Sunday, October 11, 2015

Thoughts on the Riemann Hypothesis

The Riemann Hypothesis is a mathematical conjecture that the (nontrivial) zeros of the Riemann zeta function all have real parts of 1/2 on the complex plane. Its proof (or disproof) is a millennial prize problem worth a fair amount of money and everlasting fame. Thus it has become over the 150+ years since its inception by Bernhard Riemann, a landmark for mathematic pursuit, but despite the efforts of the greatest minds in mathematics, the hypothesis remains a conjecture.

So I offer a naïve suggestion.

In the work, Principia Mathematica, Whitehead and Russell spend several pages developing the notion of cardinal and ordinal couples to the conclusion that 1 + 1 = 2. That is, there is a dimension of numbers with order arising out of the process of addition together with the concept of unity. Multiplication is another dimension related inextricably to that of addition as the replication of the now existent numbers by each other along the dimension of order. The primes are simply byproducts of this relation.

The Riemann Hypothesis seems complex, but it is a result of the structure of multiplication over addition and nothing more. Two dimensions are required to solve polynomials--the expression of addition and multiplication together--and so the complex plane is two-dimensional (the imaginary number, i,  is just a symbol that marks the relation). No more and no less. The fact that multiplication is an operation quasi-independent of addition necessitates two dimensions for the expression of the solution of a polynomial expression once we have allowed that numbers have a structure that is ordinal.

My guess is that we will eventually find that the density of the primes is nothing more than a representation of the structure of addition and multiplication expressed in spatial dimensions similarly to how we view fractals--apparently complex, but exactly only the generating formula and nothing more in its essence.

The Riemann Hypothesis could not exist without the Euler product formula which is itself an expression of the Sieve of Eratosthenes which is nothing more than an expression of the artifacts of multiplication over the generated dimension of addition.

In short, the Riemann Hypothesis, is nothing more and nothing less than a statement that there is a midpoint created by the new ordinal relation of 1 + 1 and it occurs at 1/2 the distance between the unity of multiplication and the unity of addition. It cannot be anything different and it cannot be anything more or less.

Of course, this is just a thought.

Wednesday, June 10, 2015

Against the Export-Import Bank

The Export-Import (Ex-Im) Bank is a taxpayer sourced fund with the ostensible purpose to give the federal executive power to improve the trading prospects of selected US companies in respect to their foreign competition. The benefit to the taxpayer is then supposed to be a return on investment through localizing the creation of the exported wealth (keeping that economic activity local) while focusing the receipt of the trade for that export into a discrete stream of foreign currency which can be attributed directly to that trade.

The benefit of subsidizing free trade for local economic benefit is nonsense on its face. We can consider the fallacy of the benefit of the transaction using the ideas of conservation of mass (price) and the efficiency of the information streams.

1. Subsidizing a trade is a reduction of price of the traded good below the actual market price. By itself, the subsidy transmits economic inefficiencies to the supposed beneficiaries of the trade and sets up structures within the local creation of the traded wealth which result in an apparent price below the market price. This difference in wealth is dissipated through inefficient consumption (e.g. an extra yacht for the CEO of Boeing that he lets rot in the bay through lack of use together with the idea that the extra unused yacht is a good thing).  In short, it makes the US in actuality less competitive. This is the free trade principle in conservation of mass (price) terms.

2. However, there is an argument that if a central authority can direct a game against a foreign competitor, he may be able to direct the transactions so as to win market share and increase the leverage of the local (although necessarily inefficient) company by an effective monopoly. Therefore the immediate inefficiency would be worthwhile in order to create a subsequent greater net positive increase in wealth once the foreign competition is weakened through lack of capital investment. The best example of this is a technical efficiency in the production of a high tech asset such as an aircraft that the foreign competition cannot match through economies of scale, (e.g. the US can build planes with fiber composites at less per airframe in actuality than the foreign competitor because of the US investment in the process).

