## 1. MATHEMATICS

#### Subsections

- 1. mathematics - general
- 1.1 proofs
- 1.2 statistics and statisticians
- 1.3 mathematicians
- 1.4 mathematics poetry
- 1.5 mathematics quotes
- 1.6 mathematic puns
- 1.7 mathematical terms
- 1.8 mathematical exams
- 1.9 learning mathematics
- 1.10 mathematical tests to take
- 1.11 mathematical equations
- 1.12 topology
- 1.13 numbers
- 1.14 set theory
- 1.15 functions

### 1.1 PROOFS

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#### PROOFS THAT P

                      (attributed to Hartry Field)

    Davidson's proof that p: Let us make the following bold conjecture: p

    Wallace's proof that p: Davidson has made the following bold conjecture: p

    Grunbaum:  As I have asserted again and again in previous publications, p.

    Morgenbesser: If not p, what? q maybe?

    Putnam:  Some philosophers have argued that not-p, on the grounds that q.
    It would be an interesting exercise to count all the fallacies in this
    "argument".  (It's really awful, isn't it?)  Therefore p.

    Rawls:  It would be a nice to have a deductive argument that p from
    self-evident premises.  Unfortunately, I am unable to provide one.  So
    I will have to rest content with the following intuitive considerations
    in its support: p.

    Unger:  Suppose it were the case that not-p.  It would follow from
    this that someone knows that q.  But on my view, no one knows anything
    whatsoever.  Therefore p.  (Unger believes that the louder you say
    this argument the more persuasive it becomes.)

    Katz:  I have seventeen arguments for the claim that p, and I know
    of only four for the claim that not-p.  Therefore p.

    Lewis:  Most people find the claim that not p completely obvious and
    when I assert p they give me an incredulous stare.  But the fact
    that they find not-p obvious is no argument that it is true; and I
    do not know how to refute an incredulous stare.  Therefore p.

    Fodor:  My argument for p is based on three premises:
    (1) q
    (2) r
    and
    (3) p
    >From these, the claim that p deductively follows.

    Some people may find the third premise controversial, but it is
    clear that if we replaced that premise by any other reasonable
    premise, the argument would go through just as well.

    Sellars's proof that p:  Unfortunately, limitations of space prevent
    it from being included here, but important parts of the proof can be
    found in each of the articles in the attached bibliography.

    Earman:  There are solutions to the field equations of general
    relativity in which space-time has the structure of a four-dimensional
    klein bottle and in which there is no matter.  In each such
    space-time, the claim that not-p is false.  Therefore p.

    Kripke:

                          OUTLINE OF A "PROOF" THAT P [footnote]

                                    Saul Kripke

    Some philosophers have argued that not-p.  But none of them seems to me
    to have made a convincing argument against the intuitive view that
    this is not the case.  Therefore, p.

    [footnote].  This outline was prepared hastily--at the editor's
    insistence---from a taped transcript of a lecture.  Since I was
    not even given the opportunity to revise the first draft before
    publication, I cannot be held responsible for any lacunae in the
    (published version of the) argument, or for any fallacious or
    garbled inferences resulting from faulty preparation of the
    typescript.  Also, the argument now seems to me to have problems
    which I did not know when I wrote it, but which I can't discuss
    here, and which are completely unrelated to any criticisms that
    have appeared in the literature (or that I have seen in manuscript);
    all such criticisms misconstrue the argument.  It will be noted
    that the present version of the argument seems to presuppose the
    (intuitionistically unacceptable) law of double negation.  But the
    argument can easily be reformulated in a way that avoids employing
    such an inference rule.  I hope to expand on these matters further
    in a separate monograph.

    Routley and Meyer:  If (q & not-q) is true, then there is a model for p.
    Therefore p.

    From: "Randall D. Wald" <randy@rwald.com>
    Fermat:
    I have a truly marvelous demonstration of this proposition which this
    margin is too narrow to contain.

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    From: "Eric T. Ferguson" <e.ferguson@antenna.nl>
    Theorem: 1 is the smallest positive number
    Proof:
    Let the smallest postive number be called x.

    x squared is also positive, therefore
    	x^2 >= x

    We can divide both sides by the positive number x

    therefore
    	x >= 1

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    Theorem : All positive integers are equal.
    Proof : Sufficient to show that for any two positive integers, A and B,
       A = B.  Further, it is sufficient to show that for all N > 0, if A
       and B (positive integers) satisfy (MAX(A, B) = N) then A = B.

       Proceed by induction.

       If N = 1, then A and B, being positive integers, must both be 1.
       So A = B.

       Assume that the theorem is true for some value k.  Take A and B
       with MAX(A, B) = k+1.  Then  MAX((A-1), (B-1)) = k.  And hence
       (A-1) = (B-1).  Consequently, A = B.

