Alright, let's now, look at non euclidean geometry and the challenge it presents for Kant arguments.
So I'll first start by explaining what non euclidean geometry is where it came from and then explain
why this is a problem for Kant.
So let's go back to Euclid. Five postulates.
The problem is with this number five, the parallel postulate. So recall that this postulates state
that any straight line, there's a straight line. And any point not on that line is the point
determine a unique to just one straight line that's parallel to the first. So that's this blue one
here.
Now this one was like, one of these postulates is not like the others people were immediately
uncomfortable with this one. A bit suspicious of it. It doesn't seem as neatly intuitive or natural
as the other four.
Sorry. Yeah, this is the the one we're going to look at him.
So, as we've seen a postulates or an axiom to statement that acts as the starting point for a theory
and it's not something that can be proven from a previous result, but it should then be very
plausible or self-evident.
Now, Euclid himself may have actually been a bit suspicious of this one because actually proved his
first 28 propositions without making use of it. You only use the the first for proper.
Axiom hume's to derive, the first 28 propositions, and yeah, other great mathematicians following
him over the centuries and Millennia, they struggled with it, they didn't really think that it fit.
So they were trying to find maybe a proof of it will show that it could be derived from the other
four accent.
And none succeeded and it's a pretty great history. I recommend that these two YouTube videos. Well,
one is a playlist on the history of euclidean geometry and seeing some of these people that wrestled
with this. Yeah, I really recommend these. I mean, it's not so relevant for this this lecture, but
as a background or if you're procrastinating the, it's really cool stuff.
So although Euclid in Geometry, could not be proven though, still seemed inconceivable that it could
be false. We still kept using it a relying on. It still seems to just fit, so naturally with
everything we're doing and everything we're experiencing, how could it be wrong?
Well, a Hungarian mathematician a bully. I he was one of these great thinkers who became pretty
obsessed with this fifth postulate.
And actually, he's father, who was also a mathematician interested in it recognize this and actually
wrote to his son and said, you must not attempt. This approach to parallels. I know this way to the
very end. I have traversed. This bottomless night with extinguished, all light, and joy in my life,
son. I entreat you leave the science of parallels alone. Learn from my example,
But he did not and he kept pursuing and checking and seeing what sense you can make out of this
parallel postulate. So he wanted to set out to prove it to put it on more rigorous footing to show
that it's necessary. And one way of doing proofs in mathematics is a like a proof by contradiction
by assuming if you assume that this
A false and then show that by assuming it's false, you led to a contradiction then you know that it
must be true.
So you wanted to show that if we deny this postulates that were LED with a, a inconsistent set for
an inconsistent geometry, and then it just won't work and thereby show that this parallel.
Postulates is necessary.
So what he did was try a assuming that there isn't just one unique straight line. Parallel to the
first line that goes through this point. Instead, he assumes that. Well, if we think that there are
many parallel lines that go through the point, do we end up with a consistent geometry or not?
So here's a depiction these blue lines. Now actually parallel to the white line.
So this is the set of Axiom as you would end up with by assuming instead this new Axiom of many
parallels assuming that for any line and any point there are many lines that are parallel and go
through that point and actually what will the I showed was. In fact, this does give us a consistent
geometry.
Where I discovered that the original fits postulates, a a is independent of the other axioms and
that a different, but consistent geometry can be constructed without assuming the original V
posture.
But you end up. Yes, with some very strange unintuitive geometry.
So he wrote to his father, I have discovered such wonderful things that I was amazed out of nothing.
I have created a strange new universe. So a body is work was eventually published as an appendix to
a mathematics. Textbook was his father, but actually before it got published, there were similar
work also published by a Russian mathematician a, a little bash SQ.
In 1829, who worked on this independently and unknown to bolyai-lobachevskian a bird. This type of
geometry at once.
So we now call it boy, I love love a a skin or hyperbolic geometry so using the original for Axiom
but now with the fifth postulate replaced by the many parallels, postulates you end up with
completely different theorems from those of euclidean geometry. For instance, the ratio of the
circumference of the circle to its
It is is more than 2 pi geometric figures. Do not have the same shape as one another without being
the same size and strikingly the internal angles of a triangle, add up to less than a hundred eighty
degrees.
So yes, it is a definitely, a strange new world that he's created. You can see here, this is a
representation of the geometry. Having this saddle shape. This green thing is a triangle. You can
see that if we add up the internal angles, they're going to be less than a hundred and eighty
degrees. And these two lines with the right angle between them
These are example of parallel lines, you can see that they eventually diverge from one another.
