Principia Mathematica to *56 (Cambridge Mathematical Library) by Alfred North Whitehead (PDF)

Description

The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will wish to refer to the complete edition). It contains the whole of the preliminary sections (which present the authors’ justification of the philosophical standpoint adopted at the outset of their work); the whole of Part 1 (in which the logical properties of propositions, propositional functions, classes and relations are established); section 6 of Part 2 (dealing with unit classes and couples); and Appendices A and B (which give further developments of the argument on the theory of deduction and truth functions).

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⭐The notation of PM is hard to read by anyone who learned logic post 1960, say. The typesetting is archaic. Hundreds of theorems are proved, but it is not clear wherethey all lead. Russell and Whitehead are guilty of a number of major philosophical confusions, such as use and mention, between meta- and object language, and their confused notion of “propositional function.” Their choice of axioms can be much improved upon. The PM theory of types and orders is a complicated horror; Chwistek, Ramsey, and others later showed that it could be radically simplified. R & W think they can substitute the intensional for the extensional, and ultimately define sets and relations in logical terms. PM does not have a clue about model theory or metatheory. There is no hint of proofs of consistency, completeness, categoricity, and Loewenheim-Skolem. In this sense, the fathers of modern logic are Skolem, Goedel, Tarski, and Church. And Goedel did indeed prove that there must exist mathematical truths that cannot be proved true using the axioms of PM, or any other finite set of axioms.But this is still one of the greatest works of mathematics and philosophy of all time. The long prose introduction is a philosophical masterpiece. The collaboration between Russell and Whitehead may be the greatest scientific collaboration in British history. Whitehead, who was trained as a mathematician, went on to become one of the shrewder philosophers of the 20th century, and supervised Quine’s PhD thesis. PM’s treatment of the algebra of relations (a brilliant generalisation of Boolean algebra thathas not received the study it deserves) is perhaps the most thorough ever.Mathematical logic is indeed the abstract structure that underlies the digital electronics revolution. And PM is still perhaps the greatest work of math logic ever penned.

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⭐Bertrand Arthur William Russell (1872-1970) was an influential British philosopher, logician, mathematician, and political activist. In 1950, he was awarded the Nobel Prize in Literature, in recognition of his many books such as

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⭐, etc.Russell (who actually wrote the book) said in the Preface to this 1910 book, “The mathematical treatment of the principles of mathematics, which is the subject of the present work, has arisen from the conjunction of two different studies… On the one hand we have the work of …Cantor and others on such matters as the theory of aggregates. On the other hand we have symbolic logic, which… thanks to Peano and his followers, acquired the technical adaptability and the logical comprehensiveness that are essential to a mathematical instrument for dealing with what have hitherto been the beginnings of mathematics. From the combination of these two studies two results emerge, namely (1) that what were formerly taken … as axioms, are either unnecessary or demonstrable; (2) that the same methods by which supposed axioms are demonstrated will give valuable results in regions… which had formerly been regarded as inaccessible to human knowledge. Hence the scope of mathematics is enlarged both by the addition of new subjects and by a backward extension into provinces hitherto abandoned to philosophy.“The present work was originally intended by us to be comprised in a second volume of

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⭐… But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed: moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have no arrived at what we believe to be satisfactory solutions. It therefore became necessary to make our book independent of ‘The Principles of Mathematics.’ We have, however, avoided both controversy and general philosophy, and made our statements dogmatic in form. The justification for this is that the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics… hence the early deductions … give reasons rather for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises.” (Pg. v)He continues, “A very large part of the labor involved in writing the present work has been expended on the contradictions and paradoxes which have infected logic and the theory of aggregates. We have examined a great number of hypotheses for dealing with these contradictions; many such hypotheses have been advanced by others, and about as many have been advanced by ourselves… it gradually became evident to us that some form of the doctrine of types must be adopted if the contradictions were to be avoided. The particular form of the doctrine of types advocated in the present work is not logically indispensable… We have particularized, both because the form of the doctrine which we advocate appears to us the most probable, and because it was necessary to give at least one perfectly definite theory which avoids the contradictions. But hardly anything in our book would be changed by the adoption of a different form of the doctrine of types.” (Pg. vii)He notes, “The symbolic form of the work has been forced upon us by necessity: without its help we should have been unable to perform the requisite reasoning. It has been developed as the result of actual practice, and is not an excrescence introduced for the mere purpose of exposition.” (Pg. vii)The Introduction states, “The mathematical logic which occupies Part I of the present work… aims at … diminishing to the utmost the number of the undefined ideas and undemonstrated propositions… from which it starts… the system is specially framed to solve the paradoxes which, in recent years, have troubled students of symbolic logic and the theory of aggregates; it is believed that the theory of types, as set forth in what follows, leads both to the avoidance of contradictions, and to the detection of the precise fallacy which has given rise to them.” (Pg. 1)Russell explains his Theory of Logical Types: “An analysis of the paradoxes to be avoided shows that they all result from a certain kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. Thus, for example, the collection of PROPOSITIONS will be supposed to contain a proposition stating that ‘all propositions are either true or false.’ It would seem, however, that such a statement could not be legitimate unless ‘all propositions’ referred to some already definite collection, which it cannot do it new propositions are created by statements about ‘all propositions.’ We shall, therefore, have to say that statements about ‘all propositions’ are meaningless. More generally, given any set of objects such that, if we suppose the set to have a total, it will contain members which presuppose this total, then such a set cannot have a total. By saying that a set has ‘no total,’ we mean, primarily, that no significant statement can be made about ‘all its members.’ … In such cases, it is necessary to break up our set into smaller sets, each of which is capable of a total. This is what the theory of types aims at effecting…” (Pg. 37)He suggests, “That the axiom of reducibility is self-evident is a proposition which can hardly be maintained. But in fact self-evidence is never more than a part of the reason for accepting an axiom, and is never indispensable. The reason for accepting an axiom as for accepting any other proposition, is always largely inductive, namely that many propositions which are nearly indubitable can be deduced from it, and that no equally plausible way is known by these propositions could be true if the axiom were false, and nothing which is probably false can be deduced from it. If the axiom is apparently self-evident, that only means, practically, that it is nearly indubitable; for things have been thought to be self-evident and have yet turned out to be false.” (Pg. 59)Russell addresses a famous paradox that he discovered in Frege’s work: “In order to solve the contradiction about the class of classes which are not members of themselves, we shall assume… that a proposition about a class is always to be reduced to a statement about a function which defines the class, i.e., a function which is satisfied by the members of the class and by no other arguments… Hence a class cannot, by the vicious-circle principle, significantly to be the argument to its defining function… Hence a class neither satisfies nor does not satisfy its defining function, and therefore… is neither a member of itself nor not a member of itself. This is an immediate consequence of the limitation to the possible arguments to a function which was explained [earlier]… Hence the contradiction which results from supposing that there is such a class disappears.” (Pg. 62-63)Those who are interested in mathematical theory and symbolic logic may find this book “absolutely ESSENTIAL reading”; but others may nevertheless find certain “gems” contained in its narrative sections.

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⭐This is one of the silliest mathematical books ever to have been written. Poincare gave the perfect put down, “Logic is sterile but mathematics is the most fertile of mothers”. Proving that 2 + 2 = 4 imagine that! ABSURDUM NATURALIUM RERUM REVERITATEM PER FALSAS CAUSAS DEMONSTRARE — a maxim from the Renaissance writer Ramus which means “It is ridiculous to demonstrate by false reasons the truth of natural things”. from Sebastian Hayes

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