Lecture 2- Logic and Mathematics

Lecture 2

Computing leads onto technical maths and logic. Frege, Whitehead and Russell talk of logic and mathematics. They use the idea of natural numbers, words used to count things. Count is to create an abstract category or group. Creating words and abstract symbols for plural categories (plural = more than one) requires a system of number system of words.

The three fundamental attitudes towards languages were rejected.

1.     Natural- not mind dependant- J'S Mill empirically observed (exist independently).

2.     Platonic- intuitions of a harmonic platonic other world (e.g. pythagoreanism- Russell started this as numbers come from another world.) Take them a priori as a number and a noumena. This is Platonism which Russell and Frege converted to.

3.      Syntax- logical system using rules of inference to alter meaning rules for modifying the meaning of one logical object to another e.g. words and numbers.

The Numerical Naturalism/ Evolutionary psychology

The apes and stone age tribes appear to be able to judge empirical plurality typically:

0-    absence of a thing (e.g. they could tell if there was only a banana in a room)

1-    One banana

2-    (maybe) a lot of bananas/ unlimited amount of bananas

These are known as natural numbers and only use the idea of  ‘one thing’ ‘possibly many things’ and ‘more than one thing’ which we use when calculating from the mind. People that are advanced use cultures small number of words functionally which are different to large number of words. (Logic- relations to empirical pluralities) use image in head to count roughly how many people are in a room at a glance. Natural numbers are smaller numbers and ones that you don’t count. Whereas lots of people in a stadium is abstract.

Stuart Mill rejected this idea as he said you can count grains of sand on a beach. We use words pluralities which aren’t the same as numbers to describe an amount. For example the stadium is empty etc. 7,434 is a predicate symbol of more basic symbols organised according to known syntax. It can be analysed as a predicate ‘analytic philosophy’.

The first attitude is that syntax can be learnt or be innate says Chumski and is complicated and impossible for children to learn. You are supposed to be innate which is a rejection of Locke. You are born with syntax according to Chumski but it is subtle compared to anything in technology like the human syntax.

Attitude – Pythagoreanism/ Platonism states that prime numbers are not divisible and are pre-existing eternal supernatural forms. They are necessary in pre conditions for consciousness- ‘the logos’.  All other numbers are rational combinations of prime numbers, (contra-Kant- ‘existence is not a predicate’ for Platonism existence is a predicate of numbers and other external forms.

There is a religion of triangle and worship in Russell’s pythagoreanism.  Numbers are a communication of an alien strange world that doesn’t occur anywhere in nature as a thing in itself. The language you have got is evidence that you must be good simply as it cannot be thought up or arise naturally. Prime numbers have a special religious significance as they are things from eternity and perception. The number one will always be known as the number one.

The Orphic religion which was Nietzsche Apollonian religion was a pre-Socratic religion ‘birth of tragedy’. It was the irrationality in Ancient Greece. The God of music was Orphic and the Judaism religion believed that trumpets have special powers.

Three is known as a magic number and also has a special religious significance. It is the first plural prime number, art uses the ‘rule of thirds’ and music uses ‘the three chord triad, whereas drama is set into a beginning, middle and end. In Christianity the number three is big, the father, sun and holy spirit. Jesus rises on the third day is all based on the number 3. Greeks feared numbers zero and one as they said they are naturally impossible as dualism cant be one thing so everything started on the number two.

Other primes have religious and even magic significance such as Frazer and the golden bough, Islam Arab neo-Platonism exhibits cults around the none plural prime (one) but also five and seven (pentagrams). Islam are obsessed with the number one as they believe there is only one God Allah.

Aristotle’s physics was a matter of solid geometrical shapes that differentiated air from water was the number of faces of solid geometrical fundamental object. This is all Orphism according to Nietzsche, Russell and Fazer.

‘One’ and ‘not one’ are special categories that are different logical categories ‘exist’ or ‘doesn’t’.  In the Odysseus and Cyclops he says ‘my name is no-one’ and Cyclops replies ‘no one is there’. No one is someone or something.

