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Abe's Blog

This is an eclectic blog. Not only will I talk about technologies, engineering, nanotechnology, but also about education, music, art, and other human endeavors. After all, humans are not only engineers or doctors.

Bertrand Russell's Set Theory Paradox

Posted January 25, 2017 5:31 PM by amichelen
Pathfinder Tags: education Logic math paradox

Bertrand Russell

Gottlob Frege

In June 1902, Bertrand Russell, the great British mathematician and logician, sent the statement of a paradox to his friend Gottlob Frege, a German philosopher, logician and mathematician. Frege had been working for more than 10 years writing his monumental work “The Foundations of Arithmetic” and was finishing the final chapter of the second volume of this two-volume treatise. Russell and Frege had been friends for many years and Russell encouraged his friend to write a book about the mere foundations of arithmetic based on the set theory of Cantor. Frege obliged, but one day Russell found a contradiction of the work Frege was proposing in his book. This contradiction simply destroyed the very principle of his friend’s logic. Frege was devastated; his work of over ten years was simply irrelevant. Russell’s letter to Frege terminated the labor of more than ten years. Frege sank into a deep depression, while Russel tried to repair the damage by constructing a new theory of logic that would be immune to the paradox. He couldn’t. The paradox appeared again in his new theory.

The Russell paradox has been popularized in many ways. One of the best known of these was given by Russell in 1919 and concerns the plight of a barber of a certain village who has enunciated the principle that he shaves all those persons of the village who do not shave themselves. The paradoxical nature of the situation is realized when we try to answer the question: “Does the barber shave himself?” If he does shave himself, then he shouldn’t according to his principle; if he doesn’t shave himself, then he should according to his principle.

Since the discovery of the above contradictions, additional paradoxes have been produced in abundance. These modern paradoxes of set theory are related to several ancient paradoxes of logic, such as:

  1. Eubulides, of the fourth century B.C. is credited with making the remark, “The statement I am making is false.” If Eubulides statement is true, then, by what it says, the statement must be false. On the other hand, if Eubulides’ statement is false, it follows that the statement must be true. Then Eubulides’ statement can neither be true nor false without entailing a contradiction.
  2. Epimenides, who himself was a Cretan philosopher of the sixth century B.C., is claimed to have made the remark, “Cretans are always liars.” A simple analysis of this remark easily reveals that it, too, is self-contradictory.

The existence of paradoxes in set theory, like those described above, clearly indicates that something is wrong. Since their discovery, a great deal of literature on the subject has appeared, and numerous attempts at a solution have been offered. For some mathematicians there seems to be an easy way out. One has merely to reconstruct set theory on an axiomatic basis sufficiently restrictive to exclude these known antinomies. This is simply a procedure that avoids the paradoxes (putting your head in the sand.)

24 comments; last comment on 02/02/2017
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Analog Filters Video Lecture

Posted December 07, 2016 12:00 AM by amichelen

Introduction

In today's entry I am a introducing a presentation video related to the general topic of Analog Filters. The presentation is a general overview of analog filters, without the rigor of mathematical equations or design methods. This first video includes the following concepts:

· Definitions

· Signal representation

o Time domain

o Frequency domain

· Ideal filters

· Practical filters

· Filter types

· Frequency domain characteristics

o Butterworth

o Chebyshev

o Elliptic

o Bessel

· Filter realizations

o Passive filters

o Active Filters

· Specifications

Filters are electrical circuits designed to remove, attenuate or alter the characteristics of electrical signals; in particular these devices reduce the magnitude and the phase of unwanted signals with certain frequencies. For example noise (normally at a frequency of 60 Hz in electronics circuits) is always present, and it is desirable to suppress the noise from the system. We can achieve this by passing the system signal (voltage and noise) through a filter. If the filter is designed to suppress or attenuate the magnitude of the noise, the output of the filter will contain only (or mostly) the system signal. For another example consider a typical radio receiver (the radio in your car); by tuning to a particular radio station you are selecting one signal while attenuating the signals of the other radio stations. This process is accomplished by mean of a filter.

Several categories of filters exist, but the main distinction is between analog and digital filters. Analog filters are designed to attenuate signals in analog systems, while digital filters attenuate digital signals in digital systems. These notes will concentrate in the study of analog filters, leaving digital filter for another occasion. The study of filters entails the use of complex mathematical techniques such as z-transform, Laplace Transform, convolution, recursion, and others that will be discussed in later articles. This module present filter behavior without engaging the reader through advanced mathematics and complex techniques.