Unfortunately, in this game there can be no long term winner. If either side gains a monetized technological advantage, then the monetary value of that edge will induce that player to maximize profits at the expense of technological advancement. Eventually, that player will become technologically stagnant as focus on the measurable good (money) attributed directly to the trade dominates the allocation of its resources. Meanwhile, because technology is interconnected and nonlinear in its advacement, other players may discover the means by which to nullify the measured technological edge of the leading player by happenstance if not by design. With this undisclosed technological revolution, a trailing player may take the lead.

For the leading player to reliably maintain the position advantage, it must continue to innovate in necessarily unprofitable ways at a rate that entirely swamps the innovation rate of its competitors. Thus, technological gains are ephemeral since this cost is usually exorbitant and cannot me rationalized for the discrete benefits of the trade. The transitory cost of these gains together with the indirect benefits of the transaction that is subsidized minus the inefficiencies of the subsidy is the equilibrium price difference of the transaction which ultimately goes against the subsidy.

In short, even in the best case, the subsidy reinforces measurable failure at the expense of immeasurable innovations and gains in other areas of the economy. The best strategy is one that is naive in the directed allocation of profits as it is that strategy that maximizes innovation across all areas of the economy. The paradox is that in the face of technological innovation, long term investment in the future should be divorced from the optimization of short term gains.

Thus I am against the use of taxpayer funds in the Ex-Im bank except for the case of preserving a strategic defense capability.

Tuesday, May 05, 2015

On "isms"

An "ism", such as racism, sexism, ageism, nationalism, etc. is a shorthand term we use to describe the thinking of people as if that thinking put them in a group themselves.  The people in these groups which we ourselves created by the use of the shorthand are then typically labeled as "ists" as if they functioned as automatons according to our categorization. When we do this, we are very often attempting to justify ourselves by describing what we reject.  Unfortunately, when we extend concepts that we created for our own convenience to the categorization of other thinking people, we are enforcing a relation that is counterproductive to the elimination of the "ism" that we criticize.

In short, calling others racists or sexists or communists etc. tends to sharpen the boundaries against free thinking that we are against. Why?

Useful information is like a virus. It spreads and affects us to the degree that we value it's utility. If the information is contradicted by experience, it loses its veracity and hence some degree of its utility. Curiously, humans do seek out and value thoughts which are often in contradiction with empirical reality when those ideas are useful to our emotional well being. This last statement is simply saying that denial is a natural stage in learning just as it is in the stages of grief.

However, unlike religious convictions that are neither provable or refutable, the utility of beliefs in contradiction to experience is transitory for any learning being that evolves towards it's own tangible benefit. Denial yields to anger to bargaining and ultimately to acceptance of the new experienced truth.  That is, of course, while we allow ourselves to evolve. The ideology of the "isms" act as a learning impairment. When we label others with a derogatory group membership as "racists" etc. we are actually defining in our mind a justification against learning.  What we are not doing is teaching the object of our derision. Whatever the good that we might offer it is effectively nullified by the symmetrical onus that we confer since all people want to feel good about themselves while they think about the things around them.

Learning is its own joy. To the degree that our information is useful, the new found utility of the information is a source of empowerment and so naturally appeals to all. The solution of all the pathologies of the isms is found through mutual learning--not by name calling.

So when you next hear of someone being called an "ist" ask whether one is solving a problem or contributing to it.

Sunday, January 05, 2014

Protection and Privacy

As we discuss here, privacy is the property of the individual to exist in a state of personal knowledge without that knowledge being controlled or influenced by forces outside that individual. Privacy is not a state of knowledge that is necessarily unknown to others, it is simply that they have no means of utilizing that private knowledge in directing a force that affects the individual. This knowledge can be shared outside of the individual, but it cannot be acted upon.

If it were not for privacy, the system of the whole would utilize the useful knowledge of the individuals to reach a more optimal overall state. This state would be cemented in equilibrium and impervious to adaptation except for the influences of knowledge and forces entirely external to that system. In other words, the system could not self-regulate, it could only respond.

