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.

A bit of light reading ...

Bertrand Russell ‘A History of Western Philosophy’ is not your typical light reading, carrying it around is a work out in itself. Unfortunately as a champion of slow reading, working through book one resulted in many late nights, and a few too many cups of coffee. Caffeine fuelled I managed to work my way through it, and just in the nick of time.

Part one concerns Pre-Socratic philosophers, based in the city of Miletus in the region of Ionia, who created a foundation of philosophy for Socrates, Plato, and Aristotle. The first Pre-Socratic philosopher we encounter is Thales. As a Greek philosopher, scientist, and mathematician he theorised that everything in the universe is made up of water. What was also found to be shocking was that he made no accounts of ‘the Gods’, which at the time were central to civilisation. However his upmost achievement was the successful prediction of a solar eclipse in 585 BC.

Anaximander, who was thought to be the student of Thales, argued that ‘the boundless’ is the source of all things, and it is an accumulation of all the elements which makes up the universe, not just water. Anaximander was criticised by Aristotle for never giving an explanation of what he meant by ‘the boundless’.

Pythagoras is the next philosopher to cross our path, but unlike Thales and Anaximander he was more focused on the idea of form, rather than matter. He wanted to use mathematics and geometry to explain the universe and all that is contained within it. However I cannot deny that I am pleased that not all of his theories have made it through the generations, I don’t think I could survive as a student without Heinz baked beans. With Parmenides there is very little to say on him, other than he believed that everything comes from one substance.

 It then leads us to Heraclitus, who is considered to be the most important of all the Pre-Socratic thinkers, and possesses a pessimistic view of human nature. Writing his doctrine ‘On Nature’ he stated that the world is made up of a conflict of opposites and is in a constant flux, coining the famous phrase ‘you can’t step into the same river twice’.

Part Two focuses upon Socrates, Plato, and Aristotle. These familiar names were surprisingly reassuring, bringing back memories from A-level philosophy, a subject I felt so grateful to escape back in June.

Socrates was the teacher of Aristotle, and one of the most influential philosophers of all time, despite never actually writing anything down. As the years went on he became absorbed by philosophy, resulting in self-inflicted poverty. Unfortunately his life was cut short as he was accused of impiety, and had to end it himself by drinking hemlock.

Plato is a well-known student of Socrates, and most of what we know of Socrates has come from Plato’s writings. His most well renowned piece of work is ‘The Republic’, which seems to lay out his plans for a utopian society. He covers the concept of justice, politics, ethics, and what is knowledge?

Aristotle was a student of Plato, who argued about ethics, politics, physics and logic. Unlike Socrates and Plato, who believed that knowledge was innate, Aristotle argued that knowledge is gained empirically. Like Socrates he was charged with impiety, however unlike Socrates, he ran off and escaped his fate.

Part Three tells us about Stoicism. Their main belief is that only Zeus is granted with immortality, the other Gods were created at the beginning of the Cosmos, so are not as high in status. If you lived a ‘good’ and virtuous life you would be granted with immortality and a place among the Gods.

For the Epicureans, the soul and death affected their view of the afterlife. They stated that the fear of death is the main thing that plagues the human race, not death itself.

After the Hellenistic period, the population fell under the sway of Aristotelian teachings again, where no part of the human soul possesses immortality.

I feel it is safe to presume that you are damn right knackered from reading all of this, and I am shattered from writing it. I bid you adieu.