Modernism, Postmodernism, and Digitality
History as a Vector Space
Sigmoidal Curve
One important point is to change people’s views on history. Often times the argument is based on the idea that trends will continue upward logarithmically, that is by ex. this is what Attenborough was getting at because economists often think in this manner. However, biologists think of equations that go up by ex but then slow down as various pressures on the situation assert themselves. In this book the way of thinking of this is called sa igmoidal curve: it goes up by ex but then slows down as the other forces come into play. It will be shown by the following graph:
We saw earlier how there were significant divisions based on time or the lack of such.
This is not just a supposition or a postulant but comes out of the equations which are borrowed from biology. With every Dirac delta function a new function is then introduced, and because it is recursive it introduces to Dirac Delta functions as potentially part of its existence. These delta functions may have to wait for a specific time, such as the Dirac delta function which makes reproduction possible but there are other functions that send out imaginary points such as a written declaration in the form of a manifesto. This means that the beginning of a cycle will correspond with ex. however, when opposing structures begin to act against the Dirac delta function the function then slows in response to the countervailing force. This then means it is not logarithmic but as this book terms it sigmoidal. The details of a particular Dirac Delta function will be specific to the countervailing functions which is why there are many forms of sigmoidal function.
To show this we will start with an idea called the Metropolis-Hastings Algorithm. The first step is to explain what is going on. Imagine that an individual is in a Markov Chain of the states of a particular Cultural System. Imagine that the individual appears random in the next state of the Markov chain: that is from the outside we cannot accurately deduce which state the individual will select. This is a Laplace transform: we do not know which state probabilistically selects which chain will be selected by iterating the selection over the distribution that the Markov chain produces. We can do this by having an arbitrary individual asking one other individual which place the individual would go to were key in the same predicament. In other words, it is asking what the other person would do if he were in the situation. This is a common form of “double ask” question: “What would you do if you were in my situation?” then have the individual decide whether to accept or reject the offered choice based on the algorithm value.
This can be mathematically formed as follows, assuming that g(x) is proportional to a general p(x) where p(x) contains imaginary values that are inside of an individual’s head so we can only sample g(x) by watching individuals move:
Select an initial value for x0
For i= 1… n repeat the iterative process, where * is the proposed outcome of q, or the proposed location:
x* = q(x*,xi-1)
Compute
width the proposal we will then sort out whether the proposal is accepted:
And repeat the steps until there is a sufficiently large n.
Since this was originally a physics paper published in 1953, let us unpack this mathematical solution. What we are doing here is finding a position in the total number of positions and then moving it to a better position, labeled α, or moving it to the new position which is not clearly better but may be no worse but only with probability α. If the position is clearly worse than rejected try a different position. We do this iteratively for some large numbers n.
But this is not the full algorithm that we are going to take. Instead, we have to imagine that an individual will go through this cycle many times and each time get a little bit older. That is the “time” value in the algorithm is an imaginary time, while the individual goes through the cycle by the clock and calendar. This means that we need to elaborate the algorithm to capture that not only is time passing, but the probabilities will not be any other individual but those individuals who match the individual by the algorithm.
So, for example, let us take a college graduate. They will not look at what CEOs do but other members of their cohort. This means that the i will change over time as they get further and further along their career or other elements that will affect their decision change. For example, when computers became more accessible to individuals then more and more of them would consider changing to a computer rather than other means. This means that the individual who is selecting takes a slightly different Markov Chain than the previous iteration. But that means that some changes will be more universal. For example, a stock market crash leading to the Great Depression will have larger effects on more individuals.
Cultural Systems are affected by major changes or minor changes that have a significant impact on a few people. For example, at one time a lamp lighter was a potential occupation, but then electricity replaced gas, and it was no longer tenable. This would not affect a large number of people, but it would affect a very small group of people to a larger degree. What Cultural Systems do is that small changes can add up to a very large change when the amalgam of occupations changes as one after another the gas which is previous becomes rare.
So let us add in the changes to the Cultural System. This means that for any particular i, there will be a lifecycle with the individual instance to those people who are most able to direct them. This will mean those individuals who are older than i, and those people who are in the same group as i as “peer group” competitors. The weight of this is to develop a subset of the Markov chain which the i can access of the total in the Markov chain and then place the Metropolitan-Hastings algorithm to find out which one will be selected and repeat with the Markov chain subset.
This means that
Markov Chain sub-selection
Metropolis-Hastings Algorithm some number of times. And then go back to 1.
However, the Markov Chain is also controlled by the number of individual i’s that flow through it in section 2. This means that one needs to know about 2, but that means that the MHA needs to have a difference from normal. Normally we want an MHA to be in a condition of Detailed balance, which requires three conditions:
A given i do not change if the direction of time is reversed.
Equilibrium is constant under time reversal.
The entirety of i must be distinguishable from all of the other i’s which are tested.
