When was principia written

ZF and PM cannot simply be compared in terms of their theorems. Not only are there different axioms in the two theories, but the very languages in which they are expressed differ in logical power. As Quine remarks in his study of the logic of Whitehead and Russell, it would seem that after a certain point the body of PM makes use of extensional higher-order logic in a simple theory of types:.

In any case there are no specific attributes [propositional functions] that can be proved in Principia to be true of just the same things and yet to differ from one another. The theory of attributes receives no application, therefore, for which the theory of classes would not have served. Once classes have been introduced, attributes are scarcely mentioned again in the course of the three volumes.

Quine here hints at the view of PM that is widely shared among mathematical logicians, who see the ramified theory of types, with its accompanying Axiom or Reducibility, as a digression taking logic into a realm of obscure intensional notions, when instead logic, even if expressed in a theory of types, is extensional and is comparable to axiomatic set theory presented with a simple hierarchy of sets of individuals, sets of sets individuals, and so on.

It is certainly true that the the remainder of PM is devoted to the theory of individuals, classes, and relations in extension between those entities. Thus the ontology of these later portions is a hierarchy of predicative functions arranged in a simple theory of types. This has led one interpreter, Gregory Landini , to argue that only predicative functions are values of bound variables in PM.

The only bound variables in PM, he asserts, range over predicative functions. This is a strong version of a view that others such as Kanamori have expressed, going back to Ramsey , namely that the introduction of the Axiom of Reducibility has the effect of undoing the ramification of the theory of types, at least for a theory of classes, and so a higher-order logic used for the foundations of mathematics ought to have only a simple type structure.

In the summary of the later sections of PM that follows below, it will appear that in fact the symbolic development follows very closely that of PoM from ten years earlier. To remind the reader of the change from talking of propositional functions to relations in extension, two further notational alterations are introduced. The obvious limitation of this notation is that it is not readily extended to three place relations, adding a third variable, say z. The notions of the subset relation and the intersection and union of sets are defined in PM exactly as they are now albeit with different terminology.

The complement of a set of a given type is the set of all entities of that type that are not in the set. There is no class of all classes of whatever type.

This is in common with axiomatic set theory which holds that there is no set of all sets. If there is a binary relation which has a unique second argument for each first argument, i.

The definition of a monadic functional term then is:. The diligent reader will find that this presentation does not follow PM exactly. The practice of reading the argument of a relational function as the x and the value as the y is so well established that we have taken a liberty with the actual definitions in PM. A series of notions are defined in a way quite familiar to the modern treatment of relations as sets of n -tuples:.

The notions of the domain , range , and field of a relation are also given a contemporary definition and so also the notions of the domain , range and field of a function. Note that it is possible that a relation can have its domain in one type and range in another.

This adds complications in the theory of cardinal numbers when a relation of similarity equinumerousity holds between classes of different types. In his survey of PM, Quine complains that this last pages of Part I is occupied with proving theorems relating redundant definitions of the same notions. Thus PM defines the notion of domain and range and then introduces notions that again define the same classes, which are proved to be equivalent.

PM, In contemporary logic with the notation of set theory used above, there is no need for a special symbol for this notion, as it is written as:. So the cardinal number 1 is the class of all singletons. There will be a different number 1 for each type of x.

Frege, by contrast, defines the natural number 1 as the extension of a certain concept, namely being identical with the number 0, which itself is the extension of the empty concept of not being self identical. This construction is named the von Neumann ordinals. Similarly, the number 2 is the class of all pairs, rather than a particular pair.

In the type theory of PM there will be distinct couples for the types of y and x. Even with homogenous pairs there will be distinct classes of pairs for each type, and thus a different number 2 for each type. The same notion applies to relations. It is a relation in extension, which is the analogue of a property in extension or class. A relation in extension has a distinction between the first and second elements due to the order of the defining relation.

The closest in contemporary language would be:. Given the definition of extensions of relations this is the version of the no-classes theory for relations. After attending classes of Russell the year before, and having several discussions, Norbert Wiener proposed the following definition in modern notation :.

This section is little used in Volume I. The special consequences for this notion when dealing with relative types of cardinal numbers is the topic of the Preface to Volume II, which was added after the first volume was already in print.

The delay due to working out these details partially explains the three year gap between the publication of Volume I in , and the remaining volumes II and III in Difficulties arise with respect to the definition of cardinal numbers when the relation of similarity they involve is one that has a domain and range in different types. The proof here explicitly follows the proof by Ernst Zermelo from The one-one mapping is constructed in stages.

But some elements in the range of S will have already been mapped by R. See Hinkis for a history of the many different proofs of this theorem. The writing of this preface delayed the publication of the second volume of PM, as Whitehead and Russell struggled over the complications it raised.

