Logic and Mathematics continued

Epistemology ‘the theory of knowledge’: developed by two Empiricists: Mill and Newman
John Stuart Mill
In Mill's 'System of Knowledge' he states that all knowledge derives from experience, including mathematics. Mill believed that logical propositions were a posteriori, affirming his support towards empiricism. 

On mathematics... Mill said that each number involved an assertion of a physical fact: one being singular, two being a pair, twelve being a dozen etc, 
and that the numbers two, three, four etc all denote physical phenomena, and connote a physical property. 

But Mill admits that senses find it difficult to distinguish big amounts. Smaller numbers are easier. He says that '102 apples and 103 apples' would be difficult to distinguish. 

Mill’s thesis: arithmetic is an empirical science. 
He claims that a principle like: ‘The sums of equals are equals’ is an inductive truth. 
(An inductive truth is a generalization based on a individuals experience.) 
But the inductive truth isn't always true. For one pound weight is not equal to another. 

John Henry Newman 
He was part of the same empiricist tradition as Mill, he believed that the only direct contact we have with things outside of ourselves is through our senses. 

Reason does not perceive anything, the exercise of reason is to assert one thing on the grounds of another thing. When we reason Newman said we use two functions:
  1. Interference 
  2. Assent (agreement) 
To infer...you must start with a proposition which then you led you to a conclusion to why your statement is true. 

With assent (agreement) you can conclude to the wrong conclusion if your evidence is wrong. 

Newman makes a distinction between simple assent and complex. Simple accent may be unconscious and rash. Complex accent involves three elements; it must follow on proof, it must be accompanied by a specific stones of intellectual contentment and it must be irreversible. 

Agreed among philosophers is this: If i know p, then p is true. But I may be certain that p and p be false. 

Frege on Logic, Psychology and Epistemology
‘Begriffsschrift’ is Frege’s book on modern logic. 

Frege wanted epistemology to be linked to logic rather than the empiricist tradition to link it with psychology. 

Frege adapted Kant’s distinction between a priori and a poseriori knowledge. How do we first come to believe a proposition and how we come can justify it. 

Frege said we need knowledge a priori ( a priori knowledge does not rely on experience)

A priori truths are known before experience, they are true by definition
e.g All Bachelors are unmarried men or All Triangles have 3 sides- geometry is a priori 

There is no such thing as an a priori mistake as we can only know what is already true, otherwise we don't know anything at all. 

Frege's definition of a posteriori knowledge is that it's a judgement based on the fundamental ground of proposition that we believe to be true, in other words, the evidence.

A mathematical proposition must be justified by maths says Frege. It cannot be a psychological matter of process. A mathematician has a sensation and mental image when calculating. But images are not what arithmetic is about. Mathematicians may associate the different images with just one number. 

-Arithmetic is concerned with the truth of the proposition not the psychology. 

Simply Frege explains Psychology concerns the cause of our thinking and mathematics is the proof of our thoughts. 

In Frege’s late essay ‘Thoughts’ he maintains the distinction between logic and psychology. He highlights the ambiguity inherent in the statement that logic deals with ‘laws of thought’ 

Both Descartes and Frege accept a division between a public world of physical things and a private world of human consciousness, they appeal to the third world in away that rejoins what was separated.


Natural numbers: are the words used to count thing.

 ‘to count’ is to use an abstract category creating words and abstract symbols for plural categories (10+) needs a system of number words and logical syntax to put together for further numbers. 

Attitudes towards languages and syntax number systems: 

  1. They are natural and can be empirically observed
  2. They are instincts of a harmonic platonic other world. Like pythagoreanism.
  3. They are abstract logical objects constructed from syntax (Frege) 

Syntax: a set of rules of modifying a meaning of one logical object or another adjectives and nouns have syntactical forms. 

Evolutionary psychology/ Numerical naturalism:

Apes and stone age ancestors appear to be able to jude simple empirical plurality like this:
0= absence of an object ( no bananas) 
1= one object ( a banana)
2= a lot of bananas

Other empirical plurality: in everyday life you walk into a room and if there is more than one person in that room you can notice that using simple plurality using logical relations. If there is 6+ people in that room you may have to count. 

Addition and multiplication: plurals of plurals.
If a football ground is full you wouldn’t count how many people are there you would say its nearly full or busy. 

Pythagoreanism/ Platonism
focus on prime numbers. The prime numbers are seen as logo’s- languages of God. Platonists would say prime numbers show necessary pre conditions for consciousness. 

Orphic Religions:
Nietzsche’s god of music, harmony of the spheres.
The behavior towards the number three is abnormal, it is treated differently than a word. In Christianity three has significance ‘the father, son and holy spirit,’ rises on the third day’ 
The greek’s fear zero and no.1- counting began with two and saw it as though it is naturally impossible to have nothing.
In Islam they love the number one. There is only one Allah. 

Frege’s method. 

His axiom: ‘ all things that are identical are equal to themselves’ 
  • he followed that all things that are pairs are identical to all other pairs. 

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