Paths To Knowledge (dot Science)

What is actually real in Objective Reality? How do you know? Now, prove it's real!

Archive for the ‘Math’ Category

Amazing Mechanical Computers From the 1700’s

Posted by pwl on September 28, 2013

“The story of Pierre Jaquet Droz and his sons is one of the most moving in the history of Horology. Born in 1721, Pierre Jaquet Droz, master of time in the Age of Enlightenment — mechanical genius, avant-garde creator of jewellery watchmaking and composer of poetry and dreams — is one of the most fascinating figures of the period.
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Posted in Awesome, Get some perspective people, Good science attitude, Hard Science, History, Ignorance to Knowledge, Majestic Universe, Making a REAL Positive Difference in the World, Math, Parallel Computing, People|Science, Rational Thinking, Science Education, Science Info Educational Videos, Science Shows, Video, WOW!!! | Tagged: | 6 Comments »

The Mean Median, A Mode of Confusion

Posted by pwl on January 14, 2010

Not to be mean but to clarify a conversation about global warming where there was a mode of confusion about the meaning of “mean” when “median” was meant by the other party. (I apologize for being in pun mode but there are some mean puns in there that were not meant to be avoided). [:)]

The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode).

This confusion was the case in the conversation. It’s important that all parties get the same meaning for terms being used. This primer is to clarify for anyone who wants to comprehend these basic yet important statistical terms.

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Posted in Math, Rational Thinking, Science Education | 1 Comment »

NASA JPL Satellite Eyes On the Earth 3D – Gravity Balls and Friends

Posted by pwl on December 10, 2009

Very cool tool from visualizing data from NASA satellites in 3d in a web browser.

What I’m interested in is not just the visualization aspect but the actual data behind the visualizations since as we know from the Climategate confirmations of scientific fraud in climate science visual images can be highly misleading towards the alleged scientists point of view. Raw data please. All manipulations MUST be FULLY documented with the software source code that made the changes and detailed reasons listed for all adjustments.

If we are to raise our knowledge, skills and competence in using the scientific method to study the Earth, Moon, Sol and other relevant systems we must do some basic learning. Hard science requires making use of hard data without cheating and with showing ones data and any adjustments with justifications and open source code for auditing and proper open peer reviews!

Now let’s see what we can learn from this 3d puppy. My favorite is the GRACE Gravity satellites.

I’ve often wondered what impact the uneven gravity has on the Earth’s climate systems and if the gravity effects are taken into account in the so called climate models. As you can see from the above video the Earth isn’t even an oblate spheroid, it’s a really bumpy place when it comes to gravity. This must impact the weather and thus the climate systems as the atmosphere and water and ice move about.

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Posted in Adventure, Awesome, Big Brother Planetary Control System, Climate Science, Climategate, Complex Systems, Evil Walks the Earth and Carries a Big Stick, Exercise for the Reader (that's you), Get some perspective people, Graphics, Gravity, Hard Science, Hard Science Required, Humbled by Nature, Ignorance to Knowledge, It's weather AND climate!, Learning about Science Organizations, Majestic Universe, Math, Proofs, Proofs Needed, Rational Thinking, Reality Based Environmentalism, Science Ficition, Science Missions, Science over Propaganada, SkyNet Battlefield Earth, Total Control Over Our Lives, Video, WOW!!! | 4 Comments »

The Most Important Conversation to have with a Roman Catholic or other Christian

Posted by pwl on September 6, 2009


Over a period of 28 hours or so this long weekend I had a conversation with someone, CK, I met on Face Book who shared a common technical interest. We had a number of other exchanges with each other over the past few months but this one was different partly for it’s brutal honesty and partly since CK choose to jump out of what was likely the most important conversation of his life.

I managed to save a copy of the conversation moments before CK deleted my access (it was on his wall) and thus ending the conversation with us two. Likely he’s busy gathering agreement with his friends that I’m an evil no good person when in fact it’s likely that I’m one of his best new friends he’s ever had. Be that as it may, here is a non edited transcript (spelling corrections were made). Oh, from what I can tell CK is a young adult male with a gay lover and a roman catholic upbringing.

