This study is concerned with the origination of the logic of symbolic mathematics as investigated by Edmund Husserl and Jacob Klein. The ‘logic’ of symbolic mathematics at issue here is that which allows everyone—from barely literate school children to master mathematicians—to employ sense-perceptible letter signs, without a second thought, in a “mathematical” manner of social empires hack tool download. The content of mathematics, like the content of its logic, is immaterial to its topic, which is how it has come about that such signs are self-evidently perceived to represent an “indeterminate” conceptual content as readily and unproblematically as, for example, the perception of the color and shape of this book.
What is responsible for social empires hack tool download this topic is uncontroversially referred to as ‘formalization’. What formalization is, however, is controversial. At one extreme, formalization is understood as the employment of letter signs or other marks to, at the very least, “stand for” or “symbolize” any arbitrary object or content—“whatever”—belonging to a certain “domain.” Let ‘3’ stand for the number of any arbitrary objects whatever; let ‘X’ stand for any arbitrary number whatever; let ‘S’ stand for any arbitrary subject member of any proposition whatever—all these expressions are examples of formalization, and when “interpreted” in a manner that finds nothing especially problematic to speak of here, these examples illustrate pretty much all that is needed—or the minimum needed—to begin formalization. At the other extreme is the view that formalization is the fulcrum for an unprecedented transformation in how the science of the so-called West forms its concepts, a transformation that is as all-encompassing as it is invisible to this day—especially to those who study the history of this science or are engaged in scientific inquiry.
The present study has as its subject matter the latter understanding of formalization. In it we shall investigate the major work of its first proponent, the twentieth-century historian and philosopher of mathematics, Jacob Klein. The work in question is entitled “Die griechische Logistik und die Entstehung der Algebra,” which was originally published in two parts in 1934 and 1936 and then in English translation as Greek Mathematical Thought and the Origin of Algebra in 1968.1 Klein’s major thesis there is that the history of the transformation of what he calls the “conceptuality” of the most basic concept employed by science, that of number, from a non-conceptual and nonlinguistic multitude of determinate things to a concept that is identical with a symbolic language, is inseparable from the meaning of the symbolic employment of letter signs—from the most elementary, such as ‘2’, to the most universal, say, ‘X’.
This history is important for Klein because it discloses that the conceptuality of the symbolically transformed concept of number represents, in a paradigmatic way, a radically different (and philosophically significant) apprehension of things from how they were apprehended before that transformation. Prior to it, things were apprehended “directly,” first through the senses and then through the employment of concepts that were apprehended as different from both the social empires hack tool download things whose apprehension they permitted and from the language that made use of concepts in order to bring about this apprehension.
What is responsible for social empires hack tool download this topic is uncontroversially referred to as ‘formalization’. What formalization is, however, is controversial. At one extreme, formalization is understood as the employment of letter signs or other marks to, at the very least, “stand for” or “symbolize” any arbitrary object or content—“whatever”—belonging to a certain “domain.” Let ‘3’ stand for the number of any arbitrary objects whatever; let ‘X’ stand for any arbitrary number whatever; let ‘S’ stand for any arbitrary subject member of any proposition whatever—all these expressions are examples of formalization, and when “interpreted” in a manner that finds nothing especially problematic to speak of here, these examples illustrate pretty much all that is needed—or the minimum needed—to begin formalization. At the other extreme is the view that formalization is the fulcrum for an unprecedented transformation in how the science of the so-called West forms its concepts, a transformation that is as all-encompassing as it is invisible to this day—especially to those who study the history of this science or are engaged in scientific inquiry.
The present study has as its subject matter the latter understanding of formalization. In it we shall investigate the major work of its first proponent, the twentieth-century historian and philosopher of mathematics, Jacob Klein. The work in question is entitled “Die griechische Logistik und die Entstehung der Algebra,” which was originally published in two parts in 1934 and 1936 and then in English translation as Greek Mathematical Thought and the Origin of Algebra in 1968.1 Klein’s major thesis there is that the history of the transformation of what he calls the “conceptuality” of the most basic concept employed by science, that of number, from a non-conceptual and nonlinguistic multitude of determinate things to a concept that is identical with a symbolic language, is inseparable from the meaning of the symbolic employment of letter signs—from the most elementary, such as ‘2’, to the most universal, say, ‘X’.
This history is important for Klein because it discloses that the conceptuality of the symbolically transformed concept of number represents, in a paradigmatic way, a radically different (and philosophically significant) apprehension of things from how they were apprehended before that transformation. Prior to it, things were apprehended “directly,” first through the senses and then through the employment of concepts that were apprehended as different from both the social empires hack tool download things whose apprehension they permitted and from the language that made use of concepts in order to bring about this apprehension.