Type of paper: Essay

Topic: Mathematics, Theory, Logic, Education, Knowledge, Motivation, Reduction, Career

Pages: 5

Words: 1375

Published: 2020/12/27

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Russell was undoubtedly one of the great logicians of the twentieth century. His theories, though not always sound and sometimes quite strange, have painted a picture of mathematics which few have dared attempt prior to him. Russell’s great preoccupation was to find a satisfactory logical base for mathematics. This tricky and highly complex challenge had Russell changing his stance on various points and tenets of mathematics and logic several times over the course of his career – if there is a solution to the problem which Russell had set out to solve, it could not be discovered in one lifetime.
But one must stop and consider why Russell had such a fascination with this particular form of enquiry. Aside from a personal appetence towards both philosophy and mathematics, Russell had several motivating factors which caused him to look to Logicism to solve the problems he set out to solve. Before Russell, there had been numerous logicians who had attempted to reduce mathematics into a few basic axioms of logic, but none of them had been quite as thorough as Russell. Much of his success can be attributed to his initial skirmishes with the dominant philosophic school in Cambridge in the early part of his career. The ‘Absolute Idealist’ school held that the entire universe was constituted of parts which were in and of themselves absolutely essential. The removal of any of these parts, or even a change in any of their attributes or properties would result in the universe not existing at all. This absolutist stance was not in itself repugnant to Russell, though he did have problems with many of its tenets.
His refutation of absolute idealism began with his acceptance of the three basic premises of philosophic realism (a stance which he took jointly with his colleague G.E. Moore) – that all everyday objects exist, that there exist relations between these objects, which are ‘abstract’ objects, and that every object of thought has a real object to which it refers. These three ‘axioms’ defined ‘truth’ – any proposition must be judged based on its adherence to these three facts of existence. The problem, however, was with negative existentials – how could something not exist and yet be conceived of? This conundrum drove Russell to formulate more theories and rethink his philosophies.
This point has implications in his theory of mathematics because at the heart of his enterprise (it can be argued) is the question of whether mathematics is a body of essential truths in and of itself (meaning that mathematics constitutes an essential part of truth which forms the base for other ‘knowable’ things), or is it simply a body of knowledge which we know is true (in the same way as we know physics or chemistry contains truths)? Early on, Russell developed the idea that mathematics was a form of symbolic logic. Much of his early career was spent attempting to reduce mathematics to this form of logic and he met with considerable success. The most likely motivation for this was to create a kind of ‘abstract’ elegance in mathematics – if the subject could be reduced to symbolic logic it would rely not on the many concepts and axioms of arithmetic, as is the case with fields like algebra, but would be built on a very small, very secure base of a handful of logical axioms. This would be achieved by ‘translating’ mathematics into a very small set of ‘marks’. This reduction would be analogous to Frege’s work on reducing language into propositional atoms and ‘augment|function’ of generalization.
Russell’s great advantage over all his predecessors who attempted this kind of reduction was the availability of new instruments for analysis, in particular, the theory of abstract relations and it will later be seen how this helped him formulate his particular brand of ‘logicism for math’. However, at this point, a small digression must be made to present the classical motivation for logicism. Russell did, at various points in his career, accept and reject these points, though, it can be argued, that they did, by and large, remain with him throughout his life. Firstly, logicism was a system of justification. The old epistemological idea that the fewer ‘imponderables’ one used in an explanation, the more likely that explanation is to be true is clearly present in Russell. Mathematics is troublesome, to say the least, when trying to find justifications because there is almost no way to test and prove empirically many of its parts or tenets (some of them simply cannot be tested). If mathematics were reducible to logic alone, then the axioms of the subject would be seen as nothing more than consequences of purely logical experimentation.
