A comprehension of the content of Riemann's habilitation dissertation is essential for substantial understanding and progress in all important fields of thought. This presentation by Riemann, often treated (wrongly) as a mathematical presentation, reflects his views on the physical universe and the human mind, and gives us useful insight into understanding social processes. In this video, we'll cover the content of Riemann's presentation, and demonstrate an important fact of mental life: human beings do not sit outside the universe, investigating it from a fixed, stable location – rather, creative mental activity is itself a universal power, and must be itself considered by anyone seeking a unified physical view of the world. Let's dive in.
On the Hypotheses Which Underlie Geometry
Now, you've probably heard that if you add up the angles in a triangle, you get 180 degrees, or two right angles. You've also heard that two parallel lines never cross, even if you extend them infinitely. That's quite a claim. This triangle on a sphere has three right angles, and these lines, which seemed parallel, actually do cross when they are extended far enough.
What would our geometry teacher say! Well, you'd probably hear the response: “The rules all work fine if the lines are straight.” But, let me ask: what does it mean for a line to be straight? Can you think of a definition? Is it, perhaps, the shortest distance between two points? If so, the lines on our sphere are straight. “But the sphere is curved!” comes the objection, “make your lines in space.” Fine – but how will we make them them? Maybe we should use a ruler, although, how would we know that our ruler is straight? Maybe we should use a beam of light, a physical process. But that won't work, because light bends.
You see, the problem here is that without realizing it, we all have hypotheses about the nature of space itself, and we have preconceptions about constructions in space, such as making parallel lines. The now-famous faker Euclid didn't question whether his assumptions were true: he simply wrote out the geometry that corresponded to those assumptions (including the flatness of space), without showing that they were valid! More egregious than any specific wrong assumptions Euclid may have made, was the fact that his axioms found their basis only in a priori thinking of his imagination, not in real physical experiment.
Up until Riemann's day, the hypotheses underlying space and geometry had not been examined in a general way, nor was it recognized that these foundations were actually hypotheses. To make his thoughts clear, Riemann had to offer a general concept of what he called “manifolds” of various numbers of dimensions and reveal the possible curvatures of these manifolds. Then, he could return to the shape of the actual space we inhabit, and decide how – and upon what basis – it would be distinguished from other possible imaginary spaces. This isn't something to be settled by logic; the answer can only be reached by a continued path of experiment.
So, let's delve into the concept of manifolds in general.
Manifolds in General
Start with magnitude – what is it? Riemann says that a magnitude is a general concept that has multiple specific instances, or ways of being, or specializations. For example, “length” is a single concept, that has multiple specializations, such as “two inches,” “three miles,” or “five kilometers.” They are all specific lengths. Taste is another example: “salty”, “sweet”, or “oregano-flavored” are specific instances of the general concept of “taste”. Shoe size, temperature, location – these are all magnitudes.
Now we can distinguish two different kinds of magnitudes, those that change continuously and those which change in discrete jumps. As an example, the tones that can be played on a cello are continuous, while those on a piano are discrete. There is no key on the keyboard between a B and a C, but a cello can vary between them.
We can also consider whether a magnitude's particular specification (or sometimes called “mode of determination”), whether this particular specification requires one or multiple values. One example is position on the earth. All lengths can be arranged by size, and you can always say which one is shorter and which longer, but the same is not true for position. A location has both a latitude and a longitude, and while positions could be arranged by latitude, or by longitude, they cannot be arranged by position. The position of New York isn't larger or smaller than the position of Houston. Such magnitudes (or “manifolds”) for which the fixing of position requires two specifications, are said to be doubly extended. For example, specifying the location on any of these different surfaces requires two values: here you see the sphere, the plane, the monkey saddle, and the catenoid.
What's more, the location on a single surface can be specified in several different ways. Here are a variety of different coordinate systems being used to indicate the movement of a spot on a flat plane. While the coordinates are curved, the plane is not.
Now, if we move beyond surfaces, a position in space is triply extended. There are many ways of identifying a location in space, but all require are least three specifications. For example, we could give the XYZ position – the latitude, longitude, and altitude – or we could use cylindrical coordinates – or catenoidal ones. Now, these are all different ways of describing a location in space, like the variety of ways of indicating the motion of the spot on the plane. They don't, as coordinate systems, indicate space as curved.
