Seminar Paper – Mathematics, Logic and Language
History of Western Philosophy, Chapter 31:
Russell wrote a body of ground-breaking work in the field of logic up until the age of about 40, after which he took a bit of a breather and wrote light-hearted stuff like the History of Western Philosophy instead. In his chapter on The Philosophy of Logical Analysis, he speaks about the intellectual divide between men who are more influenced by empirical sciences and men who are mainly inspired by mathematics.
He describes Kant and Plato as being among the empiricists; Locke and Aristotle belong to the mathematician party. Russell says that modern philosophy sets out to remove Pythagoreanism from the principles of maths, and to combine empiricism with the deductive parts of human knowledge.
Georg Cantor worked on 'infinite' numbers – and says to think of them in terms of a collection, and that an infinite collection is one that has parts containing as many terms as the whole collection contains. The Mandelbrot set is a visual representation of this idea; infinitely recurring self-similar fractal/
Russell says that it was Frege that truly defined numbers. He said that in the past, they had been clumsily assigned the meaning of their own plurality – as seen in the Pythagorean idea of the number 3 containing a triad, some kind of mystical property. Frege went on to demonstrate that mathematics is nothing but a prolongation of pure deductive logic. This helped to pave the way that most philosophical reasoning can be reduced to syntax. Rudolf Carnap thought that all philosophical problems were really syntactical, and that by trying to avoid problems with the syntax you either solved your quandary or rendered it insoluble; an extreme view perhaps,
Russell claims that the problem regarding accidentally assigning existence to an impossible object i.e. the golden mountain can be properly circumnavigated simply by using good syntax. “There is no entity c such that 'x is golden and mountainous' is true when x is c, but not otherwise”
He goes on to say that, though maths is not empirical, it is neither strictly a priori knowledge about the world; he argues that it is merely verbal knowledge and does not deserve the 'lofty' place it has been given. Offers physics as the best level of understanding one can have about the world – philosophers can now think of particles occupying a space in space time, and speak of quantum 'events' instead of just particles.
He also argues that the deeper understanding of physics and the physiology of the way our minds perceive events throws 'a new light on the problem of perception.' He tries to definite the word 'perception' – it must be an effect of the object perceived, and must more or less resemble the object if it is to be a source of any knowledge of said object. He says the the first condition can only be fulfilled if this effect is adhering to laws which are independent from the rest of the world – that is, in terms that I can understand, an effect which is scientifically demonstrable in a controlled environment. Physics can back this up – 'seeing the sun' for example, can be verified by study in to light waves and the biology of the eye, so that we can prove that we are 'seeing' the sun. Russell goes on to say that despite this, our personal knowledge of the physical world is merely abstract and mathematical.
In conclusion, modern analytical empiricism is for Russell a way of separating the search for truth and the search for ethics and value. The muddling of the two branches of philosophy have significantly impeded human progress as people have attempted to impose their own beliefs in philosophy and bent the rules of mathematics and logic in order to validate themselves; St Thomas Aquinas, Kant, etc. He believes that logical analysis is the best way to uncover truth, and that truth is the only thing a philosopher should concern themselves with.
Philosophy in the Modern World:
John Stuart Mill was the first British empiricist to work on formal logic. He rejected Hobbes' version of nominalism, which is the two-name theory of proposition – the theory that a proposition is only really true if the subject and predicate are the same thing, e.g. Superman is Clark Kent. Mill said that this only worked where the subject and predicate were both proper names. He developed the idea that a name could be pretty much anything – any word that contains the attributes of the subject it is trying to describe – therefore, 'wise' and 'old man' could denote 'Socrates' as part of a proposition. He accepts that every proposition is, in effect, a conjunction of names – though 'name' in its broadest sense as explained before. Although this sounds pointless, it is the connotations in these names that go about trying to establish a truth value – the proposition 'all men are mortal' tells us that the attributes of animality and rationality found in men are always accompanied by the attribute of mortality.
Mill wrestles with the idea that the conclusions of many propositions are actually contained within the premiss, and that no new knowledge is actually being derived from them. He thinks that syllogistic logic does not contain a genuine inference.
Induction is the process of moving from the specific to the general – deriving a certain truth from a set of particular instances. Mill says that inferences can be deductively valid without necessarily being informative or bringing to light any new knowledge, which is the nature of most syllogisms. He is interested to know how human beings can make the connection between two connoted attributes (like in 'All men are mortal'.) (A machine would think that the class 'men' also belong to the class 'mortal')
He set out five 'canons' to guide the inductive discovery of cause and effect; the first two are:
Method of agreement – If phenomenon A occurs in the presence of A, B and C and also in C, D, and E, then we are to conclude that C – the only common feature – is causally related to F.
Method of disagreement – if F occurs in the presence of A, B and C, but not in the presence of A, B and D, then we have to conclude that C is causally related to F as it was the only feature differing in the two cases.
Mill thinks that we are always applying these canons to everyday life: If we hear that a man has died after being shot in the heart, then we conclude that it was the gunshot that killed him because he was in perfect health before the wound was inflicted upon him. The wound is the only circumstance that has changed.
Frege attempted to show that all arithmetic was deducible to a purely logical premisses, and his logical methods were the polar opposite to Mill's. Mills conclusions were drawn from a posteriori propositions, whereas logic for Frege was a priori and also analytic.
Frege developed a system to overcome the weaknesses inherent in syllogistic logic that had been uncovered over the past couple of centuries – problems such as the syllogism only being designed to cope with subject-predicate sentences, and being unable to account for inferences in which words like 'all' or 'some' were included.
He replaced the ideas of subject and predicate with the words argument and function instead, which allowed grammarians and logicians a more flexible way of analysing a sentence. The boundary between what is the argument and what is the function can be moved in the sentence, so 'Wellington defeated Napoleon' can be analysed in a number of different ways. It gives them room to mark out logically relevant similarities between two sentences.
Frege also added new notations to logical sentences in order to express generality. These are symbols which are added to the structure of the proposition in order to mark out that a certain function is always true or false no matter what the argument is; this include notations for 'all' 'none' 'some' and 'if', which was arguably the most important element.
To make a statement like this, logicians would write: (x)(x is mortal) ...which is technically false, as it means that everything, everywhere is mortal, and Frege believed that all objects were nameable; therefore, this statement is attempting to say that the number 10 is mortal, which is wrong.
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