Frege and a mention of Russell on Logic and Mathematics

Frege’s intent when he set out writing Begriffschrift, which was published in 1879, was to address the inefficiencies in mathematical notation, which he felt were insufficiently precise as well as mathematical proofs not being rigorous enough. Thus, Begriffschrift was a new method of mathematical notation that aimed to show more precisely what it was that a mathematical statement aimed to show, making it obvious how one step in a mathematical proof succeeded another.

Frege thought that mathematical statements, as with any other statement, relied upon logical premises, which can often cause problems. To address this problem Frege sought to show that all mathematics was derived from pure principles of logic. In order to prove this, Frege did two things, the first of which was to show that all mathematical concepts (numbers) can be derived from pure principles of logic:

To define the concept of numbers in purely logical terms, Frege stated that all numbers (not including 0 or infinity) could be summarised as follows: The number 2 for example is representative of a pair of things, which, when replicated, is identical to all other pairs of things.

The next step for Frege was to apply Aristotelian logic to mathematics in a general sense, however, this method was later proven to not be applicable by Gödel, although he created premises from which most arithmetic was deducible.

One of Frege’s most important theories was his rejection of Husserl and the German Idealist logic, which suggested that logic was a theory of judgment that was effected by human psychology. Frege, as an empiricist refuted this notion, stating that in order to be valid, mathematical proofs cannot rely upon the human psychology but must be objective in order to be considered valid. This theory can easily be seen as influencing the later theories of the logical positivist movement which considers all metaphysics as unnecessary absent of empirical proof.

Frege also sought to differentiate between sense meaning and empirical referential meaning outside of mathematics, stating that an objects referential meaning was the object itself so in the example of the evening and morning star, which is actually Venus, the referential meaning would be that which refers to the specific star as an empirical value so saying that the evening star and morning star are one and the same in a referential sense means that Venus is Venus, which is uninteresting, whereas the sense meaning denotes the its contribution to the sense of the sentence, so saying that the evening and morning star are one and the same in this sense denotes the mathematical discovery that whilst appearing as different stars they are in fact the same planet, which is of interest.

This study of logic and meaning has heavily influenced modern philosophers such as Michael Dummett and indeed philosophy itself, as with the rise of science, philosophy lost its ability to compete in terms of the discovery of knowledge and instead concerns itself with the definition of logic and the meaning behind concepts.

Russell’s early work saw him unknowingly repeat what a fairly anonymous Frege had already discovered. However, it wasn’t entirely in vain, as through his research, Russell found a paradox within Frege’s theories discovering self-defeating propositions following Frege’s logic. This had a profound impact on Frege, who believed the entirety of his work to be abolished; he never completed the 3^rd volume of Begriffschrift. Russell, whilst repeating Frege’s work in what some believe to be a less impressive manner, not expressing proofs in as rigorous a way as Frege, Russell and Whitehead’s Pricipia Mathematica popularised these theories, which has meant that an increased uptake in the subject has caused it advance in leaps and bounds since then.