Now, go up a level – just as the plane and the catenoid have different characteristics, not just different coordinate systems. For example, you can't put a plane on a catenoid – you'd have to bend it and stretch it. Can you think of a different space? That is, can you think of a three-dimensional space that isn't flat? It's difficult not to simply think of a curved object in space when you're pondering this question. We'll return to this question later, with some more ammunition for strengthening our imaginations of curved spaces.
As another example, how many dimensions are there in color, as perceived by the human eye? If you are familiar with the way color is represented on televisions and computer monitors, you know that all colors are created by varying amounts of roughly red, green, and blue light, making it three-dimensional. There are many other ways of representing color, such as YUV, L*a*b* color, or Hue-Saturation-Lightness (seen here), but they all use three-dimensions. This triple extension is not a character of light itself, but it's a characteristic of the human eye, which contains three different types of color receptors. This can trick you – mixing red and yellow light together may look orange to us, but do not become orange light – the red and yellow can be separated back out with a prism. Another quirk of vision is that the color magenta is not really between red and violet in the color spectrum – it does not wrap around except for the way our mind puts together the experience of the senses. While color is three-dimensional for us, for birds, which have four different color-receptor cells, color is four-dimensional!
This leads us to an example of a magnitude with so many dimensions you can't even count them up! The example is light itself, which has an infinite number of specifications to indicate it exactly. This comes up when matching paint, where the three dimensions of color on a monitor just aren't enough – the paint may match under a certain kinds of light, but not others. You can see how the match is better or worse, depending on the kind of light.
Here you have the color curve for a certain color of professional light filter – for each possible color in the spectrum, the filter transmits a certain percentage of light – so each of the infinite number of colors in the rainbow has its own value of transmittance – actual light (as opposed to perceived color) doesn't have three, but an infinite number of dimensions! Here, you see the perceived color of a changing color curve – there is so much in light that our eyes cannot distinguish!
Armed with these general concepts of magnitudes, we can now join Riemann in investigating the different metric relations that manifolds are susceptible of, and how they can be determined. What makes a sphere different from a plane?
Possible Metric Relations
To start our investigation of the internal characteristics of manifolds, we'll start with two-dimensional curved surfaces, and for this, we'll use the approach of the great scientist Carl Gauss, who chose the topic for Riemann's lecture, and was delighted to hear it. Gauss had developed a completely general method for investigating curved surfaces, with specific techniques.
For example, the sphere is curved while the plane is flat. Here's another shape, known as a monkey saddle. Unlike the plane, it is curved, but it's not curved the same way as the sphere is curved. How is it different? Can we quantify its curvature? As a first technique for doing so, we'll introduce the normal – it's a direction at each point on the surface that points directly away from it, perpendicularly. We'll use the normals of a surface to measure how curved it is. To do this, Gauss mapped the normals onto an auxiliary sphere – he kept only the direction of the normal, but not its location. You can also imagine the monkey saddle as being incredibly tiny and at the center of the sphere, just like we on the earth pointing at distant stars. Look at the directions of the normals as we move around on the surface. Sometimes, when we move to the right on the surface, the direction the normal points, moves to the left. The sphere serves for us as a kind of 3-dimensional compass, allowing us to indicate spatial directions, just like the rim of an ordinary compass tells us our direction on the two-dimensional surface of the earth.
Gauss's first way of measuring the curvature of a surface was to take a region on the surface, and compare it to the size of the corresponding region on the auxiliary sphere. The larger the area on the sphere, the more curved the region on the surface. Here, the red region is several times more curved than the blue region. To measure the curvature at a specific point, he would shrink the region until it was infinitesimally small. If we use this technique, we find that a cylinder has no curvature at all – it is called flat by Gauss's technique! As we cover this quadrilateral region on the cylinder, the region traced out by the normals is just a straight line, with no area: zero curvature.
Gauss's next technique for measuring curvature uses what are called osculating circles. Just as any two points imply a direction by connecting them and drawing a line through them, any three points form a circle. So if we pass a plane through a surface, we form a curve, and there is a circle that best fits that curve at the given point. Here you can see the series of osculating circles for a given point on the surface.
Gauss demonstrated that the most extreme osculating circles are always on planes perpendicular to each other, and showed that by multiplying the radii of the two circles and taking the inverse, you get the same measure of curvature that we got earlier with the normals. Again, we find that a cylinder has no curvature: one extreme osculating circle is the radius of the cylinder, while the other appears as a straight line, with an infinite radius. One divided by the product of these radii, is zero.
