# Logic and Mathematics

Frege:

1876: Begriffschrift (trs: ‘concept script’)

This is ‘analytic’ or ‘symbolic’ logic which replaces the syllogistic logic of Aristotle.

Aristotle:

A = man

[b] = mortal

X = Socrates

All [A]s are [b]  {axiom; apriori}

X is [a]

Therefore

X is [b]

Frege’s propositional syllogism. He discovered that IF is the most powerful word in logic and removes the need for geometric axioms which, according to Hume and Kant, can only exist in mathematic (e.g. Euclid’s Pythagorean Theorem). His logic, sometimes called ‘fuzzy logic’, can provide conditional truths. Provisional truths correspondent more closely to sense experience, especially in Kant’s epistemology.

IF  X = A; then X is [b] {argument} : If Socrates is a man then Socrates is mortal [is true]

AND  X = [a] {datum} : Socrates is a man

THEN X = [b]: Socrates is mortal

This is a closed logical system which process data from ARGUMENTS into FUNCTIONS via OPERATORS.  There is no need for axioms.

In the example:

“IF…. (etc) is the ARGUMENT

THEN… (etc) is the FUNCTION

The structure of the proposition is THE OPERATOR

The FUNCTION varies according to the DATUM and not the ARGUMENT (and there are not axomatic truths).

This breakthrough eventually leads to TURING and the writing of “computer language” based on entirely on Frege’s propositional logic. Modern logical language “computer code” is has many subtle innovations so that its various OPERATORS can produce much more useful FUNCTIONS than proving that if Socrates was a man then he was a mortal.

Example:

IF  X = A; then X is [b] {argument} : If five minutes have passed then traffic lights must be switched to red [is true]

AND  X = [a] {datum or variable} : five minutes have passed

THEN X = [b]: then the traffic lights must be switched to red.

This is the sort of thing that is going on in the microchip circuits of computers all the time – but with far more complex operators, expressed as computer programme lecturing.

The revolutionary aspect is the replacement of axioms with operators. The data – or variables (which could be “trees” of further logical operators reaching incredible complexity; or could be simple datum such as “the traffic lights are now switched to red”) fuels the whole system. The operators, importantly, are what Kant would called APRIORI truths – truth by definition, and both universal and necessary. Valid Apriori propositions  (e.g. all triangles have three sides) can never be false in any universe which corresponds to human perceptive apparatus (Kant: “intuition”).

Frege creates something very like Platonic perfect forms – Plato’s veneration of mathematics and geometry as proof of a perfect and eternal realm; and his scepticism about the reliability of sense perception (which was disputed by Sextus Empiricus, then carried into the modern era by Descartes, then disputed by Hume and Mill, then revived again by Liebniz and Kant and then in our own time perfected into an orthodoxy by Frege and Russell).

The “ghost in the machine” is not Hegel’s WorldSpirit or God; the ghost is the eternal, universal and necessary propositions of Fregean logic, grounded on Kant’s materialist metaphysics of the limitations of human sensory perception.

Frege on Arithmetic:

The Grundlagen der Arithmetik (1884) published around the same time as Thus Spake Zarathustra; and about the same time as the first Impressionist exhibition in Paris; and the premier of the Ring Cycle. This was the birth of “high modernism” and the “belle epoch” of Europe, to be fulfilled in the next generation in Vienna, with logical positivism, post-impressionism, abstract expressionism. It 100 years since the French and American revolutions, Kant and Mozart.

Logical proof of arthimetical operations like “plus” and “equals”. E.g. 2 + 2 = 4. How to prove that this is a universal logical necessity and not an empirical generalisation. You can not check empirically that all pairs of pairs (2 + 2) is exactly equivalent to all other pairs of pairs which you might be please to call “4”.

Then there is the problem of “zero” which is an empirical impossibility. There is no natural zero (as Kant showed; the existence of space and time is a necessary precondition for all and any congnition).

Frege in the Begriffschrift discusses the ambiguity of propositons such as “nobody knows everybody” and “any child can master all the languages” and how this differs from “all children can master any language”. The problems of QUANTIFIERS, which have to be added to the  basic operators to enable the FUNCTION (or truth value) of OPERATIONS to be valid.

The Logical Proof Of “ZERO”

Unknown to the greeks; “impossible” to Plato: “something can not come from nothing”.

Frege starts the logical proof of all numbers by proving zero.

ZERO is the class of all objects which are not exactly equivalent to themselves. In other words it is total “number” of all propositions which have the truth value of “false” when Aristotle’s law of non-contradiction is stated.

For example this class would include the proposition (as it would be stated using Begriftscrift) Venus is not venus: true (p | p : T).

It turns out that there are “zero” propositions of this type.

Therefore the total “number” of propositions in the class “objects which are NOT the exact equivalent of themselves” is “Zero”; or rather Zero a SIGNIFIER (or symbol) of the class of all things which are not equivalent to themselves.

Having proved the zero logically, empiricism in arithmatic can dispensed with (except in everyday pragmatic use) CLOSED logical definition of quantifiers can be added to the OPERATORS in logic.

Moving on to the non-empirical, non-naturalistic proof of “ONE”…

ONE is a signifier of the class of all classes which are not equivalent to themselves.

This builds on the breakthrough definition of of zero. Zero (remember) is the class of things that are not equivalent to themselves. There is only one such class. So ONE is the number of possible numbers that could be zero. There is only one number which could be zero. So ‘One’ is derived logically from zero.

As it happens BINARY arthimatic is all you in fact need for any arithmetical calculation; and all computer programming is done using binary which ony needs ZERO and ONE.

0 0 0 0 1 = 1

0 1 0 0 1 = 2

0 0 1 0 0 = 3

[or any nominal value you wish to attribute to the ARRAY (the above is not nominally accurate, just and illustration)].

Frege nevertheless logically defines two, three, etc.

Two is the class of classes whose members are not equivalent to themselves AND the class of whose members are not equivalent to themselves.

So for Frege: 0 + 1 = 2 (the infinite hotel paradox). LOGICAL number sequences begin with 0 as the first number, 2 as the first number, 3 as the second number. This is because there is nothing mystical or fantastical about ZERO. It is a logical object like any other.

In 1903 Bertrand Russell published The principles of Mathematics.

The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself.

Frege had a nervous breakdown when he discovered this; Russell pointed it out. Russell worked on the problem for 20 years; but gave up and wrote pot-biolers like History of Western Philosophy and also set up Campaign for Nuclear Disarmament.

Sense and Reference

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

Cartesian Skepticism

Frege showed certain knowledge is not possible. Logical propositions can be true or false at the same time.

“The Morning Star is not the Evening Star.”

Wittgenstein and Language

Wittgenstein moves frege on; there is only language, reality is only language, reality is merely a collection of statements of facts. He wrote the notes for Tractatus while he was a soldier during World War I.

“The world is the totality of facts, not of things made up of the world.”

Similarly, nihilists see a world consisting of facts but devoid of values. Thus, nothing is morally wrong or right. This view is also highly held by existentialists who challenge the idea of there being a meaning or purpose to life.