I hope this helps when talking with religious delusionals of all sorts but christians in particular since we deal with dispensing the christian miracle mythologies here.

I’ll add comments and expand my responses and even critique my own responses over the next few weeks or so as I have time to reread the transcript below and think longer it.

All the best and live long and prosper in peace. Your feed back and comments are appreciated.


ps. This Jesus and Mo comic is appropriate since CK’s solution for “resolving the incompatibilities” between science and religion is to become insane by fully embracing “cognitive dissonance” and not working to resolve the conflict even though he claims to work towards that end. More on this in the conversation. He also, as you will see, negotiates his way between these two incompatibilities by accepting part of Objectivism (it’s basic three axioms of existence, identity and consciousness) while maintaining his “faithful belief” in his invisible super friends even though acknowledging that objectivism provides “proof” that there are no gods other than those in human minds. Thus CK’s religious driven insanity.

CK: just wants to find SOMEONE who shares his philosophies… lol

PWL: What philosophy are yours?

CK: Except for the stance on matters of sexuality and religion, I’m a staunch objectivist. Go Ayn Rand!

I believe cognitive dissonance is the greatest evil in the universe.

That’s a good start to the summary. In essence, I believe you must accept a small set of axioms as true, and then use inductive logic based on that. If only I could put my axioms in words… lol

PWL: I am also a big fan of Ayn Rand.

People do work well together and can in many circumstances work towards their mutual benefit.

What do you mean by cognitive dissonance?

I’m confused by your sexuality considering you seem to be very catholic… seems mutually incompatible to me.

I’m confused by your religious stance and Ayn Rand, objectivism proves that your mythical god doesn’t exist and is just an artifact of your mind wanting comfort or what not. So you’re not a staunch objectivist if you believe in the mythical invisible super beings and thus the supernatural realm.

The real world doesn’t care about our philosophies since it doesn’t care about anything.

They are not axioms if they are not in words…

BUT then Leaps of Faith know no boundaries or limits and ignore any that are pointed out. That is the essential nature of a leap of faith, it destroys actual rational thinking with the emotional non-rational leap of faith.

“Cognitive dissonance is an uncomfortable feeling caused by holding two contradictory ideas simultaneously. The “ideas” or “cognitions” in question may include attitudes and beliefs, the awareness of one’s behavior, and facts. The theory of cognitive dissonance proposes that people have a motivational drive to reduce dissonance by changing their attitudes, beliefs, and behaviors, or by justifying or rationalizing their attitudes, beliefs, and behaviors.[1] Cognitive dissonance theory is one of the most influential and extensively studied theories in social psychology.” – Cognitive Dissonance.

So if you believe “cognitive dissonance” is the greatest evil in the universe why do you practice it?

Why do you consider it evil?

MD: wow :{

CK: Sexuality: Rand didn’t exactly believe they should wait until they found the right one, and then stay with them the rest of their life. Yes, objectivism does reject the almighty.

When I say cognitive dissonance, I mean holding two apparently exclusive beliefs wihout batting an eye — it’s an offense to reason. In my case (objectivist Catholic), although you’d think they are exclusive, but the reason I believe they are not is that I do not blindly believe — I observe the positive effects of faith, I observe the evidence of miracles, and I make judgments for myself. I consider the beliefs and try to reason and justify them.

If I were to just say, “well, it is” THEN faith and reason would be exclusive. But to examine faith and try to reason about it, why you believe it even when you know science, then you have used reason. You just use inductive logic rather than deductive logic.

PWL: Actually you must use actual repeatable verifiable evidence not logic or some belief stricken dogma system…

As the saying goes you are mistaken about a great many things CK.

There are no real positive effects of faith that are not available without faith!