However, in systems that allow for individual privacy, adaptation can occur within the system as information is released by will of the individual at states and times of that individual's choosing. Thus a small dose of information at the proper moment might swing the system into an entirely different trajectory from which it might have ever evolved if all information was shared. The momentum of the initial change in state could see the evolution through.

In order for this form of privacy to exist within the system, the individual must be protected from influences within the system that tend towards system-wide equilibrium. In the least, freedom of conscious must be allowed to exist. The individual must be in a way sovereign to himself, but not necessarily independent of others. He must be allowed to self-organize, i.e. to think, learn, feel, and forget autonomously. So that the system of the whole might be more sensitive to learning and evolving itself, the individual must have freedom of action in addition to freedom of conscious---all while remaining interdependent on others.

It follows that such sovereignty cannot exist without a shared respect for original life, liberty, and the pursuit of "fill in the blank", whether it be happiness, industry, love, or whatever are the shared values of the system. But always, there must be respect for life. Without that, there is nothing. 

Sunday, December 01, 2013

Privacy and Protection

There are two models for scalable information-theoretic systems that are generally applicable to human systems: the small-worlds model and the hierarchal command and control model.

In the case of the hierarchal model, there are no limits on an individual's ability to absorb and process information. However, in the small-worlds model, a connectionist node is saturated with specialized, local information and a small number of connections outside of that immediate proximity. The flattening of a small-worlds network scales negative exponentially with the removal of intermediate layers while the hierarchal model scales linearly. In short, the small-worlds model is realistic for human systems whereas the hierarchal model fails catastrophically with small changes in scaling of human systems. Hierarchal human systems only exist in the real world as abstractions.

In our personal lives, we may want to have a degree of privacy that is disproportional to the good that we would like to gain from our interdependence with others. Thus there is a natural tension for the individual. For the system, there is no such tension. The system finds equilibrium at all points and does not hold itself against imbalances. A large positive imbalance in the individual is immediately offset by smaller negative imbalances in many others within the local neighborhood. The wants and needs of the individual are washed away by the wants and needs of the many.

Fortunately, the information from the individual does not scale in the same ratio as the individual's ability to absorb and process information. The flow of information from the individual is asymmetric to the same scaling factor as the connections of the small-worlds model. The specialized local nodes of the small-worlds model report up the net discrepancies of  relatively similar subordinate nodes amplified by their own processing filters.  Thus the individual acting through the local connectionist node can transmit information throughout the system to the degree in which it is a useful innovation throughout several layers rather than to the degree to which it is "voted" by the aggregate of the nodes in the local and subordinate connected layers.

In equilibrium, the usefulness of an individual's information dies out as quickly as under the hierarchal model. The virtue of the small-worlds model is in its ability to adapt to changing information on short time scales.

The destruction of privacy of the individual is akin to the substitution of a hierarchal model rather than a small-worlds model. In short, it is an unnatural and statist substitution that leads to systems that are incapable of adaptation on the same time scale as natural exogenous inputs. Hierarchal systems fail to adapt readily and are most often made obsolescent by competitors and outside disturbances. Thus, to safeguard the system at large, the privacy of the individual must be safeguarded.

Tuesday, January 08, 2013

Outliers: Those not Covered by the Second Amendment

Clearly, in any time, it would be considered immoral to force those who cannot form moral thoughts to take on the responsibilities of the whole. Was it ever the intention that the Second Amendment (or for that matter, the rest of the Constitution), that we should apply its obligations to those that were infirm, adolescent, or insane? No, the Constitution is a ongoing social contract and like any contract, the parties must come to it with volition and the capacity for that volition.

In the area of controls, we would call a system that requires an exogenous input for its stability to be "conditionally stable", i.e. that it is itself unstable in nature. These systems are never left to their own devices for a slight perturbance can cause dynamics that can quickly overcome any realistic control. Think of a wrecking ball being held at the pinnacle of a mountain by your finger, for example.

