Logic and Mathematics - Revision Notes

Natural Numbers=Words used to count things

3 Basic Approaches to Number:
1) They are natural and can be empirically observed (MILL)
2) They are institutions of a perfect and harmonic platonic world (PYTHAGORAS, DESCARTES)
3) They are abstract logical objects, constructed purely from syntax (FREGE)

1) Numerical Naturalism:
Stone age tribes appear to be able to judge simple empirical plurality. “one thing”, “more than one thing”, “lots of things” are all the numbers they need. If you walk into a room and see one person you don’t physically count that one person, you can just categorise it in terms of plurality. Most people can get up to six or even seven before they physically would have to count how many people there are.  A large number like 7,246 is just a predicate symbol of more basic symbols, organised according to known syntax.  Realistically you would just say “there are a lot of people” or “the room is full”.
MILL went beyond his predecessors claiming that not only all science, but also all mathematics is derived from experience. The definition of each number contains the assertion of a physical fact. Every number (2, 3, 4 etc.) denotes physical phenomena and connotes a physical property of that phenomena. E.g. “two” denotes a pair of things, and connotes what makes them pairs. Two apples are physically distinguishable from three apples. They are a different visible and tangible phenomenon.

Mill doesn’t make it clear exactly what the property is that is connoted by the name of a number, and Mill also admits that the mind has some difficulty distinguished between 103 apples and 104 apples.

2) Pythagoreanism/Platonism:
Prime numbers are pre-existing, eternal, supernatural forms.  They are necessary preconditions for consciousness. This goes against KANT’S theory “existence is not a predicate”, for Platonism existence is a predicate of numbers.  Prime numbers exist in a non-human dimension, just like the perfect form of an object exists in the realm of the forms. These things are eternally true.
There is a special religious significance to the number three. Three is the magic number.  Rule of thirds, three part drama, three chord triad etc.
PYTHAGORAS and all the Greeks only regarded plurals as natural numbers, so began counting with two. “One” and “not one” were different logical categories. FREGE later points out this can cause a problem in logic, “there is no one on the road” does not mean the road is empty.

PROBLEM OF NOTHING AND ZERO – Introduction of zero came from India after the fall of Rome.  This is difficult because zero=nothing=something. This falls under ARISTOTLE’S law of contradiction. LEIBNIZ solves this law of contradiction by stating that an object can contain its own negation. Modern philosophers of mathematics have now asserted that zero is in fact a natural number.

3) Numbers as Logical Objects:
The problem of nothing and zero remained unsolved for 1000 years until FREGE.
He links logic and arithmetic in an overall system of philosophy of language. He attempted to demonstrate the logical basis for numbers therefore refuting Platonism. He also rejected MILL’S numerical empiricism, you cannot find zero in nature.
FREGE’S method:
Axiom= all things that are identical are equal to themselves (definitional, a priori, deductive truth).
- All things which are pairs are identical to other pairs.
- We assign a nominal symbol to this class of pairs (e.g. two)
- “One” is the class of all things not associated with other things.
- “Zero” is the class of all possible objects that are not equal to themselves.  “Null class”
- Therefore “zero” is defined into existence as a logical object.

Epistemology

Mill- Went beyond his predecessors by claiming that not only all science, but also all of mathematics is derived from experience. The axioms of geometry and the basic principles of mathematics are all founded on sensory experience. Mill sees the fact that the sum of equals is always an equal as an inductive truth, or a law of nature.

Newman- Belonged to the same empiricist tradition as Mill and disliked the German metaphysics, he described it as “a vain system of words without ideas”. Newman stated that the only direct acquaintance we have with things outside of ourselves is through our senses, so to think we could have “direct knowledge” of immaterial things is just ludicrous. Even our senses don’t allow us to be far out from ourselves e.g. we need to be near things to touch them. We cannot think or act without accepting that some truths aren’t intuitive and are not clearly demonstrated, yet they are still sovereign.
The difference between knowledge and certainty:
Knowledge= if I know “p” then “p” is true.
Certainty= I’m certain of “p” but “p” may still be false.
To be certain of something is to believe that it is true. Anyone who loses certainty off a statement was never really certain of it.

C.S. Pierce- Tried to create an epistemology appropriate for an age of scientific enquiry. Pierce observes that inquiry always originates in doubt and ends in relief, the only object of enquiry is to settle opinion.
4 methods commonly used:
- Tenacity
- Authority
- A priori method
- Scientific method
It is possible for “p” to be true even if every person on earth believes “p” is false.
Belief has 3 properties:
1. Something we are aware of
2. Appeases the irritation of doubt
3. Establishes a habit

Frege- Believed epistemology had been given a place in philosophy when it really should have been assigned to logic. A proposition may be thought of without being true, and a proposition may be true without being thought of. Frege created a doctrine of 2 separate worlds, one interior and private, and the other exterior and public. Physical things in the public world are accessible to all of us , but then there is also the private inner world of sense impressions and ideas. Frege also argued that there must also be a “third realm”, the realm of objective thought.

Russell- What is immediately known in sensation is different from any real object. Sense data are the only things we can be really certain of. What we immediately experience is an inference from what is immediately known. There is no proof that life isn’t just a dream but that doesn’t mean that it’s wrong. Knowledge of what is known by acquaintance is the only way we can have knowledge of things we have never experienced. E.g. how we have knowledge of “Europe” even though it’s too big to be comprehended by the senses.

Wittgenstein – Private language argument- there is no way of identifying terms of consciousness without referring to the public world. Doubt needs ground, genuine doubt will result in a change of someone’s behaviour. In order to express doubt that “p” one must understand what is meant by saying “p”.