Now, before we get to Gauss's third method, which will be the most important for Riemann, let's take up a specific historical example: figuring out the size of the earth. To our knowledge, this was first discovered in the third century BC, by the onetime librarian of Alexandria, Eratosthenes of Cyrene. He had noticed that on the day of the summer solstice, the sun appeared directly overhead in Aswan, Egypt. Then, he measured the shadows on the same day of the year in Alexandria. By assuming the sun was so far away as to make its rays parallel, and by combining the angle of the shadows with the distance between the two cities, he estimated the circumference of the entire earth as 250,000 stadia, which was a remarkably accurate estimate. A characteristic of the entire planet was determined by making measurements in a small area.
Now we're ready to distinguish two different categories of surface characteristics: extrinsic characteristics and intrinsic ones. All the examples given so far were extrinsic characteristics, which use external objects and positions as references. To contrast intrinsic characteristics, let's pose this: how could Eratosthenes have measured the size of the earth if the atmosphere was constantly cloudy, like on Venus? If he only had the surface of the earth, and no extrinsic sun to help him, what techniques would lie open to him to discover what the characteristics of the earth's surface were?
Just to make it a little harder, let's say that we ourselves are two-dimensional creatures, rather than three dimensional. A popular example of this is in the book Flatland by E. A. Abbot. This author writes of a world in which only two dimensions exist: the residents are lines, triangles, quadrilaterals, and other polygons, with the leaders having many sides and approaching circular form. But what if there was a mistake? What if Flatland were really Sphereland? Each shape, moving around on a vast sphere, wouldn't notice one spot to be different than another, yet they could still get clues that something wasn't quite flat. Two such clues are the application Pythagorean theorem and displacing directions.
So, the Pythagorean theorem: the Pythagorean theorem relates three squares, arranged to form a right triangle. Everyone has heard in school that square A plus square B equals square C, but do you know that it's really true? Here, let's form two larger squares by adding a number of equal triangles. These two larger squares are the same size and area. Removing four equal triangles from both, what remains should also be equal – so the A and B squares do have the same size as the C square. But as we saw earlier, this is certainly not true on a sphere. Remember our triangle with three right angles – which side is A, which B, and which C? The Pythagorean theorem certainly doesn't hold here. The way it must be modified is a clue to the Surfaceland polygons – it's a technique for discovering the shape of their world. But remember, like us trying to imagine curved space, how could they imagine a curved surface? The anomalies tell them what's happening, even though they can't visualize a sphere, since it is three-dimensional while they are two dimensional.
A second technique involves direction. You could walk through a town or a building while keeping a sense of which way north is by keeping track of all the turns you've made. Here is an example on a plane. Now let's do the same thing on a sphere. We're moving around on the sphere, while always keeping the pointer in the same direction when turning. Now, when we get back, we aren't pointing in the original direction anymore! Let's see that again – although the pointer wasn't twisted along the way, its direction still changed.
So both of these two techniques – the Pythagorean theorem and moving directions – these are intrinsic to the surface: they do not make any reference to anything outside the surface, including even the concept of space outside it. In summary, anything the polygon creatures can do or learn is intrinsic.
Gauss then makes two amazing points: one of them is that none of the intrinsic characteristics change if the surface is bent around, so long as you don't stretch it. That is, bending a plane into a cylinder and back doesn't change anything in the surface itself: the distances between points, the shortest line between two points, angles, etc., everything remains the same. The Surfaceland residents would never notice a difference. Similarly, the catenoid and the helicoid have the same intrinsic characteristics, and are formed from bending one into the other. Gauss's second breakthrough was to come up with a way to measure the curvature at every point intrinsically, like the flat polygons could, meaning that the surface could be its own complete world, and giving it a certain shape in space (such as the cylinder versus the plane) is unnecessary. We don't need normals or osculating circles. The surface can be understood, from within.
This breakthrough is key for Riemann's examination of curved spaces. Triply extended manifolds (such as space) can be curved! But now, we can only think of intrinsic curvature, since we can't step outside of space from a fourth dimension to look down on it, as we can look down on doubly extended surfaces. Instead, we are in the same shoes as our polygon friends for determining the shape of space around us. Riemann, in his paper, gives the most general possible way of determining curvature from within a manifold. It is the anomalous characteristics of action and motion, that define the curvature.
With all this possibilities, what is the shape of actual space, as opposed to a mathematical daydream?
Understanding Curved Spaces
As we apply these considerations to the actual shape of physical space, let's first contrast the two different concepts of unboundedness and infiniteness, which are often confused. Take a sphere as an example – motion on the sphere meets with no boundaries, finds no limits, and yet the sphere is not infinite – it has a total size that is measurable, while motion is unbounded. While space itself appears unbounded in every respect, we cannot conclude from this that it is infinite! Space might be finite.