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Posted in Adult Supervision Required, Awesome beyond awesome, Bad Science Attitude, Belief Stricken, Believe it or your a denier!, Big Brother Planetary Control System, Biology, Ethics in Science, Evil Walks the Earth and Carries a Big Stick, Exercise for the Reader (that's you), Gay, Get some perspective people, Hard Science Required, Holding those who run the world responsible for their crimes against humanity, Human|Ape, Humbled by Nature, Ideas Crazy Enough to Have a Chance, Ignorance to Knowledge, Intelligent Designer - Yeah Right, Invisible Friend Crowd, It's a good thing that your god(s) die with you, Majestic Universe, Math, My Invisible Friend Needs Me For His-Her-Its Existence, Philosophy, Proof God Can NOT Exist, Proofs Needed, Rational Thinking, Really Funny, Religion, Scams, Science over Propaganada, SkyNet Battlefield Earth, Some Intelligent Designer, Something to think about, Terrorfying, The End is Nigh, The Ground is Falling Up!, The Sky Is Falling, To Hell With You Buddy, Total Control Over Our Lives, WOW!!!, Yeah Right, Yikes! The sky is NOT falling!, Zombie Jesus | Leave a Comment »

Rational Trigonometry and a new Foundation For Math

Posted by pwl on August 11, 2009

Two excellent video series by Norman J Wildberger. Very well done and designed to be easy to comprehend. While they start out very simple and basic they get quite deep yet remain as simple as possible. (The bonus series even gets into Relativity, very cool). Wildberger is the math teacher I wish I had all through high school and since, and now through his videos he can be our math teacher! Gotta love online education.

The first series is on the amazing new (and very old) way of doing Rational Trigonometry without using sin and cosine of Classical Trigonometry. It’s Fantastic.

“The new form of trigonometry developed here is called rational trigonometry, to distinguish it from classical trigonometry, the latter involving [cosine, sin and their fellow] functions and the many trigonometric relations currently taught to students. An essential point of rational trigonometry is that quadrance and spread, not distance and angle, are the right concepts for metrical geometry (i.e. a geometry in which measurement is involved).

Quadrance and spread are quadratic quantities, while distance and angle are almost, but not quite, linear ones. The quadratic view is the more general and powerful one. At some level, this is known by many mathematicians. When this insight is put into practice, as it is here, a new foundation for mathematics and mathematics education arises which simplifies Euclidean and non-Euclidean geometries, changes our understanding of algebraic geometry, and often simplifies difficult practical problems.

Rational trigonometry deals with many practical problems in an easier and more natural way than classical trigonometry, and often ends up with answers that are demonstrably more accurate. In fact rational trigonometry is so elementary that almost all calculations may be done by hand. Tables or calculators are not necessary, although the latter certainly speed up computations. It is a shame that this theory was not discovered earlier, since accurate tables were for many centuries not widely available.”

Exerpt from the Introduction to “Divine Proportions: Rational Trigonometry to Universal Geometry” by Associate Professor Norman J Wildberger

Here are a few videos from the Rational Trigonometry series with the full series of 46+ episodes here. These first episodes lay the corner stone of Rational Trig, and it only gets better from there!!! Watch them all a number of times and do the examples. Amazing!

Why classical trigonometry is hard:

The problem is that defining an angle correctly [in Classical Trigonometry] requires calculus [whereas Rational Trigonometry doesn’t]. This is a point implicit in Archimedes’ derivation of the length of the circumference of a circle, using an infinite sequence of successively refined approximations with regular polygons. It is also supported by the fact that The Elements [Euclid] does not try to measure angles, with the exception of right angles and some related special cases. Further evidence can be found in the universal reluctance of traditional texts to spell out a clear definition of this supposedly ‘basic’ concept.” – NJ Wildberger

“Quadrance measures the separation of two points. The easiest definition is that quadrance is distance squared.

… quadrance is the more fundamental quantity, since it does not involve the square root function. The relationship between the two notions is perhaps more accurately described by the statement that distance is the square root of quadrance.

