Logic and Mathematics

Frege, Russell and Whitehead

Creating words and abstract symbols for plural categories requires a system of number words I.E 'symbols' and a logical syntax ( a syntax is a logical system using rules of inference to alter meaning of symbols) for combining these numbers to words. These are altered however, due to rules of modern language.

There are three fundamental attitudes towards languages, including syntactical number systems such as arithmetic but especially numbers.

1. They are natural and can be empirically observed e.g Mill
2. They are intuitions of a harmonic perfect platonic other word eg pythagoreasnism- Descartes, geometry.
3. They're abstract logical objects which are constructed purely from syntax ( Frege, early Russell)

For Frege, numbers are created using syntax

Numerical naturalism/evolutionary psychology. 

Apes and stone age tribes appear to be able to judge simple empirical plurality, typically:

If you are a monkey.

0: absence of a thing ( eg a banana.) 

1= one banana/enough bananas 

2= maybe a lot of bananas/unlimited. 

1 thing, more than one thing and plenty of things. These are the only numbers they need. Even for people from advanced cultures small number words are functionally different to large number words.

For example, if you came into a room and there is one person, you don’t count the one person even with three you can categorise that as a simple plurality. Most people will go up to maybe six or seven objects in a group before counting, using logic relations to the empirical pluralities.

Predicate (In logic and grammar) is the result of a process of a calculation.Subject Verb- Predicate. The result of the action of a verb. Verbs are operators in Frege logic. 

Syntax- is it learnt or innate? Chomsky. Its complicated it would be impossible for a child to learn it, it must be innate. Against John Locke who thought nothing is innate we learn things. Human syntax is subtle compared to syntax on a computer. On a computer it used predicate logic. Fuzzy Logic vs Neat Logic.

Attitude 2: Pythagoreanism/Platonism

Prime numbers are pre-existing external supernatural forms- necessary pre- conditions for consciousness- the 'logos'. All other numbers are just rational combinations of prime numbers (contra Kant). For Platonism existence is a predicate of numbers and other external forms. Primes exist in a non-human dimension just as the ideal objects of aesthetic perfection and the ratios of geometry. These are eternally true and ultimately mysterious and part of the panoply of Orphic religion (Nietzsche's Birth of Tragedy)

The special religious significance of the prime number three, people behave abnormally to numbers. You wouldn’t say you have a magic word, but with numbers people do. E.g you have the rule of thirds. Three quad triads in music. A beginning, middle, end. Three stages. etc. Other primes have religious and even magic significance, The Greek’s feared no.1 and zero. Greek counting began with two. zero would not be something they entertain. It is naturally impossible to have nothing, nothingness cannot be an object. Islam love the no.1. There is only one substance, allah. 

Aristotle's physics was a matter of solid geometrical shapes; what differentiated air from water was the number of faces of the sold geometrical fundamental objects.

Pythagoras and all Greeks regarded only plurals as natural numbers and so they began counting with 'two'. Frege had the same problem in logic. There is no one on the road, does not mean the same thing as the road is empty and anyway the road is not empty because it contains at least the road. 

Special Problem of nothing and zero

The introduction of the concept of zero came from India. Entire Arabic numeral system was introduced in the middle ages after the fall of Rome. It is a very difficult concept because zero = nothing which = something. This is central to Aristotle's 'Law of Contradiction'.

Modern Philosophers of mathematics have thus asserted that zero is a neutral number, logically derived as 1-1=0

'Nothing' is a philosophic absurdity  that someone like Heidegger would asset. Also the qualitative have a differential gap between 0=nothing and 1= something as big as the universe. 

Zero is nothing and nothing is something. The first rule of Aristotle's logic was not to contradict the moon is the moon the sun is the sun. therefore the sun isn't the moon. 

The problem here is the consistently. 

0+1=0 but 0x1=0 so what does +1 mean? 0+1=1 1+1=2

plus 1 is doing different things all the time. it is not a stable syntactical object. Frege solves that. Numbers are platonic entities, not known as things in itself.

Attitude Three: numbers as logical objects

The problem of zero and 'nothing' remained unsolved until Ferge (1848-1925) decided to link logic and arithmetic in an overall system of philosophy of language with arithmetic as a special case of language. He wrote this in the book: 'The foundations of Arithmetic'. This was then adapted by Russell and Whitehead as an attempt to demonstrate the logical basis for numbers.

Frege's Method

All things that are identical are equal to themselves. ( This is apriori, deductive.) 

