Ali's Journal

The complex Fourier series provides two independent images, a real image and an imaginary image. This means there should be two independent proj lines in probabilistic proj geometry. However, these two images are linked, and modifications in one (for example the real part of the Fourier series) impact the other image (i.e. the imaginary part). These two images together form a complete image.

This is another mechanism for developing evolution mechanism. In the immigration plan, I work on, I said that any plan on this issue should consist of arguments from both sides. If the right develops a plan that pleases right-wing voters, once the right loses the election left will discard that and will develop an entirely different plan. The evolution of plans is not possible in this situation. But if two independent images are developing at the same time that form one entity then these two images will interact with each other to form one coherent entity.

It’s known that the efficiency of heat engines is limited. So a portion of energy will be wasted. The same holds true in collective intelligence. A complex system contains several components. The process of gathering the info that is distributed among the system into a coherent knowledge entity is comparable to that process. Inevitably a portion of info will be lost in the process and the aim is to minimize that so that the collective intelligence gathered from the system provides the best possible description of the system.

As an example, A painting might contain several shapes, etc. To produce that painting those elements should be placed alongside each other in a particular fashion. A better example would be a Puzzle that contains several pieces. Pieces are dispersed among a complex system and the task in collective intelligence is to retrieve those pieces and place them together in a coherent entity. The top-down approach works for very simple systems but in hugely complex systems info is so dispersed among the system that the top-down approach (someone who knows every detail of everything) is not doable.

The same ideas can be used in the wave probability. It was said yesterday that protective lines should be waves. But those waves are produced by different basic sine waves as specified by its Fourier transform. Every basic sine captures one characteristic of the scene to be observed (think of it as different layers in a neural network). So the task in probabilistic observation is to define such a wave proj line that minimizes the lost info so that the best possible description of the scene is provided.

This is a section from the article I’m writing:

If an unbiased coin is tossed several times and most of the time it lands head, psychologically one expects that the more the ratio N(H)/N approaches toward 1 for large N, p(H) should tend to zero in the next rounds (N(H) is the number of times the coin lands head). As explained before, this is a false belief in the probability theory. But since humans possess memory this experiment will involve the psychological element when the coin is tossed numerous times.
Imagine a fixed object in front of you on the desk you work every day. Contrary to what most people might believe if that object is modified slightly it might not be noticeable immediately. That’s because that object is seen numerous times every day, so after some time, the brain constructs it automatically. That reduces the energy required to notice every detail of the environment. So brain should assign variable probabilities to different elements in the environment. Elements with higher probabilities are more likely to be noticed. As an example, if someone is injured in some situation, the next time that person is placed in a similar situation, for evolutionary reasons, the probability of noticing the object that caused the injury is much higher. This type of probability assignment procedure is based on memory and therefore is totally different from the conventional probability theory because previous occurrences influence the next probabilities.
In fact, this is not strange at all. For instance for an unbiased coin, p(H) = ½. In other words, psi = ½ phi_1 + ½ phi_2 where psi represents tossing an unbiased coin, phi_1 represents head and phi_2 represents tail. To make things similar to quantum mechanics and since this is an arbitrary representation for now, define a discrete probability as psi:= A_1phi_1 + A_2phi_2 + … where phi_i refers to possible occurrences and p(phi_i) = (A_i)^2. Before observing the outcome of an experiment, psi represents that experiment clarifying what occurrences might occur and with what probability for each. Once the outcome of the experiment is observed psi collapses into one of the states (i.e. a particular phi_i). In other words, psi’=1*phi_i (for some i). So when the probability of phi_i is increased to 1, the probability of phi_j (j != i) collapses to zero. So based on this interpretation observation (i.e. psi’) and the probabilistic state (i.e. phi) are fundamentally the same. The only difference is that for psi’, p(phi_i)=1. The implication is that the transition from psi to psi’ is continuous.
This interpretation provides a common ground for the development of the conventional probability theory and the one that contains memory. This method requires that psi is governed by a symmetry so when p(phi_i) changes, simultaneously, all p(phi_j) should also change so that sigma (n=1 to max(j, i)) p(phi_n) =1. In this particular example psi’ changes the value of p(phi_i) to 1 and sets all other probabilities to zero. In real-world examples, for instance, in the one mentioned above, probabilities of objects in an environment can get any value (other than zero and one). The object that caused the injury may get higher probability but for all sorts of reasons it might not be perceived by that person in a particular situation. To clarify, this method adds the capability of changing the probabilities of possible occurrences. Then observation is merely the limit point (when the probability of an outcome equals 1) of the probabilistic state.
The mathematical tool for observation is projective geometry. The process of observation involves the point at infinity. Imagine a very long straight road and trees on both sides. When this path is observed trees gradually shrink and eventually the entire space turns into one point. So the mechanism of observation cuts a portion of the space. Based on what was said about probability previously, this mechanism can be expressed by probabilistic terms as well. Meaning that the probability of observing the portion of the image that is beyond the point at infinity is zero. So observation assigns probabilities 0 and 1 to elements in space. Observable elements that are included in the image get the value 1 and other elements get the probability of zero. This interpretation requires that observation should be the limit point of a process. Put differently, there should be distinct possible outcomes (i.e. images) and when the projective plane (or P) is formed and observation is complete, one specific image will be formed on P. This makes projective geometry very similar to social perceptions as described briefly previously. Because different viewpoints in the social context are due to observing the same things in different ways. This discussion suggests adding a probabilistic phase to projective geometry. Then the formation of an image on P is the limit point of the probabilistic phase.
It is easy to express these ideas in mathematical terms. In fact, projective geometry already includes the probabilistic phase. All it takes to add this phase is to remove P. Define the set of all slopes in the projection space as S:= {S_theta; for all theta in [0, pi/2]}, Also define each slope as S_theta:= {A_theta,r; for all r in R-extended}, where R-extended= {R U inf}. So A_theta,r refers to a specific point on a specific slope. Therefore, S_theta is an equivalency class, because A_theta,r for a specific r, and for all r in R-extended refers to the same element on P. By construction that element is the intersection of S_theta and P. Different selections of P pick different elements from S_theta as the representative element of the class. In the conventional construction of the projective geometry P (or z=1) is fixed. But when the probabilistic phase is added P turns into a set defined as {P_r; for all r in R-extended). So z=1 is only one element of the set P. When the probabilistic phase is added z can get any real value. It’s beyond the scope of this article to show that each element in the set P refers to a specific frame of reference as explained in special relativity.
In this construction, P contains flat projective planes only. However, based on the probabilistic phase, P gets different properties. So P is not necessarily a line, instead, it is a curve defined by parameters specified by the probabilistic phase. Each element in P, therefore, is a wave with a specific probability.
To conclude, in the conventional projective geometry, the image formed on P is fixed. This doesn’t let any sort of gradual evolution of the image formed on P. However, inspired by mechanisms viewpoints in the social context evolve and deform over time, by adding the probabilistic phase to projective geometry, this mathematical structure will contain the capability of gradual evolution.