Unfinished Math Thoughts

Experts in education have pointed out that a problem with math education is that students never actually see real math in process. They don't get to see the incomplete ideas, the reevaluation of errors, the incomplete leads, and the powerful insights that lead to new thinking. All our education systems shows students is formal expressions of ideas that have been completed long ago by someone else.

Although this site already contains some incomplete ideas, as a mathematical hobbyist, I thought perhaps I should show ideas still in process, thoughts that are still incomplete.


Last Updated: May 2013

Pure Math

The mathematician moves ideas from the right brain to the left brain. He starts by creatively thinking about about the big picture, the patterns, the relationships. As he processes that thinking he develops symbols, language, formal logic and conclusions.

However, math education rarely reaches the right brain. Educational standards and traditional methods start with the formal language and methods that have already been developed. Students are asked to memorize these formalities. The students do not get involved in the creative and discovery processes that make up real math and real science. They are tested on the formalities, not the deeper understanding or creative processes. They do not get to actually see what real math is.

Goldbach's Conjecture

We have started to examine Goldbach's Conjecture using pattern recursion. We looked for the arrangement of holes (potential prime pairs.) Perhaps, we should have been looking for the arrangement of zeros (convergence of all critical primes.) For example, we can look at the list of holes for G = 2 and critical primes = 2,3,5,7. Each pair below (except S) adds to G and has the critical primes in their factor list.

S G                
1 0 -28 -40 -70 -138 -168 -180 -208  
1 2 30 42 72 140 170 182 210  

This list hints at the equivalents for G simply by adding the absolute values. D1 equivalents for G = 2 are 58, 82, 142, 278, 362, and 418.

Recently finished



Comparing the Last Two Recessions

In late 2000 a slow down became apparent. By mid-2001 the economy was in decline. Annual growth did not return to positive until December 2003. Then again, by mid-2007 a slow down became evident. By early 2008 the economy was in decline. The economy bottomed out in mid-2009 and returned to positive annual growth by mid-2010.

The 2008 recession was much deeper than 2001 recession, but return to positive growth happened quicker. Just 19 month after Obama took office the economy returned to positive growth. But it took about 35 months after Bush took office for the economy to return to positive annual growth. So far average annual growth after the recovery proved to be almost identical under Obama as it had been under Bush.

Employment & Unemployment

A more important measure than unemployment is employment. More important than employment is the participation rate. Participation measures what portion of the population is actually involved in earning an income. Participation rate can be broken down by segments of the populace to see who is affected. When we look at the participation rate for adult males we see a disturbing pattern. Participation has been dropping linearly since 1956. The ability of the American economy to create jobs for men has been falling steadily for nearly 60 years.
What is going wrong? How do we fix it?

Estimates of unemployment contain too many factors to reliably provide the information we want to see. That's why this site has tracked employment instead of unemployment. But one can put the employment and unemployment numbers back together to see the problems.

The official counts show a drop in the total workforce in 2001 followed by sluggish growth throughout the decade. We have no reason to believe that the number of people needing jobs actually dropped or that the population quit growing.

We can extrapolate the average workforce growth rate from the proceeding years to estimate the probable real workforce. When we do this we see that real unemployment grew to about 10% and remained high throughout the Bush administration. The real unemployment rate was already near 14% the day Obama took office. You may find similar analysis at Shadow Stats.

Private Sector Employment

Private sector growth has been declining linearly for the last 30 years. The trend line declines. The maximum points have been declining since 1978. The depth of recessions has been getting progressively deeper. If this trend continues, average growth will be negative by 2013. Maximum growth will be negative by 2020.

Is this trend a real consequence of a real factor within our economy? It is imperative to figure this question out.
Some claim the problem is caused by taxes or unions. But this entire era has been characterized by tax cuts and union busting. So those things are not the cause. They might be part of the problem.




