Saturday, 25 February 2012

Modern Mathematics, Logic and Language - Seminar Summary

The earliest examples of number systems came from apes and stone age tribes. The systems adopted at this point in time were fairly basic, consisting of 'one thing', 'more than one thing' and 'many things'. Later on, the Greek and Roman systems for counting depended entirely on numeral symbols and did not contain zero. In these systems, one was not regarded as a number, either. Interestingly, most people will identify up to 8 objects in a group before having to consciously count them out. 

Bertrand Russell famously stated that there are the same amount of even numbers as there are whole numbers. There is a clear link here with the concept of infinity. If there are an infinite amount of numbers, then this proposition must be correct. In his book, 'The History of Western Philosophy', Russell explains this hypothesis in further detail, writing: "Whatever finite number you mention, there are evidently more numbers than that, because from 1 up to the number in question there are just that number of numbers, and then there are others that are greater". Russell defined number, in general, as 'the class of classes similar to a given class'.

Above: An ancient Babylonian mathematical tablet

The concept of zero (or nothing)originally came from India and much later via Islam. The thought behind the figure is that zero is nothing, which is something. This means that zero is nothing and something at the same time. From a philosophical standpoint, however, 'nothing' is an absurdity.

Number systems have always had a close link with the concept of magic. Numbers were seen as a type of magic as the numbers were free floating, perfect platonic forms, attributed with these unexplainable properties. Even today, for example, people have an idea of what they see as a 'lucky' or 'unlucky' number.

It's worth mentioning Frege when it comes to mathematics and logic. Frege developed a system of propositional logic and also saw logic as a discipline that sorted out good inferences from bad inferences. Frege felt that philosophers stuck in empiricist tradition had confused logic with psychology. He worked on making it easier to understand the structure of language, sentences and propositions. Frege felt, for example, that every sentence was made up of two parts: The function (the first part of the sentence) and the argument (the second part of the sentence). 

Russell wrote about Frege and his logical theory, writing: "From Frege's work it followed that arithmetic, and pure mathematics generally, is nothing but a prolongation of deductive logic". Frege felt that epistemology had a fundamental role in philosophy, also stating that when syntax errors were avoided, philosophical problems were solved or shown to be unsolvable.

Frege's work was developed by Russell and Alfred North Whitehead, who wrote Principia Mathematica. The book was released in three volumes in 1910, 1912 and 1913 and aimed to explore the subject of formal logic. It was written as a means of popularising modern mathematical logic and to this day remains one of the most influential texts on logic ever written.  

We now come to Peano's Axioms. Peano, born in 1858, was interested in defining natural numbers in terms of sets . He published his ideas in 1889 in a book titled 'Arithmetices principia, nova methodo exposita' and these were later developed in Principa Mathematica. His axioms were as follows:

1) Zero is a natural number. The 'number' zero can be used to count.

2) X = X. Every number is its own equivalent. (If A is a number, the successor of A is a number)

3) Every natural number has a successor number (Implies numbers are an infinite series)

4) There is no natural number whose successor is zero. (Negative numbers are not real i.e they cannot be used for counting)

5) (Induction axiom) If the successor of N is equal to the successor of M, then N is equal to M for all numbers in all series

John Stuart Mill was a British philosopher born in 1906. He was an empiricist whose system of logic fell into two principles parts: formal logic and the methodology the natural and social sciences. Mill felt that maths was derived from experience and was also associated with the idea of nominalism. Nominalism was the name given to a two-name theory of the proposition. The theory that a proposition is true if and only if subject and predicate are names of the same things. All names denote things. In logic, however, connotation is prior to denotation.

Mill also discussed inference, the term given to the act of drawing conclusions from what is assumed or known to be true. There are two kinds of inference: Real and verbal. Real inference is when we infer to a truth. It is a truth which is not already contained in the premisses. Knowledge of the language alone, in this instance, is enough to allow us to derive the conclusion from the premise. The second type is called verbal inference, which brings us no further knowledge about the world.

During the seminar we discussed the link between infinity and mathematics. We mentioned the Mandelbrot set, a fragmented geometric shape that is infinitely complex. The term came from Benoit Mandebrot in 1975 and described the fractal, which is made by 'copying an altering an input image' The set itself is obtained from the quadratic recurrence equation: . I personally found the video above an interesting watch and feel that It's a nice way of linking to Russell's views of infinity.


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