The Principles of Mathematics
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The Principles of Mathematics (PoM) is a book written by Bertrand Russell in 1903. In it he presented his famous paradox and argued his thesis that mathematics and logic are identical.[1]
The book presents a view of the foundations of mathematics and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others.
In 1905 Louis Couturat published a partial French translation[2] that expanded the book's readership. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were printed in 1938, 1951, 1996, and 2009.
The Principles of Mathematics
consists of 59 chapters divided into seven parts: indefinables in
mathematics, number, quantity, order, infinity and continuity, space,
matter and motion.
Title page of first edition
| |
| Author | Bertrand Russell |
|---|---|
| Translator | Louis Couturat |
| Country | United Kingdom |
| Language | English |
| Series | I. (all published.) |
| Subject | Mathematics, Logic |
| Genre | Foundations of mathematics, Symbolic logic |
| Publisher | Cambridge University Press |
Publication date
| 1903, 1938, 1951, 1996, and 2009 |
| Pages | 534 (first edition) |
| ISBN | 978-1-313-30597-6 Paperback edition |
| OCLC | 1192386 |
| Website | http://fair-use.org/bertrand-russell/the-principles-of-mathematics/ |
The book presents a view of the foundations of mathematics and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others.
In 1905 Louis Couturat published a partial French translation[2] that expanded the book's readership. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were printed in 1938, 1951, 1996, and 2009.
Contents
Contents
In chapter one, "Definition of Pure Mathematics", Russell asserts that :
There is an anticipation of relativity physics in the final part as the last three chapters consider Newton's laws of motion, absolute and relative motion, and Hertz's dynamics. However, Russell rejects what he calls "the relational theory", and says on page 489 :The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.[3]
- For us, since absolute space and time have been admitted, there is no need to avoid absolute motion, and indeed no possibility of doing so.
Early reviews
Reviews were prepared by G. E. Moore and Charles Sanders Peirce, but Moore's was never published[5] and that of Peirce was brief and somewhat dismissive. He indicated that he thought it unoriginal, saying that the book "can hardly be called literature" and "Whoever wishes a convenient introduction to the remarkable researches into the logic of mathematics that have been made during the last sixty years [...] will do well to take up this book."[6]G. H. Hardy wrote a favorable review[4] expecting the book to appeal more to philosophers than mathematicians. But he says :
- [I]n spite of its five hundred pages the book is much too short. Many chapters dealing with important questions are compressed into five or six pages, and in some places, especially in the most avowedly controversial parts, the argument is almost too condensed to follow. And the philosopher who attempts to read the book will be especially puzzled by the constant presupposition of a whole philosophical system utterly unlike any of those usually accepted.
Second edition
In 1938 the book was re-issued with a new preface by Russell. This preface was interpreted as a retreat from the realism of the first edition and a turn toward nominalist philosophy of symbolic logic. James Feibleman, an admirer of the book, thought Russell’s new preface went too far into nominalism so he wrote a rebuttal to this introduction.[7] Feibleman says, "It is the first comprehensive treatise on symbolic logic to be written in English; and it gives to that system of logic a realistic interpretation."Later reviews
In 1959 Russell wrote My Philosophical Development, in which he recalled the impetus to write the Principles:- It was at the International Congress of Philosophy in Paris in the year 1900 that I became aware of the importance of logical reform for the philosophy of mathematics. ... I was impressed by the fact that, in every discussion, [Peano] showed more precision and more logical rigour than was shown by anybody else. ... It was [Peano's works] that gave the impetus to my own views on the principles of mathematics.[8]
- The Principles of Mathematics, which I finished on 23 May 1902, turned out to be a crude and rather immature draft of the subsequent work [Principia Mathematica], from which, however, it differed in containing controversy with other philosophies of mathematics.[9]
- The Principles inaugurated contemporary philosophy. Other works have won and lost the title. Such is not the case with this one. It is serious, and its wealth perseveres. Furthermore, in relation to it, in a deliberate fashion or not, it locates itself again today in the eyes of all those that believe that contemporary science has modified our representation of the universe and through this representation, our relation to ourselves and to others.[10]
- Peano's symbolic notation took Russell by storm in 1900, but Russell’s Principles was still in unrelieved prose. I was inspired by its profundity [in 1928] and baffled by its frequent opacity. In part it was rough going because of the cumbersomeness of ordinary language as compared with the suppleness of a notation especially devised for these intricate themes. Rereading it years later, I discovered that it had been rough going also because matters were unclear in Russell's own mind in those pioneer days.
