scratching surfaces

[AntonioLao]

Non-Euclidean geometries were created after failures to prove Euclid’s fifth parallel postulate using the other four postulates, 5 common notions, 23 definitions, and 28 theorems. The five postulates are: (1) to draw a straight line from any point to any point, (2) to produce a finite straight line continuously in a straight line, (3) to describe a circle with any center and distance, (4) that all right angles are equal to one another, and (5) if a straight line falling on two straight lines makes the interior angles in the same side less than two right angles, the two straight lines, if produced indefinitely meet on that side on which are the angles less than the two right angles. The Egyptian Ptolemy made the 1st attempt in the first century. The Persian Nasir-Eddin (1201-74) made the 2nd recorded attempt, next came Wallis, Saccheri, Fenn, Playfair, Legendre, Klügel, Lambert, Schweikart, Taurinus, and many others between the mid 16th century and the end of the 18th century. It is believed that no major discovery of mathematics is the work of one person. But at best, a decisive step or proof may be credited to an individual. This cumulative development of mathematics also applies to the discovery of non-Euclidean geometries. Nonetheless, the futile efforts to prove the fifth postulate became known as the foremost scandal of mathematical incompetency for mathematics in general and for geometry in particular.


The first break came in 1799 where and when Carl Friedrich Gauss was finally convinced that the parallel postulate cannot be deduced from the other axioms. He began the serious development of a new and possibly applied geometry based mostly on his practical experiences in land surveying and geodesy. This was the birth of spherical geometry. Its practical applications are in spherical trigonometry. Without it global maritime commerce would not have been possible. Thus begins the era of scratching surfaces of given spheres. They are mathematical idealization of the more realistic ellipsoidal volume of planet earth whose quadric equations in a proper coordinate system is x²/a²+y²/b²+z²/c²=1. Another branch known as hyperbolic non-Euclidean geometry was credited to the independent publications of Lobatchevsky and Bolyai in the early part of the 19th century whose proper quadric equations are the hyperboloid of one sheet and the hyperboloid of two sheets given respectively by x²/a²+y²/b²-z²/c²=1 and x²/a²+y²/b²-z²/c²=-1. Their practical applications could be found in space-time travel into the past or into the future.

[Lloyd Gillespie]

Hi Antoneo, here's a link to a book discussing the Poincare' conjecture: http://www.amazon.com/Poincares-Priz...owViewpoints=1
And these also: http://www.amazon.com/ref=pe_606_13337660_pe_ar_hd
Donald Duck in mathland: http://www.youtube.com/watch?v=P_ssR7M5Px0

IMO, general probability math = The only generally true possible at limits +...
IMO, complex number math = False at number limits + due to combinatorial explosions to infinities...
Complex number math --> M=1+IEE --> oo or Math = 1 + Infinite extensions and entanglements to infinities, never to be solved, absolutely accurately, mathematically_Only generally with probability maths and general geometries...

I'm a mathematical skeptic Antoneo, due to the fact that some 27 years ago, while trying to get to the bottom of economic maths, I discovered that near impossibility, that is iff using complexity maths. You may not be aware that economics is based in the math of physics from Boltzmann's ergodicity maths, which since last Sept. `08 has verified all us heterodox economists, as being correct in stating for over 100 years, that only universal probability maths could solve for the most complex problems, due to what's stated above. If one truly investigates ergodicity, one will find that Boltzmann forgot to mention the other half of the axiom of ergodicity__Not only do things tend toward equilibrium_They also tend away from equilibrium_and herein lies maths' greatest complexity faults(ignoring the total facts of)...

Complexity maths will always contain hidden variables approaching true infinities, therefore un-resolvable, except by re-normalizations of some sorts, or by reverting back to Huygen's original statistical probability maths. IMO, we in the economics community, as well as those in the physics community, are left with only Godel's and Tarsky's incompletenesses at the complex levels(higher order formal maths), though the simpler 1'st order maths are complete and reliable, therefore I choose the general geometries and probabilities of...

IMO, complexity maths will always contain these hidden variables to truly un-resolvable infinities, yet the Universe and nature contain no hidden variables unto themselves...

Will the academic math communities, and others, ever recognize these facts...rrr

Just my opinions...

[AntonioLao]

You may not be aware that economics is based in the math of physics from Boltzmann's ergodicity maths

As a believer of the theory of individual excellence I am not buying the law of averages and its statistical implications that are being used by economists to predict the laws of supplies and demands. On the contrary, I do believe that the prosperity of any human society is based on the industry of many individual's drive for excellence in whatever they do as an expert of their respective field. In other words, the success of any human civilization is dependent on group efforts.

[Profpat]

"You may not be aware that economics is based in the math of physics from Boltzmann's ergodicity maths, ..."

