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.

Logic and Mathematics

Natural Numbers- words used to count things.

Three basic approaches to number:

1)       Natural and can be empirically observed. -  Mill.

2)      They are institutions of a harmonic, perfect, platonic world. – Descartes.

3)      They are abstract logical objects constructed purely from syntax – Frege.

Syntax= logical system using rules of inference to alter the meaning of symbols.

Numerical Naturalism
Apes and stone age tribes appear to be able to judge empirical plurality, typically;
0 = Absence of a thing
1 = One banana/enough bananas
2 = A lot of bananas/unlimited bananas

“one thing” “more than one thing” and “lots of things” are the only number system they need.

Small number words are functionally different to large number words. If you come into a room and there is one person, you don’t consciously count that person, but people struggle once they get past 6/7 and have to physically count. So the number 7434 is a predicate symbol of more basic symbols organised according to known syntax, and as a predicate it can be analysed.

Pythagoreanism/Platonism
Prime numbers are pre-existing, eternal, supernatural forms. Necessary pre-conditions for consciousness. All other numbers are just rational combinations of prime numbers. Prime numbers exist in a non human dimesion and they are eternally true and ultimately mysterious. E.g. the religious significance of the number 3.

Pythagoras and (and all the Greeks) regarded only plurals as natural numbers so they begin counting with “two”.

Problem of nothing and zero
The concept of zero came from India, much later via Sufi Islam. The entire Arabic numerical system was introduced in the middle ages after the fall of Rome.
Zero =Nothing
Nothing = Something
Problem of the law of contradiction was solved by Leibniz’s monads that an object can contain its own negation. Modern philosophers have thus asserted that zero is a natural number. Logically derived as 1-1=0 “nothing” is a philosophical absurdity. The qualitative differential gap between 0=nothing and 1=something is as big as the universe.

The problem of nothing/zero remained unsolved until Frege.

Numbers as logical objects
Links logic and arithmetic in an overall system of philosophy of language, with arithmetic as a special case of language.

Axiom- all things which are identical are equal to themselves.
It follows – all things which are pairs are identical to all other pairs, regardless what they are pairs of.
The class of all pairs contains all pairs and this can be given a purely nominal symbol e.g. “two” a word or a numeral it doesn’t matter.

Epistemology
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.

He also claims that “the sum of equals are equals” is an inductive truth, or a law of nature I the highest order. Inductive truths = generalisations based on individual experiences.

Newman belonged to the same empiricist tradition as Mill. Newman believed that the only direct acquaintance we have with things outside ourselves comes through our senses, to think we have faculties for direct knowledge of immaterial things is merely superstition. Our senses don’t let u go very far out of ourselves, we have to be near things to touch them and we can neither hear nor touch things from the past or future. Reason is our way of proceeding from things which are perceived to things that are not. The exercise of reason is to assert one thing on the grounds of another.

Newman identified two different aspects that are exercised when we reason; inference (form premises) and assent (to a conclusion). These two things are very different and assent may be given without grounds, or based upon a poorly constructed argument.  Arguments may be better or worse, but assent either exists or it doesn’t. Locke argued that there is there cant be a concrete assent, as the assent must be conditional. Newman didn’t agree with this stance, there is no such thing as degrees of assent. However he still argued that assent based on evidence that is merely intuition may still be valid, e.g. all of us are certain that we will one day die but we cannot give exact evidence to support this.

There is a difference between knowledge and certainty. If I know “p” then “p” is true, but I may be certain of “p” but “p” can still be false.