Now, as another consideration: let's ask whether objects are independent of position. Take as an example, the difference between an orange and a watermelon, or some other kind of fruit. On the orange, the skin at once location can be moved to another without stretching – there's no difference at all, the parts are all the same. But this isn't the case with our other example. The almost-flat region has a certain relationship between circumference and area, but here, the same circumference has a greater area! So stretching would be involved in moving from one place to the other. In three dimensions, this would be like moving an orange in space, and having the insides get stretched bigger, even while the skin on the outside stays the same size.
Maybe we want to avoid that. So, we could do that by first hypothesizing that objects are independent of position – that would mean that every portion of space has the same curvature as any other. Now, if that measure were even slightly positive, then space would be finite. This would show up in astronomy, where the same star could appear to be in two opposite directions due to the curved nature of the intervening space, as you see here. Setting out in any direction, you will return to your starting point. Even if we didn't go all the way around, the same star could appear on multiple different paths, appearing as a halo, rather than a point.
But we don't see duplicated stars or such halos in astronomy, even when we look far away. So if space were uniform, it would have to be flat, or almost entirely so. But what if space isn't uniform? What if objects aren't independent of position? What if spatial relations change from place to place? Then we couldn't infer anything about relationships in the small from what we have discovered in the large from astronomy. In fact, in Riemann's day, breakthroughs were being made in chemistry and electromagnetism, by hypotheses about the nature of activity on the very small. The metric relations in the small could have all sorts of characteristics, so long as on very large scales, the curvature evened out to the near-zero curvature inferred from astronomy. The apparent flatness of geometry on the astronomical scale, as understood in Riemann's day, has no inherent truth on the micro-scale.
So how can we figure out the small-scale metric relationships? In what way is space curved, and, more importantly, how can we discover why it is curved as it is?
“In a discrete manifold, the basis of metric relations is contained in the concept of the manifold itself, while it must come from elsewhere in the case of a continuous manifold.”
To clarify that, whenever you name or conceptualize a discrete manifold, such as “the keys on a keyboard” or “the people in a room”, you've already given the means of measurement with the conception of the manifold. But, in a continuous manifold, such as length or position, you're given no idea of what the space is like, or how measurements ought to be made. Riemann continues:
“Either then, the reality underlying space must form a discrete manifold, or the basis of metric relations must be sought for outside it, in the binding forces that operate upon it.” Yes, exactly! The basis for anything, its sufficient reason for being as it is, rather than otherwise, does not lie in making many observations of it!
It seems like Riemann's investigation won't be ending with a final conclusion! In fact, Riemann finishes his lecture with the limits of armchair mathematical theorizing. While studies such as his can remove unjustified presumptions, they cannot make affirmative conclusions. Riemann closes: “This leads us into the domain of another science, the realm of physics, into which the nature of the present occasion does not permit us to enter.”
We'll now leave mathematics behind, and venture into the realm of physics, the realm of reality.
“The Realm of Physics”
Consider Johannes Kepler and the birth of astrophysics. Kepler entered a field that had been studied on the basis of understanding nature from the standpoint of the senses, very explicitly. His predecessors had put forward various models for the planetary system, based upon “saving appearances” – that is, their goal was to cause their models to present the same impression to the senses as the planets do. Kepler proved that even as they chased appearances, the models were always wrong, because of their method. And, furthermore, he knew that even a wrong hypothesis could look like the truth. He wrote about his own Vicarious Hypothesis:
“Further the lack of any perceptible difference in effects between the as-yet unknown true hypothesis and the false one assumed by us does not make the effect identical. For there can be a small discrepancy which the senses do not perceive.”
There will always be things we have not measured. Even if our model matches observations perfectly, that does not mean it is true – not just because the observations will get better in the future, but because matching observations, although necessary, is never itself the standard of truth. Although Kepler made a working mathematical model for the planets, he was not content without a hypothesis of the purpose lying behind their motions! Why do they move the way they do? In his New Astronomy, he hypothesized that a power in the sun caused the motions. In his Harmonies of the World, he details the harmonic principles of composition that required the specific orbits displayed by the planets. In this case, cause is purpose – it is a why, rather than a what or a how. Newton's inverse-square generalization may describe how motion changes from moment to moment, but not why, as Newton himself admitted. Kepler demonstrated the failure of mathematics, and succeeded as a scientist! His simple, beautiful question: “why so, rather than otherwise?” moved him beyond inductive generalizations of a world as according to sense-impression, and into a conception of the world as composed on the principles of beauty, as rigorous as they are free.