Types of Analog Filters

Filters are broadly classified according to the type of frequencies that the filter is able to suppressed or attenuate. In this regard, there are four main categories:

· Low-pass filter. This type of filter attenuates or suppresses signals with frequencies above a particular frequency called the cutoff or critical frequency ( ). For example a low-pass filter (LPF) with a cutoff frequency of 40 Hz can eliminate noise with a frequency of 60 Hz.

· High-pass filter. This is a filter that suppresses or attenuates signals with frequencies lower than a particular frequency - also called the cutoff or critical frequency. For instance a high-pass filter (HPF) with a cutoff frequency of 100 Hz can be used to suppress the unwanted DC voltage in amplifier systems, if so desired.

· Band-pass filter. A filter that attenuates or suppresses signals with frequencies outside a band of frequencies. This is the general type of filters used when tuning radio or TV signals.

· Band-reject, or Notch filter. A filter that attenuates or suppresses signals with a range of frequencies. For instance, we can use such a filter to reject signals with frequencies between 50 Hz and 150 Hz.

Response Characteristics

The frequency response of any filter (LPF,HPF,PBF,BRF) can be designed by properly selecting the circuit components. The characteristics of filters are defined by the shape of the frequency response curve; the most important response shapes are named after a researcher who studied the particular filters. There are filter of type Butterworth, Chebyshev (types I and type II) , Elliptic (or Cauer), and Bessel, to mention the most important. These filter types are named after the British researcher Stephen Butterworth, the Russian mathematician Pafnuty Chebyshev, the German scientist Wilhelm Cauer, and the German mathematician Friedrich Bessel, respectively. Each one of these filters types has a particular advantage in certain applications. The following figure shows the characteristics of four low-pass filters, each one of three-poles and cutoff frequency of 10. Note the different types of shape represented by the frequency responses.

The video presentation is located here: Analog Filters

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Hypatia of Alexandria – Part 3

Posted November 15, 2016 10:12 AM by amichelen

(Note: This is last part of Hypatia of Alexandri)

Personal life

If we have little knowledge of her scientific achievements, we know even less about her personal life. Some rumors had it that Hypatia married the philosopher Isidore of Alexandria (even if Isidore was born after her death) and that she was a follower of the Hellenistic (pagan) gods. There is no proof of these legends. On the contrary, she always showed a rational posture in relation to the Hellenic tradition and made sure she kept herself away from the constant disputes between Christians and pagans in the Alexandria of her time. Hypatia participated, however, in municipal politics. She was respected for her ethic values, and pagan and Christian politicians asked her advice in matters related to the city’s management. Although she was not Christian, she was not pagan either in the sense that she never worshiped Hellenistic gods or engaged in pagan rituals, but may have engaged in reciting Hellenic prayers or hymns during her teachings.

Hypatia was the teacher and friend of Orestes, a converted Christian, who was the Roman governor (prefects) of the province of Egypt, and a staunch defender of the rights of both Christians and pagans to practice their cultural activities and beliefs. However, on October 17, 412 the fate of Alexandria changed when Ciryl was elected Bishop (Patriarch) of Alexandria, who declared an enemy and heretic of any person that did not consider Christianity as the only acceptable religion. Even many Christians of Egypt were opposed to Cyril’s intolerant ideas and a “war” between Orestes and Cyril ensued. Alexandria descended into a total chaos and extreme violence. Cyril took the lead in this war and, that there were more Christians in the city, he ordered the organized mobs to kill the Jews and pagans associated with Orestes. The fight against paganism ended in the destruction of the Serapeum a smaller library remnant of the great Library of Alexandria, destroyed during the invasion of Alexandria by the Emperor Aurelian in AD 270. It is said that Hypatia and her students rushed to the Serapeum to salvage scrolls and books before their destruction. Who knows if some of those books and scrolls were the last existing works of Aristotle and other Greek authors, copies of which we have today? Hypatia may have been the responsible for the surviving texts so dear to us today.

