Thinking about the new maths (of the sputnik crisis) and 'set theory' brought some other connections to mind, quite apart from the empty set as mathematical ontology of space. I have posted this piece of 'set theory' before as a sort of 'mathematics of space'.https://en.wikipedia.org/wiki/Empty_set#Philosophical_issuesDarling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."
The 'New Maths' refers to the changes in maths education and America brought about by the notion that 'the soviet union were pulling ahead in the space-race'. At this point 'Set theory' was introduced into the curriculum and maths was 'reconfigured'. I will post this horrific piece of ideological terrorism once more.http://www.math.rochester.edu/people/faculty/rarm/smsg.html
One thing that even non-mathematical people know (especially social scientists) is that the Zermano-Fraenkl version of set theory, known as ZFC, allows a person to reconstruct 'the entirety of mathematics' on the basis of sets (how convenient if you happen to belong
to one of those sets. Tough luck if you dont.)
Here is 'set theory' and its Zermano-Fraenkl form.https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryZFC is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of ZFC refer only to pure sets and prevent its models from containing urelements (elements of sets that are not themselves sets).
What does wikipedia mean when it says 'all entities in the universe of discourse are such sets'?https://en.wikipedia.org/wiki/Domain_of_discourseIn the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
(Is this where we live now? In the domain of discourse?
or in the Von Neumann universe?https://en.wikipedia.org/wiki/Von_Neumann_universe
What is very strange is that, in the context of 'the space-race', the powers that be were able to present 'set theory' as a step forward, as a progression, the objective movement of history. In some respects, at least in its public presentation, it was suggested to be supplanting other axiomatic systems, like Euclidean geometry or basic number theory.
I am deeply uncomfortable about this notion: As I said in another thread, I am fairly uncomfortable about set theory in general. I think that there is a lot that has to be said about Georg Cantor and his 'transfinite numbers' (this notion that infinity is a mathematical matter, as opposed to also a metaphysical one and, dare I say it, a theological one), as well as the more general questions about trying to uncover the ultimate teleology of a given piece of mathematical abstraction. Is it meaningful to say that 'all mathematics can be reconstructed from x'? Is this any different (or more meaningful than an argument between, say Thales saying 'its all made of water' and Heraclitus saying it is all made of fire? All mathematics has already been 'reconstructed' on the principle of logical axioms by Betrand Russell, is it even a noteworthy idea to say that it can be once more reconstructed through observations (because observations are all they are) about given 'mathematical objects' belonging to some or other set or superset, or set of all sets? Is it desirable? Is it progress?https://en.wikipedia.org/wiki/Principia_Mathematica
Incidentally, Gödel did not believe that such a general operation is possible, or logically meaningful.
I personally do not really take a position on Gödel's incompleteness theorem, but I suspect that he was on our side.
A simple paraphrase.Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.
Georg Cantor, originator of set theory (even wikipedia is willing to call him a quack, as was Wittgenstein.) https://en.wikipedia.org/wiki/Georg_Cantor
As I said before, I dont accept his theory of transfinite numbers. I think they are a psyop aimed at God.https://en.wikipedia.org/wiki/Transfinite_number
Tell me this is not just a bunch of metaphysical gobbledegook dressed up in fancy verbiage and symbolic notation that the layman, or even maths student, is terrified to criticise.https://en.wikipedia.org/wiki/Aleph_number
I think that Aleph numbers are quite literally the formal expression of asking how many angels can dance on the head of a pin.
Back to the general subject of set theory, its relationship to formal logic and the universe (the real one, not the Von Neumann one) at large. Consider this. it is a time honoured trick to see if a political speech consists of truistic, empty nonsense by reversing the meaning - 'we should leave our children a filthy, unsafe, economically devastated non-country, in which different ethnicities live together in mutual hatred and suspicion, divided by shared anti-values of disrespect, intolerance, enforcement of pop-cultural-homogeneity, arbitrary violence, meaningless continental abstraction and the good old satanic principle of 'persecute thy neighbour', because children are NOT the future.'