Pierre de Fermat demonstrated the value of purpose as a scientific concept. The refractive bending of light as it moves from one material to another had puzzled thinkers for centuries – why does light bend the way it does when it enters water, for example? Fermat discovered the principle of least time. Among all the paths light might have taken from its origin to its destination, the paths it follows are those that make the journey the quickest. In this case, least time is nothing sense-perceptual. It has the character of a motive, not an appearance.
Gottfried Leibniz demonstrated that the assumption of an a priori, independent space and time let to absurdities. When the Newtonian Samual Clarke tried to demonstrate God's great power in the free choice that God had in deciding where to make everything at the moment of creation, Leibniz responded that that's no decision at all. Moving the universe two feet to the right, is the same as moving space two feet to the left. The relation between objects wouldn't change at all, nothing would ever know. It has absolutely no meaning. The absurdity arises from the assumption of a space prior to, and independent of, things to be related in space. There is no absolute space.
This is also seen when you compare Descartes's laws of motion with those of Leibniz. Descartes's biggest problem (well, one of his biggest problems), is that he believes in absolute motion and absolute rest, which drove him to conclusions that are really nuts. Leibniz knew that motions are only relative, while the cause of the motions can be real and absolute. Physics must be the foundation of geometry, and that was the basis of Leibniz's development of the infinitesimal calculus.
Take Albert Einstein. His theory of relativity discards distinct geometric time and space, instead using the physical process of light propagation to give a physical meaning to space-time, and, in doing so, showed how space-times differ for different observers. They exist as action-spaces, not geometric spaces. Physical action is primary, and geometries are created to reflect our hypotheses of the true relations between unfolding actions in the universe. You've got to make geometry match the physics, not the other way around. Today, Einstein's general theory of relativity is the most commonly cited example of curved space-time (although not always correctly).
Now move to biology, and beyond. Vladimir Vernadsky's passionate search for understanding the nature of life and cognition led him to hunt for geometries capable of expressing activities of life that he knew simply could not exist in a Euclidean space. One example is the chirality, the handedness of living processes. Pasteur and Curie had demonstrated that unlike abiotic processes, living processes showed a preference between left and right-handed versions of the same molecule, a preference which could not exist in simple Euclidean space. Vernadsky also wrote much about the different kind of living time distinct from abiotic time. In evolutionary living time, for example, before and after are not merely distinguished chronologically, as before being not-after and after being the opposite of before, but rather after is fundamentally different than before, being a time in which higher developments of new life processes exist. This is seen much more strongly in human time. In our economic time, the power of the human species – and we are ourselves a physical force – changes categorically with new discoveries of principle. Economic times differ qualitatively, not quantitatively. And such human time doesn't just “happen” like the ticking of a clock, it has to be created through discovery and driven by passion! This is the spacetime of economic development.
So Euclid wasn't just wrong in his specific axioms; any different geometry starting with geometric axioms, rather than the principles that shape real physical action, would err as well. With Riemann, “geometry” itself completely changes its meaning – it isn't the stage upon which events unfold, it's the shape of action itself!
What can we say about the process of gaining knowledge about this shape? Is our goal an ultimate understanding of nature, which we observe, hypothesize, and approach, as if asymptotically, although never reaching this ultimate knowledge? No: we are part of what we're studying!
Consider the most powerful of physical forces: the human mind. Creative thought is a physical force: it has physical effects just like electromagnetism, plasma, biological processes. A true Riemannian geometry, based firmly on the principles that lie behind perceived appearances, must take creative mind into account. Our goal isn't a final geometry of the world out there: it must include the developing powers of reason – the kernel of economic development. There must be no separation between physics and the study of the mind. Physical science that rejects Mind can never find true principles, but will always be stuck in a bog of statistical induction and correlation based on the senses. Similarly, social philosophy or sociology which isn't informed by the study of fertile creative thought will become a study of neuroses or pedestrian irrelevancies, with a complete lack of any useful direction.
There is only one world to discover and act on. Mind discovers, mind acts, mind creates. Riemann brings us into reality, and shows that the principles underlying reality cohere with the mind. While he concluded his lecture with the need to abandon mathematics for physics, to truly achieve Riemann's program, we must go beyond physics to economics; we must include the progressing development of the powers of the human mind. Thus, the importance of Riemann for economic science.