The difficulties arise from the typical ambiguity of terms and formulas of the theory of types. Without assuming the Axiom of Infinity for individuals, there is no guarantee that a given constant designates a non-empty class in a given type. The notions of cardinal numbers are developed in full generality, extending to infinite cardinals.

The Summary to section A introduces the notion of homogenous cardinals , which are classes of similar classes whose members are all of the same type. Cardinal Numbers are classes of equinumerous similar classes. We can add a notion of the number of a class to allow for a direct comparison with Frege:.

Landini argues that this section of PM is confused. This qualification is hidden in PM by the use of expressions for functional relations that are sometimes undefined.

It is more recognizable to contemporary set theory by the equivalent definition subject to the same exception when :. For the finite natural numbers, special notions need to be defined first.

For the proof of the Peano Postulates it is necessary not only to define 0, but also the notion of successor. For Frege the notion of weak predecessor of a number is defined, thus 0 and 1 are the predecessors of 1, while 0, 1 and 2 are the predecessors of 2, etc.

The successor of n is then defined by counting the predecessors of a number, in terms of the definition of number, it is the number of the class of predecessors.

This definition would not work for PM, where each number would be of a higher type, as it is defined as a set containing that number. There will in fact be natural numbers for each type, thus a set of all pairs of individuals of type 0, a set of pairs of sets of type 1, etc.

There is no one type, however, at which there are all of the natural numbers sets of equinumerous sets of that type without an assumption that there are infinitely many members of some one type. The solution in PM is to guarantee that for each finite set of n individuals of type 0, there will be some object not in that set, which can be included in the set defining the successor. That such a new individual can be found is guaranteed by the Axiom of Infinity, which in effect asserts the existence of distinct individuals of any finite number.

Instead it is an additional hypothesis, to be used as an antecedent to mathematical assertions upon which it depends. This Axiom of Infinity asserts that all inductive cardinals are non-empty. The results are not proved separately, but as they appear in a development of various results about natural numbers.

The use of recursive definitions is justified by a theorem proving that they describe a unique function. At this point, after pages in Volume II, the reader will see how to compare the logicist reduction of arithmetic in PM with rival accounts of Frege and of contemporary set theory. Frege completes his development of the natural numbers at page 68 of the Volume II of his Basic Laws of Arithmetic published in , which follows the pages of Volume I that had been published in So both Frege and the authors of PM took great pains to prove more advanced theorems only after a chain of closely argued lemmas based on their own formalized symbolic logic.

Frege ends his deductions of the laws of arithemtic with results about the notions of 0, Successor, and the principle of Induction which include the Peano Axioms. He does not consider arithmetical functions, such as addition or multiplication, and thus does not define the successor of a number n as the result of adding 1 to n.

Indeed the analysis of identity sentences is the starting point of his introduction of the theory of sense and reference in , yet Frege does not diverge from his project enough to show how such an identity would be proved. Frege also does not construct the general theory of the arithmetic of cardinal and ordinal numbers that occupies PM for much of Part III. Admittedly the system of PM is an indirect and cumbersome system to develop if the theory of Arithmetic were the only goal in mind.

Firstly, however, the system of the ramified theory of types is independently interesting for the foundations of logic that it provides.

The results in set theory will seem primitive, as the results are dated to around , at just the point when axiomatic set theory began its extraordinary development.

Whitehead and Russell were not active contributors to set theory and so PM should not be studied for later technical results that may have been anticipated here. There is, however, one result concerning two notions of infinity that appears to originate with PM. It will be infinite if and only if it is not inductive.

A class is Dedekind Infinite Reflexive if and only if it can be put in a one to one correspondence with a proper subset of itself. The Inductive and Reflexive notions of infinity coincide if one assumes that axiom of Choice. This result does not assume the axiom of Choice. George Boolos 27 describes the details of this argument and quotes J. Littlewood as saying:. He [Russell] has a secret craving to have proved some straight mathematical theorem. Perfectly good mathematics.

As use of the Axiom of Choice is explicitly indicated, and many results do not use it, the unique contribution to set theory of PM may be in its indication of what can be proved without assuming Choice.

Relation Arithmetic is the study of the generalization of cardinal and ordinal numbers to classes of similar classes where similarity is based on an arbitrary relation. The mapping S is an isomorphism between the relations P and Q. A relation number will then be a class of relations that are similar to each other.

Relation Arithmetic then generalizes the notions of cardinal arithmetic, such as sum and product, to arbitrary relation numbers. Russell himself expressed regret that the material in Part IV was not more carefully studied by his contemporaries Russell The sums of ordinal numbers are are studied in Tarski , but there has been little interest in the more general notion of Relation Arithmetic presented in these sections of PM.

See Solomon This is the generalization from classes to relations of the fact that the cardinality of the relative product of two classes is the cardinality of the class of ordered pairs of elements taken one from each. It is possible to prove results that show the differences between products and sums of relations and of numbers.