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    From: Benjamin.J.Tilly@dartmouth.edu (Benjamin J. Tilly)

    Theorem : All numbers are equal to zero.
    Proof: Suppose that a=b. Then
    a = b
    a^2 = ab
    a^2 - b^2 = ab - b^2
    (a + b)(a - b) = b(a - b)
    a + b = b
    a = 0

    From: Chris Trevino <fred.trevino@worldnet.att.net>
    And Furthermore if a + b = b, and a = b, then b + b = b, and 2b = b,
    which mean that
    2 = 1

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    From: Michael_Ketzlick@h2.maus.de (Michael Ketzlick)

    Theorem : 3=4
    Proof:
    Suppose:
            a    +    b    =    c

    This can also be written as:

         4a - 3a + 4b - 3b = 4c - 3c

    After reorganising:

         4a + 4b - 4c = 3a + 3b - 3c

    Take the constants out of the brackets:

         4 * (a+b-c) = 3 * (a+b-c)

    Remove the same term left and right:

                4 = 3

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    From: "Ken Payson" <kpayson@worldnet.att.net>
    Theorem: 0 = 0
    Proof:
    You could proove by induction that x + 0 = x for all x
    When x = 0 we have 0 + 0 = 0.  0 + 0 = 0 so we can make a substitution on
    the left hand side and get 0 = 0

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    From: Benjamin.J.Tilly@dartmouth.edu (Benjamin J. Tilly)

    Theorem: 1$ = 1c.
    Proof:
    And another that gives you a sense of money disappearing...

    1$ = 100c
       = (10c)^2
       = (0.1$)^2
       = 0.01$
       = 1c

    Here $ means dollars and c means cents. This one is scary in that I
    have seen PhD's in math who were unable to see what was wrong with this
    one. Actually I am crossposting this to news:sci.physics because I think
    that the latter makes a very nice introduction to the importance of
    keeping track of your dimensions...

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    From: "Brijesh  " <birjoo@mailcity.com>
    Theorem: 1$ = 10 cent
    Proof:
    we know that
        $ 1 = 100 cents
     divide both sides by 100
        $ 1/100  =  100/100 cents
    =>  $ 1/100 = 1 cent
      take square root both side
    =>  squr($1/100) = squr (1 cent)
    =>  $ 1/10 = 1 cent
      multiply both side by 10
    =>  $1  = 10 cent

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    From: clubok@physics11 (Kenneth S. Clubok)

    Theorem: 1 = -1 .
    Proof:
     1    -1
    --  = --
    -1     1

           1            -1
    sqrt[ -- ]  = sqrt[ -- ]
          -1             1

    sqrt[1]   sqrt[-1]
    ------- = -------
    sqrt[-1]  sqrt[1]

    1=-1 (by cross-multiplication)

    And here's my personal favorite:

    Use integration by parts to find the anti-derivative of 1/x.  One
    can get the amusing result that 0=1.  (Until you realize you have to put
    in the limits.)

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    From: jreimer@aol.com (JReimer)

    Theorem: 1 = -1
    Proof:
    1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = 1^ = -1

    Also one can disprove the axiom that things equal to the same thing
    are equal to each other.

    1 = sqrt(1)
    -1 = sqrt(1)
    therefore 1 = -1

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    From:waaben <nissa@bouldernews.infi.net>
    Theorem: 1 = -1
    Proof:
    x=1
    x^2=x
    x^2-1=x-1
    (x+1)(x-1)=(x-1)
    (x+1)=(x-1)/(x-1)
    x+1=1
    x=0
    0=1

    => 0/0=1/1=1

    would you like for me to produced another rabbit :)

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    From: "Gerben Dirksen" <gerben47@hotmail.com>

    Theorem:
    1 + 2 + 4 + 8 + 16 + ... = -1

    It is well known that in computers the number -1 is represented by FFFFFFFF (hexadecimal) or 1 + 2 + 4 + 8 + 16 + ...

    Proof:

    Let x = 1 + 2 + 4 + 8 + 16 + ...

    => 2x =     2 + 4 + 8 + 16 + ...
    -------------------------------- -
      -x = 1

       x = -1

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    From: kdq@marsupial.jpl.nasa.gov (Kevin D. Quitt)

    Theorem: 4 = 5
    Proof:
    -20 = -20
    16 - 36 = 25 - 45
    4^2 - 9*4 = 5^2 - 9*5
    4^2 - 9*4 + 81/4 = 5^2 - 9*5 + 81/4
    (4 - 9/2)^2 = (5 - 9/2)^2
    4 - 9/2 = 5 - 9/2
    4 = 5

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    From: baez@guitar.ucr.edu (john baez)

    Theorem: 1 + 1 = 2
    Proof:
    n(2n - 2) = n(2n - 2)
    n(2n - 2) - n(2n - 2) = 0
    (n - n)(2n - 2) = 0
    2n(n - n) - 2(n - n) = 0
    2n - 2 = 0
    2n = 2
    n + n = 2
    or setting n = 1
    1 + 1 = 2

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    From: Elie Accari <elie.accari@darbeirut.com>

    The aim is to proove that 1 = 2.

