Transcript Philosophy of Mathematics

Philosophy of Mathematics
“Schools” of
thought
Logicis
m
Intuitionism
Mathematical
Realism
1
What is Math?
Is math just numbers?
Is math just equations?
Einstein’s Logic Riddle
There are 5 houses each with a different color. Their owners, each with a unique
heritage, drinks a certain type of beverage, smokes a certain brand of cigarette,
and keeps a certain variety of pet. None of the owners have the same variety of
pet, smoke the same brand of cigarette or drink the same beverage.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Can words be math?
The Brit lives in the red house.
The Swede keeps dogs as pets.
The Dane drinks tea.
Looking from in front, the green house is just to the left of the white house.
The green house's owner drinks coffee.
The person who smokes Pall Malls raises birds.
The owner of the yellow house smokes Dunhill.
The man living in the center house drinks milk.
The Norwegian lives in the leftmost house.
The man who smokes Blends lives next to the one who keeps cats.
The man who keeps a horse lives next to the man who smokes Dunhill.
The owner who smokes Bluemasters also drinks beer.
The German smokes Prince.
The Norwegian lives next to the blue house.
The man who smokes Blends has a neighbor who drinks water.
Can music be math?
Can you hold math?
Who owns the pet fish?
2
Math = The Universe
Math is a process used to
investigate the world around us.
Math exists because it is needed
by the sciences
Mathematical “truths” are only
empirically true like Newton’s
Laws of Motion
So does this mean 1+1 may not
= 2?
3
Mathematics = Logic
All math concepts can be written as
propositions and tested.
Proven math statements are logical
tautologies and thus are always
true.
Questions like “Does god exist?” are
logically meaningless and have no
value in mathematics.
So are there limits to logic?
Gödel proved that any logical
system of thought is incomplete.
New Yorker, Mar 5, 2001, pg. 78.
4
Math = The Mind
Do imaginary numbers like
exist in the “real world”?
We can imagine infinity but
does it exist?
Wormholes exist
mathematically but are they
real?
Is time just a 4th dimension we
don’t see and are the
dimensions predicted by string
theory “real”?
5
Beauty in Math
Elegant
method
s
Beautif
ul
results
6
An elegant Chinese
Proof Chou Pei’s
Euclid’s Pythagorean
Proof
Proof
7
Isn’t this a beautiful
result?
=e
1
iπ
8
Is it Math or Art?
9
Is the use of math
always ethical?
E= mc² Should the possible
negative consequences be
considered?
Can abstract mathematics do
harm to real people?
10