At the same time Cyril started a campaign of defamation against Hypatia. For Cyril the influence of Hypatia in the upper echelon of the imperial and municipal politics was a threat to him and to the Church. He declared her to be a dangerous “witch” totally dedicated to black magic, and the creator of atheists.

A writer of that time (John, Bishop of Nikiu) describes the incitement against Hypatia by the clergy: “And in those days there appeared in Alexandria a female philosopher, a pagan named Hypatia, and she was devoted at all times to magic, astrolabes and instruments of music, and she beguiled many people through (her) Satanic wiles. And the governor of the city (Orestes) honored her exceedingly; for she had beguiled him through her magic…”

Another writer (Damascius) explains in more details: Thus it happened one day that Cyril, bishop of the opposition sect [i.e. Christianity] was passing by Hypatia’s house, and he saw a great crowd of people and horses in front of her door. Some were arriving, some departing, and others standing around. When he asked why there was a crowd there and what all the fuss was about, he was told by her followers that it was the house of Hypatia the philosopher and she was about to greet them. When Cyril learned this he was so struck with envy that he immediately began plotting her murder and the most heinous form of murder at that.”

In fact, in March of 415, Hypatia was brutally killed by a mob of Christian fanatics lead by Peter the Reader. The mob stopped her chariot, dragged her to a church, stripped her of all clothes and killed her with ceramic shells. They destroyed her body and burned it in a public plaza as an example to people who deviated from the teaching of the Church. Hypatia was murdered by people who felt threatened by her knowledge, her scholarship and her profound scientific knowledge.

Whom to blame for her murder?

For some, like Voltaire, she was killed because she came to symbolize learning and science which the early Christians identified with paganism. However, she taught many Christians, including Orestes and Synesius of Cyrene who later became an important bishop. For others, her murder was a consequence of the bitter fight between the civil power represented by Orestes and the ecclesiastic power represented by Cyril, and not as a confrontation between Christianity and paganism.

Her murder represents the start of the decline of Alexandria. Cyril’s mobs not only assassinated a great scientist, but also provoked the flight of the most brilliant scientists and philosophers from the city. Few years after Hypatia’s death Alexandria and the rest of the Western world entered into the dark ages. This lasted until the Renaissance of the 14th century. Ten centuries of darkness!

Alejandro Amenabar, the great Spanish director recreated her life in a movie called Agora. The following YouTube link is a video with scenes from this movie:

https://www.youtube.com/watch?v=G4yj3uXS2uM

"Death of the philosopher Hypatia, in Alexandria" from Vies des savants illustres, depuis l'antiquité jusqu'au dix-neuvième siècle, 1866, by Louis Figuier.

35 comments; last comment on 11/19/2016
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Hypatia of Alexandria – Part 2

Posted November 08, 2016 12:00 AM by amichelen

(This is the second part of Hypatia of Alexandria)

The ideas developed by the Pythagoreans represent a solid contribution to the scientific knowledge of the times. The most important contribution to cosmology was the idea that the Earth and the other planets move around the Sun (Sun-centered system, or heliocentric), making the Earth simply another planet. This is an important contribution of 4th-century science at a time when biblical ideas put the Earth at the center of the solar system, following the Ptolemaic Earth-centered principle, as indicated in the following figure.

This vision of the universe was not new. Seven centuries before – 3rd century B.C. – the ancient Greek astronomer Aristarchus of Samos was the first to develop a heliocentric model of the solar system. Unfortunately his original work was lost and we only know about his work is through references to it by Archimedes and Plutarco.

In the Ptolemaic system (geocentric) each planet moves in a small circle called the epicycle that moves in a large sphere called a deferent. The stars moved in an outer celestial sphere around the planetary system. In order to explain the actual observation of the movement of the planets, Ptolomy used three tricks: (1) locate the Earth out of the center of the large sphere, putting us in an “eccentric” position; (2) then create the epicycle; (3) then define a point – not in the center of the large sphere either– called the “equant”, as illustrated. These three tricks – the epicycle, the “eccentric” Earth location, and the equant – allowed Ptolomy to mathematically explain the movement of the Sun and the rest of the planets around the Earth in a very accurate manner. Just as a curiosity here, the Ptolemaic model was so mathematically accurate that today planetariums are built based on this principle. A planetarium consists of a projector, one motor or gear to move the projector around in a big circle (the deferent), and a gear or motor moving in a small circle, following the large circle, to mimic the epicycle. In this way the sky is viewed from a stationary Earth. It is certainly amazing to build our planetariums using a model developed almost 2,000 years ago. It is even more impressive to know that this fake model was accepted and not seriously challenged by science and society for over 1,300 years!