In diagrams, small rectangles along the sides of a triangle indicate that quadrance, not distance, is being measured … .” – NJ Wildberger

Spread measures the separation of two lines. This turns out to be a much more subtle issue than the separation of two points.” – NJ Wildberger

The spread s between the two lines is the ratio of quadrances. The spread also works out to “square of sine of angle”.

“The spread between two lines is a dimensionless quantity, and in the rational or decimal number fields takes on values between 0 and 1, with 0 occurring when lines are parallel and 1 occurring when lines are perpendicular. Forty-five degrees becomes a spread of 1/2, while thirty and sixty degrees become respectively spreads of 1/4 and 3/4. What could be simpler than that? Another advantage with spreads is that the measurement is taken between lines, not rays. As a consequence, the two range of angles from 0◦ to 90◦ and from 90◦ to 180◦ are treated symmetrically.” – NJ Wildberger

You can find a couple freely available pdf chapters of the book “Divine Proportions: Rational Trigonometry to Universal Geometry” here. Make sure to check out Divine Proportions Overview Chapter 1 as it covers the basics with examples.

You will find some additional pdf papers by the professor here.

The second series is on the Foundations of Math. An excellent review and introduction for everyone at any age.

Here are a few videos from the Foundations of Math series with the full series of 35+ episodes here.

There is also a series of advanced Math Seminars on the topic of Hyperbolic Geometry is Projective Relativistic Geometry.

Posted in Awesome beyond awesome, Exercise for the Reader (that's you), Graphics, Hard Science, Humbled by Nature, Ignorance to Knowledge, Math, Norman J Wildberger, Proofs, Rational Thinking, Science Education, Science Info Educational Videos, Science Shows, Video, WOW!!! | 12 Comments »


Posted by pwl on July 3, 2009


Calculus. (Latin, calculus, a small stone used for counting) is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series, and which constitutes a major part of modern university education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.

Historically, calculus was called “the calculus of infinitesimals”, or “infinitesimal calculus”. More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus and join calculus.

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Posted in Awesome, Fun, Get some perspective people, Math, Science Education, Video | 1 Comment »

How do we know global warming is a problem if we can’t trust the U.S. temperature record?

Posted by pwl on May 10, 2009

This report on the US Temperature Surface Stations (pdf) raises serious questions about the whole premise underlying the “man cause global warming crisis”. IF the underlying data is corrupted and inaccurate how can any conclusions of science be drawn from them?

Serious questions are raised and need to be answered. Heads must roll. They do that too in science.

Executive Summary
Global warming is one of the most serious issues of our times. Some experts claim the rise in temperature during the past century was “unprecedented” and proof that immediate action to reduce human greenhouse gas emissions must begin. Other experts say the warming was very modest and the case for action has yet to be made.

The reliability of data used to document temperature trends is of great importance in this debate. We can’t know for sure if global warming is a problem if we can’t trust the data. The official record of temperatures in the continental United States comes from a network of 1,221 climate-monitoring stations overseen by the National Weather Service, a department of the National Oceanic and Atmospheric Administration (NOAA). Until now, no one had ever conducted a comprehensive review of the quality of the measurement environment of those stations.

During the past few years I recruited a team of more than 650 volunteers to visually inspect and photographically document more than 860 of these temperature stations. We were shocked by what we found. We found stations located next to the exhaust fans of air conditioning units, surrounded by asphalt parking lots and roads, on blistering-hot rooftops, and near sidewalks and buildings that absorb and radiate heat. We found 68 stations located at waste water treatment plants, where the process of waste digestion causes temperatures to be higher than in surrounding areas. In fact, we found that 89 percent of the stations – nearly 9 of every 10 – fail to meet the National Weather Service’s own siting requirements that stations must be 30 meters (about 100 feet) or more away from an artificial heating or radiating/reflecting heat source.