Ivor Grattan-Guinness made an in-depth study of Principles. First he published Dear Russell – Dear Jourdain (1977)[13] which included correspondence with Philip Jourdain who promulgated some of the book’s ideas. Then in 2000 Grattan-Guinness published The Search for Mathematical Roots 1870 – 1940 which considered the author’s circumstances, the book’s composition and its shortcomings.[14]
In 2006, Philip Ehrlich challenged the validity of Russell's analysis of infinitesimals in the Leibniz tradition.[15] A recent study documents the non-sequiturs in Russell's critique of the infinitesimals of Gottfried Leibniz and Hermann Cohen.[16]
See also
Notes
The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify.The quotation is from the first page of Russell's introduction to the second (1938) edition.
- Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y.
References
- Stefan Andersson (1994). In Quest of Certainty: Bertrand Russell's Search for Certainty in Religion and Mathematics Up to The Principles of Mathematics. Stockholm: Almquist & Wiksell. ISBN 91-22-01607-4.
External links
- The Principles of Mathematics – Free searchable full text versions in PDF, ePub and HTML formats
- The Principles of Mathematics – Online text (scan of original)
- The Principles of Mathematics – Full text at the Internet Archive
"The number 3 multiplies itself through the system as a perfect square. It bounces from position 3, to 6, to 9, to 12. All multiples of 3 are found in these positions." N. Tesla
1 1 2 3 5 8 13(1+3) 4(3+4) 7(4+7) 11(1+1) 2(1+2) 3(2+3) 5(3+5) 8(5+8) 13(1+3) 4(3+4) 7(4+7) 11(1+1) 2(1+2) 3(2+3) 5(3+5) 8(5+8) 13(1+3) 4(3+4) 7(4+7) 11(1+1) 2(1+2) 3(2+3) 5(3+5) 8(5+8) 13(1+3) 4(3+4) 7(4+7) 11
https://anindependentmindknotlogic.blogspot.com/2018/11/fibonacci-sequence-changes-and-pattern.html?zx=b51d2cb1ded4fc42
Figure 1: 7 + 11 + 2 = 20(2+0) 2 = 2.0
20(2x0) 0 = 0.0
Figure 2: 7 x 11 x 2 = 154(1+5+4) 10 = 1.0
Figure 3: π(1+1+1) = 3 = 3.141
"11 is the top left prime position. It cascades out to the left and circles back around the system." N. Tesla
Figure 5: 11(1+1) 2
Genesis 7:9 KJV: There went in two and two unto Noah into the ark
Genesis 7:12 KJV: And the rain was upon the earth forty days and forty nights.
1 John 5:7 KJV: For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one.
Hebrews 6:1 KJV: Therefore leaving the principles of the doctrine of Christ, let us go on unto perfection; not laying again the foundation of repentance from dead works, and of faith toward God
What is mathematics but timing, it is tone, in tone the timing would be the number itself, a sort of address.
If we were to look at Genesis as a record of science instead of missing this piece or look at N. Tesla's words as the missing piece or just say 1,2,3 equals for, would that increase the relativity to our existence by extending time and not multiplying problems? Why is Noah in the Book of Genesis when Adam and Eve are meant to explain everything? Is that the missing parallel line that we should cross, a line that we'll evolve a little bit more in our understanding and possibly find the figure is the missing piece:
As this identifies the intersection from which different angles can cross!! The bibical viewers must admit that three in one is pi as the number π demonstrates I I I in its signature and it is one letter/number.
The Principles of Mathematics consists of 59 chapters divided into seven parts: indefinables in mathematics, number, quantity, order, infinity and continuity, space, matter and motion.
In chapter one, "Definition of Pure Mathematics", Russell asserts that :
"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself."
The Principles of Mathematics - Wikipedia
https://en.wikipedia.org/wiki/The_Principles_of_Mathematics
de·nom·i·na·tor
/dəˈnäməˌnādər/
noun
Mathematics
noun: denominator; plural noun: denominators
- the number below the line in a common fraction; a divisor.
- a figure representing the total population in terms of which statistical values are expressed.