The second motivation (which was mentioned earlier as likely to be the prime motivation for Russell) is the problem of knowledge and ‘knowability’. Mathematics is a field which (usually) takes for granted that it is a built on a-priori knowledge. While the proposition might not seem problematic to a student of metaphysics (or at least it is problematic in a different way), to a logician, such as Russell, the problem is very real and very fundamental. It can be argued that to a logician, claiming a-priori knowledge is as good as claiming the existence of carnivorous cows or round squares. Testable facts must be formed and tested in order for a field to attain logical validity. The point here is that, in reducing mathematics to logical axioms, Russell was essentially reducing it to a set of linguistic propositions.
These linguistic propositions consist of logical words such as and, or, not, some, all, etc. The issue, then, is then simply a matter of choice – one simply decides that the most relation in mathematics are explained (or encapsulated) in certain words (which are of our choosing). Scott Soames has noted that this high degree of reduction to a linguistic level is problematic in that it begins with trivial truths and to reconstruct the vast, complex nexus of concepts, methods and tenets of mathematics from something so simple is a daunting task (one which Russell and Whitehead did with considerable success in their enormous work Pincipia Mathematica).
Finally, the third motivation for logicism stems from ontology. Mathematics, in and of itself (without the reduction brought about by logicism) demands a separate existence for a huge variety of objects such as natural numbers, irrational numbers, even imaginary numbers. The problem here is that each entity is ‘disjoint’ insomuch that for one to interact with the other, a set of rules and methods must be applied. This heavy ontological baggage makes mathematics, to put it bluntly, inelegant. Natural numbers were ‘ground zero’ for mathematics, but with Russell and logicism, even these were reducible to an even simpler forms, i.e. to set of certain types.
In Russell’s case, the particular use of logic resulted in the formation of the famed ‘Russell’s Paradox’. The paradox is built around a particular axiom of infinity – considering the problem of the number of objects, a question arises as to whether or not infinity can be understood as a genuine logical axiom. Russell attempted to solve this problem by developing his theory of types. The theory of types can only be understood if the paradox itself is made clear (or unclear). Russell conceived of a set of all and only those things which are not members of themselves. Mathematically, it is represented as ‘~x∈x’. Let y be the name for this strange set. If this is the case and the rule for the set mentioned earlier applies, y is a member of itself if and only of it is not a member of itself. It must obey the same mathematical equation as stated earlier: ‘~y∈y’. The paradox is apparent and Russell formulated his theory of types in the hope that it will solve this problem.
The theory of types is so called because of Russell’s formulation of four different types of sets. Each level contains a defined kind of set – Type i contains concrete individuals, type ii contains sets of individuals, type iii contains sets of sets of individuals, and type iiii contains sets of sets of sets of individuals. The point of this division into types is so that one can find an adequate level at which to perform the long sort after reduction of arithmetic to logic. As Soames has pointed out, no matter how one cuts this case, the idea of infinity simply cannot be adequately explained by use of pure logic. The ideas repel each other.
Unfortunately Russell had clearly failed in this particular enterprise – his results were unusable. It can be argued that Russell’s theory of types did not vindicate the epistemological aspects of logicism and its application to mathematics because it did not give mathematics either justification or proof that many aspects of mathematics were knowable through anything but a-priory knowledge. This meant that there could be no justification for mathematics in this quarter either – there are some areas of mathematics which can be known only through a-priory knowledge. Finally, this theory does seem to hold some validity on the ontological argument because it does reduce numbers to sets of sets and sets of sets of sets, etc. which does clearly lighten the existential baggage of mathematics. But a score of one out of three is not promising, to say the least. Russell, though ingenious and brilliant, was ultimately dumbfounded by infinity.

Work Cited

Soames, Scott. Philosophical analysis in the twentieth century, volume 1: The Dawn
of analysis. Vol. 1. Princeton: Princeton University Press, 2009. Print.

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WePapers. (2020, December, 27) Russell’s Logicism Essay Samples. Retrieved January 22, 2022, from https://www.wepapers.com/samples/russells-logicism-essay-samples/
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