It follows all things which are pairs are identical to all other pairs regardless of what they are pairs of.The class of all pairs contains all pairs and this can be given a purely nominal symbol eg two a word or numeral, it does no matter.

Larger numbers can be built as logical constructs along the lines of the class of all things which are pairs of pairs we can attach any symbol we like to this the conventional one would be ‘four’. 

One is the class of all things which are not associated with other things

Zero as a class of all possible objects which are NOT equal to themselves. There are no such objects by definition. See Axiom. An Axiom is a thing which is identical and equal to themselves.

Russell (1872-1970)

It’s possible for a sentence to make sense but have no reference. It makes sense but there is no point of reference. This conversation would make sense but there are no reference points.

New Semester, New WINOL role

As the new semester begins things are gradually beginning to pick up where we all left off before the Christmas holidays since we are still ahead of East London Lines (our student compeitiors) both nationally and globally despite our Christmas break.WINOL currently has an Alexa ranking of 454,851 globally compare to East London Line who have a ranking of 654,024. The UK Alexa ranking for WINOL is 26,496 and East London Lines are behind with a ranking of 56,868. This is a promising start to the new  year and we can only keep improving on these figures.

I've made the transition from news to features and am thrilled to have been given the role as beauty editor for WINOL's online fashion magazine 'Absolute:ly'. So far we're preparing to start a nail art competiton to attract even more attention to the site. We will be giving away a selection of nail varnish to the lucky winner.
Visit www.winol.co.uk and select the Absolute:ly tab at the top of the page for details on how to enter from Monday the 4/2/13.

Seminar Paper on Phenomenology and Existentialism

Freud and Psychoanalysis

·         Freud was the greatest influence on Anglo-American philosophy even though he was technically a scientist and not a philosopher.

·         Freud was born into an Austrian family of non-observant Jews in 1856. Later in Vienna, Freud trained as a doctor and went on to study brain anatomy. In 1895 Freud published a booked based on ‘hysteria’, this was to be the first analysis of mental illness which led him to develop his idea of psychoanalysis.

·          In 1900 he published the most important book of all his works: ‘The interpretation of Dreams’ in which Freud argues that dreams are repressed sexual desires.

·         Freud summed up psychoanalytic theory in two fundamental theses:

1.       The greater part of our mental life, whether it’s our feelings or thoughts, is unconscious.

2.       Sexual impulses are supremely important and can cause mental illness if suppressed. Sexual instincts are basic human instincts.

·         Towards the end of his life Freud focused on the ‘Mental Apparatus’ which includes the personality being divided into the Id, ego and the superego. As long as the ego is in harmony with the Id and the superego then all is well but if not then mental illness is inevitable.

Husserl’s Phenomenology

·         Edmund Husserl’s life can arguably be compared to Freud since both men were born into Jewish families in Austria and also both men devoted their lives to scientific study of the human mind.

·         Husserl believed that the data of the conscious comes in two kinds:

1.       Physical e.g. colours and smells

2.       Mental phenomena e.g. thoughts

·         In his logical investigations of 1900 he argued that logic cannot be derived from psychology since there is a sharp distinction between logic and psychology.

·         Husserl also believed that there are two essential parts to a thought:

1.       That it should have a content and

2.       A possessor

For example, a dog. When you see a dog you know it is a dog and then the thought that you have that the dog is a dog is your own thought therefore you are the possessor of the thought.

·         Concepts are therefore, in the ‘Logical Investigations’ defined on the basis of psychological terms. This means that Husserl was able to disown his earlier psychologism and make a clear distinction between psychology and logic. A line was also drawn between psychology and epistemology (the study of knowledge and justified belief). Husserl managed to do this through a reinvention of psychology as a new discipline of Phenomenology.

·         Phenomenology was developed during the 20^th century by a group of collaborators in Munich who created the ‘Phenomenological Movement’.

·         Phenomenology is :

1.        The science of phenomena as distinct from that of the nature of being.

2.       An approach that concentrates on the study of consciousness and the objects of direct experience.

·         Phenomenology is not the same as Phenomenalism. A Phenomenalist believes that nothing exists expect phenomena and also that statements about such things as material objects have to be translated into statements about appearances.

·         Husserl deliberately left open the possibility that there is a world of non-phenomenal objects although, these objects are of no interest to him- we have absolute immediate knowledge of the objects of our own consciousness while we have only distorted knowledge about the external world.