Electric arc
An electric arc between two nails
Techniques for arc suppression can be used to reduce the duration or likelihood of arc formation.
In the late 1800s, electric arc lighting was in wide use for public lighting. Some low-pressure electric arcs are used in many applications. For example, fluorescent tubes, mercury, sodium, and metal-halide lamps are used for lighting; xenon arc lamps have been used for movie projectors.
Contents
History
Natural Lightning is now considered an electric spark, not electric arc.
The first continuous arc was discovered independently in 1802 and described in 1803[6] as a "special fluid with electrical properties", by Vasily V. Petrov, a Russian scientist experimenting with a copper-zinc battery consisting of 4200 discs.[6][7]
In the late nineteenth century, electric arc lighting was in wide use for public lighting. The tendency of electric arcs to flicker and hiss was a major problem. In 1895, Hertha Marks Ayrton wrote a series of articles for the Electrician, explaining that these phenomena were the result of oxygen coming into contact with the carbon rods used to create the arc. In 1899, she was the first woman ever to read her own paper before the Institution of Electrical Engineers (IEE). Her paper was entitled "The Hissing of the Electric Arc". Shortly thereafter, Ayrton was elected the first female member of the IEE; the next woman to be admitted to the IEE was in 1958.[8] She petitioned to present a paper before the Royal Society, but she was not allowed because of her sex, and "The Mechanism of the Electric Arc" was read by John Perry in her stead in 1901.
Overview
Electric arcs between the power line and pantographs of an electric train after catenary icing
Electricity arcs between the power rail and electrical pickup "shoe" on a London Underground train
An arc between two electrodes can be initiated by ionization and glow discharge, when the current through the electrodes is increased. The breakdown voltage of the electrode gap is a combined function of the pressure, distance between electrodes and type of gas surrounding the electrodes. When an arc starts, its terminal voltage is much less than a glow discharge, and current is higher. An arc in gases near atmospheric pressure is characterized by visible light emission, high current density, and high temperature. An arc is distinguished from a glow discharge partly by the approximately equal effective temperatures of both electrons and positive ions; but in a glow discharge, ions have much less thermal energy than the electrons.
A drawn arc can be initiated by two electrodes initially in contact and drawn apart; this can initiate an arc without the high-voltage glow discharge. This is the way a welder starts to weld a joint, momentarily touching the welding electrode against the workpiece then withdrawing it till a stable arc is formed. Another example is separation of electrical contacts in switches, relays or circuit breakers; in high-energy circuits arc suppression may be required to prevent damage to contacts.[9]
Electrical resistance along the continuous electric arc creates heat, which ionizes more gas molecules (where the degree of ionization is determined by temperature), and as per this sequence: solid-liquid-gas-plasma; the gas is gradually turned into a thermal plasma. A thermal plasma is in thermal equilibrium; the temperature is relatively homogeneous throughout the atoms, molecules, ions, and electrons. The energy given to electrons is dispersed rapidly to the heavier particles by elastic collisions, due to their great mobility and large numbers.
Current in the arc is sustained by thermionic emission and field emission of electrons at the cathode. The current may be concentrated in a very small hot spot on the cathode; current densities on the order of one million amperes per square centimeter can be found. Unlike a glow discharge, an arc has little discernible structure, since the positive column is quite bright and extends nearly to the electrodes on both ends. The cathode fall and anode fall of a few volts occur within a fraction of a millimeter of each electrode. The positive column has a lower voltage gradient and may be absent in very short arcs.[9]
A low-frequency (less than 100 Hz) alternating current arc resembles a direct current arc; on each cycle, the arc is initiated by breakdown, and the electrodes interchange roles, as anode or cathode, when current reverses. As the frequency of the current increases, there is not enough time for all ionization to disperse on each half cycle, and the breakdown is no longer needed to sustain the arc; the voltage vs. current characteristic becomes more nearly ohmic.[9]
Electric arc between strands of wire.
An electric arc has a non-linear relationship between current and voltage. Once the arc is established (either by progression from a glow discharge[10] or by momentarily touching the electrodes then separating them), increased current results in a lower voltage between the arc terminals. This negative resistance effect requires that some positive form of impedance (as an electrical ballast) be placed in the circuit to maintain a stable arc. This property is the reason uncontrolled electrical arcs in apparatus become so destructive since once initiated, an arc will draw more and more current from a fixed-voltage supply until the apparatus is destroyed.
