HCJ Semester 2, Lecture 2
Frege, Russell, Whitehead - logic and mathematics
Natural numbers, these are words used to count things. To 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 words (‘symbols’), and a logical syntax for combining these number-words (symbols) to imply further or predicate number-words.
Three fundamental attitudes towards languages (including syntactical number systems, such as arithmetic), but especially numbers
Three fundamental attitudes towards languages (including syntactical number systems, such as arithmetic), but especially numbers
- The natural and can be empirically observed (e.g. Mill and most ordinary folks)
- Platonic. They are intuitions of harmonic perfect platonic other world (e.g. pythagoreanism, Descartes, geometry)
- They are abstract logical objects, constructed purely from syntax (Frege, early Russell)
Russell started as a platonist.
SYNTAX - logical system using rules of inference to alter meaning of symbols (e.g. words numbers). It is basically a set of rules to modify the meaning of one logical object to another. Adjectives and nouns have syntactical forms. Numbers are created by Syntax according to Frege.
Numerical Naturalism/ Evolutionary Psychology
There are said to be three areas of numbers. Apes and stone-age tribes appear to be able to judge simple empirical plurality , typically;
Numerical Naturalism/ Evolutionary Psychology
There are said to be three areas of numbers. Apes and stone-age tribes appear to be able to judge simple empirical plurality , typically;
0 = absence of a thing e.g. banana
1 = one banana/enough bananas
2 = (maybe) a lot of bananas/unlimited bananas
“one thing” and “more than one thing” and possible “many things”. These are the only numbers they need. Even for people from advanced cultures small number words are functionally different to large number words. If you come into a room and there is one person, you don’t count the one person; even with three you can categorize that as a simply plurality. Most people will go up to maybe six or seven objects in a group before counting, using logic relations to the empirical pluralities.
Addition and multiplication - empirically - are plurals or plurals, systems of storing and communicating information. You would not go to a football ground and look at the crowd and say there are 37,879 people here. You could go to a football ground and say there were four people there, without counting. Or ‘the ground was empty‘ (the empirical zero, called ‘the null class’) then; ‘there were a few people in the ground‘ (relative to capacity) and ‘the ground was full’. These are ‘natural numbers‘ or ‘simple pluralities’.
So the number 7,456 is a predicate symbol of more basic symbols organized according to known syntax (logical rules of inference). And as a predicate it can be analyzed (operation similar to division and calculating number squares and roots) - “analytic philosophy” - the paradigm of analysis.
Technology - basic logical language - computers, Facebook etc
Discuss - the limits of logical modeling of human intelligence e.g. predictive texting (fuzzy logic vs neat logic.
Noam Chomsky - innate knowledge (look over English Lang notes from A Level)
Human syntax is incredibly subtle compared to computer syntax
Addition and multiplication - empirically - are plurals or plurals, systems of storing and communicating information. You would not go to a football ground and look at the crowd and say there are 37,879 people here. You could go to a football ground and say there were four people there, without counting. Or ‘the ground was empty‘ (the empirical zero, called ‘the null class’) then; ‘there were a few people in the ground‘ (relative to capacity) and ‘the ground was full’. These are ‘natural numbers‘ or ‘simple pluralities’.
So the number 7,456 is a predicate symbol of more basic symbols organized according to known syntax (logical rules of inference). And as a predicate it can be analyzed (operation similar to division and calculating number squares and roots) - “analytic philosophy” - the paradigm of analysis.
Technology - basic logical language - computers, Facebook etc
Discuss - the limits of logical modeling of human intelligence e.g. predictive texting (fuzzy logic vs neat logic.
Noam Chomsky - innate knowledge (look over English Lang notes from A Level)
Human syntax is incredibly subtle compared to computer syntax
Attitude 2: Pythagoreanism/Platonism
This is the view that numbers are so strange that they don’t occur in nature. Not known in a Kantian world. Simple plurality does occur in nature.
Prime numbers (not divisible - they don’t resolve into whole 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 - “existence is not predicate” for Platonism existence is a predicate of numbers and other external forms. Primes exist in non-human dimensions, just as the idea, objects of aesthetic perfection and the ratios of geometry. These things are externally true, ultimately mysterious and part of the panoply of Orphic (pre-socratic, more of a general movement, Orpheous is the God of music) religion (Neitzsche’s Appolonion religion - pace, the birth of tragedy)
Orphic Religion - pythagorean. You couldn’t eat beans in this religion as they thought they were fetus’s, you can’t walk on the pavement, you must never leave the impression of your body on your bed. Cult like ideas. Worship mathematics. Chanting geometrical ideas etc. Contemplating the eternal truth
Prime numbers are rational.
Similar to Plato and the cave analogy. Things for the domain on perfection.
The special religious significance of the prime number three (the first plural prime).
Three is the ‘magic’ number. Art - rule of thirds. Music - three chord triad. The three part drama with a beginning, a middle and an end.
Other primes have religious and even magic significance (Frazer, the golden Bough) Islam (Arab neo-platonism) exhibits cults around the none plural prime (one), but also five and seven (pentagrams)
Aristotle’s physics was a matter of solid geometrical shapes; what differentiated air from water was the number of faces of the sold geometrics fundamental objects.
The special religious significance of the prime number three (the first plural prime).
Three is the ‘magic’ number. Art - rule of thirds. Music - three chord triad. The three part drama with a beginning, a middle and an end.
Other primes have religious and even magic significance (Frazer, the golden Bough) Islam (Arab neo-platonism) exhibits cults around the none plural prime (one), but also five and seven (pentagrams)
Aristotle’s physics was a matter of solid geometrical shapes; what differentiated air from water was the number of faces of the sold geometrics fundamental objects.
