Frege, Gottlob (1848–1925)
Authors: George Alexander , Heck Richard 
Published: September 2018

A German philosopher-mathematician, Gottlob Frege was primarily interested in understanding both the nature of mathematical truths and the means whereby they are ultimately to be justified. In general, he held that what justifies mathematical statements is reason alone; their justification proceeds without the benefit or need of either perceptual information or the deliverances of any faculty of intuition.

To give this view substance, Frege had to articulate an experience- and intuition-independent conception of reason. In 1879, with extreme clarity, rigour and technical brilliance, he first presented his conception of rational justification. In effect, it constitutes perhaps the greatest single contribution to logic ever made and it was, in any event, the most important advance since Aristotle. For the first time, a deep analysis was possible of deductive inferences involving sentences containing multiply embedded expressions of generality (such as ‘Everyone loves someone’). Furthermore, he presented a logical system within which such arguments could be perspicuously represented: this was the most significant development in our understanding of axiomatic systems since Euclid.

Frege’s goal was to show that most of mathematics could be reduced to logic, in the sense that the full content of all mathematical truths could be expressed using only logical notions and that the truths so expressed could be deduced from logical first principles using only logical means of inference. In this task, Frege is widely thought to have failed, but the attempted execution of his project was not in vain: for Frege did show how the axioms of arithmetic can be derived, using only logical resources, from a single principle which some have argued is, if not a logical principle, still appropriately fundamental. In addition, Frege contributed importantly to the philosophy of mathematics through his trenchant critiques of alternative conceptions of mathematics, in particular those advanced by John Stuart Mill and Immanuel Kant, and through his sustained inquiry into the nature of number and, more generally, of abstract objects.

In the course of offering an analysis of deductive argument, Frege was led to probe beneath the surface form of sentences to an underlying structure by virtue of which the cogency of inferences obtains. As a consequence of his explorations, Frege came to offer the first non-trivial and remotely plausible account of the functioning of language. Many of his specific theses about language – for instance, that understanding a linguistic expression does not consist merely of knowing which object it refers to – are acknowledged as of fundamental importance even by those who reject them.

More generally, three features of Frege’s approach to philosophical problems have shaped the concerns and methods of analytic philosophy, one of the twentieth century’s dominant traditions. First, Frege translates central philosophical problems into problems about language: for example, faced with the epistemological question of how we are able to have knowledge of objects which we can neither observe nor intuit, such as numbers, Frege replaces it with the question of how we are able to talk about those objects using language and, once the question is so put, avenues of exploration previously invisible come to seem plausible and even natural. Second, Frege’s focus on language is governed by the principle that it is the operation of sentences that is explanatorily primary: the explanation of the functioning of all parts of speech is to be in terms of their contribution to the meanings of full sentences in which they occur. Finally, Frege insists that we not confuse such explanations with psychological accounts of the mental states of speakers: inquiry into the nature of the link between language and the world, on the one hand, and language and thought, on the other, must not concern itself with unshareable aspects of individual experience.

These three guiding ideas – lingua-centrism, the primacy of the sentence, and anti-psychologism – exercised a commanding influence on early analytic philosophers, such as Wittgenstein, Russell and Carnap. Through them, these ideas have been spread far and wide, and they have come to create and shape analytic philosophy, with whose fathering Frege, more than anyone else, must be credited.