Ali's Journal

In the OOP course, I explained that dimensions could be more complex than axes in R^n for example where points on the x-axis are specified by a real number. For example, in a room that contains several lions and tigers, it can be said that the space of objects is 2 dimensional. Lions and tigers define two distinct axes for this space. The thing is it’s not easy to organise all lions for example in a line (similar to the line of real numbers where all numbers could be placed in a line). The axis defined by lions is more complex. For this purpose, it should be clarified what a lion actually is.

The human brain more or less automatically distinguishes lions from other animals. But if one wants to do the process consciously it becomes a very complicated task. Probably that person should write a list of all the properties of lions. Some properties are not straightforward. For example, behavioural variations cannot easily be notated. Imagine a notation method that identifies all possible such variations. When all properties of lions or tigers are precisely defined (that is all possible variations for all properties could be notated by some notation systems) then it’s possible to construct the dimension defined by lions. However, This “dimension” is a type of space. Every lion then will be a vector in that space.

The same method can be used for memories. There should be some type of tag assignment that organizes memories. Related memories are organized under one tag and so on. Based on the previous discussion, it’s possible to consider memories as entities (similar to lions for example). Then each memory will be a vector in that space. That space is flexible. It means it is reshaped over time. This is true about memories. Not only some details of memories will be lost over time but also they can be reinterpreted once new memories are added.

In R^n axes are always fixed. But in the space of memories, the base vectors can be redefined and so the space will be reshaped over time. As a result, all vectors contained in that space will also be modified. To clarify if lions acquire new traits or some existing traits change over the span of thousands of years, the space that contains lions will also be modified. The same should be true about the space of memories.

The question that arises here is that how consciousness is linked with the formation of this space. I know nothing about consciousness myself, but before working on that topic, the mathematical aspects of the article should be clarified more.

based on wave probability, if a coin is tossed N times (N>>1) and never lands the tail, the probability that in the next round, it lands the tail should be higher than 1/2 (at the limit, N -> inf, the probability tends to 1).

But standard probability theory is different because each experiment is independent and therefore the probability is always 1/2. These two approaches are vividly different. However, the rule of large numbers can be used to link these two approaches. If N>>1, then n_t ~ N/2 and n_h ~ N/2 (where n_t refers to the number of times the coin lands the tail). As N -> inf, n_t -> N/2.

So if for some large N, the coin never lands the tail, then the rule of the large number should somehow deform the space so that the probability that in the next rounds, the coin lands the tail is higher.

UPDATE 1:

In other words, there should be a symmetry that deforms space so that n_h and n_t converge to N/2 (as N -> inf). The wave probability therefore requires mechanisms for this purpose. This distinguishes the wave probability from the standard probability theory.

It was said yesterday that a wavepacket can be considered one wave plus fuzziness. Also, in a wavepacket, a wave is localized. The same mechanism can be used in probability and observation.

It was said yesterday that each experiment is one observation. Each observation is a localized version of the probability wave. So wavepacket is very suitable for observation in probability. It means to observe an outcome in an experiment, the probability wave should be blurred so that it turns into a localized wavepacket so one outcome can be observed.

A mathematician trying to construct a structure using OOP is comparable to a game designer who designs the environment and gameplay of a game. That mathematician sets up all necessary tools and designs the environment where that particular construct is contained. The same method will be used in developing wave probability. It means at first, scenarios should be written about how different elements will be in the structure and then trying to construct them using OOP. So this is the scenario phase of constructing the wave probability. Evidently, this construction and scenario is not unique.

To estimate the probability distribution of something (call it P), the experiment should be repeated several times. Denote N as the number of times that experiment is repeated. if N is small then the variance around the mean is large, when N is large the variance shrinks. Each experiment can be considered to be one observation. Several observations are then required to produce an exact image of that thing. Precision here refers to low probability. In fact probability and observation are opposite concepts. when something is observed there is no probability. When a coin is tossed, the probability it lands head is 1/2 but when the outcome is observed, it has certainly landed head or tail, no probability. In fact, when observation has not occurred the outcome is a superposition of head or tail but when the outcome is observed, the wave collapses into either head or tail. This is similar to quantum mechanics.

About the previous example about the distribution of P, each experiment is one observation. but each observation alone provides a vague image of P, which is why the variance of P is so large when N is small. To re-construct this using OOP this scenario can be used (again this scenario is not unique):

P is a n-d wave rotating around itself. Each time its image is captured (i.e. after each experiment – equivalently each time the wave is observed) a blurred image of the wave will be produced. That is similar to a wave packet. In this scenario a wave packet is not a combination of several waves rather it is a wave with an added fuzziness. When more and more images are taken, the combination of these blurred images produces a somehow accurate image of the 3d wave. More images (i.e. large N), produce less probability (i.e. less inaccuracy). In this scenario, waves come with intrinsic rotations and motions. that links wave probability to SR. Note that SR and probability both depend on the concept of observation.

UPDATE 1:

Spin in the wave probability is similar to the same phenomenon in quantum mechanics. Due to the connection between SR and the wave probability, it’s beneficial to express a constant velocity in the form of spin. An object moving with constant velocity retains its velocity forever. It’s better to express that as an intrinsic rotation (or spin). Then it’s easier to connect SR to the wave probability.

In quantum mechanics, fermions are based on the Pauli exclusion principle. In Probability theory, possible outcomes for something are complements and the sum of all probabilities equals 1. To illustrate that similar graphs to the ones used in the set theory can be applied.

When the wave functions of fermions do not overlap, the Pauli exclusion principle is not used for them (so for example different tosses of a coin wouldn’t make the wave inference effect explained yesterday). The same ideas can be extended to the probability theory. Different outcomes can be considered as fermions or bosons (based on the situation – if they are independent or not).

UPDATE 1:

Complement is a term used in set theory. However, in probability complements exclude each other so it’s better not to use that word in probability. I used the word to link it to set theory.

UPDATE 2:

The inability to observe plays a role in probability. When a box contains invisible objects (visibility means they cannot be sensed by human senses), they cannot be picked.

The point at inf is a vital part of observation, this is also true in probability. A coin is hypothetically tossed an inf number of times and that’s how the probability has been calculated.