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.