Logic and Mathematics- Lecture Notes

HCJ Lecture 2

Frege, Russell, Whitehead- logic and mathematics.
(Mathematic and technology language) Very much on the analytic side based on logic and logical positivism. (Frege is german, not anglo saxon.)

Solecistic logic. 

Natural numbers, these words used to count for things. Natural logic from the mathematical field of logic.  

Three fundamental attitudes towards languages, including syntactical number systems such as arithmetic, but especially numbers

  1. they are natural and can be empirically observed. 
  2. They are intuitions of a harmonic, perfect platonic other word. Numbers are platonic. Pythagorus. Bert Russell started as a platonist- numbers came from a different world. You would never know the noumena of a number it can never be a thing in itself. Platonism. 
  3. They are abstract logical objects, constructed purely from syntax. (Frege)

Syntax a set of rules of modifying a meaning of one logical object and another. Adjectives and nouns have syntactic forms. Syntactically altered, Large house- Blue house. 

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. 

ZERO one and two are strong natural numbers. 

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 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. 

You can say in a footage ground. it could appear to be empty. You could say there are a few people or a “crowd of people” This is known as simple 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. PC programming was a Very slow progress to begin with. 

Horrie: “Philosophy degrees today are in a lab- trying to put people’s brains into equations.”


Beans- Fetus, strange cult like ideas. But they worships maths. chanting geonomitary.

Prime numbers are pre existing, external supernatural forms. Numbers are a Unknown kantian world. They are not in nature simple plurality. Prime numbers are “Logos’ evidence there must be god, language of God. Prime numbers are invisible. if you divide them you do not get a whole.  Similar to plato’s cave, things for the domain of perfection and eternity. 

Orphic religion, in ancient greece- Nietzsche’s appolonian religion, pace the birth of tragedy. God of music, the harmony of the spheres. The eternal ratio of thirds on strings and harp. Trumpets- judaism. 

All other numbers are just rational represent geometrical ratio’s. 

The special religious significance of the prime number three. Three is the magic number. People behave abnormally to numbers. You wouldn’t say you have a magic word, but with numbers people do. 
the numbers three- you have the rule of thirds. Three quad triads in music. A beginning, middle, end. Three stages. 

In Christianity three is massive. Father, son, holy spirit. Rises on the third day. Coincidence? 

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. 

No. 5 and no.7 are important. pentagrams. 

Arisitotle physics was a matter of sold geometrical fundamental objects. 

This is all ophism and also the codified religion of pythagoreanism. 

Pythagoras regarded only plurals as natural numbers so they began counting with two. “one” and “Not One” were different logical categories. The odyssey- the cylops, “my name is no one” “No one is there” Frege cites the same problems 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. 


Zero as a operator came from India. Arabic letters are where numbers came from. 
It was avery difficult concept because zero+nothing BUT nothing is something. This is contra to and breaks Aristotle's law of contradiction, the foundational axiom of all logic. Problem of the law of contradiction solved by Leibniz monads, that an object can “contain” its own negation. Modern philosophers of mathematics have thus asset that zero is a natural number, logically derived as 1-1=0 

“Nothing” is a philosophic absurdity like somebody like Heidegger would asset Also the qualitative differnteraly gap between 0=nothing and 1= something as big as the universe. 

 Aristotle huge step on the way. Yes we know things but here is a way to know how you knew these things. Why do you know the swan is white? 
All swans are white- proposition
This bird is a swan therefore it is white. 
knew these things. Why do you know the swan is white. 
BUT The australian black swan cleared that out. 

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 entireties, not known as things in itself. 

Attitude today 

Numbers are logical objects- Frege Revolutionary thought. ‘The Grundlagen.’ this is a third approach of numbers- LOGICAL OBJECTS

Book ‘the foundations of arithmetic.’ 
Arithmetic is just a language according to Frege  

Adapted by Russell and Whitehead as an attempt to demonstrate the logical basis for numbers arithmetic and mathematics thus refuting platonism and numerological mysticism. 

Rejected Mill’s numerological empiricism. you cannot find zero in nature. and then there is the + 1 increment. Not coming from a platonic therefore numbers must be derived from logic. in his book ‘equations’ 


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. 

Bertrand Russell 1872-1970


The philosophy for language- Wittgenstein 

Frege moved logic to syllogistic logic. 

-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 

Frege solves that with a new form of language the Beriffsschrift. 


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