#dedekind

Frege’nin Mantıksız Mantıkçılığı

Ah şu zavallı Batı adamı… Hakikatin “merkezî karargâhı” kurulmadığı zaman, hakikat arayışınızın tek usulü “deneme yanılma”dan ibarettir. Batı bilim telakkisinin bütünü “deneme yanılma” yoluyla inşa edilmiştir.

Modern Matematik felsefesinin babası Gottlob Frege’nin, Aritmetiğin Temelleri adlı eserinde Leibniz’in görüşlerinden oldukça faydalandığı görülmektedir.

Lei…

We have seen two approaches to finding an answer to the question "What is a natural number?" The first is the one by Dedekind: starting from a Dedekind-infinite set he made a set that behaves exactly the way we want the natural numbers to behave. The second is the one by Peano: a list of properties that the (set of) natural numbers should satisfy.

Both approaches complement each other; in the language of Mathematical Logic: Peano formulated the axioms of arithmetic and Dedekind made a model for the resulting theory.

The only gap in this whole argument is Dedekind's first step: Take a Dedekind-infinite set, that is a set S and an injective map φ from S to itself that is not surjective. In Wass sind und was sollen die Zahlen? there is a beautiful proof of the existence of this kinds of sets (Satz 66 on page 357). That is to say, beautiful in the sense of the images it evokes but not in a mathematical sense, and by today's standards wholly inadeqwuate. To some extent that is not Dedekind's fault: Set Theory was in its infant stage and Dedekind's example, "Meine Gedankenwelt, d. h. die Gesamtheit S aller Dinge, welche Gegenstand meines Denken sein können" ("the world of my thoughts, that is, the totality S of all things that can be the subjects of my thinking"), was considered a ligitimate set.

What later became apparent was that "There is an infinite set" actually had to be an extra axiom/assumption. The question then becomes is how to formulate that assumption mathematically and in an as simple as possible way, and by `as simple as possible' I do not mean `easily understood' but rather `using the simplest possible set-theoretical tools'. The following formulation was worked out by Von Neumann.

Consider what we need: a set, a map and a special point. The simplest possible point there is is ∅, the empty set. We also have a map that yields a kind of `successor': φ(x)=x∪{x}, the union of x and {x}. (In Set Theory every individual is a set, so this map is well-defined.) It is a goed exercise to show that φ is injective.

The Axiom of Infinity postulates the existence of the desired set: There is a set S with the following two properties

∅ belongs to S

for every x we have: if x belongs to S then so does φ(x)

To this set, this map and this point we apply Dedekind's procedure; the set of natural numbers, N, is the smallest subset of S that meets the above requirements.

This N is the standard set of natural numbers. The smallest element of N is ∅ and we also call it 0. So 0=∅. The next number is {∅} or {0} and we denote that 1. Then comes 1∪{1} and thet is {0,{0}} or {0,1}, this set then is 2. Continuiung in this way we find that 3={0,1,2}, 4={0,1,2,3}, etcetera.

Set-theoretically this is quite economal/elegant, every natural number is de set of its predecessors; so we have m<m exactly when m∈n, the element-of relation gives us the less-than relation.

The question has been posed explicitly here: “What are numbers?”. The answer that was given there may not feel completely satisfactory: “We do not know what numbers are but we do know how they behave”.

Practically everyone will say: “we use them every day, so they are there!” That is, taken very strictly, not true; what we use is a system of notations and abbreviations that represent and describe something that we all agree upon. However, who ever observed that thing that we denote by `the number 3’: let me know where that was. The idea that we can stick a label like `3’ on something and that almost everyone in the world will associate that with the same notion is quite exceptional.

Mathematics has functioned quite well without a proper definition of, say, `natural number’. That nobody noticed this was probably due to the aforementioned collective act of abstraction (or conspiracy if you will). One of the first who realized that mathematics did not have a `standard’ set of natural numbers was Richard Dedekind. In his Wass sind und was sollen die Zahlen? he took matters in hand and created a set of natural numbers.

That `creation’ went as follows. He took a (Dedekind-)infinite set, S say. That means that there is a map, φ, from S to S that is injective but not surjective. Thus we have a point a in S that is not of the form φ(s) with s in S. What Dedekind showed was that in this situation there is a subset N of S that satisfies

if s in N then also φ(s) in N

N is the smallest set that satisfies 1. and that contains a

a≠φ(s) for all s in N

φ is injective

There are a few things in this list that were said before but what Dedekind showed in the rest of his booklet was that these four properties suffice to have this N serve as the/a set of natural numbers. The point a plays the role of 1, we can define addition and multiplication in such a way that all requirements we have of the natural numbers are met (and φ(s) is just s+1).

This then means that every Dedekind-infinite set S carries such an N within itself and that all those N’s can serve as `set of natural numbers’. This appears to lead to a huge variety of natural numbers but we have a solution for that: those N’s are isomorphic. That means: if N and M are both sets of natural numbers, with special elements a and b respectively, and their respective maps φ and ψ, then there is a bijective map f from N to M that satisfies f(a)=b and f(φ(s))=ψ(f(s)). Everything we can do with N we can do with M and vice versa. Or, as a mathematician would express it: up to isomorphism there is only one set of natural numbers.

We reject others based in what we reject in ourselves. We accept others based on what we want to accept in ourselves. This is how universal compassion is connected to self acceptance and eternal love.

Fordism is the material practice that created the need for compliance.  Training people to follow exact processes is what public school is.

The creation of similitude in products however, also creates a similitude in subjectivity qua consumerism.  As explored here: The Gollum Effect we can see how formalization of products = the mechanization of subjectivity.  Indirectly, our proliferation of attorneys and laws is due the formalization of production and consumption...the need for "correct" procedure demonstrates that products are actually material epistemes for the alignment of our consumption as a knowledge milieu and as praxis.  Bear witness to how being in the know also means being in the know of material process (how to correctly eat sushi) as well as knowledge generated by products (various kinds of vacuum cleaners, and their history as put forth by hipsters, trivia and game shows).  The process of production/consumption belies boundaries of which attorneys are there to formalize social arrangement; reinforce the status quo -- even in the case of discrimination or accessibility, what's being enforced is the consumerist ideology that anyone should be allowed to participate if they can afford to pay/work.

An extension of this thought belies the actual social ideology; the ideology of merchant capital.  Biological drives are captured for the smoothening and extension of capital flows through advertisement and alignment of labor-subjectivity to consumer-subjectivity post WW2.  In this sense, Marxism, spreading prolitariat subjectivity worked in favor of merchant capital by allowing workers to fully identify with their work situation.  This identification was later shifted into an identification with one's white collar jobs, in which technical know-how replaced identity as a marker of one's status, verified on the one hand by increased access to resources (as a consumer) and on the other hand, assured place in the techostructure through the extension of the megalithic corporate ladders that exist today. The capture of biological drives, however, isn't as simple as advertising lifestyle predicated on a techne based milieu of products (think about the dads that stand round with their beers admiring one's car and talking about the product specificiations or the moms sharing recipes -- process based production, often now sprinkled with the insertion of consumerist product, use this brand of chocolate not this one for this fudge).  Biological drives are notoriously atonic, that is, biology lacks inflection until an external marker or a structural lag provides a cue to act.  Since product specification is too refined for biology to care money plays a role as well in refining the drive state.  Money works like Dedekind's cut for real numbers.  In this cut, money provides inflection so that people can objective a decision making strategy for a various situation based on their ranking in the capital exchange marketplace.