Mathematics in Logic and Language

Natural numbers - words used to count things. To count is to create an abstract category or group.
On average, natural numbers go up to 7.
A number such as 73,962 is abstract and mysterious, impossible to visualise or recognise.

Any complex number can be analysed

Analytic philosophy - idealist or synthetic - breaking things down to find their logical basis.

Plurality - more than one
Plurals of plurals. Eg: 9 = 3 x 3

'Special words' - have been used in the past by tribes to count stuff.
Eg- 'one thing', 'more than one thing' and 'many things'
All that would be needed is those 3 phrases.

Syntax and Grammer - Symbols = progress

Zero = Nothing = Something which cotnradictrs Aristotles law of contradiction, the foundation of all logic.
Bertrand Russell says '0 (nothing) is impossible'.
0 is not nothing, for example it can mean 'there is nobody in the room'.
0 + 1 = 1 but 0 x 1 = 0 ---- Could this be considered a contradiction?

The first manipulable number was 2, in Greece.
Greeks believed the number 3 had magical powers. Even basing a religion on worshipping the number 3.

Geometry - Platonic theory of numbers
The world consists of geometrical shapes.
Pie is a retio derived from a geometrical idea.

Music - Every note is defined by a ratio length.
Harmonisation.
Eg- From middle C

RUssell thought that number and arithmetic were neither Platonic ideal forms, nor empirical generalistion.

Principia Mathematica
Founder - Campaign for nuclear disarment

His intellectual career began as a dedicated Hegelian idealist, he retains some of this in social theories, especially his popular scientific work.

Mathematics appeared to be a contradiction of idealism, because numbers appear to have an objective in some sense, and their nature is not apparatnly determined or affected by the act of obseriving them.

Peano's Axioms
0 is a natural number (it is not nothingness)
x = x - Every numver is its own equivalent
Every natural number jas a successor number.
But no natural number has 0 as a successor.

Russells Paradox
The key terms Russell uses are class and belonging to a class and similarity

Number in general is 'the class of classes similar to a given class'

So number three is a word nominally corresponding a logical class composed of all possible classes.

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Frege and Language

Language - Sense and Reference

According to Frege there are 2 different kinds of signification:-
1- Sense of an expression - The particular mode in which a sign presents what it designates.
2- Reference of an expression - The object to which is refers.

Account of meaning - there are items at 3 levels:-
1- Signs
2- Their sense
3- Their references

Frege believed in a weel-regulated language, every sign would have only one sense.

He also felt thart what is lost in translation is 'the colour' of the text. 'Colour' is important for poetry but not for logic: it is not objective in the way that sense is.

- The sense of a word is what we grasp when we understand the word.
- The sense of a sign is something that is the common properly of all users of the language.
The reference of a sentence is determined by the reference of the parts of a sentence.
Frege does agree though that there can be sentences that are lacking references.

A Young Frege

Many philosophers have accepted a difference between predication and assertion.

Pragmatists hold a slightly different view to Frege, they believe that if something is widely believed then it can be correct in a sentence. Such as 'Hitler is Evil' - Pragmatists would accept this sentence because it is widely believed and therefore accepted. However, Frege would state that evil isn't defined and therefore doesn't accept the sentence.

Semiotics -
1 - Natural Signs
Eg: Clouds = a sign of rain

2 - Iconic Signs [Signified by resembling their objects]
Eg: Sculptures, paintings and maps

3 -  Symbols [Determined by conventions]
Eg: Uniforms and traffic symbols

Iconic signs have two essential features:-
1- Should share with its object some feeatures that each could have if the other did not exist.
2- The method of interpreting this feature should be fixed by convention

Charles Sanders Pierce had his own theory's about Semiotics. He saw them as split into three.
1- Syntactics
2- Semantics - Study of the relationship between language and reality.
3- Pragmatics - The study of the social context and the purposes and consequences of communication.

Pierce focuses on meaning and truth. He believed that truth and not just meaning relied on thier consequences.

Thier is both Logical and Emotive language.

Logical is logical proposition.
Emotive langaue can range from religion, art, poetry, love affairs and hatred.
Emotive language usually can't have a reference.

As journalists - people often try to pass off emotion as fact.

'Capitalism is evil' - not verifiable.
Can try to identify evil by saying what you think it is. If you define evil then it would be accepted by Frege.

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