    Notice the following:

                      1 = 1^1 = 1
                  2 + 2 = 2^2 = 4
              3 + 3 + 3 = 3^2 = 9
          4 + 4 + 4 + 4 = 4^2 = 16

    and in general:

         x + x + ..... + x = x^2
         \___ x times ___/

    Derive in place:

         1 + 1 + ..... + 1 = 2 x
         \___ x times ___/

           1 * x = 2 x

    Simplify by x which is not zero, 1 = 2.

    Note the formula is valid for pure fractionals too, for example:

    0.05 + 0.05 + 0.05 + 0.05 + 0.05 =
    0.01 * (5 + 5 + 5 + 5 + 5) = 0.01 * 5^2 = 0.05 ^2 = 0.25

    and thus it is valid for any real number x because we can always write x as
    a sum of a pure integer u and a pure fractional v: x = u+v. For example
    x=3.8 => u=3 and v=0.8. So we are not deriving in a discrete space, think
    somewhere else.

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    From: Elie Accari <elie.accari@darbeirut.com>

    Problem: Prove that 90 = 110.

    Demonstration:
    Hypothesis:
              ^
    1) Angle CBA = 90 degrees
              ^
    2) Angle DAB = 110 degrees
    3) Segment BC = Segment AD
    4) O is the intersection of the two perpendiculars in their middle
    (mediator) of segments AB and CD.
    5) My major is not graphic design :)

                                   _____________-------| C
        ___________----------------|                 . |
     D \ . .                       |               .   |
        \    .                     |              .    |
         \     ....                 |            .     |
          \        ..               |           .      |
           \         ...            |          .       |
            \           ...          |        .        |
             \             ...       |      ..         |
              \               ...    ||   ..           |
               \                 .... | ..             |
                \                     O                |
                 \               . . .|. . .           |
                  \        . . .      |     . . .      |
                   \  . . .           |          . . . |
                  A \_________________|________________| B

                         ^       ^
    Consider the angles CBA and DAB
     ^     ^     ^
    CBA = CBO + OBA
     ^     ^     ^
    DAB = DAO + OAB

            ^       ^
    Angles OBA and OAB are equal because the triangle AOB is isosceles (OA=OB).

    Consider triangles AOD and BOC
    They have: AD = BC by construction
               OA = OB, O belongs to the mediator of AB
               OD = OC, O belongs to the mediator of CD

    Then triangles AOD and BOC are congruent, and the corresponding angles are
    equal and we have:
     ^   ^
    CBO=DAO
    Getting back to the equalities:
     ^     ^     ^     ^     ^     ^
    CBA = CBO + OBA = DAO + OAB = DAB

    90 = 110

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    From: magidin@uclink.berkeley.edu (Arturo Viso Magidin)

    Theorem: In any finite set of women, if one has blue eyes then they
    all have blue eyes.

    Proof. Induction on the number of elements.

    if n= 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

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    From: Zorro
    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.

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    From:         Jeff Erickson <jeffe@cs.duke.edu>
    Theorem: All numbers are boring.
    Proof (by contradiction):
            Suppose x is the first non-boring number.  Who cares?

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    From: daniel@hagar.ph.utexas.edu (James Daniel)

    Aren't multi-valued functions fun?  Once you realize what's going on,
    though, you can make them into silly proofs pretty much without thinking.

    Here's one I just made up:

    Object: to prove that  i < 0  ( that is, sqrt(-1) < 0  )

    Well, ( .5 + sqrt(3/4)*i )^3 = (-1)^3

                    (most would assert this to be a false statement -- mostly
                     cuz they'll get the math wrong.  It's a true statement.
                     It's the next statement that is false.)

    which means that .5 + sqrt(3/4)*i = -1

    So then      1 + sqrt(3)*i = -2

                 sqrt(3)*i = -1

                 i = -1/sqrt(3)

    Therefore i is a negative number.  QED.

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    From: julison@cco.caltech.edu (Julian C. Jamison)

    Theorem: All numbers are equal.
    Proof:
    Choose arbitrary a and b, and let t = a + b. Then
    a + b = t
    (a + b)(a - b) = t(a - b)
    a^2 - b^2 = ta - tb
    a^2 - ta = b^2 - tb
    a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4
    (a - t/2)^2 = (b - t/2)^2
    a - t/2 = b - t/2
    a = b

    So all numbers are the same, and math is pointless.

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    From: pfc@math.ufl.edu (P. Fritz Cronheim)

    This one is from Jerry King's _Art of Mathematics_

    16/64=1/4 by cancelling the 6's.  Here the result is true, but the method
    is not.  Do the ends justify the means? :)_

    From: ejones@hooked.net (Earle Jones)

    Try 19 / 95 -- just cancel the nines!

    From: Urban Ingelsson <urbin545@student.liu.se>
    19/95=1/5=1999...9/999...95
    have as many nines as you want to, just have the same amount on both
    sides of the division line.
    Same accounts for
    16/64=1/4=1666...6/666...64

    The proof is a fun 5 minutes of calculation that I hope you take as a
    challenge.

