Download Non-Euclidean geometry and consistency

Non-Euclidean geometry and
consistency
Euclidean Geometry
Remember we said that a mathematical
system depends on its basic assumptions –
its axioms.
These should be self-evident.
a+b=b+a
Euclidean Geometry
Axioms of
Euclidean
Geometry
Euclidean Geometry
1. It shall be possible to draw a straight line
joining any two points
Euclidean Geometry
2. A finite straight line may be extended
without limit in either direction.
Euclidean Geometry
3. It shall be possible to draw a circle with a
given centre and through a given point.
Euclidean Geometry
4. All right angles are equal to one another.
Euclidean Geometry
5. There is just one straight line through a
given point which is parallel to a given line
Non-Euclidean geometry
The last axiom of Euclid is not
quite as self evident as the
others.
In the 19th century, Georg
Friedrich Bernard Riemann
came up with the idea of
replacing Euclid’s axioms with
their opposites
Non-Euclidean geometry
• Two points may determine more than one
line (instead of axiom 1)
• All lines are finite in length but endless (i.e.
circles!) (instead of axiom 2)
• There are no parallel lines (instead of axiom
5)
Non-Euclidean geometry
People expected these new axioms to throw
up inconsistencies….. But they didn’t!
Non-Euclidean geometry
Among the theorems that can be deduced
from these new axioms are
1. All perpendiculars to a straight line meet
at one point.
2. Two straight lines enclose an area
3. The sum of the angles of a triangle are
grater than 180°
Do these make sense?!
1. All perpendiculars to a straight line meet
at one point.
2. Two straight lines enclose an area
3. The sum of the angles of a triangle are
grater than 180°
Do these make sense?!
They do if we imagine space is like the
surface of a sphere!
1. All perpendiculars to a straight line meet
at one point.
2. Two straight lines enclose an area
3. The sum of the angles of a triangle are
grater than 180°
Non-Euclidean geometry
On the surface of a sphere, it can be shown
that the shortest distance between two
points is always the arc of a circle. This
means in Riemannian geometry, a straight
line will appear as a curve when represented
in two dimensions.
Although these look curved, you can be
sure the airlines are following the
shortest route to save money!
Straight lines in Riemannian
geometry
Once we have clarified the meaning of a
straight line in Riemannian geometry, we
can give a meaning to the three theorems
given earlier.
All perpendiculars to a straight
line meet at one point.
Lines of longitude are perpendicular to the
equator but meet at the North pole
Two straight lines enclose an area
Any two lines of longitude (straight lines)
meet at both the North and South poles so
define an area.
The sum of the angles of a
triangle are greater than 180°
General relativity
According to Einstein’s
general theory of
relativity, the Universe
obeys the rules of
Riemannian geometry not
that of Euclid. According
to Einstein, space is
curved!
Consistency
It would seem that it is easy to have a
system of mathematics that is consistent.
Not so!
Set theory
At the heart of set theory is a contradiction
Set theory
A feeling for the contradiction can be found
in the following story;
Set theory
A barber had an affair
with a princess. The
king was very angry
and wanted the barber
executed. The princess
begged for his life and
the king agreed,
provided that………
Set theory
……the barber went back to his village and
only shaved all the inhabitants that did not
shave themselves.
Set theory
“That’s easy” said the
barber.
Is it?
Set theory
Another example is to imagine catalogues
in a library. Some catalogues are for
novels, some for reference, poetry etc.
The librarian notices that some catalogues
list themselves inside, some don’t.
Set theory
The librarian decides to make two more
catalogues, one which lists all te
catalogues which do list themselves, and
more interestingly, a catalogue which lists
all the catalogues which do not list
themselves.
Set theory
Catalogues
which list
themselves
Catalogues
which do not
list themselves
Set theory
Should the catalogue which lists all the
catalogues which do not list themselves be
listed in itself?
If it is listed, then by definition it should
not be listed, and if it is not listed, it
should be listed!
Gödel’s incompleteness theory
Kurt Gödel (1906-1978) was able to prove
that it is impossible to prove that any formal
system of mathematics is without
contradictions.
Mathematicians’ certainty is an
illusion!
That’s it!
Thanks for your attention. Good luck next
year!