Gottlob Frege (1848-1925)

Frege, a German mathematician and philosopher, founded the discipline of logic in its modern form.

In publishing his best known book Begriffsschrift, Frege invented the very first predicate calculus, formalized the system of proofs, and formed the basis of what he would later devote his life to: an unsuccessful attempt at showing that mathematics was reducible to logic (Bertrand Russell showed that Russell's paradox made one of his axioms contradictory).

Despite this, Frege's work was a significant advance in the field of logic, an area that had been basically stagnant since the medieval revival of Aristotelian logic . He is thus called by many as the "father of modern logic". Frege had also made a significant contribution to the philosophy of language by suggesting that an expression had both a sense and a denotation -- an idea that spawned significant research for a century. Russel, Carnap, and Wittgenstein were some of the many thinkers strongly influenced by Frege.

Frege was a conservative German, which at that time meant that he was quite distrustful of foreigners and was anti-semitic. Frege wrote little after 1903, apparently depressed at the poor reception of his ideas.

Frege, Gottlob,

Without doubt, Gottlob Frege (1848-1925) deserves to be counted amongst the most important and influential philosophers and logicians of the past two centuries. In his early Begriffsschrift (Concept Script), he single-handedly invented the propositional and predicate calculi (even though the unwieldy notation he used for their expression was soon understandably abandoned by others), taking the first major leaps forward in formal logic since Aristotle. He also made, in Grundlagen der Arithmetik, a valiant and ambitious attempt to prove the thesis of logicism in mathematics: the belief that the whole of mathematics can (contra Kant and others) be shown to be a branch of logic. This involved providing a definition of number in terms of the logical notion of a class, or set. Fascinating as his attempt was, it was ultimately proved by Bertrand Russell to be doomed to failure, relying as it did on an axiom that entailed the existence of the set of all sets that are not members of themselves. What has come to be known as Russell's Paradox shows that such a set is an impossible item. That is, clearly the set is either a member of itself, or it is not. If it is a member of itself, then it is not a member of itself, and if it is not a member of itself, then it is a member of itself.

However, it is chiefly in connection with the sense/reference distinction in the philosophy of language that Frege's name is known to modern-day students of philosophy. The introduction of this distinction in the seminal paper 'Über Sinn und Bedeutung' ('On Sense and Reference') is motivated by a puzzle about the nature of true identity statements of the form 'a = b'. To see what this puzzle is, consider the following two sentences:

  1. Hesperus is Hesperus
  2. Hesperus is Phosphorus

Since the names 'Hesperus' and 'Phosphorus' both refer to the same object (the planet Venus), both (1) and (2) are true. And yet whilst (1) is clearly trivially true, (2) is not: it had to be empirically discovered by astronomers to be true. Thus, (1) and (2) differ in informational content. Yet the puzzle is that since Hesperus is Phosphorus, we could be forgiven for supposing that (1) and (2) should be alike in meaning, and hence should both convey precisely the same information.

Frege's solution is that whilst the names 'Hesperus' and 'Phosphorus' share a reference, they nonetheless differ in sense. The reference of an expression E is characterised by Frege as the extralinguistic item which E picks out (in our example, Venus); the sense of E, on the other hand, is explained - not completely unmysteriously - to be E's reference's 'mode of presentation'. He further portrays E's sense as that aspect of its meaning that determines, or fixes, what E's reference is; in other words, the sense of E is what needs to be grasped if we are to understand what object E picks out.

Crucially, it is quite possible, and indeed frequently occurs, that two expressions E1 and E2 might offer a different mode of presentation of one and the same object, or (which is perhaps to say the same thing) that the reference of E1 might be determined in a different way to that of E2. And just this is the case, so Frege argues, in our Hesperus/Phosphorus example. As a result of the disparity in sense between 'Hesperus' and 'Phosphorus', it is possible to know that Hesperus is Hesperus (since Hesperus has the same sense in both its occurrences in (1)), yet simultaneously not to know that Hesperus is Phosphorus.

Frege takes great pains to point out that sense is most definitely not something subjective, perhaps akin to an idea awakened in us when we hear an expression. Since the sense of E must be grasped by anyone who is to know what E stands for, it clearly must be something publicly accessible, something thoroughly objective.

It is not only proper names such as 'Hesperus' and 'Phosphorus' that are to be thought of as having senses and references. Predicates and sentences, too, have attached to them a 'mode of presentation' of the objects to which they refer. In Frege's view, the references of predicates are functions such as 'x is round' that take the reference of a proper name as argument. Frege goes on to claim that the reference of an entire indicative sentence is a truth value: either 'the True' or 'the False', depending on the proposition (viewed as an eternal, self-subsistent object) which constitutes the sentence's sense. According to Frege, then, all non-synonymous declarative sentences are simply different modes of presentation of the two truth-values.

Although not widely recognised in his own lifetime, over the last century or so Frege's theory of sense and reference has provided an indispensable philosophical springboard for an immense amount of work on the nature of meaning. This work has done much to address, and occasionally clarify, the relationship between the world and the language we use to think and talk about it. It is beyond question that, without Frege, the contemporary philosophical landscape would be very different - and much less featureful.

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