    From Seo Sang Hyun(SKC8563@hitel.net) / Korea

    also 26/65 = 2/5, and 49/98 = 4/8. As Urban Ingelsson states,
    266...6/666...65 = 2/5, 499...9/999...8 = 4/8.

    and 16/64, 19/95, 26/65, 49/98 are all cases that satisfy a such
    property. It is easy to prove!

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    From: Wei Kiet Soo <ahleksoo@nw.com.au>
    This reminds me of my maths teacher's demonstration last year:

    sin x = 6n
    canceling out the ns, six = 6 ...:)

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    From: afetrmath2@aol.com (Soroban)
    Another false proof for a correct result:

    Reduce:   1 - x^2      1 - x
             ---------  =  -----      [cancel exponents]
             (1 + x)^2     1 + x

    From: Lynn Killingbeck <killbeck@pointecom.net>

    This works for all n.
    Reduce:   1 - x^n      1 - x
             ---------  =  -----
             (1 + x)^n     1 + x

    Proof is by induction.
    For n=1,
            (1-x^1)/(1+x)^1 = (1-x)/(1+x), by cancelling the '1' in the exponents.
    For n=2, ditto by cancelling the '2' in the exponent.

    It's true for n=1, and from there for n==>n+1=2, so must be true for all
    n. QED

    [Putting in the QED makes it a valid proof! Have you ever seen a
    formal published proof that ends with QED be wrong? Don't let JSH know
    that secret!]

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    From: mathwft@math.canterbury.ac.nz (Bill Taylor)

               sin x + sin 3x + sin 5x
              ------------------------- = tan 3x
               cos x + cos 3x + cos 5x

    ...which is proved by cancelling and/or averaging everything in sight!

    Oddly enough, though filthy to prove using old-fashioned trig formulae,
    it becomes almost as easy to "see" as the fake proof, when re-interpreted
    as the real and imaginary parts of the appropriate complex exponentials.

    Altogether cool in every respect.

    Oh and yes, this one *does* work for large values of 5...

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    Theorem: n=n+1
    Proof:
    (n+1)^2 = n^2 + 2*n + 1

    Bring 2n+1 to the left:
    (n+1)^2 - (2n+1) = n^2

    Substract n(2n+1) from both sides and factoring, we have:
    (n+1)^2 - (n+1)(2n+1) = n^2 - n(2n+1)

    Adding 1/4(2n+1)^2 to both sides yields:
    (n+1)^2 - (n+1)(2n+1) + 1/4(2n+1)^2 = n^2 - n(2n+1) + 1/4(2n+1)^2

    This may be written:
    [ (n+1) - 1/2(2n+1) ]^2 = [ n - 1/2(2n+1) ]^2

    Taking the square roots of both sides:
    (n+1) - 1/2(2n+1)  = n - 1/2(2n+1)

    Add 1/2(2n+1) to both sides:
    n+1 = n

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    Theorem: log(-1) = 0
    Proof:
    a) log[(-1)^2] = 2 * log(-1)
    On the other hand:
    b) log[(-1)^2] = log(1) = 0
    Combining a) and b) gives:
    2* log(-1) = 0
    Divide both sides by 2:
    log(-1) = 0

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    Theorem: ln(2) = 0
    Proof:
    Consider the series equivalent of ln 2:
    ln 2 = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 ...
    Rearange the terms:
    ln 2 = (1 + 1/3 + 1/5 + 1/7 ...) - (1/2 + 1/4 + 1/6 + 1/8 ...)
    Thus:
    ln 2 = (1 + 1/3 + 1/5 + 1/7 ...) + (1/2 + 1/4 + 1/6 + 1/8 ...) -
            2 * (1/2 + 1/4 + 1/6 + 1/8 ...)
    Combine the first to series:
    ln 2 = (1 + 1/2 + 1/3 + 1/4 + 1/5 ...) - (1 + 1/2 + 1/3 + 1/4 + 1/5 ...)
    Therefore:
    ln 2 = 0

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    Theorem: 1 = 0 = -1/2
    Proof:
    Consider the infinite series: 1 - 1 + 1 - 1 + 1 - 1 + 1 -1 + 1 ...
    Pair the terms:
    a) (1 - 1) + (1 - 1) + (1 - 1) + ... = 0
    Pair the terms differently:
    b) 1 - (1 - 1) + (1 - 1) + (1 - 1) + ... =  1
    Combine a) and b):
    1 = 0
    From: ad@dcs.st-and.ac.uk (Tony Davie)

    c) -S = -1 + 1 - 1 + 1 - ...
          = 1 + S
    So 2S = -1 and S = -1/2
    combine with a) and b):
    1 = 0 = - 1/2

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    From: Andreas Jung <ajung@informatik.uni-rostock.de>
    Theorem: e=1
    Proof:
                                        1
                  2 pi i              ------        1
                  ------              2 pi i      ------
           1      2 pi i     [ 2 pi i]            2 pi i
    e  =  e   =  e        =  [e      ]        =  1        =  1