·         Immanent perception provides the subject matter of Phenomenology; this is because immanent perception is a person’s immediate acquaintance with their own current mental acts and states. Immanent perception is self-evident. Whereas transcendent perception is a person’s perception of their own past acts and states of physical objects and events and of the contents of other people’s minds.

·         Only Consciousness has absolute ‘being’ since all other forms of ‘being’ depend upon consciousness for their existence and so Phenomenology is the most basic of all disciplines- all items that are its subject matter provide the data for all other branches of philosophy and science.

·         By the mid of the 20^th century all had changed amongst philosophical thinking, English and foreign philosophers went their separate ways with their philosophies. In Europe Existentialism (which is a philosophical theory or approach that emphasises the existence of the individual person as free and responsible, determining their own development through acts of the will.) was fashionable and was led by Jean Paul Sartre in France and Martin Heidegger in Germany.

The existentialism of Heidegger

·         Martin Heidegger (1889-1976), was one of Husserl’s many pupils.

·         Heidegger published the ‘Sein Und Zeit’ was published in 1927 which translates as ‘Being and Time’ in English. Within the book Heidegger claims that Phenomenology up to that point had been half-heartedly studied. He adopted consciousness as the subject matter of Phenomenology and so the first task was to study the concept of being (Sein) since ‘Being’ is the starting point of all philosophy.

·         Heidegger invented words to help him explain his philosophy in greater depth. ‘Dasein’ meaning ‘being-here’ or ‘existence’, is the most important of all the words Heidegger created.

·         Thinking is the only way of engaging in the world. ‘Dasein’ is prior to the distinction between thinking and willing.

·         He emphasises the temporal nature of ‘Dasein’- think of it not as a substance but as the unfolding of a life:’ Our life isn’t a self-contained, self-developing entity’

·         The activity of ‘Dasein’ has three fundamental aspects:

1.       Attunement- the situations into which we are thrown manifest themselves as attractive or alarming etc. and we respond to them with moods of various kinds.

2.       Discursive- It operates within a world of discourses among entities that are articulated and interpreted for us by the language and culture that we share with others.

3.       Understanding- in a special sense- that is to say, its activities are directed towards some goal.

These 3 aspects all correspond to the past, present and future of time.

·         Heidegger’s opponent Gilbert Ryle admitted at the end of Heidegger’s ‘Being and Time’ that he had nothing but admiration for his ‘Phenomenological analysis of the root workings of the human soul’.

·         Concepts and judgements can be thought of as instruments for coping with the world. However, there are more primitive instruments, things that are tools in a literal sense. E.g. A carpenter relates to the world by using a hammer- doesn’t need to be thinking about the hammer to be using it well. Consciousness of the hammer may get in the way of the concentration on his project that is his true engagement with reality.

·         In 1929 Heidegger succeeded Husserl as professor of Philosophy of Freiburg and in 1933 became Rector of the university.

The Existentialism of Sartre  

·         In contrast to the right- wing existentialism of Heidegger, Jean-Paul Sartre developed a form of existentialism that was more left-wing.

·         Sartre was born in Paris in 1905-1980.Later in life he taught philosophy in high schools from 1933-1935 where he began to develop his own philosophy. His first essays are detailed studies based on the philosophy of the mind in the Phenomenological mould.

·         In 1943 he published ‘Magnus Opus’ ‘Being and Nothingness’ which was greatly influenced by Heidegger.

·         ‘Being’ for Sartre is what underlies all the different kinds and aspects of things that we encounter in consciousness. To say that an object is without a cause isn’t to say that it is its own cause, the object is just simply there. Sartre call this ‘gratuitous’.

·         In Sartrean existentialism, ‘being in itself’ (en-soi) refers to objects in the external world- a mode of existence that simply is. It is not conscious so it is neither active nor passive and contains no potentiality for transcendence (go beyond the limit). This mode of being is relevant to inanimate objects, but not to humans, who Sartre says must always make a choice.

·         One of the problems of human existence for Sartre is the desire to obtain ‘being in itself’, which he describes as the desire to be God -this is a longing for full control over one's destiny and for absolute identity which is only attainable by achieving full control over the destiny of all existence. The desire to be God is one of the ways people fall into ‘bad faith’.

Derrida

·         Jacques Derrida possessed hostility towards Phonocentrism, the idea that sounds and speech are superior to written language. He believed this to be the case because speech is closer than writing to the thinking that is idealized as the ultimate, transcendental object of signification.

·         Phonecentrism was part of his attack which he called the ‘Metaphysics of Presence’  and was mainly focused on targeting the philosophical work of Husserl.