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.

Ethics

There are many moral philosophers, dating back to Plato and Aristotle, which have treated happiness as the supreme good. However Kant in his “Groundwork” stated that duty, not happiness, was the supreme moral motive.

Bentham is one of the philosophers who does identify happiness as the supreme moral principle, but Bentham identifies happiness as pleasure. Bentham is a Utilitarian so he argues that pleasure is the main cause of all action, and an action should be considered in regards to the amount it increases pleasure or reduces pain. He not only regarded happiness as being the equivalent of pleasure, but he also regarded pleasure in itself as simply a sensation. “Pain and pleasure are what everybody feels to be such”. The relation between an activity and whether it causes pleasure or pain is just cause and effect. The value of every pleasure is the same, regardless of what has caused it, so the happiness you feel when you see a fine piece of art, and the happiness you feel when you when you itch that scratch you couldn’t reach, are the exact same. The important thing is the quantity of pleasure or pain caused, not the value of the action. Bentham offered systems for measuring pleasure and pain which are laid out in his “Felicific Calculus”. There are seven elements that have to be considered;

1.       Intensity

2.       Duration

3.       Certainty (how likely is it that the pleasure will happen?)

4.       How soon it will occur

5.       Fecundity (how likely is it that it will result in a subsequent series of pleasures?)  

6.       Purity (how likely is it that it will subsequently cause pain?)

7.       Extent (how many people will it affect?)

Number 7 is the most important one as the main foundation of utilitarianism is “the greatest happiness for the greatest number”. Bentham contrasted utilitarianism with asceticism, which is to judge an action by the amount it reduces happiness. However Bentham admitted that this principle of self-inflicted misery could never be upheld by any living thing.

John Stuart Mill was also a utilitarian, but in his “utilitarianism” he laid out that some kinds of pleasure are more valuable than others. This solved the criticism that utilitarianism reduces humans to being no better than swine by simply following our pleasures. It is this ability to distinguish between higher and lower pleasures which sets us apart from the animals, “it’s better to be a human dissatisfied than a pig satisfied”.

Schopenhauer  links his ethics to his metaphysical view that the world of experience is merely an illusion, and true reality (the thing in itself) is the universal will. Morality is a matter of training the character. The will is free from eternity to eternity but everything in nature, including human nature, is determined by necessity. If you could get complete knowledge of a persons character and the motives which drive them, then you could predict all of their future conduct. We believe we are free to decide because prior to the choice we have no knowledge of how “the will” is going to decide. The belief we can decide for ourselves is merely an illusion. Our wills can never change but there are different degrees of awareness of will. Repentance or regret never comes from the will, but from a greater sense of self awareness. Schopenhauer believes that it is unlikely we will ever be content. The will by nature is insatiable. The basis of all will is need and pain, we suffer until our needs are satisfied, but if the will becomes satisfied and loses desire then life becomes boring.

Kierkegaard’s moral system is similar to Schopenhauer’s in that they both take a pessimistic view. However whereas Schopenhauer’s ethics aims towards erasing individuality, Kierkegaard aims to put the individual in full possession of their own character as a unique creature of God.

Nietzsche says that history shows two types of morality; master morality and a morality for the herd (the poor and weak). Nietzsche argued that the revolt of the slaves triumphed with Christianity, “the success of Christianity lead to the degeneration of the human race”. To save the human race we must reverse the values of Christianity. Life forces us to establish values, and human life is the highest form of life so far, but it has sunk back to the levels of those which had preceded it. We must bring life to a new level past the system of master and slave, the Ubermensch (superman). The Ubermensch would be the highest form of life, the will to live. The will to live must not favour the weak, it must favour power, “man is a bridge, not a goal”. The Ubermensch will not be achieved through evolution, but only through the exercise of the will. Nietzsche encouraged war, war is an education in freedom.    

Seminar Paper - Logical Positivism, Wittgenstein and Karl Popper

In the early 20^th century Wittgenstein and Schlick met weekly, and were soon joined by other philosophers such as Carnap and Waismann, they were referred to as the “Vienna Circle”. After Wittgenstein left to work on another philosophical manuscript, the others developed into a self-conscious philosophical movement (logical positivists) and issued a manifesto against metaphysics, regarding it as an outdated system that must give way to science.

 They created the “verification principle” to show the difference between statements that mean something and those which don not. To verify something there must be facts, and statements which cannot be verified are just metaphysical nonsense. For example Descartes “cogito ergo sum” (I think therefore I am) is rejected because it can’t be verified, so it’s just empty words. However the verification principle came under scrutiny as it itself could not be verified.