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    From: The Human Neutrino <sirius@wam.umd.edu>
    Theorem: e=1
    Proof:
     2*e = f
      2^(2*pi*i)e^(2*pi*i) = f^(2*pi*i)
      e^(2*pi*i) = 1
    so:
      2^(2*pi*i) = f^(2*pi*i)
      2=f
    thus:
      e=1

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    From: Math Department American Collegiate Institute
    Theorem: 1 = 1/2:
    Proof:
    We can re-write the infinite series 1/(1*3) + 1/(3*5) + 1/(5*7) + 1/(7*9)
    +...
    as 1/2((1/1 - 1/3) + (1/3 - 1/5) + (1/5 - 1/7) + (1/7 - 1/9) + ... ).
    All terms after 1/1 cancel, so that the sum is 1/2.

    We can also re-write the series as (1/1 - 2/3) + (2/3 - 3/5) + (3/5 - 4/7)
    + (4/7 - 5/9) + ...
    All terms after 1/1 cancel, so that the sum is 1.

    Thus 1/2 = 1.

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    From: Kevin D. Quitt <Kevin@Quitt.net>
     25 -  45        =  16 - 36

    5^2 - 9*5        = 4^2 - 9*4

    5^2 - 9*5 + 81/4 = 4^2 - 9*4 + 81/4

       (5 - 9/2)^2   =  (4 - 9/2)^2

       (5 - 9/2)     =  (4 - 9/2)

           5         =   4

New after last time posted (February 28, 2004) mathematics
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    From: Mrinal Shenoy <mrinalf1@yahoo.co.in>

    Here is a proof for 1=3

    First we state that
    1/2 closed door=1/2 open door
    =>Closed door = Open door ......(1)

    Also,
    1/4 closed door = 3/4 open door
    Multiply by 4 both sides..
    =>1 closed door=3 open door

    But by eqn. (1),we have closed door=open door
    Therefore, 1 open door = 3 open door
    Divide by open door on both sides...
    Therefore we have 1=3
    Hence proved....

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    Theorem: It is possible to square the circle.

    Proof:
    No mathematician has squared the circle.

    Therefore: No one who has squared the circle is a mathematician.
    Therefore: All who have squared the circle are nonmathematicians.
    Therefore: Some nonmathematician has squared the circle.
    Therefore: It is possible to square to circle. [QED]

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    From: ad@dcs.st-and.ac.uk (Tony Davie)

    B) Every triangle is isosceles (and hence equilateral)
    ...........................A.*
    ............................/|\
    .........................../.|.\
    ........................../../.\
    ........................./..|...\
    ......................./....|...\
    ....................../...../....\
    ...................../.....|......\
    ..................../......|......\
    ................../......../.......\
    ................./........|.........\
    ................/.........|.........\
    ..............Z*-........./.........\
    ............./...--......|..........-*Y
    ............/......--....|.......---..\
    .........../.........--.P/...----.....\
    ........../...........--*----..........\
    ........./........---...|.-.............\
    ......./.......--.......|...--..........\
    ....../.....--..........|......--........\
    ...../...--.............|.........--......\
    ..../.--................|............--....\
    ../--...................|...............--.\
    B*----------------------*-------------------*C
                            X

    Proof: Consider triangle ABC. Let the bisector of angle A and the
    perpendicular bisector of BC meet at P. Drop perpendiculars from P onto AB
    meeting it at Z and on AC meeting it at Y. Join PB and PC.

    Triangles APZ and APY are congruent (2 angles and corresponding side) so
    AZ=AY and ZP=YP.

    Triangles PXB and PXC are congruent (2 sides and included angle) so PB=PC

    Hence triangles ZPB and YPC are congruent (right angle, hypotenuse and side)
    So BZ=CY

    So BZ+ZA=CY+YA
    QED

    C) Consider the sentence S = "If this sentence is true then God exists".
    This could be written S = (S > G) (in the absence of an ASCII hook)

    Suppose S is true. From this, we can use Modus Ponens to show that God
    exists. Thus S |- G.

    So |- (S > G)

    So |- S

    QED

    Corrolory: G -- God exists

    [But also, of course, God doesn't exist, the Moon is made of Green Cheese,
    ...]

    Hope these are interesting. The geometry diagram took me ages!!!

