In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

The main thesis of this paper is that, Contrary to general belief, George berkeley did in fact express a coherent philosophy of mathematics in his major published works. He treated arithmetic and geometry separately and differently, And this paper focuses on his philosophy of arithmetic, Which is shown to be strikingly similar to the 19th and 20th century philosophies of mathematics known as 'formalism' and 'instrumentalism'. A major portion of the paper is devoted to showing how this philosophy of mathematics (...) follows directly from berkeley 's theory of signs and his philosophy of language, And from his discovery of an alternative to the correspondence theory of truth used by most of his predecessors. (shrink)

The project, entertained by Leibniz and others, of creating an ideal language to facilitate ratiocination, is investigated in detail. Six possible relations between the ideal language (IL) and the natural language (NL) it replaces are studied. (1) IL says exactly what NL says, but says it much more clearly. (2) IL says exactly what NL says, but does so more economically. (3) IL says exactly what NL says, but does so more succinctly. (4) IL says part of what NL says, (...) and says it more perspicuously. (5) IL says part of what NL says, and says it more perspicuously; moreover, there is an effective procedure for going from NL to IL. (6) IL says everything that NL says, plus some things that NL cannot say. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Jim Mackenzie; Tottering to Reason and Speech, Journal of Philosophy of Education, Volume 21, Issue 2, 30 May 2006, Pages 247–260, https://doi.org/10.1111/j.146.

Jim Mackenzie; Evers & Walker and Forms of Knowledge, Journal of Philosophy of Education, Volume 19, Issue 2, 30 May 2006, Pages 199–209, https://doi.org/10.

Jim Mackenzie; Evers & Walker and Forms of Knowledge, Journal of Philosophy of Education, Volume 19, Issue 2, 30 May 2006, Pages 199–209, https://doi.org/10.

The Hebrew text On the Heavens and the World, ascribed to Ibn S, is an interesting and intriguing composition. It dates from the 13th century and was quite influential. It is not a translation of any text of Ibn S known to us, but is related to the Latin De celo et mundo, which appears in the 1508 Venice edition of translations of Ibn S. The Latin and Hebrew texts differ widely and the relation between them is far from being (...) clear. Both are in sixteen chapters, the titles of the chapters are the same, but the texts are only roughly similar. The Hebrew text often offers short, incomplete summaries of the Latin arguments. On the other hand it includes many passages which have no parallel in the Latin. There are two possible explanations of the perplexing relationship between the two texts: either that there was more than one version of the Latin text, or that the translator, Shlomo ben Moshe of Laguiri wrote a kind of paraphrase. The paper shows that the second explanation is correct and offers a preliminary study of the sources and the aims of the Hebrew text. (shrink)

Historically there have been two main freewill problems, the problem of freedom versus predestination, which is mainly theological, and the problem of freedom versus determinism, which has exercised the minds of many of the great modern philosophers. The latter problem is seldom stated in full detail, for its elements are taken as so obvious that they do not need to be stated. The problem is seen as an attempt to reconcile the belief in human freedom, which is essential if men (...) are to be able to act morally, with determinism, the belief that every event is fully determined in all its details by the sum of its precedent causes. But even the meticulous Moore does not trouble to explore at length what is meant by determinism. He devotes one very short paragraph to the matter, and sums it up immediately afterwards as the view that ‘everything … has a cause’. (shrink)

The poet Philitas was so thin, they say, that he had to wear lead weights on his shoes to avoid being blown away by a gust of wind. We have two versions of the anecdote. First Aelian, Varia Historia 9.14.