This is all Orphism (according to Neitzsche, Russell and Frazer) and also the codified religion of Pythagoreanism (anthropologists often point to the Pythagorean elements in Christianity - the trinity; three people on the cross, resurrected after three days; cock crows three times, etc.
Christians - worshipers of the number three
Muslims = worshipers of the number one.
Islam is obsessed with the number 1. Only ever 1 thing, 1 substance.
The Greeks fear the number 1 and 0. Greek counting started with the number 2.
Number 5 and 7 are important. Babylonions - special significance of 12, because of the zodiac.
Pythagoras (and all the Greeks) regarded only plurals as natural numbers so they began counting with two.
“one” and “not one” were different logical categories.
The odyssey - Odyseus and the Cylops - ‘my name is no one’ and the cyclops says ‘no one is there’. Frege later cities 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 with Nothing and Zero
The concept of ‘zero’ came from India, much later via Sufi Islam. Entire Arabic numeral systems was introduced in the middle ages after the fall of Rome. It is a very difficult concept as ‘Zero = nothing. But Nothing = something.’ The is contra to Aristotle's first law of logic and contradiction. Aristotle invented Logic, and was a way of set out syntax so your beliefs will be true.
Modern philosophers of mathematics have thus 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 moon is the sun, and the sun is the sun. Therefore you know the sun isn’t the moon.
Pythagoras (and all the Greeks) regarded only plurals as natural numbers so they began counting with two.
“one” and “not one” were different logical categories.
The odyssey - Odyseus and the Cylops - ‘my name is no one’ and the cyclops says ‘no one is there’. Frege later cities 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 with Nothing and Zero
The concept of ‘zero’ came from India, much later via Sufi Islam. Entire Arabic numeral systems was introduced in the middle ages after the fall of Rome. It is a very difficult concept as ‘Zero = nothing. But Nothing = something.’ The is contra to Aristotle's first law of logic and contradiction. Aristotle invented Logic, and was a way of set out syntax so your beliefs will be true.
Modern philosophers of mathematics have thus 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 moon is the sun, and the sun is the sun. Therefore you know the sun isn’t the moon.
Aristotle
All swans are white - All men are mortal - Aristotle
This bird is a swan - therefore it is white.
Aristotle is good as you can know HOW you know something. Why do you know that that bird must be white because it’s a swan. But a problem occurred as Australia was found and they had black swans so that idea went out the window.
Organon - another word for the logic.
This bird is a swan - therefore it is white.
Aristotle is good as you can know HOW you know something. Why do you know that that bird must be white because it’s a swan. But a problem occurred as Australia was found and they had black swans so that idea went out the window.
Organon - another word for the logic.
0 + 1 = 1 but 0 x 1 = 0
BUT
So what does 1 mean? And how can it be defined?
0 + 1 = 1 (infinitely large increment)
1 + 1 = (double in size)
N + 1 (infinitely small increment)
That was all wrong attitude that numbers are platonic entities from another universe but not known as things in themselves.
Attitude three; numbers as logical objects
N + 1 (infinitely small increment)
That was all wrong attitude that numbers are platonic entities from another universe but not known as things in themselves.
Attitude three; numbers as logical objects
The problem of zero and nothing remained unresolved for 1000 years until Frege (1848 - 1925)
His Books - The foundations of Arithmetic (1884) ‘The Grundlagen’ - this is the first approach of numbers - LOGICAL OBJECTS.
His Books - The foundations of Arithmetic (1884) ‘The Grundlagen’ - this is the first approach of numbers - LOGICAL OBJECTS.
For Frege, arithmetic is just a language such as English is. They are all the same when you look at them analytically.
Links logic to arithmetic in an overall system of philosophy of language, with arithmetic as a special case of language.
Links logic to arithmetic in an overall system of philosophy of language, with arithmetic as a special case of language.
Adapted by Russel and Whitehead (Principia Mathematica) as an attempt (failed) to demonstrate the logical basis for numbers, arithmetic and mathematics - thus refuting Phatonism and numerological mysticism (Russell started as a Platonist, saying that numbers could only be observed and used in calculation, but not understood as things in themselves.
Rejected Mill’s numerological empiricism (you can not find zero in nature; the +1 increment can not be observed as empirically constant, therefore numbers must be derived from logic.
Frege had already reasoned in the same way in the previous generation.
Frege’s Method
- (*) Axiom - all things which are identical are equal to themselves. This is asserted APriori (deductive truth) - certain.
- (*) Axiom - all things which are identical are equal to themselves. This is asserted APriori (deductive truth) - certain.
- 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 and this can be given a purely nominal symbol (e.g. ‘two’) a word or a numeral, it does not 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 definitions (see (*) Axiom.
- The class of all pairs, contains all pairs and this and this can be given a purely nominal symbol (e.g. ‘two’) a word or a numeral, it does not 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 definitions (see (*) Axiom.
Bertrand Russell - 1872 - 1970
1907 - Stood for parliament as a suffragette (lost to a Tory)
1907 - Stood for parliament as a suffragette (lost to a Tory)
1913 - Principia Mathematicia (age 41 - ‘exhausted’)
1914 - Pacifist (WW1)
1940 - Sacked by New York City University for immorality
1941 - Renounces pacifism
1950 - Founder, campaign for nuclear disarmament.
Peano (same conclusion as Frege) He has the same argument as Frege about the ‘null class’.
Peano (same conclusion as Frege) He has the same argument as Frege about the ‘null class’.
Frege’s Logic
Syllogistic Logic - Aristotle
Inference - deductive (analytic/apriori) inductive (synthetic/aposterori)
Sentential logical - Frege - meaning is in the sentence as a whole
Syllogistic Logic - Aristotle
Inference - deductive (analytic/apriori) inductive (synthetic/aposterori)
Sentential logical - Frege - meaning is in the sentence as a whole
It is 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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