Frege becamethe second founder of Logic, after Aristotle. He investigated and systemised logic which led him to conclude that logic was a priori and analytic. The definition of logic in its simplest form is that it sorts out good inferences from bad. For Frege the most important part of logic was validity and invalidity of a particular form of inference, shown through syllogism. For example, a valid inference would be:
All cats have fur
Some Cats are black
Some cats have black fur
Whereas an invalid inference would be:
All cats have fur
Some cats are black
All cats have black fur
Frege devised a system to overcome the difficulty of invalid inference that lead to false conclusions, in his book Begriffsschrift he explained that in order to overcome a false conclusion you had to replace the grammatical notions of subject and predicate with new logical notions which he referred to as argument and function. For example take the statement:
“Her Majesty The Queen is Elizabeth”
The function in this statement is “Her Majesty The Queen” and the argument is “Elizabeth” as this part can be altered thus turning a true statement into a false one. For instance if I changed the argument to Sandra then the statement would read “Her Majesty the Queen is Sandra” and therefore that would be wrong. In the phrase “Socrates is mortal” the function would be “is mortal” and the argument would be “Socrates”as no matter what name was put in replace of the argument the function would maintain its truth. (x)(x = mortal) The subject in the predicate calculus is always an individual entity, never a group or class. An individual man can be treated as a subject but the class of all men must be treated as a predicate. E.g if M = predicate for “to be a man” and a = the individual “Socrates” then Ma denotes that “Socrates is a man”. a = the argument, and M can be applied to any individual.
Russell and Whitehead carried Frege’s work forward, in the work ‘The Saga Principia Mathematicia”, which contains a systematisation of logic. In brief, both propositions ‘p’ and ‘q’ must be true in order for ‘p + q’ to be true. Frege, Russell and Whitehead, all hoped that they had established that arithmetic was a branch of logic.
Epistemology
Mill’s System of Logic outlined the doctrine that knowledge is derived from experience. In contrast to Frege, Mill believed that logic propositions were a posteriori, affirming his support towards empiricism.Mill claimed that science and mathematics were derived from experience. To focus in on mathematics, Mill claimed that each number involved an assertion of a physical fact: one being singular, two being a pair, twelve being a dozen etc, the numbers two, three, four etc all denote physical phenomena, and connote a physical property. For example, two denotes all pairs of things and this connotes physical pairs of things, e.g two apples, which is different to three apples. However, Mill admits that senses find it difficult to distinguish between 102 apples and 103 apples. Mill’s thesis was that arithmetic is an empirical science.
John Henry Newman was part of the same empiricist tradition as Mill, he believed that the only direct acquaintance we have with things outside of ourselves is through our senses. Reason is the faculty of the mind that we have to use in order to discover knowledge of things external to us, such as facts and events that are beyond the range of senses. Newman states that when we reason we have to use two different functions: inference(from premises) and assent (to a conclusion) and they are very different. Assent may be given without adequate evidence or argument which can often lead to error.
The difference between knowledge and certitude is that if you know ‘p’ then ‘p’ is true. If I am certain of something, then I believe in its truth, even if my mind should let the belief drop. But what looks like certitude can always turn out to be a mistake. Take for example the criminal justice system, if a jury finds a defendant guilty, then you believe that it is true, however there are known cases in the world where this has been a mistake, a miscarriage of justice. Therefore how can you hold certainty when in the past you may have thought you were certain about an untruth?
Frege on Logic, Psychology and Epistemology
When Frege was writing his logicist works from Begriffsschrit, he was not interested in epistemology;he was more concerned with setting out the relationship between epistemology and other related disciplines. He was more anxious to show the difference in nature and role between logic and psychology; adapting Kant’s distinction between a prior and a posteriori knowledge. Knowledge is belief that is both true and justified.
If the proposition is a mathematical one the justification must be mathematical. Mathematicians have sensations and mental images that play a part in the thoughts of someone calculating, but this is not what arithmetic is about. For instance at school, when you had to mental arithmetic in class, most likely everyone had a different method and mental picture to help them process an answer; different mathematicians have different images of the same number e.g 100 or C. Psychology is interested in the cause of our thinking, whereas mathematics is interested in the proof of our thoughts.
If humans have evolved then there is no doubt that consciousness has evolved. If mathematics is a matter of sensation and ideas then we should warn astronomers away from drawing conclusions from events in the distant past. Frege brings out the absurdity of evolution by using the example of 2 X 2 = 4. How do we know that this proposition already existed in a distant past as 2 X 2 = 5.
Frege claimed that there was two separate worlds, an interior private world, and an exterior public world. Although mental images are private, thoughts are the common property to us all. Perceptible things of the physical world are accessible to us all, for example we can all see the same houses, but the inner world of senses, impressions, images, feelings desires and wishes are what we may call ideas. Frege accepts the Cartesian distinction between matter (the world of things) and mind (the world of ideas), and similarly to Descartes accepts that there needs to be an answer to idealist scepticism: the thesis that nothing exists except ideas.
