My vector idea for notation has indeed been formalized already (which is good, because this way I don’t need to make up my own notation): it’s known as “multi-index notation”. I deviate slightly from it in that I make the vector nature of the indices explicit by putting an arrow over them, but it otherwise appears to be the same.
Another Thanksgiving means another meeting with the family and another analysis of the nature of tradition. Clearly, traditions are by nature anachronisms; practices that began generally for the purpose of safety or survival but endure even when that need is removed because of the general momentum of social thought (see the theory I proposed about society being a neural net). However, this predicts the attitude of society to tradition; on the individual level, it is still something that requires empirical observation rather than an abstract theory.
It is interesting to observe how, at least in my family, tradition is a fiercely guarded aspect of individual social identity. To threaten one’s traditions is to threaten one’s self, therefore objective analysis (as if such a thing existed!) becomes absolutely impossible. To even suggest that one examine one’s traditions invites debate.
Now, I should point out that it isn’t Thanksgiving itself that I’m speaking of here. A day of companionship, reflection, and thanks is a welcome thing in almost any social framework; there certainly aren’t enough other days designated for this purpose. It’s simply the lack of “objective” reasoning being applied to tradition in general that is appalling. It represents a method by which one’s society/community can dictate one’s behavior; like all such methods, blindly following without applying one’s own reasoning as a filter deprives one of an individual identity. In essence, it coerces the individual to the ends of the society.
This is one of facets of the constraint function acting upon the social optimization process.
In general, individuals can be seen as optimizing a specific (very complicated) constraint function. That is, people in general desire security, health, wealth, fame, etc. and will actively work towards these goals. Now, this is interesting because not everyone is optimizing on the same constraint; however, there are a set of common traits that will on average always factor into the optimization. At the level of an individual, aspects of the constraint that do not fall within this average area are noise, and will have little or no significant impact on the overall optimization.
So we have a bunch of people essentially performing regression on some unknown but deterministic constraint. What happens when we connect them (a “social network” formed by interaction with others)?
We get a neural network, of course! Thus society is, in a very strong sense (because humans are so much better at intelligent behavior than computers are at the moment), intelligent.
Treating society as a neural net, we can extend some properties of neural nets formally to society:
1. The “Social Limit Theorem” – as more people interact and participate in a society, the society becomes capable of modeling more and more complex problems; its appearance becomes more “intelligent”. It’s merely an extension of a well-known property of a neural network, but can be rigorously proven with bias-variance decomposition. The consequence of this is overfitting and “brittle” behavior, as in a traditional neural net; the society becomes unable to adapt to new situations / patterns easily. This leads to the rather pervasive and positively deplorable social inertia that we are unfortunately exposed to on a daily basis. It is the reason an entrenched sociological philosophy, of any sort (political, economic, ethical, environmental, etc.) cannot easily change. It also explains why the ideals of one society (in effect, the pattern it has learned) do not necessarily work as well in other societies; the model does not generalize well to new problems due to the complexity of the fit.
2. Formalization of the “linking postulate” (and others among my sociological postulates) – There is a clear dependency between the overall behavior of society and the behavior of the individual nodes with high weights (influential people) because the individual variance of the optimization will be more clearly expressed as the node’s weight factors more into the overall decision of the network. This has the same type of effect on overall weight propagation as changing an influential node in an abstract neural network from linear to sigmoid would, for example.
3. If the constraint can be discovered, the overall behavior of the society could conceivably be represented as an abstract neural network (with a degree of error proportional to the overall variance from the mean, probably modeled by a normal distribution), though this may be computationally intractable due to the sheer size, number of interactions, and overall complexity of the optimization. Still, it may be possible to obtain a practical approximation.
4. This answers my previous question of how a society composed of primarily individualistic members could exhibit a fairly optimal behavior on the scale of the entire society while simultaneously fulfilling the individuals’ goals fairly well. The weights are modified as necessary for the optimization of the entire network; this optimization is performed by the individuals attempting to optimize their own goals. For example, people going to work do so to achieve financial stability and monetary gain. However, the amount of pay they receive depends on their benefit to their employer, which itself depends on the profitability of the organization, which depends on the organization’s benefit to the society. Thus, so long as society’s constraint ties local optimization to global optimization, the society will continue to progress.
There are some other consequences of this as well, but I have to get back to my dissertation.
Today’s random thought: tensors have an integer rank, but what would happen if we extended the notion of a rank into the entire domain of reals (or even to complex numbers)? What would it mean for a tensor to have a rank of 2.5? Would the tensor have a fractal structure? What about a tensor of rank i?
Not the sort of question I have the time to chase, but an interesting one.
Yeah. I told you so?
Manifold learning techniques such as SDE have the ability to extract data from a high dimensional space and describe it in terms of its degrees of freedom. Thus low level concepts such as “collection of pixels” become integrated into higher-level concepts such as “teapot rotated at this angle”.
In other words, this is how you teach a system abstraction. Thus, use of something of this nature may be a necessary component of an artificially intelligent system. The only problem is that current methods may be computationally infeasible for this use. Of course, approximation would be a good idea here.