Uses
An electric arc can melt calcium oxide
Carbon arc lights were the first electric lights. They were used for street lights in the 19th century and for specialized applications such as searchlights until World War 2. Today, low-pressure electric arcs are used in many applications. For example, fluorescent tubes, mercury, sodium, and metal halide lamps are used for lighting; xenon arc lamps are used for movie projectors.
Formation of an intense electric arc, similar to a small-scale arc flash, is the foundation of exploding-bridgewire detonators.
A major remaining application is in high voltage switchgear for high-voltage transmission networks. Modern devices use sulphur hexafluoride at high pressure in a nozzle flow between separated electrodes within a pressure vessel. The AC fault current is interrupted at current zero by the highly electronegative SF6 ions absorbing free electrons from the decaying plasma. A similar air-based technology has largely been replaced because many noisy units in series were required to prevent the current re-igniting under similar supergrid conditions.
Electric arcs have been studied for electric propulsion of spacecraft.
Guiding the arc
Scientists have discovered a method to control the path of an arc between two electrodes by firing laser beams at the gas between the electrodes. The gas becomes a plasma and guides the arc. By constructing the plasma path between the electrodes with different laser beams, the arc can be formed into curved and S-shaped paths. The arc could also hit an obstacle and reform on the other side of the obstacle. The laser-guided arc technology could be useful in applications to deliver a spark of electricity to a precise spot.[11][12]Undesired arcing
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Undesired arcing in electrical contacts of contactors, relays and switches can be reduced by devices such as contact arc suppressors[13] and RC Snubbers or through techniques including:
- immersion in transformer oil, dielectric gas or vacuum
- arc chutes
- magnetic blowouts
- pneumatic blowouts
- sacrificial ("arcing") contacts
- damping materials to absorb arc energy, either thermally or through chemical decomposition
An electric arc over the surface of plastics causes their degradation. A conductive carbon-rich track tends to form in the arc path, called "carbon tracking", negatively influencing their insulation properties. The arc susceptibility, or "track resistance", is tested according to ASTM D495, by point electrodes and continuous and intermittent arcs; it is measured in seconds required to form a track that is conductive under high-voltage low-current conditions.[14] Some materials are less susceptible to degradation than others. For example, polytetrafluoroethylene has arc resistance of about 200 seconds (3.3 minutes). From thermosetting plastics, alkyds and melamine resins are better than phenolic resins. Polyethylenes have arc resistance of about 150 seconds; polystyrenes and polyvinyl chlorides have relatively low resistance of about 70 seconds. Plastics can be formulated to emit gases with arc-extinguishing properties; these are known as arc-extinguishing plastics.[15]
Arcing over some types of printed circuit boards, possibly due to cracks of the traces or the failure of a solder, renders the affected insulating layer conductive as the dielectric is combusted due to the high temperatures involved. This conductivity prolongs the arcing due to cascading failure of the surface.
Arc suppression
Arc suppression is a method of attempting to reduce or eliminate an electrical arc. There are several possible areas of use of arc suppression methods, among them metal film deposition and sputtering, arc flash protection, electrostatic processes where electrical arcs are not desired (such as powder painting, air purification, PVDF film poling) and contact current arc suppression. In industrial, military and consumer electronic design, the latter method generally applies to devices such as electromechanical power switches, relays and contactors. In this context, arc suppression uses contact protection.Part of the energy of an electrical arc forms new chemical compounds from the air surrounding the arc: these include oxides of nitrogen and ozone, the second of which can be detected by its distinctive sharp smell. These chemicals can be produced by high-power contacts in relays and motor commutators, and they are corrosive to nearby metal surfaces. Arcing also erodes the surfaces of the contacts, wearing them down and creating high contact resistance when closed.[16]
See also
References
- "Lab Note #106 Environmental Impact of Arc Suppression". Arc Suppression Technologies. April 2011. Retrieved October 10, 2011.
External links
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Wikimedia Commons has media related to Electric arc. |
- "High Voltage Arcs and Sparks" Videos of 230 kV 3-phase "Jacobs Ladder" and unintentional 500 kV power arc
- High Voltage Arc Gap Calculator to calculate the length of an arc knowing the voltage or vice versa
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