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Is there always one right answer?
Mathematics as an AOK
Is There Always One Right Answer?
Let's see. Try selecting
a figure from the five
shown on the right
that is different from
all the others at least
in one respect.
Select a shape before
you proceed further.
Is There Always One Right Answer?
If you chose figure b),
congratulations! You've
picked the right answer.
Figure b) is the only one
that has all straight
lines. Give yourself a
pat on the back!
Is There Always One Right Answer?
Some of you, however, may have chosen figure
c), thinking that c) is unique because it's the only
one that is asymmetrical. And you are also right!
c) is the right answer.
A case can also be made for figure a): it's the only
one with no points. Therefore, a) is the right
answer.
What about d)? It is the only one that has both a
straight line and a curved line. So, d) is the right
answer too.
And e)? Among other things, e) is the only one
that looks like a projection of a non-Euclidean
triangle into Euclidean space. It is also the right
answer.
In other words, they are all right depending on
your point of view.
Reference
R. von Oech, A Whack on the Side of the Head,
Warner Books, 1990
Does 1 + 1 always equal 2?
1 + 1 = 10 (in binary system)
100 ml of vodka and 100 ml of orange juice only gives you a 190 ml cocktail.
Put one rabbit and another rabbit in a box and come back in a couple of
months, you’ll have more than two rabbits
Put one live ninja assassin and another live ninja assassin in a room and come
back in a couple of hours, you may not have two live ninja assassins
I cup of flour and 1 table spoon of oil gives one lump of dough (it will taste
awful and not cook properly but it is dough).
What conditions have to be true for 1 + 1 = 2?
Does 1 + 1 always equal 2?
With regard to mathematics, it must be repeated here that
many students continue to cite that mathematical
statements are, as a rule, justified empirically – so 1 + 1 = 2
is proven by re-arranging apples. Students should see from
their own mathematical experiences that most things they
know (matrices, functions, groups, sets) cannot be justified
in this way, and even for the simple 1 + 1 = 2 case, the
“proof” involving apples is about apples, not numbers.
TOK Subject Report, May 2011, IBO, page 8 - 9
So, is there only one right answer?
Well, that depends on your axioms.
BUT in mathematics the contradictory of a mathematical
statement is necessarily false (e.g. if 2+2=4 then 2+2 can
not equal 3).
One complication
Euclid ‘Things that are equal to the same thing are equal to each other’.
But 0 ÷ 5 = ∞ and 0 ÷12 = ∞, so does 0 ÷ 5 = 0 ÷12?
This is Spinal Tap
“Our amps go up to 11!”
Same questions in maths:
Why do we have base 10 system for counting?
Why do we use per cent?
Why are there 360° in a circle?
Why do we go (x, y)? Why not (y,x)?
AXIOMS!
Most of our geometry is based on Euclid‘s axioms.
Euclid’s axioms
A point is space has no size.
The shortest distance between two points is a line. A line
has no thickness.
A line extends in both directions forever.
Parallel lines never meet (or there is just one line through a
given point that is parallel to another line)
All right angles are equal to each other.
But these aren’t realistic!!!
Most of our algebra is based on Peano‘s axioms.
Peano’s Axioms (Arithmetic)
Reflexive:
x = x always
Symmetrical: If x = y then y = x
Transitive:
If x = y and y = z, then x = z
But these can only be applied
to reality in ideal circumstances
Most of our discrete mathematics (probability, logic,
combinations & counting principles) is based on these axioms.
De Morgan’s axiom (Probability)
P(A’) = 1 – P(A)
Boolean Axioms (Logic)
Propositions (or statements) can either be true or false.
True statements can be given a value of 1.
False statements can be given a value of 0.
But these are just arbitrary selections.
So what is the right answer?
What works best in the situation ????
Yes, but it has to be internally consistent and valid
AND externally verifiable
Gödel’s incompleteness – In any system, it is possible to formulate an expression
which says ‘this is unproveable in this system’. Therefore, we need axioms
because we just need to declare some things as true or valid so we can get on
with the rest of math. Therefore, all of mathematics is a paradox.
Russel’s Paradox – ‘There is a barber in a village that shaves all the men who
do not shave themselves. Does he shave himself?’. ‘You must answer ‘yes’ or
‘no’. Will the next word you say be ‘no’?’. ‘All Australians are liars’.