The philosophy of mathematics

Mathematics in itself is an intriguing area of philosophy as it is potentially the basis for a great many things, primarily those of logic, understanding and overt meaning. Unlike language it has a far more straightforward existence and all that it truly relies upon to differ is the context of the time, whereas language is far more subjective.

The logic principles of mathematics are many and varied. Firstly they are based in the context of their time. For example, to modern audiences it is accepted that zero and one are the first two positive numbers to exist. To an Ancient Greek audience they are to be understood as a completely different ideological principle, that being that zero is representative of there being nothing present and thus can not be counted as a number would be, and one is a singular existence that warrants no recognition past the fact that there is an solitary item present, whether it is a person, animal, building or something else.

Furthermore, if we hail back to even earlier times the accepted means by which to count things would be "one thing", "more than one thing" and "many things". Even within modern audiences there are dramatic differences between smaller and larger numbers logically speaking. Smaller numbers, those which we do not necessarily have to count to recognise, are known as "Natural Numbers". These would be numbers which are minds can automatically recognise without counting and typically run up to seven (although there are extreme examples of more or less). Even a number as simple as 17 would not be considered a natural number as the average human mind can not automatically recognise the presence of such a number.

At its core basis mathematics is a language constructed for the purpose of grouping particular qualities together, whether it be a general genus, such as six cats, or a more specific aspect of a particular object, such as 42 windows. Yes there are many variations of this principle, such as measurements of distances or the quantifying of objects' capacities, but it all ultimately boils down to the same ideal: grouping the aspect of something into a definitive number. This is what addition and multiplication in essence are. Essentially non-natural numbers (numbers which we do not automatically recognise) are the equivalent of creating words from standard letters. As such they have their own particular syntax, though a much simpler and more direct one than that of language. Division and subtraction are means of analysis of the structure of numbers. They are a means by which to understand how precisely a number can be formed. Numbers such as 827 are predicates of simpler numbers and as such can be analysed through the use of division and calculation of square roots.

It is reasonable to say that the beginnings of any form of numerology begin in the assorted hieroglyphs of the ancient world. Although termed differently, modern numbers are for all intents and purposes, simply a newer version of hieroglyphs. This is one of the other reasons why zero was not present within numerology, that a figure had not been present to represent it until it had come from India via the Sufi Islamic language (although this is a minor reason compared to the ideology of being a different philosophical imperative altogether). The issue of the concept of zero as a number is that it contradicts Aristotle's imperatives that something can not contain its own negation, which the existence of zero effectively asserts. This issue is resolved by Leibnitz's monads which argue that an item can contain its own negation. Modern philosophers have reasoned that zero is a natural number and that the differential between 0 being equal to nothing and 1 being equal to something is as big as the universe itself.

Because of the perfection they represent, their impossibility to be anything other than what they are, numbers were once seen as a form of magic, particularly the numbers 3, 7, 12 and 13. They float in perfect Platonic forms. Out of context these numbers might seem strange, but when given the context of geometry, music or architecture it is far clearer to understand, as these particular numbers have in many ways a balance. A harmony in themselves which can be best displayed within mediums such as music. There is a beauty within them, and beauty is only possible through a balance and perfect ratio.

The logical underpinning for mathematics pioneered by Frege was based initially in his work on language. Frege argued that empirical data for arithmetic is unreasonable as it is not something that can be tested like you would the weight of something or the time it takes for something to reach point B from point A. Two water droplets added to each other result in just one larger water droplet. His argument, based in Plato, is that numbers are perfect and relatable to each other, but have no corresponding objects, that is to say that there is no one particular thing which has two of something or that can be grouped specifically as only two.

According to Russell numbers are neither empirical data nor Platonic ideal forms but are instead the result of a-priori propositions which are definable by a limited set of axioms or premises. Russell's ideology is that numbers are not "things in themselves" as Platonists believe, nor are the empirical generalisation as was the dominant opinion of the 19th century, fronted by Mill. Numbers are in effect fractions of the whole, retained from Hegel's teachings is that to understand the elements within the system one must first understand the system as a whole. Numbers only hold meaning upon being considered as elements of the whole.

Although retaining the empiricist view to a certain extent, it did not allow Russell the full extent of explanations necessary, those being the answers to the following questions:

1. What is a number?
2. What is 'a number'?
3. What is meant by arithmetic operators such as addition and subtraction?

Guiseppe Peano, the Italian mathematician, showed that numbers could be deduced from very specific axioms:

1. Zero is a natural number and can be used to count. To use it in counting is along the lines of saying that there is no one in the room. It is not nothingness in the metaphysical sense.
2. X=X. Every number is its own equivalence, which is further sub-divided into five types of equivalence
3. Every natural number has a successor
4. No natural number has the successor of zero. That is to say that negative numbers are non-existent as they are not a means by which to count
5. If the successor of N is equal to the successor of M, then N is equal to M for all numbers in the series

Zero, number and successor all remain undefined by Peano. Russell wished to create definitions for these through objective or further axiomatic means. As such he used the terms class, belonging to a class and similarity. Thus, a number is a means by which to establish the definition of a class or group, such as a 3 cats, 3 dogs or 3 philosophers. 

This ultimately resolves issues such as illogical additions, such as 3 cats + 4 dogs = 5 catdogs, as we have definitive specifications. Such a sum would be resolved through the simplification of using similarity, such as 3 animals + 4 animals = 7 animals.