In history, we need to distinguish between time in the real historical system and “time” in the MHA: time passes in a Cultural System but in the selection of the proposed solution it is an imaginary time. But this means that within the MHA, we can use a Laplace transform: it is the decision by an individual i to select which location on the Markov chain that i is going to go to next. This means that the MHA is in detailed balance only when selecting a particular location or detailed balance is recursive and with each iteration of the function must be reset.
Since the detailed location is set for each member of i, the detailed balance only needs to be statistically satisfied rather than logically satisfied. That is, each member of i needs to think that detailed balance will be maintained whether or not it actually is maintained. What this means is that an extremely unlikely selection is impossible because that would be statistically impossible. That means that all of the molecules in a black box will not go to the same location just have “pure chance” because there is a total number that can fit in any particular space by the Pauli Exclusionary Principal and similarly, all of the i’s cannot all go to the same location because they cannot all be help by the staff or be alone with everyone there.
This reflexivity of functions in history can be overwhelming if there is an outside event that is not foreseen. A simple example is that on one particular day on 8th April, 2024 a solar eclipse occurred in North America. The normal detailed balance was disrupted because a large number of people went out of their way to see the solar eclipse. This meant that the normal boundaries of statistically detailed balance were overwhelmed by the outside force of the solar eclipse.
But these events are extremely unlikely and will because the statistically detailed balance will, over time, equal the logically detailed balance. That is, locations that do not run a profit will go out of business, and so on. This means that we can invoke a number of layers which corresponds to raising a flag to a higher level that the statistically detailed balance is farther and farther out of accord with the logically detailed balance.
So, to conclude:
Individuals select an MHA-accepted location. This is done however many times.
The Markov chain will be reorganized so that the logically detailed balance will be changed by the statistically detailed balance.
This changed Markov balance is then used as the new Markov chain for the next iteration.
The difference is that the MC is only statistically detailed and balanced. If the result is concentric, the chain repeats and if it is not, the MC is reorganized so that the statistical detailed balance is equal to the logically detailed balance at some period. For example, a person may rent a space for a set period of time. When that period of time is up, then the leaser checks whether the amount of money is equal to the set value for that period of time. If it is not sufficient, the leaser checks with the local ordinances as to whether he can remove the tenant. This shows the level of fact mentioned above: the leaser may not be able to do as the leaser likes because of legal restrictions. The law uses the statistically detailed balance to set a restriction on how long the leaser must wait.
We shall call this a Markov restructuring MHA or MRMHA.
But there is another wrinkle in this and that is that over time the i’s go through a lifecycle that ends with their deaths. And so we need to modify not only the Markov chain for individuals but also for the individuals participating in the Markov chain. So:
Individuals select an MHA-accepted location. This is done however many times.
The Markov chain will be reorganized so that the logically detailed balance will be changed by the statistically detailed balance.
This changed Markov balance is then used as the new Markov chain for the next iteration.
Over several cycles, the Markov chain will be modified to a particular set of individuals.
The final step is that over time the individual’s age is outside of the statistically detailed balance because that is only for when the Markov Chain is in statistical balance before the Markov Chain before be Markov chain is rebalanced. This means that the individuals’ group is slowly aging and therefore slowly adapting to locations while her proposal/acceptance is only for the instantaneous Markov Chain.
This means that the individual sigmoidal curve is analogous to the group sigmoidal curve though obviously at different timescales. That is the individual first learns and dreams in a visionary mode, they then find a revolutionary mode which overthrows some aspect of their existence, they then use that to build their reputation, and finally, they go to the late stage to hold on to the benefits that they have accrued. This is why there are multiple Cultural Systems in play: one can organize the visionary, revolutionary, evolutionary, and late stages around linear algebra because it is the linear algebra that forms the sigmoidal curve.
This means that the visionary aspects are free to evolve in any mode which is appropriate, but they do not have a wider reach because they are visionary. This mean means that a group of people branch off because they feel that they have a different visionary idea, such as the visionary groups that wanted a different relationship with God in the 1800s. they left the organized states to follow whatever vision seemed appropriate to them. A few managed to hold on even to this day and one became a force in the nature of the Latter-day Saints. This pattern is repeated over and over again when the visionary mode strikes a people.
When the visionary turns to revolutionary, new constraints appear but also the ability to fight the organized state of affairs. Most forms die off, but a few forms have the potential to organize themselves in order to fight the organized Cultural System, which includes “the state.” The main advantage of the revolutionary versus the visionary is the recursive ability to organize large groups of people by revolutionary means. As we have seen this can be artistic, scientific, or personal as well as social or governmental: the key focus is to organize a body of people in a particular action. For example, the discovery of radio waves organized a great number of people to transmit information from where they are to someplace else and the ability of others to listen for the transmission.