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#### Methods of Mathematical Proof

This is from _A Random Walk in Science_, R.L. Weber and E. Mendoza (ed.)(by Joel E. Cohen?): To illustrate the various methods of proof we give an example of a logical system. THE PEJORATIVE CALCULUS Lemma 1. All horses are the same colour. (Proof by induction) Proof. It is obvious that one horse is the same colour. Let us assume the proposition P(k) that k horses are the same colour and use this to imply that k+1 horses are the same colour. Given the set of k+1 horses, we remove one horse; then the remaining k horses are the same colour, by hypothesis. We remove another horse and replace the first; the k horses, by hypothesis, are again the same colour. We repeat this until by exhaustion the k+1 sets of k horses have been shown to be the same colour. It follows that since every horse is the same colour as every other horse, P(k) entails P(k+1). But since we have shown P(1) to be true, P is true for all succeeding values of k, that is, all horses are the same colour. Theorem 1. Every horse has an infinite number of legs. (Proof by intimidation.) Proof. Horses have an even number of legs. Behind they have two legs and in front they have fore legs. This makes six legs, which is cer- tainly an odd number of legs for a horse. But the only number that is both odd and even is infinity. Therefore horses have an infinite num- ber of legs. Now to show that this is general, suppose that somewhere there is a horse with a finite number of legs. But that is a horse of another colour, and by the lemma that does not exist. Corollary 1. Everything is the same colour. Proof. The proof of lemma 1 does not depend at all on the nature of the object under consideration. The predicate of the antecedent of the uni- versally-quantified conditional 'For all x, if x is a horse, then x is the same colour,' namely 'is a horse' may be generalized to 'is anything' without affecting the validity of the proof; hence, 'for all x, if x is anything, x is the same colour.' Corollary 2. Everything is white. Proof. If a sentential formula in x is logically true, then any parti- cular substitution instance of it is a true sentence. In particular then: 'for all x, if x is an elephant, then x is the same colour' is true. Now it is manifestly axiomatic that white elephants exist (for proof by blatant assertion consult Mark Twain 'The Stolen White Ele- phant'). Therefore all elephants are white. By corollary 1 everything is white. Theorem 2. Alexander the Great did not exist and he had an infinite number of limbs. Proof. We prove this theorem in two parts. First we note the obvious fact that historians always tell the truth (for historians always take a stand, and therefore they cannot lie). Hence we have the historically true sentence, 'If Alexander the Great existed, then he rode a black horse Bucephalus.' But we know by corollary 2 everything is white; hence Alexander could not have ridden a black horse. Since the conse- quent of the conditional is false, in order for the whole statement to be true the antecedent must be false. Hence Alexander the Great did not exist. We have also the historically true statement that Alexander was warned by an oracle that he would meet death if he crossed a certain river. He had two legs; and 'forewarned is four-armed.' This gives him six limbs, an even number, which is certainly an odd number of limbs for a man. Now the only number which is even and odd is infinity; hence Alexander had an infinite number of limbs. We have thus proved that Alexander the Great did not exist and that he had an infinite number of limbs.

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    Theorem: a cat has nine tails.
    Proof: No cat has eight tails. A cat has one tail more than no cat
    Therefore, a cat has nine tails.

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    From: rmaimon@husc9.Harvard.EDU (Ron Maimon)

    Theorem: All dogs have nine legs.
    Proof:
    would you agree that no dog has five legs?
    would you agree that _a_ dog has four legs more then _no_ dog?
    4 + 5 = ?

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    From: sld1n@cc.usu.edu
    Prove that the crocodile is longer than it is wide.

    Lemma 1. The crocodile is longer than it is green:
    Let's look at the crocodile. It is long on the top and on the bottom, but
    it is green only on the top. Therefore, the crocodile is longer than it is
    green.

    Lemma 2. The crocodile is greener than it is wide:
    Let's look at the crocodile. It is green along its length and width, but it
    is wide only along its width. Therefore, the crocodile is greener than it
    is wide.

    From Lemma 1 and Lemma 2 we conclude that the crocodile is longer than it
    is wide.

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    From: Anth
    Proof: Girls are evil:

    First we state that girls require time and money :-

     Girls = Time x Money

    And we know that time is money :-

     Time = Money

    Therefore :-

     Girls = Money x Money

     Girls = (Money)^2

    And because 'money is the root of all evil'

     Girls = (Evil)^1/2 x (Evil)^1/2
     Girls = Evil

    From: geenius@enteract.com (Geenius@Wrok)

    Note:
    A problem with this proof occurred to me immediately: Since evil is
    negative, the square root of evil must be imaginary, which would mean that
    money is imaginary, and therefore, by definition, so is time.

    Then I realized that, in my life at least, that's pretty much true.

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    From: rdownes@aol.com (RDownes)

    Proof: Girls are good:

    First we state that girls require time and money :-

     Girls = Time x Money

    And we know that time is money :-

     Time = Money

    Therefore :-

     Girls = Money x Money

     Girls = (Money)^2

    And because 'money is the root of all evil'

     Girls = (Evil)^1/2 x (Evil)^1/2

    But evil is negative, and hence
    (Evil)^1/2=i(|Evil|)^1/2

    Multiplying the two imaginaries gives -Evil,

     which is of course GOOD!

