-1 =0
Therefore x = 1
Therefore 1 = 0
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26. Three men go to a hotel to
rent a room. The desk clerk
says there is a 3 bed suite for
30 dollars. Each man pays 10
dollars and go to the room.
The hotel manager questions
the desk clerk. The hotel
manager tells the desk clerk,
the room is only 25 dollars,
and gives the clerk 5 dollar
bills to pay the men back.
On the way to the room the
clerk realizes 5 dollar bills
can’t be split 3 ways.
Dishonestly, he keeps 2 dollars
and gives each man back one
dollar. Now the 3 men paid
Rs.27.00 for the room, and the
clerk has 2 dollars. Where did
the other dollar go?
27.00 + 2.00 = 29.00!
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27. “Mathematics is the language with
which God has written the
universe.”
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28. Thank You
K.THIYAGU
Assistant Professor
Department of Education
Central University of Kerala, Kasaragod CC-BY-SA (Paradox) 28