When in 1543 Nicolaus Copernicus’ treatise De revolutionibus orbium coelestium was published posthumously, a new conception of the universe was brought back. Copernicus was a student of Domenico Maria de Novara, an astronomy professor in Bologna, who had a critical position of the Ptolemaic system. He taught Copernicus about the Pythagorean ideas of a Sun-centered model, where there are no epicycles.

Hypatia never accepted the Earth-centered model. All her life she supported the heliocentric model, and her observations in the commentaries of Theon Ptolemy's Almagest prove that she clearly rejects the geocentric Ptolemaic theory in favor of the heliocentric model. It is very possible that Copernicus read the Almagest with Theon and Hypatia’s comments when he was in Florence studying the Ptolomy works, given that the only copy of the Almagest was in the Medici library of that city, a library that Copernicus visited for his research. If this is the case, this would imply that Hypatia may have had a direct influence in the development of the Copernican Revolution.

1 comments; last comment on 11/10/2016
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Hypatia of Alexandria – Part 1

Posted November 01, 2016 12:00 AM by amichelen

(Note: This is part 1 of a three series blog.)

I would like to write my first blog by introducing a woman, who, many years ago in Alexandria, excelled in philosophy and science, even surpassing many scientists and philosophers of the time. Being an educated woman in the fourth century was not a trivial matter. Hypatia, sometimes called Hypatia of Alexandria, was the daughter of the philosopher and mathematician Theon of Alexandria, and was born in the fourth century (around the year 370), a historic period during which scientists were debating about the position and role of the Earth in the Universe.

Hypatia is one of the first woman scientists that we know about. She was an important philosopher of the Neoplatonic school of Alexandria and made important contributions to astronomy and mathematics. Her father and teacher Theon was a professor at the great Library of Alexandria.

Hypathia Attacked by the Monks.
Howard Roberts (1873-77)

Hypatia’s contributions to mathematics and astronomy include collaborations with her father, who wrote extensive “commentaries” about the most important scientific books of his time. Theon studied, analyzed and edited the works of Ptolomy, in particular his book the Almagest. Theon wrote thirteen books of commentaries of the Almagest. The possible contribution of Hypatia is implied in the title of the third book of commentaries: “Commentary by Theon of Alexandria on Book III of Ptolemy's Almagest, edition revised by my daughter Hypatia, the philosopher."

Many sources like Filostorgio, Hesiquio of Alexandria, and Damascius – contemporaries – described the works and talent of Hypatia as superior to her father in scientific work.

In mathematics Theon and Hypatia produced a new version of Euclid’s Elements which has been the reference for all geometries until the 19th century. In addition, Hypatia alone wrote commentaries on the Arithmetica of Diophantus of Alexandria, one of her favorite mathematicians who is known as the father of algebra. Her book allowed Diophantus’ work to be known. Also, she wrote commentaries on the work of Apolonius of Perga who wrote the first treatise on conic sections that are important to studying celestial bodies, and Ptolemy’s works, in addition to her participation in her father’s work. Hypatia, according to her student Synesius, invented the astrolabe and maybe also the hydrometer, still in use today to measure relative densities of liquids.

Hypatia belonged to the Neoplatonist School – modeled after Plato’s Academy - which is based on Pythagorean ideas. In 400 she became the head of this school and taught the philosophies of Plato, Aristotle and Plotinus, and mathematics and astronomy to people from all over the world.

Plato was one of the philosophers who used the Pythagorean ideas as building blocks of his teaching at the Academy. According to Plato, mathematics and political sciences are the pivots of his philosophy. Ideas, for him, are more real than the material world that surrounds us. Hypatia inherited this line of thinking.

Following the Pythagorean principles of gender equality Hypatia’s classes accepted, under the same conditions, men and women; she never separated the students according to religion or gender. Her classes were models of complete ethnic, religious and gender diversity. This fact attracted intellectuals from all over the known world who came to Alexandria to study philosophy, mathematics, astronomy, ethics and religion under her mentoring.

9 comments; last comment on 11/04/2016
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