Best false inductive "proofs"?
Perdita Stevens
for a good selection of false inductive "proofs", the funnier the better.
You know the sort of thing, "all horses have one leg", and so on. I'd be
grateful if you could post or mail me your favourites!
Perdita
--
`How do I feel?' he cried. `Well, I don't know how to say it. I feel, I feel'
-- he waved his arms in the air -- `I feel like spring after winter, and sun
on the leaves; and like trumpets and harps and all the songs I have ever
heard!' JRR Tolkien
Ludwig Plutonium
p...@dcs.ed.ac.uk (Perdita Stevens) writes:
> As part of an attempt to teach students mathematical induction, I'm looking
> for a good selection of false inductive "proofs", the funnier the better.
> You know the sort of thing, "all horses have one leg", and so on. I'd be
> grateful if you could post or mail me your favourites!
The funniest is Math Induction itself. It is false. The Math Induction
Postulate in the Peano Axioms is as fake as the idea of the set of all
sets.
Show your students the P-adics and then they can appreciate how fake
Math Induction really is.
94th ELECTRON OF 231PU
Very crude dot picture of 5f6, 94TH ELECTRON
\ ::| :./.
.\::|::/.:
_ _
(:Y:)
- -
::/.|.\.:
:: /.:| :\.:
/ | \.
LP
One of those dots is the Sun with 9 smaller dots around it.
Look in a chemistry textbook or quantum physics textbook of the
electron cloud dot picture.
franklin
the proof that all odd numbers >1 are primes...
Math student: 3 is prime, 5 is prime, 7 is prime, the result follows by
induction.
Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is ... experimental error
11 is prime, 13 is prime, result by induction.
Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, result by
induction.
Computer scientist: 3 is prime, 3 is prime, 3 is prime,...
:-) :-) :-)
Dave
Arturo Viso Magidin
Perdita Stevens <p...@dcs.ed.ac.uk> wrote:
>As part of an attempt to teach students mathematical induction, I'm looking
>for a good selection of false inductive "proofs", the funnier the better.
>You know the sort of thing, "all horses have one leg", and so on. I'd be
>grateful if you could post or mail me your favourites!
>
My personal favorite.
Theorem: In any finite set of women, if one has blue eyes then they
all have blue eyes.
Proof. Induction on the number of elementes.
if n=0 or n=1 it is immediate.
Assume it is true for k
Consider a group with k+1 women, and without loss of generality assume
the first one has blue eyes. I will represent one with blue eyes with
a '*' and one with unknown eye color as @.
You have the set of women:
{*,@,...,@} with k+1 elements. Consider the subset made up of the first
k. This subset is a set of k women, of which one has blue eyes. By
the induction hypothesis, all of them have blue eyes. We have then:
{*,...,*,@}, with k+1 elements. Now consider the subset of the last k
women. This is a set of k women, of which one has blue eyes (the next-to-last
element of the set), hence they all have blue eyes, in particular
the k+1-th woman has blue eyes.
Hence all k+1 women have blue eyes.
By induction, it follows that in any finite set of women, if one has
blue eyes they all have blue eyes. QED
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@uclink.berkeley.edu
mag...@math.berkeley.edu
Zorro
here, but there it goes...
Theorem:
All positive integers are interesting.
Proof:
Assume the contrary. Then there is a lowest non-interesting positive
integer. But, hey, that's pretty interesting! A contradiction.
QED
I heard this one from G. B. Thomas, but I don't know whether it is due to
him.
Zorro
Benjamin J. Tilly
Zorro writes:
> I don't know whether the term "mathematical induction" is at all appropriate
> here, but there it goes...
>
> Theorem:
> All positive integers are interesting.
>
> Proof:
> Assume the contrary. Then there is a lowest non-interesting positive
> integer. But, hey, that's pretty interesting! A contradiction.
>
> QED
(yawn) Only if you have the right mindset....
Ben Tilly
Ron Maimon
|> As part of an attempt to teach students mathematical induction, I'm looking
|> for a good selection of false inductive "proofs", the funnier the better.
|> You know the sort of thing, "all horses have one leg", and so on. I'd be
|> grateful if you could post or mail me your favourites!
|>
All dogs have nine legs:
would you agree that no dog has five legs?
would you agree that _a_ dog has four legs more then _no_ dog?
4 + 5 = ?
--
Ron Maimon
+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=
You say you don't want it
You don't want it
You say you don't want it
and then you slip it right in - Black Flag
+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+=+
Bill Taylor
|>
|> Theorem:
|> All positive integers are interesting.
|>
|> Proof:
|> Assume the contrary. Then there is a lowest non-interesting positive
|> integer. But, hey, that's pretty interesting! A contradiction.
Nice try. But sorry, :) see my standard sig...
-------------------------------------------------------------------------------
Bill Taylor w...@math.canterbury.ac.nz
-------------------------------------------------------------------------------
The smallest uninteresting natural number is *not* thereby interesting;
...but it *is* very meta-interesting !
-------------------------------------------------------------------------------
Mattias Hartikainen
that a number is the smallest with some property. Also, of course all numbers are
interesting; we wouldn't be mathematicians if we didn't think so, would we?
One could allow different degrees of interest, though, so that what people generally
mean by interesting would be associated with things with more advanced properties
than this. The property of being (a candidate for) the lowest non-interesting number
would be assigned a low degree of interest, but enough to be called interesting in
this new, extended sense.
------>8------------>8------------>8------------>8------------>8------
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( ) Mattias Hartikainen _ _
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UUUU UUUUU UUUU etx...@tn.etx.ericsson.se ~~ ~~ ~~
md87...@nada.kth.se (for yet a while... ;-)
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All of the above are expressions of my own mind, and have nothing to
do with my employer and/or university. And yes, I *am* a frog!
Ohn Christian
: As part of an attempt to teach students mathematical induction, I'm looking
: for a good selection of false inductive "proofs", the funnier the better.
: You know the sort of thing, "all horses have one leg", and so on. I'd be
: grateful if you could post or mail me your favourites!
Here are two of them:
1) Fermat's theorem by induction. We know x^3+y^3=z^3 has no nontrivial
integer solutions. Now, let n=>3 and assume x^n+y^n=z^n has none either and
look at
x^(n+1) + y^(n+1) = z^(n+1)
taking derivatives:
(n+1)x^n + (n+1)y^n = (n+1)z^n :-):-):-)
simplifying:
x^n + y^n = z^n
we get a contradiction with our induction hypothesis. So Fermat is true for
all n.
2) A method to catch a lion in the desert. Consider, for each n, the following
statement:
P(n) : 'It is possible to catch n lions in the desert.'
Of course, P(n) is true for large enough n, because the lions are then
so tightly packed together that it is easy to catch them. But now,
P(n) implies P(n-1) ('cause if you catch some lions, you can always
release one of them). Hence, P(1) is true. :-):-):-)
Christian