Frege concludes that either our ideas which are the object of our awareness is false or all knowledge and perception is restricted to the range of our ideas. For example if we live in a materialistic world where we want a fast car, then our perception will be bound by this idea, whereas someone who may be a a nature lover may not pay attention to the fast car as it is not an idea of relevance to them, and therefore would notice the willow tree. Picture a filter, if we held up a colour filter in front of our eyes, our self consciousness would only filter the ideas that we are interested in and therefore bound our knowledge to these ideas.
Frege concludes that either our ideas which are the object of our awareness is false or all knowledge and perception is restricted to the range of our ideas. For example if we live in a materialistic world where we want a fast car, then our perception will be bound by this idea, whereas someone who may be a a nature lover may not pay attention to the fast car as it is not an idea of relevance to them, and therefore would notice the willow tree. Picture a filter, if we held up a colour filter in front of our eyes, our self consciousness would only filter the ideas that we are interested in and therefore bound our knowledge to these ideas.
If there is no owner of ideas, then there are no ideas, there cannot be an experience without someone to experience it. For instance if my hand touches a hot pan, then that pain and burn is felt, what is felt therefore must have someone feeling it; and this is an object of my thought, something that is not yet my idea.
Whereas Descartes’s ego was a non-ideal subject of thinking, Frege’s ego is a non-ideal object of thought.Its existence disproves the thesis that only what is part of the content of my consciousness can be the object of my thought. If there is no such thing as science, then Frege maintained that a third realm must be recognised; in addition to the world of things and the world of ideas. This third realm is the realm of objective thought, it is the ego as the owner of ideas which is an important part of this realm.
Frege states that we are not owners of our thoughts, like we are owners of our ideas. We do not have thoughts, thoughts are what we grasp, and what we grasp is already there, we simply just take possession of it. 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.
Mathematics and Numbers
Mathematics is the science of quantity. There are two branches of mathematics: arithmetic and geometry. Until the lecture last week, I had never given numbers much thought, so it was interesting to look deeper into arithmetic.There are natural numbers, these are where we use words to count things, there are three approaches towards them.
1. They are natural
2. They are intuitions of a harmonic perfect platonic world
3. They are abstract logical objects constructed purely from syntax
Syntax is a set of rules that modify the meaning of symbols, words and numbers.
Numerical naturalism
Numbers that are natural are those that we do not have to count. For instance when you walk into a room, there may be three people sat down at a table, we don't have to physically count one two three, as our mind can picture it, and some people can picture up to seven objects without needing to count. These small number words exist as a concept. Even apes and stone age tribes appear to be able to judge empirical plurality, simple number systems,such as “one thing” and “more than one thing” and “many things” are the only numbers they need. If you put a bunch of bananas in front of an ape they would classify it as lot of bananas, but if you put nothing in front of them they would know that there was an absence of bananas. Plurality is not exact numbers they are numerous phrases such as "few" or"many" or "more than one".
Pythagoreanism
Pythagoras was the founder of the religious movementphythagoreanism.
Prime numbers are indivisible numbers such as 3 and 5, they are only divisible by 1 and itself. All the other numbers are known as composite numbers, and these are rational numbers that represent geometrical ratio's.
Pythagoras worshipped the religion of triangles. Three is known as the magic number, think of all the examples we use in everyday like: rule of thirds, three chord triad, three part drama.
Christians are obsessive over the number three, a few examples include: The Trinity - The Father, The Son and The Holy Spirit, Jesus rose on the third day and the sign of the cross is made with three fingers.However, Muslims worship the number one, as they believe in unity, one god, one substance: Allah. Contrasting this, the Greeks feared the number one, they disregarded it and referred to the number two as the first number.
The problem of nothing and Zero
The concept of zero came from India, zero is nothing but nothing is something, which is contra to Aristotle's law of contradiction; this was the belief that something could not be itself and another thing at the same time. The problem of this law was solved by Leibnitzs monad's, who stated that an object can contain it owns negation.
Modern philosophers consider zero as a natural number, logically derived as 1-1 = 0 "Nothing" is a philosophy absurdity, and the qualitative gap between nothing which is 0, and something which is 1 is is big as the universe.
What does 1 really mean?
If you add 1 + 1 it doubles in size, but if you have 102 apples and add one which is the example used earlier then it' s an infinitely small increment. You cannot find the noumena of numbers, they can not be "things in themselves".
The problem of zero remained unsolved for thousands of years, until Gottlob Frege. He wrote many influential books such as 'The foundations of Arithmetic' where he links logic and arithmetic.
Frege's Method
Frege's axiom is that all things which are identical are equal to themselves. He followed that all things that are pairs are identical to all other pairs, regardless of the what they pairs were of.
The class of all pairs can be given a nominal symbol - (2) whereas (1) is the class of all things that are not associated with other things. The class of (0) is the category that is empty known as The Null this is the class where all possible objects are not equal to themselves. Relating back to the axiom there are no such objects, by definition.
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