I’m posting this just for the purpose of timestamping. Today I proposed an idea to stream kernel learning techniques such as semidefinite embedding. The trick is to pass minors one row and column at a time (actually, the matrix is symmetric, so just one row) and update using incremental kernel PCA. This results in an algorithm that only needs to store N elements in memory at a time rather than N^2.
The strong anthropic principle states that any viable universe must have the capacity for observation; that is, life must evolve in it. If the multi-worlds interpretation is correct, however, we may not all necessarily be in the same universe (if “quantum immortality” is correct, it gets even weirder: people might be in the same universe at one moment in time and may forever diverge at a future branching point as their own survival takes them along different paths).
I wonder whether we can propose a “strongest anthropic principle” of some sort that roughly states that the particular universe each person (or lifeform in general) inhabits evolved specifically for that person/organism rather than for the general existence of life as a whole. The existence of multiple universes would permit it.
We could even take it further, actually, and permit free will under the assumption of determinism (disclaimer: I am not a determinist), though this treads dangerous philosophical and theological ground because it would essentially argue that we are God: if the initial state of the universe is organized for a specific organism, it may be organized for the organism’s free will, or even by the organism’s free will.
Not that I believe this, but the ideas are intriguing. It’s the ultimate philosophy of egocentrism 🙂
To quote Maslow:
“I have recently found it more and more useful to differentiate between two kinds of self-actualizing people, those who were clearly healthy, but with little or no experiences of transcendence, and those in whom transcendent experiencing was important and even central… It is unfortunate that I can no longer be theoretically neat at this level. I find not only self-actualizing persons who transcend, but also non-healthy people, non-self-actualizers who have important transcendent experiences. It seems to me that I have found some degree of transcendence in many people other than self-actualizing ones as I have defined this term…”
This is precisely the place in which Maslow’s theory can be extended to Dabrowski’s. These “transcendent experiences” likely correspond to Dabrowski’s crises. The pre-actualization crises are already explained by Dabrowski’s levels II-IV of disintegration. However, if we synthesize the two concepts, Dabrowski claims the final crisis propels an individual to self-actualization, which is characterized by inner harmony, but Maslow claims that further transcendent experiences exist!
So which is right? Well, probably both, in different senses. Dabrowski’s “level 5” is a state in which one’s behavior is completely self-constructed (based on “the third factor”, which is an individual’s drive towards growth and autonomy), which should bring about an internal peace. However, it is the nature of humans (especially those characterized by a “drive towards growth”) to continuously strive for better situations, and thus such value systems will change with time, to be replaced by value systems that the individual considers “higher” as his or her perspective, knowledge, and self-expectations shift! (It is worth noting that we avoid infinite recursion because the “meta-values” are responses to internal or external circumstances embedded within the value system itself; there is no such thing as a separate “meta-value”, which would require a “meta-meta-value”, and that a “meta-meta-meta value”, …).
Thus, we have a level 6 state, Meta-Integration, in which the value system itself becomes subject to an individual’s scrutiny. This state is likely the final resting point of the fully actualized psyche, but only because it is iterative: it represents a “punctuated equilibrium” of peaceful periods followed by intense and quite deliberately guided revisions (which are, in a sense, rapidly occurring re-disintegrations) due to rapid changes in one’s underlying values brought about by what Maslow calls “transcendent experiences”.
We can theoretically call self-scrutiny the “fourth factor”, but it’s more like an inwardly-turned version of the third. Still, the step from being certain in one’s value systems (though a healthier condition than relying upon society or self-benefit to justify one’s behaviors) to devising value systems that are internally consistent and stable, yet flexible as the individual gains new knowledge and experience is clearly a healthy one: we can never acquire the sum total of the world’s knowledge or experience, so absolute rigidity is pathological. Dabrowski himself was the one who stated that healthy people must accept the world as it is, and it is not rigid.
The step from Level 5 to Level 6 is huge, however, perhaps even to the extent of the step from Level 1 to Level 2, as it requires abandoning stability. However, it is necessary to fully achieve one’s potential, rather than to simply act as the image of one particular basis of values, even one that was self-chosen, because values are fluid. It represents the extension of one’s moral reasoning from synchronic to diachronic, as one can now envision a direction or change in a value system, and thus an internally driven future expectation. Though (barely) expressible in Maslow’s theory through his addition of “transcendent events”, this is impossible to describe in Dabrowski’s theory as Dabrowski stated it.
So how can we summarize this?
With this addition, Dabrowski’s theory falls neatly into place with Maslow’s and my own. I think it’s quite an elegant and powerful idea that unifies two major developmental theories. Ignore it at your own risk.
I also made some less elegant distinctions on Dabrowski’s first level at my Temple page.
“Sort by date”. It’s one of those features that you browse to a site and wonder “how did they miss that?”, because it just makes doing a real literature search with Google Scholar a pain.
Unless it’s just buried in some undocumented syntax? It isn’t an option in the advanced search page.
There’s also the idea of a general media similarity search that I mentioned earlier, but I’ve proposed this both during the first Summer of Code and when I interviewed at the NYC office (and I even gave them my BACH paper the second time to give them an idea of how to do it for music!), and neither time was it acted upon, so I’m going to shut up about that one. I did, however, suggest it to Yahoo!, who seems closer to having that sort of search than Google does at the moment.