The same is true of people who require drugs for their mental wellbeing. They are not themselves stable, but can act so and appear so during their maintenance. This condition might be temporary and many return to full mental health (e.g.  John Nash). Naturally, we would not allow these individuals during these conditions to serve in the armed forces and the existence of the condition exempts them from compulsory service. Fortunately, since such drugs require a prescription, there is a mechanism by which we can enforce this rule.

Let me propose that within the framework of our Constitution, we can agree on sufficient conditions for the welfare and safety of society that are also necessary to ensure the rights of the individual:

  • Every free, mentally sound American adult has the individual right to bear arms whether or not he or she chooses to practice that right. In America, we respect the rights of others whether or not we ourselves choose to partake in the freedoms of that right since the correspondent responsibilities apply to us all.
  • Those that have been adjudicated as insane, who are under sentence for felonies, or who voluntarily have been prescribed mind altering drugs for their mental wellbeing do not during that time participate in the responsibilities of the Constitution and may be denied the rights of the Second Amendment.

In the best society, we do not require paid armed guards for our childrens' schools since we are all at all times our society's and our nation's guardians. Let these disasters end with us and let us not take one more step on the path either of Germany and Russia in the first half of the last century. Let us show the world once again that the US is a nation governed by the People--rational and responsible adults. We are not a nation governed by children, but one which will protect its children.

Rights and responsibility must go hand in hand. We should insist on both at the same time.

Sunday, December 23, 2012

Systems Theory and the Second Amendment

A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.

Unlike the laws of most all nations, the US has had the benefit of a ground-up principle based Constitution with the Bill of Rights falling squarely within the principled framework. With that in mind, lets take a systemic (rather than episodic) view of the Second Amendment.

The Second Amendment was adopted not only as a guarantee of distributed rights to arm against a tyrannical central government; it was also a restatement of the citizen's duties to defend the nation. Recall that a standing army was an anomaly for the frontier republic and the civil defense was based on local militias.

So of the two imperatives for systemic success of this right, the proof of the first is that the central government does not harbor even the faint thought that it should ever be possible to intimidate or coerce to achieve aims that cannot be achieved by a free vote. The proof of the second is that we retain a culture where there are sufficient numbers of eligible citizens mindful of their obligation to defend the country that we do not need to offer much above a competitive wage to attract them into service.

Now imagine a state without the Second Amendment. Applicants to the military would be given powers that set them apart from the populace. The citizen would have no obligation to be ready to take up arms in an emergency without the express direction and provisioning of a central government. Rather than participatory citizens bearing responsibility as well as the privileges of the state, the citizens would take on the mindset of wards of a culturally foreign power.

Note that this is exactly how the economic dependent class of the country feels in regards to their economic obligations to the state. But, we dare not extend this economic dependency to the entire citizenry for the obvious reason that there would be no one left to earn the money to hand to the dependents. A similar logic for shared defense applies.

So without the Second Amendment, we might rely on universal service to culturally connect army to citizen least the army become a Praetorian Guard of the government. The instability of that situation is obvious. This relation between army and citizen exists in countries that have nothing like the Second Amendment and universal service. Take for example the UK from the perspective of a Catholic of Northern Ireland. Likewise, one might consider China and Tiananmen Square.

The conclusion of a few that weapons should not be like that of our military is misguided. The arms of the citizenry need not be as capable as those issued to the military but they ought to be so similar that they form a cultural connection between citizen and state. (Originally, they were the arms with which wars were fought! There is a reason that cannons are historically featured in the town squares. The cannons were not moved, its just that they are welded in place now.) Empirically, it is rather obvious that our army recruits tend to come disproportionately from organizations like the Boy Scouts and among the small towns and the south where the right to bear arms is seen from the civic perspective.

Today, we have an all-volunteer army, but is it so unimaginable that tomorrow the recruits might be selected based on political loyalties or other such factors? The politicization of the military has often been apparent of late at the higher ranks as well as through policy. This is a very disturbing trend.

Yet today, the Second Amendment is working for us in ways that are deeper than what is seen on the surface. Our nation exists as a shared idea. The Second Amendment is a fundamental thread that tethers us together.