Frege\'s Much-Discussed Book, the Foundations of Arithmetic,

¶ … Frege's much-discussed book, The Foundations of Arithmetic, is an influential and valuable insight into the philosophy of mathematics. A German mathematician and philosopher, Gottlob Frege was unwaveringly devoted to improving the philosophical underpinnings of mathematics and science. In the Foundations of Arithmetic, Frege delves deeply into not only an understanding of numbers, but he also looks into much larger questions surrounding meaning and truth. Frege argues against the idea that arithmetic is based on psychology, and instead notes that logic is the main underpinning of arithmetic. The Foundations of Arithmetic ultimately argues that analytic judgments govern the laws of arithmetic, and thus these laws exist a priori. This important contribution brought him great praise from many academics, but his work was not without flaws. Importantly, The Foundations of Arithmetic has been criticized for a relatively weak understanding of basic terms underlying the discussion of psychology. This is an important and weighty criticism, since much of Frege's book is based upon his rejection of the idea that logic is founded upon psychology. Despite these important criticisms of Frege's book, The Foundations of Arithmetic remains an important and insightful book into the philosophy of mathematics a century after it was first published in the original German.

Friedrich Ludwig Gottlob Frege was born in Germany in 1848. He got his doctoral degree in Gttingen, and quickly wrote his post-doctoral thesis, and became a university professor. During his long and fruitful academic career, the tireless Frege worked extensively to build up the philosophical foundations of mathematics and science. Among his important contributions is the invention of an artificial language called Begriffsschrift, which was based on logical notation (Frege Biography). Frege wrote extensively, publishing a number of other influential papers during his lengthy career, including Grundgesetze der Arithmetik, Volume I (1893), Logic (1897), Sources of Knowledge of Mathematics and the Mathematical Natural Sciences (1924/5), A Brief Survey of my Logical Doctrines (1906), and Thought (1918) (Frege Biography).

Written in 1884, The Foundations of Arithmetic remains a useful and valid analysis of mathematics and philosophy. Important in Frege's analysis is his rejection of the idea that logic is founded upon psychology. Instead, Frege effectively argues for the logicist proposal that arithmetic is founded on logic.

Frege's analysis focused on defining numbers using logical terms, but he also delved into broader questions of truth and meaning. Frege attempts to determine if the laws of arithmetic are synthetic or analytic, and whether these laws are a posteriori or a priori. He delves deeply into the concept of dumber, by beginning to determine sense of numerical identity. Ultimately, in The Foundations of Arithmetic, Frege argues that logic is based on psychology, and also notes that arithmetic is based on logic.

Originally published in German as Die Grundlagen der Arithmetik, The Foundations of Arithmetic is written in what is often a stilted and difficult style, making Frege's mostly lucid and innovative thoughts somewhat inaccessible to modern readers. For example, Frege notes,."..number is neither a collection of things nor a property of such, yet at the same time is not a subjective product of mental processes either, we concluded that a statement of number asserts something objective of a concept."

Important (in fact central) to The Foundations of Arithmetic is Frege's discussion of number. Frege's analysis of number is based on logic and context. Notes Frege, "We next laid down the fundamental principle that we must never try to define the meaning of a word in isolation, but only as it is used in the context of a proposition: only by adhering to this can we, as I believe, avoid a physical view of it."

Importantly, Frege argued that analytic judgments govern the laws of arithmetic, and thus these laws exist a priori. This analysis was likely one of the most important contributions of The Foundations of Arithmetic.

As defined in The Foundations of Arithmetic, Frege's main analysis that the laws of arithmetic are a priori has important consequences. Frege notes that the definition of the laws of arithmetic as a priori results in the following: "Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction."

Frege has a great number of influential and important admires. For example, the well-known philosopher Michael Dummet notes that Frege's work on the criterion of identity is "brilliant and philosophically fruitful." Further, Dummet gives Frege high praise for uniqueness, noting that The Foundations of Arithmetic is truly the first work of analytical philosophy (Frege Biography).

In spite of the great praise heaped upon the book, Frege's analysis is sometimes somewhat flawed. For example, Frege's analysis is often somewhat weak in terms of psychological references. He easily notes the difference between concept and object, and yet he does not clearly define sense and references. Barbosa notes that after The Foundations of Mathematics was written, Frege rejected his contextual preference, which argued that words refer to something based on context.

Frege has also been accused of distorting the ideas of other philosophers to his own benefit. Notes Barbosa, "Sometimes (Frege) distorts a little bit what others say about logic, so he argues against those thinkers more effectively." This is indeed a serious criticism, and implies that Frege was, at least in this case, not scrupulously rigorous or honest in his philosophical arguments.

Frege was always willing to correct inconsistencies in his works, and the issue of contextual preference was no exception. Barbosa notes that after Frege "wrote the book, he would reject this principle, because of his doctrine of sense and reference: the sense of the words determine the sense of the sentence; and the reference of the words determine the reference of the sentence."