Climate Change

NASA keeps monthly data. So how does the data compare to either the claims that earth is warming, or the claims that earth is cooling? Here we graph temperatures averaged across a 7.2 year span. (7.2 years was picked because there appeared to be an oscillation at roughly that frequency.) At this interval we see punctuated warming. Periods ending in 2010 are the warmest for land and the globe overall. For the ocean 2008 and 2010 roughly tie as the warmest. (See more)

One can easily find graphs showing how much CO2 is in the atmosphere. But rarely do we see graphs of annual growth in CO2. For those who have seen graphs of the temperature anomaly, annual CO2 growth will look strangely familiar.
The strong similarity between the temperature anomaly graph and the annual CO2 growth graph suggests a causal relationship. Most likely, as temperatures rise, earth's ability to absorb CO2 drops, leaving more CO2 in the atmosphere. This relationship suggests that we might reach a point of positive feedback: increased CO2 leads to rising temperatures -> rising temperatures reduces absorption -> reduced absorption increases the atmospheric CO2 -> increasing CO2 leads to rising temperatures. The correlation between temperature anomaly and annual CO2 growth needs further study!
Generalizing Peaks

Various analysts have discussed evidence that peak oil is occurring right now. Polya described the last step of good problem solving being generalizing the form to other problems. Let's take a quick look at the evidence that peak oil is now and generalize the form to other problems.

Peak Oil : In the first decade of this century the rate of production stalled even while demand rose.
Generalization: The rate of rise stalls even while forcing factors should drive continued rising.

  • Food Production: The rate of grain production stalled 30 years ago even while population has continued to rise.
  • Seasonal Cycles: At the end of the season the fruit tree slows its production, even while people are still searching for fruit.
  • Housing Bubble: Housing sales slowed in 2007 even though people still needed houses.

Peak Oil: Prices rose exponentially between 2000 & 2007 with supply & demand factors
Generalization: Secondary measures start to change rapidly, or slow down per situation

  • During the first decade prices of many commodities including food and strategic metals rose rapidly.

Peak Oil: The rate of consumption is persisting at about four times the rate discovery
Generalization: Output measures persist higher than input measures.

  • Housing Bubble: From 2000 to 2008 real estate values rose rapidly even while wages and employment stagnated.

Peak Oil: Over the last 60 years the sources of oil have moved further away, become riskier to extract, and become costlier to extract (The EROI has dropped.)
Generalization: Sources become costlier and more difficult to acquire.

  • Water Supply: Farms in various areas including the Oglalla aquifer and Southern California had had to dig much deeper wells.
  • Fishing Industry: Small fishing vessels no longer leave from coastal cities. Larger vessels travel farther to get fish. Fish formerly caught from our own coasts are now imported from other countries.

Peak Oil: Since about 1980 new discoveries of oil have become less frequent and smaller.
Generalization: Sources become smaller and less frequent.

  • Fishing Industry: Fish caught along the coast and in rivers are smaller and less common than those caught their 50 years ago.

Peak Oil: Oil suppliers have changed their behavior in two noticeable ways (1) they are now expressing excitement of low quality sources that they would have ignored in the past (eg: tar sands) (2) they are changing their economic behavior (eg: prospecting on Wall St. and buying back stock.)
Generalization: Experts and experienced workers (1) express excitement over low quality finds, (2) start looking elsewhere for opportunities.

  • Seasons: Farm workers leave specific crop behind and move to other more productive crops.
  • Fishing Industry: Fishing vessels are larger and travel farther. Some fishing vessels stopped fishing and switched to whale watching and other tourist ventures.
  • Housing Bubble: Banks bundled high risk investments with lower risk investments and sold the bundles to other banks. Some banks (Countrywide) habitually changed numbers on documents. Banks started to ask other banks to insure the risks they took on.
The forms used to identify one type of peak can be used to identify other types of peaks. The housing bubble should have been recognized very early simply by looking at the evidence for peaks. Various challenges we will face in the future (including peak oil) are already evident in the patterns if only we choose to look and prepare ourselves.



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