This site in Marysville, CA has been around for about the same amount
of time, but has been encroached upon by growth in a most serious way
by micro-site effects.

In other words, 9 of every 10 stations are likely reporting higher or rising temperatures because they are badly sited. It gets worse. We observed that changes in the technology of temperature stations over time also has caused them to report a false warming trend. We found major gaps in the data record that were filled in with data from nearby sites, a practice that propagates and compounds errors. We found that adjustments to the data by both NOAA and another government agency, NASA, cause recent temperatures to look even higher.

The conclusion is inescapable: The U.S. temperature record is unreliable.

The errors in the record exceed by a wide margin the purported rise in temperature of 0.7º C (about 1.2º F) during the twentieth century. Consequently, this record should not be cited as evidence of any trend in temperature that may have occurred across the U.S. during the past century. Since the U.S. record is thought to be “the best in the world,” it follows that the global database is likely similarly compromised and unreliable.

This report presents actual photos of more than 100 temperature stations in the U.S., many of them demonstrating vividly the siting issues we found to be rampant in the network. Photographs of all 865 stations that have been surveyed so far can be found at Surface Stations dot org, where station photos can be browsed by state or searched for by name.

Posted in Adult Supervision Required, Climate Science, Debunking Bad Environmentalism, Exercise for the Reader (that's you), Hard Science Required, Ideas Crazy Enough to Have a Chance, Intelligent Designer - Yeah Right, It's a good thing that your god(s) die with you, Math, Proofs, Proofs Needed, Science Education, TerrorForming Earth, Yikes! | Leave a Comment »

A New Kind of Science by Stephen Wolfram

Posted by pwl on May 1, 2009

Wolfram Science.

A New Kind of Science.

Posted in A New Kind of Science [NKS], Complex Systems, Humbled by Nature, Ignorance to Knowledge, Math, Stephen Wolfram | 1 Comment »

Smallest Unit of Distance and Time Confirmed?

Posted by pwl on March 12, 2009

Quantum Noise Confirms Smallest Distance? which is the Planck Length.

planck lengthFor the past seven years, [the GEO600 experiment] has been looking for gravitational waves – ripples in space-time thrown off by super-dense astronomical objects such as neutron stars and black holes. GEO600 has not detected any gravitational waves so far, but it might inadvertently have made the most important discovery in physics for half a century.

For many months, the GEO600 team-members had been scratching their heads over inexplicable noise that is plaguing their giant detector. Then, out of the blue, a researcher approached them with an explanation. In fact, he had even predicted the noise before he knew they were detecting it. According to Craig Hogan, a physicist at the Fermilab particle physics lab in Batavia, Illinois, GEO600 has stumbled upon the fundamental limit of space-time – the point where space-time stops behaving like the smooth continuum Einstein described and instead dissolves into “grains”, just as a newspaper photograph dissolves into dots as you zoom in. “It looks like GEO600 is being buffeted by the microscopic quantum convulsions of space-time,” says Hogan.

If this doesn’t blow your socks off, then Hogan, who has just been appointed director of Fermilab’s Center for Particle Astrophysics, has an even bigger shock in store: “If the GEO600 result is what I suspect it is, then we are all living in a giant cosmic hologram.”

Wow, is the Universe Digital after all?

Posted in Complex Systems, Energy, Hard Science, Math, Proofs | Leave a Comment »

Colors of Fractional Infinity

Posted by pwl on March 12, 2009

“Arthur C. Clarke presents this unusual documentary on the mathematical discovery of the Mandelbrot Set (M-Set) in the visually spectacular world of fractal geometry. This show relates the science of the M-Set to nature in a way that seems to identify the hand of God in the design of the universe itself. Dr. Mandelbrot in 1980 discovered the infinitely complex geometrical shape called the Mandelbrot Set using a very simple equation with computers and graphics.”

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Posted in Hard Science, Ignorance to Knowledge, Intelligent Designer - Yeah Right, Math, Video | Leave a Comment »

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