Sunday, July 26, 2009

Beta is Reality

I have just posted the following response to Andrew Keen's article It's Time to Bust the Beta Cult in the Internet Evolution.

Beta is Reality

Should there be such a thing called finished product?

The era of software products designed behind closed doors without dynamic user involvement is over.

a) It costs too much to develop a finished product, especially for startups, yet alone I argue that the concept of finished product is an illusion, products are never finished, and should not be finished.

b) It is too risky to ignore user feedback during development. The chances are you will be increasingly drifted away from user satisfaction. There is no way you would know what users want unless you design your product with them.

c) Increasingly users want to get involved once they experience the satisfaction of being listened. User profiles are changing, the era of passive user is over.

This is in fact what I call the evolutionary design, a concept akin to natural selection we observe in biological systems. The product evolves based on its survival value; its ability to adopt changing user requirements. In fact there is no up-front designer. Users become the nature, and they themselves design the product by selecting the fittest, fittest in terms of giving them the best satisfaction score.

The design by natural selection does not need to be perfect or completely bug free. Take the evolution of human eye for example, which is a strikingly good example to bad design, if there were an up-front designer.

"The vertebrate eye, is built "backwards and upside down", requiring "photons of light to travel through the cornea, lens, aquaeous fluid, blood vessels, ganglion cells, amacrine cells, horizontal cells, and bipolar cells before they reach the light-sensitive rods and cones that transduce the light signal into neural impulses – which are then sent to the visual cortex at the back of the brain for processing into meaningful patterns." This reduction in efficiency may be countered by the formation of a reflective layer, the tapetum, behind the retina. Light which is not absorbed by the retina on the first pass may bounce back and be detected.", ref. Wikipedia.

The beta paradigm represents evolutionary generations. Each generation, i.e. each beta life cycle changes the product's survival value, sometimes towards the extinction end, other times towards the selection end, and the reality is you would never know up-front whether your product be extinct or survive in x years time. Except that users (the nature) will survive even if it means they may end up selecting and using a different product, not yours which might have become extinct.

Therefore it seems to me that the evolutionary design (hence the beta concept) in software systems is here to stay. We humans have discovered and learned the power of natural selection in the past 150 years, why shouldn't we enjoy exploiting and using that power in commerce and in other areas of human ingenuity.

Finally, I am also thinking Beta as a meme, ie. "a postulated unit or element of cultural ideas, symbols or practices, and is transmitted from one mind to another through speech, gestures, rituals, or other imitable phenomena.", ref. Wikipedia.

Friday, July 17, 2009

Poincaré's Great Topological Papers

Henri Poincaré had learned of non-Euclidean geometry from Beltrami who first noticed that hyperbolic geometry was a constant curvature geometry in which the curvature is negative. Beltrami pointed out that a surface called the pseudosphere carried such geometry. He also discovered the disk model and realized that Riemann's conception of geometry provided the link that united the two conceptions. Poincaré worked out the details and discovered more models. Moreover, Poincaré did not stop at two dimensions. He extended this work to higher dimensional geometries. Here he is, in his own words, describing his model of three-dimensional hyperbolic space:

"Suppose, for example, a world enclosed in a large sphere and subject to the following laws: The temperature is not uniform; it is greatest at the centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows: If R be the radius of the sphere, and r is the distance of the point considered from the centre, the absolute temperature will be proportional to R2-r2. Further, I shall suppose that in this world all bodies have the same coefficient of dilatation so that the linear dilatation of any body is proportional to its absolute temperature. Finally, I shall suppose that a body transported from one point to another of different temperature is instantaneously in thermal equilibrium with its new environment. There is nothing in these hypotheses either contradictory or unimaginable. A moving object will become smaller and smaller as it approaches the circumference of the sphere. Let us observe, in the first place, that although from the point of view of our ordinary geometry this world is finite, to its inhabitants it will appear infinite. As they approach the surface of the sphere they become colder, and at the same time smaller and smaller. The steps they take are therefore also smaller and smaller, so that they can never reach the boundary of the sphere. If to us geometry is only the study of laws according to which invariable solids move, to these imaginary beings it will be the study of laws of motion deformed by the differences of temperature alluded to..

Let us make another hypothesis: suppose that light passes through media of different refractive indices, such as the index of refraction is in R2-r2. Under these conditions it is clear that the rays of light will no longer be rectilinear, but circular.... If they [the beings in such a world] construct a geometry, it will not be like ours, which is the study of movements of our invariable solids; it will be the study of the changes of position which they will have thus distinguished, and will be 'non-Euclidean displacements', and this will be non-Euclidean geometry. So that beings like ourselves, educated in such a world, will not have the same geometry as ours."
from The Poincaré Conjecture Donal O'Shea

Saturday, July 11, 2009

The Poincaré Conjecture

I've been reading this fascinating book "The Poincaré Conjecture - in search of the shape of the Universe" by Donal O'Shea.

In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds.

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold.

Around 300 BC, the Greek mathematician Euclid undertook a study of relationships among distances and angles, first in a plane (an idealized flat surface) and then in space. An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. Today these relationships are known as two- and three-dimensional Euclidean geometry. This is the type of geometry we learn in the high school.

The Poincaré conjecture concerns a space that locally looks like ordinary three dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is just a three-dimensional sphere.

A sphere can be represented by a collection of two dimensional maps; therefore a sphere is a two dimensional manifold.

An important scientific application of 3-manifolds is in physical cosmology, as models for the Shape of the Universe – the surface of the earth is locally approximately flat – it is roughly a 2-manifold, and globally the surface of the earth is a sphere. The universe, likewise, looks locally approximately like 3-dimensional Euclidean space, so the universe may be modeled as a 3-manifold, and one may ask which 3-manifold it is. More precisely, the universe is better described as 4-dimensional space-time, so this description is of a 3-dimensional spatial section.

Our genetic code allows us to see in 3 spatial dimensions. Most animals possess perspective vision evolved to enable them estimating distance, a vital tool for survival to avoid predators and approach prays in great efficiency. Simply said we have no provisions to see more than 3 spatial dimensions, therefore we have difficulty comprehending concepts such as 4-dimensional space-time. We have trouble picturing the shape of the universe as a whole.

It is not possible to get outside the universe. This is an important difference between the Earth and the universe. A rocket can leave the surface of the Earth, and we can look at the Earth from outside it. Since we see in three dimensions and since the surface of the Earth is two-dimensional, we can see our planet is bending in a third dimension and visualize its whole shape easily. However even if we could get outside the universe in an attempt to see what shape it had, since the universe is three-dimensional, we would need to be able to see in at least four dimensions to visualize the universe as a whole.

The conjecture in question is a claim made by the mathematician Henri Poincaré over 100 years ago. He said that all four dimensional spaces in which a loop in that space can be narrowed to a point are four-dimensional spheres -- there are no other possible shapes for the universe.

Until 2002, no one had been able to prove or disprove this claim. The framework for this book was the announcement by Russian mathematician Grigori Perelman that he had proven it.

If I travel on a 2-dimensional manifold or surface like Earth, I will always come back to the same point. There are no edges that I would fall. The proof of Poincaré Conjecture supported the idea of Universe is finite (a closed 3-manifold) and has no boundaries. So if I traverse the universe in one direction I will always come back to where I started.

I will have more posts on the shape of the universe, so stay tuned.