Get your own Free Home PageThis stuff, like quantum theory, is not just an obscure argument between European intellectuals, as any resident of HIroshima can tell you.
"With the appearance of Russel's paradox, the Thing that had been waiting to strike struck. The very foundations of mathematics were infected, the attempt to derive arithmetic from an intuitive and plausible form of logic doomed." The Advent of the Algorithm" by David Berlinski
Bertrand Russell himself has written a good introduction to this subject in 1919.
From Russell's Introduction to Mathematical Philosophy:
"Very few people are prepared with a definition of what is meant by "number," or "0," or "1." ...It was believed until recently that that some, at least, of these first notions of arithmetic must be accepted as too simple and primitive to be defined. Since all term that are defined are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order to have a starting point for its definitions. It is not clear that there must be terms which are *incapable* of definition: it is possible that, however far back we go in defining, we always *might* go further still. On the other hand, it is also possible that, when analysis has been pushed far enough, we can reach terms that really are simple, and therefore logically incapable of the sort of definition that consists in analysing." [1]
Hilbert's program resulted in Gödel discovering a flaw in Russell's reasoning, but first, more of Russell:
"Having reduced all traditional pure mathematics to the theory of natural numbers, the next step in logical analysis was to reduce this theory itself to the smallest set of premises and undefined terms from which it could be derived. This work was accomplished by Peano. He showed that the entire theory of the natural numbers could be derived from three primitive propositions, in addition to those of pure logic."
"The three primitive ideas in Peano's arithmetic are: 0, number, successor"
"
The five primitive propositions which Peano assumes are:
(1) 0 is a number
(2) The successor of any number is a number
(3) No two numbers have the same successor
(4) 0 is not the successor of any number
(5) Any propert which belongs to 0, and also to the successor of
every number which has the property, belongs to all numbers."
"The last of these is the principle of mathematical induction."
"It is time now to turn to the considerations which make it necessary to advance beyond the standpoint of Peano, who represents the last perfection of the 'arithmetisation' of mathematics, to that of Frege, who first succedes in 'logicising' mathematics, i.e. in reducing the logic of arithmetical notions which his predecessors had shown to be sufficient for mathematics."[1]
The main question to me has been, lately, is logic more basic than mathematics, or vice versa, or is analysis more fundamental than either. Swinton suggest that they "bootstrap".
"We shall no lay down the following definition:"
"The 'natural numbers' are the posterity of 0 with respect to the relation 'immediate predecessor' (which is the converse of 'successor')"
"We have thus arrived at a definition of one of Peano's three primitive ideas in terms of the other two. As a result of this definition, two of his primitive propositions - namely the one asserting that o is a number and the one asserting mathematical induction - become innecessary, since they result from the definition. The one asserting that the successor of a natural numner is a natural number is only needed in the weakened form 'every natural number has a successor' ".
"The number 0 is the number of terms in a class which has no members."
"Thus we have the following purely logical definition:"
"o is a class whose only member is the null class."
"The successor of the number of terms in the class a is the number of terms in the class consisting of a together with x, where x is any term not belonging to the class."
"...we have already given...a logical definition of the number of terms in a class, namely, we defined it as the set of all classes that are similar to a given class."[1]
Russell reduces all of mathematics to logic:
"We have thus reduced Peano's three primitive ideas to ideas of logic: we have given definitions of them which make them definite, no longer capable of an infinity of different meanings, as they were when they were only determinate to the extent of obeying Peano's five axioms."
"Assuming the number of individuals in the universe is not finite, we have now succeded not only in defining Peano's three primitive ideas, but in seeing how to proce his five primitive propositions, by means of primitive ideas and propositions belonging to logic. It follows that all pure mathematics, in so far as it is deducible from the theory of natural numbers, is only a prolongation of logic. The extension of this result to those modern branches of mathematics which are not deducible from the theory of natural numbers offers no difficulty of principle..."[1]
"The Advent of the Algorithm" by David Berlinski discusses the difference between Aristotle's logic and formal systems:
"Liebniz enlarged the margins of the Aristotelian system, and two centuries later, so did the English logicians George Boole, Augustus De Morgan, and John Venn...It is modern logic that is the real stuff, and this real stuff is almost entirely the creation of Gottlob Frege, the gnome of logic."(2, pg 48)
"An axiomatic system establishes a reverberating relationship between what a mathematician assumes (the axioms) and what he or she can derive (the theorems). In the best of circumstances the relationship is clear enough so that the mathematicain can submit his or her reasoning to an informal checklist..."
"It is within the context of a formal system that the checklist itself is absorbed into the structure of the system..."
"The propositional calculus is the simples imaginable formal system, and as its name suggests, is a system in which whole propositions, (or sentences) come to traffic with one another."(2, pg 50)
Berlinski writes about the flaw in Russel and Whitehead's "Principia":
"With the appearance of Russel's paradox, the Thing that had been waiting to strike struck. The very foundations of mathematics were infected, the attempt to derive arithmetic from an intuitive and plausible form of logic doomed. Russell communicated his paradox to Frege in 1903, and thereafter, the mathematicians of Europe may be seen lifting their hands to their foreheads in a gesture of delicate dismay. To everyone save for the disgruntled few who had mocked the subject form the start - Leopold Kronecker, most obviously, who may be heard snickering even now - set theory seemed at once too simple and too deep a disipline to be discarded. Like electrons in particle physics, sets are simple structures, indefinable in terms of anything simpler; like quantum electrodynamics, set theory represents a rich, intriguing and beautifull body if insights...But Russell's paradox belonged to a malignant pride of paradoxes, all ofthem similar in nature, and together they cut very deep into the mathematician's collective confidence...."[2]
Russell's paradox is usually stated: "Does the set of all sets that do not contain themselves contain itself?"