Logical Positivists thought that the true task of philosophy was to clarify non philosophical statements. Their method of clarification was to show how empirical statements are brought up from “protocol” statements (direct records of experience). The meaning of the words used in protocol statements are derived directly from a feature of the experience that the word stands for.  This came under the criticism that protocol statements appear to be private to the individual, so if meaning depends on verification and everyone is carrying out their own method of verification, how can you understand what anyone else means? Schlick responded to this problem by showing the distinction between form and content. Content of experience is what I feel or live through when I see something, and is completely private and incommunicable. Form of experience may be common to many. For example when I see a tree I cannot be certain if anyone else has the same feelings as I do when they see the tree, but as long as we can all agree that the tree is green then we are able to communicate with each other.

Wittgenstein was dissatisfied with this response and he then strove to find a solution that didn’t pose the threat of solipsism. In the 1930’s Wittgenstein showed that private experience presupposes a public world. Even the words we use to describe our most private thoughts derive their meaning from the way we use them in public and open conversations. Wittgenstein later abandoned his previous thesis he had put forward in his “Tractatus” about the relationship between language and the world saying he over simplified it. He previously held that the connection between the two only had two features; the linking of names to objects, and the matching of propositions to facts. He now saw this as a big mistake, words might look like each other but their function differs. Language is interwoven with the world in many different ways which he referred to as “language games”. Wittgenstein referred to it as a “game” not because language is trivial, but because games have a similar variety to language. Like the logical positivists Wittgenstein was hostile to metaphysics, but instead of taking a blunt attack against it like they did with the verification principle, Wittgenstein carefully made distinctions between the mix of truth and nonsense within metaphysical theories.

Karl Popper didn’t see himself as a logical positivist and was referred to as the “official opposition”. Popper thought all scientific theories couldn’t be proved because of Humes problem of induction. He thought that we needed to progress by falsifying statements instead. For example the inductive statement “all men are mortal” cannot be falsified because it is at least possible that at some point at least one person will be born and be immortal. The problem of induction is that it is unreliable. For example a turkey a Christmas – every day of the turkeys life it wakes up, eats, then goes to sleep. Therefore the turkey presumes this will go on forever. Then at Christmas the Turkey wakes up and is killed. There is no previous empirical evidence to suggest that this would happen to the turkey. This example illustrates the point that no matter how much evidence we may have, we cannot predict the future.

Popper responded to the problem of induction by saying that everything has the potential to be untrue. For example Newton’s “Principa” gave an overarching theory of how the universe worked and was believed to be an undeniable truth. But then Einstein came along and proved that Newton’s theory was wrong. We constantly strive to get as close to the exact truth as possible, but Popper argued that we have to appreciate the fact that whatever we know now has the potential to be proved wrong at some point in the future.

Popper is also well known for his political philosophy with his publication “The Open Society and its Enemies”. Popper maintained that if a political organisation is going to flourish then it must leave room for self-correction. His political philosophy can be related to his philosophy on knowledge in that, just like science is constantly progressing through the correction of flawed hypothesis, society will only progress if policies are able to be evaluated and changed.  Therefore are two things required for an open society to work;
1. The ruled have freedom to discuss and criticise the policies proposed by their rulers.
2. It should be possible to change the ruler without violence or bloodshed if the ruler fails to promote their citizens welfare.
These are the two central features of an open society and are more important to democracy than just the election of government by a majority.

Popper didn’t rule out all forms of government intervention, for example incitement of intolerance should be classed as a criminal act and they state must protect the poor from the wealthy. Therefore we cannot have a completely non-intervention state and instead Popper argued that we should have planned economic intervention of the state. Unrestrained capitalism must be replaced by economic interventionism. Popper agreed to some extent with utilitarianism that the state must minimize avoidable suffering. Instead of building a utopia the government should act as problem solvers.

In the two volumes of his publication Popper attacked two philosophers; Plato and Marx. Poppers attack on Plato was against his “Republic” where Plato stated that the experts should be in charge and knowledge meant power. Whereas in the open society those who were being ruled would also have knowledge, not just the elite and those with power.  Poppers main attack focused on Marx and his claims that he had discovered scientific laws that determined the future of the human race, working towards an inevitable result. Popper showed how the course of history had in fact falsified Marx’s predictions. Popper was very anti-teleological, he didn’t believe that we could predict the future.

Popper thought that the human race is very vulnerable when it comes to conspiracy theories. This is because we like the idea of being to explain away everything, which most conspiracy theories appear to do. An example of this is David Ickes theory that a secret group of elites rule the world. This is not an unusual claim for conspiracy theorists to make, however Icke took this one step further by claiming that those within the elite group were in fact blood drinking lizards that morphed into people. However, despite how unusual Ickes claims seemed, he still managed to create a mass following preaching to large groups across the globe.

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.