The movement of revolutionary/evolutionary is, in fact very simple: the revolutionary wins when it overthrows the evolutionary. One of the problems in history so far is to assume that the revolutionary moment will succeed and chalk it up as a failure if it does not. Most of his examples are when the Liberal Revolutions of 1848-49 failed to overthrow the conservative states that preceded them. But this is not correct, because when we are looking at the past we must see which revolution actually succeeded: it was not the liberal revolutions that succeeded but the conservative revolutions. This means that when the revolutions of 1848-9 did not, in general, succeed they were not the evolutionary moment. Instead, it was the conservative revolutions that succeeded: the Second Empire of France for example, swept aside the monarchist revolutions in favor of an imperialist state. When the Empire of Germany swept aside a collection of states led by Prussia, this meant that the evolutionary design was of a more conservative state. This means that we must accept the judgment of history whether or not we agree with it or know that it will eventually fall. The moment does not look ahead.
The last phase that is of importance for us is the late phase where the evolutionary moment is holding on by borrowing from the future. The most obvious example is when the Russian Empire collapsed into the Union of Soviet Socialist Republics. The Czar of Russia could not believe that his time was over. This may seem as if it is contradictory, but the late phase is often oblivious to its own mortality.
What this means is that the sigmoidal curve rests on the slope of the curve. This slope is what changes between visionary, revolutionary, evolutionary, and late stage. The visionary actually has the most potential, but it does not see a slope because there is too little to base a slope upon. This means that its slope is woefully underestimated because there are so many points that are not part of the visionary era that degrade the slope. Because we cannot piece aside the slope until it is obvious this means that we must accept that the slope is ill-defined. That does not mean we have no data but that the data that we do have is often misleading. But when the slope upward to the revolutionary mode takes place, we have a well-defined set which we can by least squares show that it is a linear trendline while it is in play. This was the example of color televisions where the bump occurred when the networks of the major broadcasting companies went into color. As long as there was a conflict, the slope continued upwards and when it stopped, and color televisions were the default it was a marker that the postmodern had become the dominant mode and was in an evolutionary setup.
This then means that markers of this organization will appear in their appropriate places without affecting markers of a different Cultural System which is going on at the same time. What it does do is assemble different Cultural Systems out of individual parts and it shows how different events are actually the same in a Cultural System. This means that similar marketers may indicate that the same Cultural System is in play across a variety of different media, but which have some similar points. By taking the sigmoidal curve and flattening it by linear algebra we can then determine a generic version of the sigmoidal curve.
The historian must decide what that point is after the mathematics shows a correlation. Taking σ as the slope of the sigmoidal curve:
The ranges above correspond to the Visionary, Revolutionary, Evolutionary, and Late Phase. In both the visionary and late phase, the slope ends up looking like 0, but in the late phase, the Cultural System has its disposal the ability to borrow from the future while in the visionary phase, there is no expectation of profit. The other difference is that in the early phase,e there is the expectation of future growth so there is an investment while the late phase assumes that the prophet will return to evolutionary levels. However, it should be noted that there is only one inflection point at x =0.
This is true in physics as well: the Strong Force grows stronger until a new subatomic particle forms out of the force. This too forms a sigmoidal curve that forms color consistency. This means that we have at least one example from physics of an internally driven sigmoidal curve which is the opposite of the externally driven curve from biology. But they lead to the same effect: there is a logarithmic curve that balances the force until it plateaus with a different force that peaks. Way will use this again in another manner shortly.
To take one example: when a new form of transportation becomes available, it will often be the case that new territory will be opened up for individuals to settle in. For example, in the United States when the covered wagon became common many people left the East Coast and headed for the West Coast, often because of the gold that was found in California but also because of trade with China which had previously been constrained. This then needed a new form of transportation, that is the transcontinental railroad, to ship goods from San Francisco to the east. This is an ex acceleration that produced many Fourier transforms. But at a certain point, the growth from this movement slowed. But new forms of transportation and means of using the land became available which meant that a new form of exploitation could then be introduced. This means that one ex slowed as it reached the peak with countervailing forces, but a new one could be introduced.
The question is why does the new curve appear when it does? In one sentence there is a stochastically introduction of new curves at any given point but the new curve enters a visionary process when the result of the existing curve starts to produce less value or has flaws that now are known. For example, the classical style of physics begins to show different results in the theoretical versus the experimental versions of physics, this means that Plank and Einstein began to produce papers that argued that the range of data was quantum rather than the classical notions that it was continuous.
This means that from one perspective it seems as if it is still a logarithmic curve, but when looked at from a Cultural System one can see that the initial curve slows down, but a new curve begins: we can see this by the growth of population in San Francisco:
From the San Francisco Chronicle:
https://www.sfchronicle.com/sf/article/s-f-exodus-population-recovery-data-18564064.php
one can see that there are waves of population increase in San Francisco, each wave corresponding with a new opportunity for exploitation and then the new waves of immigrants move to different regions. Again because of countervailing trends that reduce the ex growth.
This pattern of logarithmic growth until countervailing forces intervene and then new growth from a different opportunity is extremely common. We shall see it not only in cities but nations as well, and again we shall return to this in more specific circumstances in the large example.
Key Points on Curves
The sigmoidal curve is more appropriate than the logarithmic.
One can curve succeeding other curves data.