The product of relations is associative:. The Principia was ground-breaking in both what it proposed and the methods employed within it. He did this by postulating that the same gravitational force applied everywhere. The Principia not only described the world mathematically as previous works had, but used mathematics to predict how the world should behave, given a certain mathematical description of forces.

For these reasons, the Principia is widely considered one of the greatest, if not the greatest scientific text ever written. It began life, however, as a nine-page tract. Only a few hundred first editions were produced and in order to publish it, Newton had to rely on funding from a friend — two of the surviving copies of the first edition are held in the Special Collections Division of the University Library, with different provenances, along with copies of the second and third editions.

The first edition was published in and copies had a simple title page, with black ink and without pictures as shown in the image from Special Collections below. By the time the second and third editions were published in and respectively, the Principia was sufficiently renowned that publishers were able to add pictures and coloured text to the copies, knowing that the more expensive copies would still sell.

The solution contained in the tract excited Halley so much that he visited Newton again to request more of such work to present to the Royal Society. Book I of the Principia opens with a set of definitions, and Newton also introduces the concept of mass. An important contribution that he makes here is to distinguish between force and momentum.

Mobile Newsletter chat avatar. Mobile Newsletter chat subscribe. Prev NEXT. Famous Scientists. A very excited librarian holds a copy of one of the most important scientific works ever written, the Principia. Cite This! Print Citation. The work, as might have been expected, caused a great deal of excitement throughout Europe, and the whole of the impression was very soon sold. In a copy of the Principia was hard to obtain. While Newton was writing the second and third books of the Principia , a very important event occurred at Cambridge which had the effect of bringing him before the public in a new light.

James II had already, in , in open violation of the law, conferred the deanery of Christ Church at Oxford on John Massey , a person whose sole qualification was that he was a member of the Church of Rome; and the king had boasted to the pope's legate that "what he had done at Oxford would very soon be done at Cambridge. Upon receiving the mandamus Dr Pechell, the master of Magdalene College, who was vice-chancellor, sent a messenger to the duke of Albemarle, the chancellor, to request him to get the mandamus recalled; and the registrary and the bedell waited upon Francis to offer him instant admission to the degree if only he would take the necessary oaths.

Both the king and the monk were inexorable. The court and the university were thus placed on a collision course. A menacing letter was despatched by Sunderland to shake the firmness of the university—but, though humble and respectful explanations were returned, the university showed no sign of compliance, nor even of a desire to suggest a compromise.

In consequence the vice-chancellor and deputies from the senate were summoned to appear before the High Commission Court at Westminster. Newton was one of the eight deputies appointed by the senate for this purpose.

The deputies, before starting for London, held a meeting to prepare their case for the court. A compromise which was put forward by one of them was stoutly and successfully resisted by Newton.

On April 21 the deputation, with their case carefully prepared, appeared before the court. Lord Jeffreys presided at the board. The deputation appeared as a matter of course before the commissioners, and was dismissed. On April 27 they gave their plea. On May 7 it was discussed, and feebly defended by the vice-chancellor.

The deputies maintained that in the late reign several royal mandates had been withdrawn, and that no degree had ever been conferred without the oaths having been previously taken. Jeffreys spoke with his accustomed insolence to the vice-chancellor, silenced the other deputies when they offered to speak, and ordered them out of court. When recalled the deputies were reprimanded, and Pechell was deprived of his office as vice-chancellor, and of his salary as master of Magdalene. Newton returned to Trinity College to complete the Principia.

While thus occupied he had an extensive correspondence with Halley, a very great part of which is extant. The following letter from Halley, dated London, July 5th, , announcing the completion of the Principia, is of particular interest:. The last errata came just in time to be inserted. I will present from you the book you desire to the Royal Society, Mr Boyle, Mr Paget, Mr Flamsteed, and if there be any else in town that you design to gratify that way; and I have sent you to bestow on your friends in the University 20 copies, which I entreat you to accept.

In the same parcel you will receive 40 more, which having no acquaintance in Cambridge, I must entreat you to put into the hands of one or more of your ablest booksellers to dispose of them. I intend the price of them, bound in calves' leather, and lettered, to be [OCR error] shillings here.

Those I send you I value in quires at 6 shillings, to take my money as they are sold, or at 5 sh. I hope you will not repent you of the pains you have taken in so laudable a piece, so much to your own and the nation's credit, but rather, after you shall have a little diverted yourself with other studies, that you will resume those contemplations wherein you had so great success, and attempt the perfection of the lunar theory, which will be of prodigious use in navigation, as well as of profound and public speculation You will receive a box from me on Thursday next by the wagon, that starts from town tomorrow.

In and Newton seems to have had a serious illness, the nature of which has given rise to very considerable dispute.