We can also represent this hyperbolic geometry on a, to a 2d plane. And on this one, it's called
poincare, a his disc here. You can see that straight lines. They're represented here with the right
angles. So they converge at the boundaries of the disc at right, angles. So, these are all straight
line.
A look, curved due to this a representation on this flat plane.
So again, here is an example of a triangle in. This hyperbolic geometry is examples of straight
lines, and here are parallel lines going through the point together.
Okay. So yeah strange stuff. You'll also recognize this a work of a shh.
He used it in a lot in his art, these tessellation, so the figures they get smaller and smaller kind
of to the infinite edge as they move outwards in this depiction of the geometry.
But yeah, hyperbolic a geometry, wasn't the only non euclidean geometry? What happens when we
replace the parallel postulate with this postulates, that there are no parallels? So there's not
that there's one unique parallel line. There just no parallel lines possible, every pair of lines
intersect,
And if you have these five postulates, you end up with eliptic geometry.
And he's a representation of eliptic geometry. And here you can see, we have triangles with a
interior angles, add up to more than a hundred and eighty degrees and parallel lines will eventually
cross one another
So, and they're not parallel.
So here's a summary of this. You can see in euclidean Geometry, we have one parallel of this
postulates, a sum of triangles angles add up to a hundred and eighty degrees. The ratio of
circumference to the diameter of a circle is pi. Everything's nice and normal as we do in our
everyday lives.
But then, we also have these other possibilities where triangles interior angles, don't add up to
180 degrees.
so,
This shows that Euclid in Geometry is not necessary so therefore it can't be a priori as Kant
thought. This is a bit of a little, a timeline of the discovery of non euclidean geometry Bernhard,
Riemann. He actually showed that there are infinitely many possible, non euclidean geometry.
But so far this is all results in in mathematics, you know. It's certainly showed that it's not
necessary truth that the interior angles of a triangle add up to 180 degrees but it's still at the
level of mathematics and kind of not in the world. So this is already enough to destroy a, a Kant
sir.
Assumption of the, a priori nature of geometric truths of euclidean geometry. In particular, but it
gets worse.
Because in 1915 Einstein finished his general theory of relativity which showed actually that our
universe could have non euclidean geometry, so it's no longer just kind of a mathematical fantasy
that we're drawing on our papers and imagining what it would be, like actually could be our
universe, it's physically possible.
Assume that the geometry of our universe is non euclidean.
So it could end up in something like this resembling a HP Lovecraft story which is just going to
break our brains from having all this weird. Now geometry going on warping of space and time.
So, Einstein's theory of general relativity. Yes, it allows for these weird shapes of space. So what
is this Theory? It's a theory of gravity, but identifies gravity with the curvature of space-time
around, massive bodies,
So the idea is that matter with its mass it bends space-time in the vicinity and the greater the
mass of your object the more space time. Whoops. So space-time is a four dimensional field.
It's dynamical and the slogan by wheeler someone who worked very hard, a interpreting this Theory
and developing mysterious. Well he put it very eloquently space-time tells matter how to move matter
tells space-time how to curve. So it's a nonlinear Theory, you know, Matt insell, just space-time
space-time informant, doesn't matter, here's a nice picture.
A picture of this. This is supposed to be space-time curving warping around Earth. So due to the
mass of the earth and the curvature of this space-time is felt by this satellite as the pull of
gravity. So it's thanks to this curvature of space-time around the earth that keeps this satellite
in orbit.
Around.
So a matter, a affecting space-time space-time, also affected matter.
So it's not a flat euclidean geometry on this View. And but what what does the theory say about the
actual geometry of our universe? Are we in it eliptic or hyperbolic? Well, it's not determined by
the theory itself. You actually need to go out and make some observations of the universe.
A us and discover. What is the geometry of space? So a geometry is not a priori, it's empirical
relies on our interaction, observation with the world.
So Karen its argument, he starts out by saying geometrical knowledge is synthetic a priori a row he
then argue that synthetic. A priori knowledge is only possible if space is nearly a form of
intuition and not a property of subjective objects or things in themselves and then went on to argue
for this radical. Reconceptualization space is a form of intuition.
A structure that our minds impose Upon Our representations of the world.
So these discoveries in math and physics, just thinking through this all out the window, geometrical
knowledge is not synthetic, a priori. It's empirical we can go out. We need to test it.