The Hilbert program was still in place, and mathematicians working on it were able to get around Russell's paradox by a means so powerfull that the computer was invented. Turing invented the computer as a thought experiment to decide the computability problem that Gödel brought in with his "incompletness theorem". Turing then actually built a computer to decode the Nazi Enigma machine's secret codes during WWII. This stuff, like quantum theory, is not just an obscure argument between European intellectuals, as any resident of HIroshima can tell you.
"If achievement were any guide, HIlbert's name would be as well known as Einstein's, and among mathematicians it is. There are infinite dimensional Hilbert spaces in quantum physics, where they play an indispensable architectural role, and Hilbert bases, Hilbert invariants, and Hilbert integrals in mathematics itself. Hilbert revolutioized elementary geometry and unified analytic number theory...."[2]
Hilbert's program was the starting point for Gödel 's theorem. Hilbert's viewpoint was of metamathematics, and he suggested that a formal system should be consistent, complete, and decidable.
"In the autum of 1931, Gödel published a paper of twenty five pages with the title 'On Formally Undecidable Propositions of Principia Mathematica and related systems.' The reference to Russell's principia is, in fact, a diversion; Gödel 's paper is really aboutany axiomatic system in which the natural numbers may be described. It is thus a paper concerned with the oldest of mathematical ideas, the system of whole numbers."
"Within the compass of twenty five pages Gödel established that arithmetic is incomplete, the Hilbert program doomed. What is more, he illustrated that the consistency of arithmetic cannot be demonstrated by means of reasoning that is as simple as arithmetic itself. Freedom from contradiction is purchased only by systems whose own freedom from contradiction is problematic."
"And in proving this, he also brought about the advent of the algorithm, giving, for the first time, a precise mathematical description of an old but hidden idea." (pg 118, Advent of the Algorithm)
The computer is a good example of that idea. Von Neumann worked with Turing at The Institute for Advanced study in Princeton, during WWII, and he knew Gödel and Hilbert. It is because of military secrecy that Von Neumann got all of the credit for the computer after the war.
A web page by Phillip Cheng has this anecdote on Gödel
"According to Princeton legend, for his oral examination for U.S. citizenship, Gödel studied the Constitution intensely, as if it were a document of theoretical logic. Just before the interview, Gödel confided to his friend Albert Einstein that he had discovered that it was logically possible within the limits of the Constitution for the United States to be turned into a dictatorship. Einstein wisely advised Gödel not to mention this "discovery" at his citizenship interview. "
[1] Russell's Introduction to Mathematical Philosophy,1919
[2] Berlinski The Advent to the Algorithm, the 300 year Journey from an Idea to the Computer,2000
The Columbia Encyclopedia, Sixth Edition. 2001 Berttrand Russell, 3rd Earl World War I had a crucial effect on Russell: until that time he had thought of himself as a philosopher and mathematician. Although he had already embraced pacifism, it was in reaction to the war that he became passionately concerned with social issues. His active pacifism at the time of the war inspired public resentment, caused him to be dismissed from Cambridge, attacked by former associates, and fined by the government (which confiscated and sold his library when he refused to pay), and led finally to a six-month imprisonment in 1918. From 1916 until the late 1930s, Russell held no academic position and supported himself mainly by writing and by public lecturing. In 1927 he and his wife, Dora, founded the experimental Beacon Hill School, which influenced the development of other schools in Britain and America. http://www.bartleby.com/65/ru/RusslBer.html Russell's godfather John Stuart Mill : His "A System of Logic" (1843) was followed in 1848 by the "Principles of Political Economy", which influenced English radical thought. In 1851, following the death of her husband, he married Harriet Taylor, whom he had loved for 20 years. She died in 1858, and Mill, profoundly affected, dedicated to her the famous "On Liberty" (1859), on which they had worked together. In 1863, Utilitarianism was published, and his Auguste Comte and Positivism appeared in 1865. From 1865 to 1868 he served as a member of Parliament, after which he retired, spending much of his time at Avignon, France, where his wife was buried and where he died. In the year of his death appeared his celebrated Autobiography. John Stuart Mill’s philosophy followed the doctrines of his father and Jeremy Bentham, but he sought to temper them with humanitarianism. At times Mill came close to socialism, a theory repugnant to his predecessors. In logic he formulated rules for the inductive process and stressed the method of empiricism as the source of all knowledge. In his ethics he pointed out the possibility of a sentiment of unity and solidarity that may even develop a religious character, as in Comte’s religion of humanity. http://www.bartleby.com/65/mi/Mill-JS.html _____________________________________________________________ Alfred North Whitehead Whitehead’s distinction rests upon his contributions to mathematics and logic, the philosophy of science, and the study of metaphysics. In the field of mathematics Whitehead extended the range of algebraic procedures and, in collaboration with Bertrand Russell, wrote "Principia Mathematica" (3 vol., 1910–13), a landmark in the study of logic. His inquiries into the structure of science provided the background for his metaphysical writings. He criticized traditional categories of philosophy for their failure to convey the essential interrelation of matter, space, and time. For this reason he invented a special vocabulary to communicate his concept of reality, which he called the philosophy of organism. http://www.bartleby.com/65/wh/WhitehdAN.html ___________________________________________________________________-Noesis.the Mega Society magazine has some articles on metamathematics
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