Special problem of nothing and zero

Zero came from India via Islam to arrive in Europe. One, two, three and four are Arabic and as we know the Greeks had a different system with no zero included. They were like computers in this way as zero means nothing but nothing is something. Aristotle started logic with the ‘logic’ contra to Aristotle law of contradiction (foundational axiom of all logic). The object can ‘contain’ its own negation (Leibniz monads solved problem of the law of contradiction). Modern philosophers of mathematics have this assert that zero is a natural number, logically derived as 1-1=0. ‘Nothing’ is a philosophical absurdity (e.g. Heidegger) also the qualitative differential gap between 0=nothing and +1 = something is as big as the universe.

The consistency of operator +1 is a gap from 0 to everything is massive 1+1=2 that is double and in some cases can mean a lot. But adding one grain of sand to the beach isn’t changing much so +1 is different depending on the situation.

0+1=1 (infinitely large increment as we have gone from nothing to something).

1+1=2 we have doubled what we had in size.

N+1 is infinitely small increment as N could stand for the grain of sand.

Empirical phenomena is not natural and the numbers are platonic perfect entities from another world of perfectness.

Attitude three is numbers as logical objects

The problem of 0+ nothing remained unsolved for 1000 years until Frege (1848-1925). Foundations of arithmetic (1884) ‘The Grundlagen’ is the third approach to numbers as logical objects.

For Frege arithmetic is just a language and are all the same when you look at them analytically.

1-    Vocabulary of objects (words)

2-    Syntax (modifying meaning e.g. -, +)

3-    Grammar

Adapted by Whitehead and Russell- (Principia mathematica) was an attempt that failed to demonstrate logical basis for numbers, arithmetic, mathematics refuting Platonism and numerological mysticism (Russell started as a Platonist- numbers can only observed and used in calculation but not understood as things in themselves.)

He rejected Mills numerological empiricism (cannot find zero in nature) and +1 gives you an increment problem. Therefore his hypothesis was that you can’t observe so numbers must be denied by logic.

Frege’s method

Axiom are all things which are identical and are equal to themselves (asserted a priori to deductive definitional truth). The certain truth is equal and can be exchanged to work in some way. There is no case in logic and the empirical world that can prove this wrong. All things are in pairs and are identical to all other pairs (regardless of what they are a pair of).

The class of all pairs contains all pairs and this can be given a purely nominal symbol such as two, a word of a numeral it does not matter.

Larger numbers can be built up along the lines of the class of all things which are pairs as pairs. One is class of all things which are not associated with other things (one is no longer a pair as it is one.)

Zero as a class of all possible objects which are not equal to themselves. There are no such things as objects by definition so this category is empty and is known as the null category.

You have to logically define zero as it opens up the road to advance in computing and in logic. The computer never relied on an actual zero as it used Frege and his idea of a ‘null class’. The analytic philosophy was previously Frege on the logical underpinning of language (sense VS reference).

Bertrand Russell 1872-1970

1907-Stood for parliament

1913- Principia mathematica (age 41- was exhausted)

1914- Pacifist (WW1)

1940- Sacked by New York University for immorality

1941- Renounced pacifism

1950- Founder, campaign for nuclear disarment

Frege- logic (language)

Syllogistic logic is by Aristotle and includes inference deductive (analytic/ a priori) and inductive (synthetic a posteriori)

Sentential logic by Frege explains that the meaning is in the sentence as a whole. Sense and reference are where some sentences are on a  superficial level and make perfect sense but have no reference. ‘The King of France is bald’ makes sense but has no reference as there is no King. ‘The morning star is not the evening star’. This sentence makes sense but is not true as both parts have the same reference of Venus which both stars belong to.

This is all to do with computer programming and the ways that the computer can understand certain things. Some paintings make no sense as they show a light on lighting up a house when the sky is also light representing day time. The points of reference make sense as they are the sky, sun, clouds, light however when all put together it doesn’t make sense to look at.