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
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.
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.
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
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
|| 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
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.
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
|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!
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.
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.
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
- 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
- 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
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
- 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.