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.
- They are natural and can be empirically observed e.g Mill
- They are intuitions of a harmonic perfect platonic other word eg pythagoreasnism- Descartes, geometry.
- 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.
Attitude 2: Pythagoreanism/Platonism
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.
Attitude Three: numbers as logical objects
Frege's Method
Russell (1872-1970)
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.
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