I was writing the first chapter of the math course about OOP structures. However, it turned out to be more about probability.
Quantum mechanics changed the notion of scientific processes. Science has always been linked with determinism. Quantum mechanics changed it all. However, the radical change in physicists’ perspective had little impact on mathematics. Mathematicians always considered physicists’ approach “too wordy” and “not precise”. Compared to mathematical rigidity, physicists’ approximation is certainly imprecise.
It is said that mathematics is the language of nature. Then non-deterministic nature of the universe should affect the math structure too. However, it’s not merely about probability, it math should be based on more flexible structures. It should be noted that this is a shift in perspective: probability vs. flexibility. Probability is a rigid mathematical construct while flexibility is a different approach.
Consider this example. Based on probability each time a coin is tossed the probability that it lands tail is ½. No matter how many times the coin is tossed each incident is independent. On the other hand, psychologically if you throw a coin 99 times and every time it lands tail, you might think this time it has to land head. But based on probability, as a rigid mathematical construct, there is no difference between the first time you throw a coin and the 100th time. They are independent incidents. If one wants to add the psychological element to that rigid mathematical construct, different throws of a coin are psychologically not independent incidents rather they deform the probability space. What if one decides to throw a coin 100 times every morning? Should he reset the counting every day and count from 1 every morning?
The double slit experiment can clarify this issue. This experiment changed the way physicists viewed determinism. If one shoots a photon every day every hour or every year the outcome of the experiment is the same: the wave interference will appear. Should the same be true about the psychological coin example? Should a new type of probability structure contain wave behaviours? If so, approximation and wordy approaches of physicists seem to be more suitable compared to traditional rigid mathematical mindset. This course is not about probability and doesn’t aim to add wave behaviour to that. That is the subject of another study. This course aims to introduce one method to develop flexible math structures, the one that might be more suitable to describe nature.
UPDATE 1:
In the example of tossing a coin, the probability space contains all possible outcomes. However, if wave functionality is added to the probability theory then the probability space of tossing a coin can be reshaped. When the probability space keeps changing the standard probability theory cannot be used. Imagine the universe is a slope, then the physics governing that universe is a simple gravitational force explained by the Newtonian system. But if that slope is influenced by the mass of objects being placed on it, the gravitational theory will be more complicated. Now imagine a probability theory where the probability space is influenced by the elements that construct it. In the standard probability theory the probability that set A, which is a subset of the set of all possible outcomes, occurs is A/B where B is the second set (i.e. the set of all possible outcomes). But if B is not fixed and is a function of sigma A_i, where A_i sets are different possible outcomes, then things will change. For example in the experiment discussed in the previous section, every day a coin is tossed a hundred times. Then each A_i set can be the list of all outcomes every day. The probability space will then be a function of A_i sets. If a coin is tossed once, the probability space is either head or tail where the probability of each outcome is ½. But when the number of times the coin is tossed approaches inf, the probability space consists of one possible outcome half head and half tail, and this will certainly occur (i.e. 100% chance). So the rule of large numbers in the standard probability can be re-expressed as a type of reformation of the probability space. In the standard theory, probability is calculated assuming that the coin is tossed infinitely many times. This is a theoretical assumption that will never happen in the real world. In the real world, the number of times the coin is tossed is always finite. Based on the standard version the probability is 50% head and 50% tail but when the wave is added, similar to quantum mechanics, after tossing the coin infinitely many times, the outcome is a superposition of head and tail. To produce such an effect, the probability space should be deformed, so the set of possible outcomes (i.e. head and tail) gradually turns into a set with one element only: the superposition of head and tail. Then the location of an electron for example in space will not be precise as described in classical physics. A particle will be a wave spreading along the spatial axis based on some probability distribution.
To construct the probability space that can be deformed infinity should be explained. In particular, the process in which finite tends to inf realm should be clarified. It will be discussed that this process involves length contraction as discussed in SR. This means the deformation of the probability space might be governed by the same mechanisms described in SR. Then the space of possible outcomes (i.e. head and tail) gradually shrinks into one entity (i.e. superposition of head and tail). This means that the standard probability space consists of a head and tail but when the wave is added to the probability theory the set of possible outcomes will construct a more complicated probability space, similar to Mink for example.
UPDATE 2:
the wave probability space is a Hilbert space but SR should also be included. In the case of max spatial uncertainty (when a particle can be at any location with equal probability) the probability space is flat. But in other situations when uncertainty is not max the probability space deforms. That’s comparable to curved spacetime. So probability distribution refers to the deformation of the probability space. So flat space refers to max uncertainty. Then tossing a coin is also max uncertainty because the outcome is an equal chance for all possible outcomes. So tossing a coin creates a flat probability space, but repeating the process of tossing a coin changes the probability space and deforms it.
(here probability space refers to the probability function of some probability space, which is comparable to gravitational force for example)