Natural numbers, these are words used to count things. To count is to create an abstract category or group.

Apes and stoneage tribes have very simple number systems, typically “one thing” and “more than one thing” and “many things”.  These are the only numbers they need.

Even for people from advanced cultures small number words are functionally different to large number words. IF you come in to a room and there is one person, you don't count the one person; even with three you can categorise that plurality. Most people will go up to seven objects in a group before counting. Addition and multiplication - empirically - are plurals of plurals. You would not go to a football ground and look at the crowd and say there 37,523 people here. You could go to a football ground and say “there were four people there” without counting them.

Creating words and abstract symbols for plural categories (plural = more than one) requires a system of number words (‘symbols’) , and a logical syntax for combining these number-words (symbols) to imply further or predicate number-words. So the number 7,434 is a predicate symbol of more basic symbols organized according to known syntax (rules). And a predicate it can be analysed (operation similar to division and calculating number squares and roots) - “analytic philosophy” - the paradigm of analysis.  Technology - basic logical language, computers, facebook etc. Discuss - the limits of logical modeling of human intelligence, eg predictive texting (fuzzy logic vs neat logic - Stanford, etc, etc).

Ancient civilizations had hieroglyphs for numbers, and even multiples dating back to the Sumarian cuneform clay tablets, which recorded grain deliveries to the temple. Babylonian base ten and base five.

The greek and roman system depended of numeral symbols, and their system did not have zero, and one was not regarded as a number. Greek numerology (Pythagoras) regarded only plurals as natural numbers so they began counting with two. “One” and “Not One” were different logical categories.

The introduction of the concept of zero came from India,  much later via Sufi Islam. It is a very difficult concept, because: Zero = nothing = something (contra to Aristotle’s law of contradiction, the foundation of all logic). Problem of the law of contradiction solved by Leibnitz’s monads, that a object can ‘contain’ its own negation. Modern philosophers of mathematics have thus assert that zero is a natural number, logically derived as 1 - 1 = 0. “Nothing” is a philosophical absurdity, also the qualitative differntial gap between 0 = nothing and 1 = something is as big as the universe itself;  0 + 1 = 1’ but 0 x 1 = 0.

Number was seen as a type of magic, with the numbers as free floating perfect Platonic forms and attributed magical properties (especially 3, 7, 12 and 13). From the Greeks number can be derive apriori from geometry and (curiously) from aesthetics (especially music and architecture) where certain musical harmonies or spaci-al ratios (eg triangles) were held to be beautiful, and the subjective appreciation of beauty is the best proof for Platonists and Pythagoreans that the ratio is perfect, eternal, true… Thus starts the long relationship between numbers and music.

In 19th century - a quest for discovery of the logical foundation of numbers.  Steer a path between idea of number as a empirical generalization (the common sense view, but also the view of JS Mill) and Platonic idealism.

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Analytic philosophy - previously Frege on the logical underpinning of language (sense vs reference).

Frege also attempted to produce a logical analysis of mathematics. Russell was working on the same project independently. 

Arithmetic is mysterious. Many assume it to be empirical, this is the common sense view and it is not viable. For example if we have one drop of water and add it to a second drop of water the result is a single drop of water.

The alternative view is from Plato, that numbers are perfect and pure ideal entities which can be related to each other rationally, but with no corresponding objects. This may be the remnants of the Pythagorean religion. The various ratios that determine Platonic numerology are derived from aesthetics, especially from music. The relationship between music and mathematics has always been close for philosophers - Schopenhauer, Nietzsche…

Following Kant, Russell thought that number and arithmetic were neither Platonic ideal forms, nor empirical generalization, but that they were synthetic apriori propositions which could in principle defined logically from a limited set of axioms.  [more later… but first a word about Bertrand Russell.]

Bertrand Russell - 1872 - 1970
1907 - stood for parliament as a Sufferagette (lost to a Tory)
1913 - Principia Mathematica (age 41 - ‘exhausted’)
1914 - pacifist (WW1)
1940 - sacked by new york city university for immorality
1941 - renounces pacificism
1950 - founder, campaign for nuclear disarmament

His intellectual career began as a dedicated Hegelian idealist, he retained some of this in social theories, especially his popular scientific writing and broadcasting, which was strongly progressivist.

Mathematics appeared to be a contradiction of idealism, because numbers appear to have an objective existence in some sense, and their nature is not apparently determined or affected by the act of observing them.

Mathematical propositions are irreducibly relational says Russell. Numbers can not be “things in themselves” (Platonic, idealism) nor are they empirical generalization (the dominant empiricist view in the 19th century, from JS Mill’s writing on logic). This is what Russell retains from Hegel, that the whole is bigger than the sum of the parts; no isolated element of a system can be understood without first understanding the system as a whole. Individual numbers don’t mean anything, except as part of a system of numbers as a whole.

Russell embraced Mill’s extreme empiricist view in order to reject Platonic; but then found that Mill’s standpoint was not a sufficient guarantee of the truth of arithmetic propositions. Instead he wanted to reduce arithmatic and the relational values of mathematics to logic, outlining the likely logical underpinning of mathematics, so as to rebuild it on stronger foundations.

The questions is “What is number”; “what is ‘a number’” and what is meant by arithmetic operators such as addition and subtraction.

Peano had showed that all numbers could be deduced from a few axioms:

Peano’s Axioms:

1. The constant Zero is a natural number. The ‘number’ zero can be used to count. It woukld be counting of “there are no people in the room”. It is not “nothingness” in the metaphysical sense (thus zero is assumed and not proven - a significant weakness, Russell thought)

2 X = X; every number is its own equivalent (this is sub-divided into five types of equivalence)
3. Every natural number has a successor number (implies natural numbers are an infinite series. Russell objection is that the summation of an infinite series must itself be a finite number - the paradox of infinity)
4. There is no natural number whose successor is zero (negative numbers are not real ie they can not be used for counting.)
5.  If the successor of N is equal to the successor of M, then N is equal to M for all numbers in all series.
The terms “zero” and “number” and “successor” remain undefined by Peano.

Russell wanted to completed this project by providing objective or further axiomatic definitions for ‘zero’ and for ‘number’ and “successor of”.

The key terms Russell uses are “class” and ‘belonging to a class” and “similarity”.

Number in general as “the class of classes similar to a give class”

So number three is a word nominally corresponding a logical class composed of all possible classes have three members.

3 = a class containing all classes such as three cats, three dogs, three philosophers, etc

This avoids the problem of 3 cats + 4 dogs = 5 (catdogs) because the three is abstracted from the empirical basis, and is a purely logical category. The ultimate basis of this system is empirical observation, so Platonic idealism is avoid for 3 “as a thing in itself”.

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