Is this by coincidence Lloyd or by design?

I believe Economics is as old as the family or tribe. Supply and demand, a redistribution of wealth, barter and trade, and other economic functions that man has developed as he himself developed.

[Lloyd Gillespie]

As a believer of the theory of individual excellence I am not buying the law of averages and its statistical implications that are being used by [SOME]economists to predict the laws of supplies and demands. On the contrary, I do believe that the prosperity of any human society is based on the industry of many individual's drive for excellence in whatever they do as an expert of their respective field. In other words, the success of any human civilization is dependent on group efforts.[And what about when the entire group effort fails, as it has since last Sept. `08...? This is my point__And no group or single individual is speaking or being heard...]

And neither am I Antonio. That is not what I was referring to. I was stating the differences that exist between the two academic schools of mathematical thought, as relates to the true probability maths and the false...

As per__ 'If one truly investigates ergodicity, one will find that Boltzmann forgot to mention the other half of the axiom of ergodicity__Not only do things tend toward equilibrium_They also tend away from equilibrium_and herein lies maths' greatest complexity faults(ignoring the total facts of)..."

And again__ 'Complexity maths will always contain hidden variables approaching true infinities, therefore un-resolvable, except by re-normalizations of some sorts, or by reverting back to Huygen's original statistical probability maths. IMO, we in the economics community, as well as those in the physics community, are left with only Godel's and Tarsky's incompletenesses at the complex levels(higher order formal maths), though the simpler 1'st order maths are complete and reliable, therefore I choose the general geometries and probabilities of...

IMO, complexity maths will always contain these hidden variables to truly un-resolvable infinities, yet the Universe and nature contain no hidden variables unto themselves...

Will the academic math communities, and others, ever recognize these facts...'rrr

If you noticed, I'm speaking against the mathematics communities of both physics and economics, not cheering them on. I was just wondering your opinion of the historical math problems regarding both these schools of thought...?

[Lloyd Gillespie]

Complexity maths of economics only entered the picture in the middle and late 1800's Pat, as did they in physics. They weren't invented until then, by many of the voices, both Antonio and I have mentioned. Yes, there was a lot of number theory and complex geometries and algebras, but the age of true complex statistical maths and mechanics weren't really invented, until the probability and vector mechanics of J.Gibbs and others from Europe and England, in the 19th century, and sure wasn't applied semi-sucessfully to economics and physics until then. Maxwell and Boltzman were the first to apply Gibbs' vector mechanics, etc., then Jevons and L.Walras, among others in economics. Walras actually started as a physicist, switched to math and finally to economics, and was one of the first of the false equilibrium/ergodicity schools of economic maths that evolved into the mess we witness today, i.e., highly speculative corporate derivatives destroying the well-being of all...rrr

"You may not be aware that economics is based in the math of physics from Boltzmann's ergodicity maths, ..."

Is this by coincidence Lloyd or by design?

I believe Economics is as old as the family or tribe. Supply and demand, a redistribution of wealth, barter and trade, and other economic functions that man has developed as he himself developed.

[Profpat]

As a believer of the theory of individual excellence I am not buying the law of averages and its statistical implications that are being used by economists to predict the laws of supplies and demands. On the contrary, I do believe that the prosperity of any human society is based on the industry of many individual's drive for excellence in whatever they do as an expert of their respective field. In other words, the success of any human civilization is dependent on group efforts.

I agree with you Antonio, you are at the reality level of human beings, with sweat and blood and desires and ambitions and an essence towards excellence. Tied in with group efforts and you do have the foundation for an excellent economic system. Now we just have to redistribute (share) the wealth.

[Profpat]

I believe the entire financial mess was all because of greed and lack of ethics. By the oil companies, by the banks, by the government, which translates right back to me and you and We The People.

I guess, I believe the market crash itself wasn't a failure, but rather a success as a necessary correction to an overinflated market, both in housing and financial.

It may be slow but it is efficient, eventually. The real problem of course is greedy unethical people, the market just reflects that.

[Lloyd Gillespie]

Pat, what if you're wrong, and it's really the laws that are creating the greedy unethical people, created by the wrong market laws, reflecting a very un-wise democracy...rrr

It may be slow but it is efficient, eventually. The real problem of course is greedy unethical people, the market just reflects that.

[Profpat]

Pat, what if you're wrong, and it's really the laws that are creating the greedy unethical people, created by the wrong market laws, reflecting a very un-wise democracy...rrr

Well people make the laws and can change them. But now we are into the area of politics, where greedy and unethical people are usually in charge.

I do agree with having laws to remove or reduce the temptation. In accounting systems design we try our best to eliminate opportunities for fraudulent or illegal activities. But when there is corruption at the top they can override the system.

"Lead us not into temptation.." He knew human nature.