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    From: Raymond Kristiansen <raymond@spam.no>
    Proof: Girls are the root of all evil.
    if girls require time _and_ money, should that be stated

     Girls = Time + Money

    instead? This leaves us with, using the well-known fact that time is
    money which you mentioned,

    Girls = 2*Money

    Under the assumption that money is the root of all evil, it can now be
    found that

    Girls = 2*sqrt(evil)

    In words, girls are _twice_ the root of all evil, instead of just evil,
    as you proved.

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    From: "Anthony Coulter" <c17gmaster@earthlink.net>
    There have been some disputes over the initial equation. I would like to
    solve these. The correct equation is:

                G = tm   | where G = Girls, t = time, and m = money

        Girls are equal to time and money. Note that it is not negative time or
    money. Think of it this way: You can either have the girl, or you can have
    free time and money. They are equivalent.

        Also note that time AND money is time multiplied by money. The Boolean
    operator "AND" is best represented by multiplication, while "OR" is
    represented by addition. Thus, girls equal "tm" and not "t+m".

        After this point, the theorem is straightforward. Time equals money,
    and money is the square root of all evil, thus girls equal evil.

        I hope I helped to clear up any misunderstandings on the matter.

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    From: Adam Cabrera <piadam@home.com>
    Theorem: The less you know the more money you make.

    Proof:
    We know that
    a) Time is Money
    b) Knowledge is Power
    and from Physics
    c) Power = Work / Time

    By simple substitution:
    Knowledge = Work / Money
    Knowledge * Money = Work
    Money = Work / Knowledge

    It follows that as knowledge goes to 0, money goes to infinity.

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    From: "feldmann" <feldmann@bsi.fr>

    This was posted to news:sci.math , by James Patrick Ferry, a few days ago; in
    case you didn't noticed it :-)

    Those of you who have been working on a proof of Fermat's Last
    Theorem, you may cease your labors.  I have produced a proof
    whose simplicity cannot be surpassed.

          ---------------------------------------------------------
    Statement:  For any integer n >= 3, there do not exist non-zero
    integers x, y and z such that x^n + y^n = z^n.

    Proof:
          ---------------------------------------------------------

    That's right!  My proof is . . . the null proof!  This proof
    has many advantages over the proofs offered by others:

    1)  When one correctly appreciates Fermat's sense of humor, one
        sees that this is the proof he had in mind.  The margin too
        small?  Hah!  It was in the margin the whole time, but
        mathematicians, not being able to free themselves of their
        entrenched notions of "proof," were simply unable to see it.

    2)  It is concise.

    3)  It is my one and only, first and final version.

    4)  There are no gaping holes of reasoning.

    5)  There are no bizarre, non-mathematical definitions.

    Jealous mathematicians have naturally questioned my proof.  But
    none of their counter-arguments pass muster:

    > That's not a proof.  That's just stupid.

    That's not a counter-argument.  That's just histrionics.  Until
    someone either produces a counter-example or points out the
    *specific* place in my proof where it fails, I shall consider
    my proof valid.  Your emotionalism is no substitute for logic.

    > Umm, what makes this a proof of FLT rather than, well, any
    > other theorem you might care to mention?  Why not say you've

    What makes *any* proof a proof of what it proves rather than of
    something else?  The fact that it proves it.  Duh.

    > just silly.  It is hard even to talk about.  A proof has to
    > prove something.  A proof is a series of statements which
    > establish a result.  Proofs necessarily have semantic content.
    > Even nonsense proofs have syntactic content.  Your "proof" is
    > no more a proof of FLT than is a lump of tuna (which, BTW, is

    A lump of tuna?  Again, histrionics.  Again, a mathematician
    who insists it isn't fair unless you play by his rules.  Have
    you produced a counter-example?  Have you found a *specific*
    place in my proof where it fails?  Then shut up.

    It's bad enough that close-minded mathematical community won't
    recognize my brilliance.  But to heap insult and abuse on top of
    that . . . well, I shouldn't be surprised.  It's the same old
    story:  the noble and intelligent hounded by the vicious and
    ignorant.  Sigh.  Again I say, sigh.

    Copyright 1998, James Patrick Ferry

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    From: Alain Gottcheiner <agot@ulb.ac.be>
    Theorem: All odd numbers are prime
    Proof:
    Here is a set-theoretical proof of this assertion :

    1) it is well known that there is an infinite number of odd primes
    2) Test each and every prime in any order
    3) If you encounter a number you can prove being prime (which can be
    done in a finite amount of time, assuming it is), put it into set # 1
    4) If you encounter a number you cannot prove being prime, put it into
    set # 2
    5) now read all odd numbers, beginning with set #1. You won't be able to
    get to the numbers of set #2 in a finite amount of time, so you will
    encounter only prime numbers.

    Since no non-prime odd number will be found, the assertions "there
    exists a non-prime odd" cant't be proved.

    End of the proof.

    ... Wait a minut. What if, in step #5, we read the first item in 1
    second time, then the second in 1/2 second time, and so on ? Oops ...

