Bernhard Riemann

Yeah.

The next one I'm gonna do is, I think, Riemann's mapping theorem because on the 17th of September, it's the 200th anniversary of Bernard Riemann's birth.

We wait with bated breath to see if an AI will prove the Riemann hypothesis before that.

You think it will?

No, it seems out of sight, but who knows really? An AI just proved that there's a, a complexification of the six sphere.

That was out of sight.

No- nobody thought saw that coming.

Anyway, so I'm gonna put a theorem at least by the 17th of September, I shall put on Bern- uh, Riemann's mapping theorem, which is a complicated theorem.

Now we may start with this view of space according to which the position of a point may be determined by measurements in relation to any given figure, a system of coordinates, taken as fixed and then inquire what are the special characteristics of our space as manifested in the measurements that have to be made and how it differs from other extended quantities of like variety.

This path was first entered by one too early lost to science, B. Riemann of Göttingen.

It has the peculiar advantage that all its operations consist in pure calculation of quantities, which quite obviates the danger of habitual perceptions being taken for necessities of thought.

The number of measurements necessary to give the position of a point is equal to the number of dimensions of the space in question.

also

the other dimensions, because, you know, something that happened, really the start of modern physics with Einstein happened because he latched on to the insight of a German mathematician of the previous generation called Bernhard Riemann.

And what Bernhardt Riemann discovered was that if you're constructing theories to try to describe accurately what happens in the universe, then if you posit hypothetically other dimensions, you will find that you can create a model which is much more useful, much more accurate, and much better at predicting what is going to happen.

And that's weird, isn't it? Because it seems to suggest that these other dimensions are not merely hypothetical.

Now we may start with this view of space according to which the position of a point may be determined by measurements in relation to any given figure, a system of coordinates, taken as fixed, and then inquire what are the special characteristics of our space as manifested in the measurements that have to be made and how it differs from other extended quantities of like variety.

This path was first entered by one too early lost to science, B. Riemann of Göttingen.

It has the peculiar advantage that all its operations consist in pure calculation of quantities, which quite obviates the danger of habitual perceptions being taken for necessities of thought.

The number of measurements necessary to give the position of a point is equal to the number of dimensions of the space in question.

In space, the distances from three.

Or we can require, as on the earth, longitude, latitude, and height above the sea, or, as is usual in analytic geometry, the distances from three coordinate planes.

Riemann calls a system of differences in which one thing can be determined by n measurements an n-fold extended aggregate or an aggregate of n dimensions.

Thus, the space in which we live is a three-fold.

A new branch of mathematical research opened out and the possibility of non-Euclidean geometries received universal recognition.

It was to this study that Riemann devoted himself.

He imparted to it, however, an entirely new direction.

His guiding idea in working out a complete system of spherical geometry in three dimensions was that the dimensions impose limitations or restrictions on the system and determine its degree of freedom.

The number of parallel lines that can be drawn through a given point will be neither one nor many, but none.

The sum of the angles of a triangle will be greater than two right angles, and so on.

The significant part of Riemann's conception of his tri-dimensional spherical geometry, however, is not his proof that it is a workable and consistent geometry, but that seen from within it would have, for those restricted by it, the appearance of the unlimited freedom which we attach to the plane geometry of Euclid.

It would be impossible for those attached to the system to be conscious that the straight lines in which they would seem to be moving were curves.

Austin is the capital of Texas.

Riemann.

Similar.

No, they ex-identical.

Totally.

But mathematicians like Nikolai Lobachevsky, János Bolyai, and Bernhard Riemann began to play with that axiom.

They asked, "What happens if we drop the assumption that there is exactly one parallel line?"

Oh, wow.

And classical thinking would assume that changing such a fundamental truth would immediately lead to mathematical contradictions, right? The system should just collapse.

Exactly, but it didn't collapse.

When Riemann assumed there are zero parallel lines, he birthed spherical geometry.

Like drawing on a globe.

The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian coordinates in the surface.

Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed out the way to the method of Riemann of treating multidimensional, non-Euclidean continua.

Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity.

End footnote.

But speculations on the structure of the universe also move in quite another direction.

The development of non-Euclidean geometry led to the recognition of the fact that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience.

Riemann, Helmholtz.

These questions have already been treated in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them briefly here.

Now we may start with this view of space according to which the position of a point may be determined by measurements in relation to any given figure, a system of coordinates, taken as fixed, and then inquire what are the special characteristics of our space as manifested in the measurements that have to be made and how it differs from other extended quantities of like variety.

This path was first entered by one too early lost to science, B. Riemann of Göttingen.

It has the peculiar advantage that all its operations consist in pure calculation of quantities, which quite obviates the danger of habitual perceptions being taken for necessities of thought.

The number of measurements necessary to give the position of a point is equal to the number of dimensions of the space in question.

In space, the distances from three.

Or we can require, as on the earth, longitude, latitude, and height above the sea, or, as is usual in analytic geometry, the distances from three coordinate planes.

Riemann calls a system of differences in which one thing can be determined by n measurements an n-fold extended aggregate or an aggregate of n dimensions.

Thus, the space in which we live is a three-fold.