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    From: Rick <tinkety_tonk@yahoo.com>

    Twisted proof for pythagoras theorem:
      .
      |\
      | \
      |  \
    a |   \ c
      |    \
      |     \
      |      \
      |       \
      |_______x\
          b

    Pythagoras says: a^2 + b^2 = c^2

    Consider the angle 'x'.
    sin x = a/c
    cos x = b/c

    From the trigonometric identity, sin^2(x)+cos^2(x) = 1

    which gives,

    a^2 + b^2
    ---  ----  = 1
    c^2   c^2

    so a^2 + b^2 = c^2. (proved)

    I had got full points for writing this proof in my
    exam, because luckily my teacher failed to notice
    (what I noticed a day after the exam) that the
    trigonometric identity used to prove this theorem
    involves the Pythagoras thm itself.

    Its true, Ignorance (OF OTHERS) is bliss!

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    From: Colm K. Mulcahy (colm@mathcs.emory.edu)
    An application of mathematics to the Beatles

                      A Simple Proof That The Dream Is Over

                                       by

                                  Colm Mulcahy

                 [submitted to the Journal Of Algebraic Codology]

    Let G = {j, p, g, r} be a four element group, with distinguished element j.

    Note: we do not (need to) assume, as most previous authors seem to have,
            that j is the identity of this group.

    Theorem 1:   G does not exist

    Proof: Conjectured since the early 70s, Chapman gave a rather convincing
            proof of this in late 1980 [1], perhaps inspired by [3].

    Now let  G' = G \{j} - a three element set.

    Theorem 2:    G'  is not a group.

    Proof: It suffices to show that G' is not a subgroup of G.  But this
            follows from Lagrange's Theorem ([2], 9.1), as 3 doesn't divide 4
            (see [4]).

    Remarks:

       (1) The author hopes that the results presented here, while not new, may
    now reach a wider audience, thus laying to rest, once and for all, the
    absurd assertion that {p, g, r} could form a viable group.  While it is
    true that three element groups exist [5], under the conditions described
    above it is clear that the only way {p, g, r} could form a group would be
    if the binary operation on the elements were redefined.  In other words,
    and this is the key point here, *no group structure exists on {p, g, r}
    which is induced by the relationships which were present between the
    members of the original group G *.

    And it's a bit late in the day to be redefining binary operations if you ask me.

       (2) In spite of over 25 years of research by scholars worldwide (eg,
    [6]), the precise nature of the relationships between the four group
    members of G remains shrouded in mystery.  While there is convincing
    evidence that the member denoted by j played the role of group identity
    (indeed P. Erdos is rumoured to have proved a probabalistic result to that
    effect), we should be cautious before jumping to conclusions.

                                      References

    [1]      "Annihilating Operators" by Mark Chapman, Journal of Irreproducable
             Results (Vol XII, No. 8, 1980)

    [2]      "Contemporary Abstract Algebra" by Joseph Gallian  (2nd ed., 1990),
             published by D. C. Heath.

    [3]      "Happiness Is A Warm Gun" by John Lennon, The White Album, 1968

    [4]      "The Ladybird Book Of Computer Assisted Arithmetic" by A. Lenstra,
             A. Lenstra & H. Lenstra (London, 1985)

    [5]      Bruce Reznick, personal communicational (1989)

    [6]      "Monuments to Smithereens: Site Seeing In Liverpool" by Saki, Journal
             of Suburban Archeology (Vol 9, No. 9, 1999) (preprint)

    From: John Robinson (john@watever.waterloo.edu)

    It is hard to resist responding to Mulcahy's provocative proof that The
    Dream is Over [1]. I have two comments, the first somewhat tangential, but
    the second strikes at the heart of Mulcahy's thesis.

    1. First note that any group of less than six elements is Abelian. This
    means (for instance) that j*p = p*j. Songwritership would thus appear to be
    commutative - an argument maintained on artistic (and egotistic) grounds
    since the 1970s (See [2] for example).

    2. G = {j, p, g, r} does indeed have a subgroup under the same binary
    operator - though that group only has two members. If, therefore,

    j turns out not to be the identity,   (1)

    j is not its own inverse    (2)

    then one of  G1 = {p, g}, G2 = {p, r} or G3 = {g, r} is a subgroup.
    Therefore the dream may not be over.

    Mind you, this seems pretty unlikely to me. I look forward to an
    analysis of the numbered statements above.

    [1] Colm Mulcahy, "A Simple Proof that the Dream is Over",
    news:rec.music.beatles. 1990

    [2] MPL Communications, "Wings Over America", 1975 (?)

next page | Index | Comments and Contributions

#### Subsections

- 1. mathematics - general
- 1.1 proofs
- 1.2 statistics and statisticians
- 1.3 mathematicians
- 1.4 mathematics poetry
- 1.5 mathematics quotes
- 1.6 mathematic puns
- 1.7 mathematical terms
- 1.8 mathematical exams
- 1.9 learning mathematics
- 1.10 mathematical tests to take
- 1.11 mathematical equations
- 1.12 topology
- 1.13 numbers
- 1.14 set theory
- 1.15 functions

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