This is nevertheless a linguistically meaningful statement. It merely illustrates the difference between form and content, specifically in the field of normative ethics and the linguistics of political rhetoric. If we apply this 'reversal' to the idea that all mathematics can be reconstructed on the basis of Zermano-Fraenkel set-theory, what do we have? We have the idea of reconstructing all mathematics on the basis of any given mathematical object's 'non-belonging' to any kind of higher, abstract, unifying classes, as well as the non-existence of such classes. One would be left discussing the 'fiveness' of five, possibly counting on our fingers or toes, forbidden as we would be from considering individual numbers as belonging to a higher-class called, indeed numbers. Six, and seven and so on would probably be talked about with the number of petals of specific flowers, geometry would be, by necessity grounded in real, existing forms made by the hands (this is basically a glib description of Waldorf education). Such a could very quickly however, go too far and end up as either monadology (discussing mathematical and, by extension, logical and real-world problems as a series of unconnected atomistic 'truth-statements' with no logical or semantic connections between each other, in other words, the news!) or pure negation (forbidding, say for example, the logical categories of true and false as representative of a kind of 'set theory through the back door', in other words answering the famous question of Hamlet with a resounding 'not to be' and promptly drinking the hemlock.) In this sense, it is quite obvious that this Zermano-Fraenkl set theory idea is a metaphysical statement dressed up in the emperor's new clothes of symbolic logic, and not even an original one, who first spoke of objects belonging to higher classes of transcendental, analogical forms? Plato. With regards to ontology, you can run the transcendental logic of 'the higher categories' in either direction, either ending up with the forms as emanations of god and the world as mere copy, imperceptible to the limited human senses (this is basically gnosticism, or the hekhaloth) or viewing any existing 'mere thing' as belonging to an infinite hierarchy of higher classes of thing and category, eventually arriving at God, the unity. This is William Blake seeing the entire universe in a grain of sand.
Set theory is not, however, mere abstraction, or nothing more than philosophy and therefore, useless. It is the mathematical-metaphysical counterpart of Object Oriented Programming. It is not for me to go into an explanation of exactly what OOP is. Any layman can see the connection.https://en.wikipedia.org/wiki/Object-oriented_programming#Composition.2C_inheritance.2C_and_delegation
If we think of the changes in programmer mentality that were brought about by the paradigm shift from procedural to OOP, it is analogous to the difference between waiting for the pronouncement of the vatican and being given permission, by Luther, to interpret the bible oneself. Even in the case of OOP's object-inheritance model, there is nothing fixed about this. It is not 'the answer', it is just a 'way of doing things', more than anything in to avoid having to reinvent the wheel. A development situation in reverse can very easily arise whereby one would begin with the idea of an object as specific instance and then work the way back up the inheritance class the other way round, arriving at rather than emanating from the 'transcendental' class Object (with its methods, arguments etc Note the language that is used here 'method' and 'argument' are terms more associated with philosophy). There also exist programming paradigms in which completely other notions are foregrounded (LISP based languages, for example) and these are good for different things. Hell, around the time of the Millennium bug nonsense, it came out that a hell of a lot of business architecture still ran on Cobol. So it all basically comes down to pragmatism, which is not surprising. Unlike, in say mathematical speculation, or philosophy, in programming you have a time limit and an actual goal: the production of usable software.
Philosophical pragmatism is the 'American philosophy'. Common-sense, down to earth, Terra Firma, not 'lost in space' or 'head in the clouds'! (wink)https://en.wikipedia.org/wiki/PragmatismConsider the practical effects of the objects of your conception. Then, your conception of those effects is the whole of your conception of the object.
In other words, any conception of a given object contains a pragmatic a consideration, a practical and teleological consideration of the matter at hand. To quote V.I. Lenin, What is to be done?
Speaking of pragmatism as a political and moral philosophy (what works, in other words), what were the 'pragmatic' reasons for the introduction of set-theory mathematics (and for that matter, set-theory ontology and ethics, i am sure you know what i am getting at) into education, in the context of the fake space-race? I have a theory. In introducing set theory into mathematical teaching in America, the perps introduced not just abstraction, but a specific, platonism-in-reverse metaphysical way of seeing the world. One that is associated with higher-level-computing and the logic of the computer age, with virtual reality, with our fake world-worldliness and, by extension with belief in our fake moon voyage and our fake pictures of the earth. It not only robbed children and adults of their common-sense reasoning, leaving them to have their intellects raped in every imaginable way by the high-priests of science, it actually introduced a form of new religious-perception through the backdoor, a kind of worship of the demiurge, the material deceiver, that is to say, Satan. The funny thing is that the people who are most deceived by all this go around proclaiming their atheism, as if this shows how clever and 'enlightened and freethinking' they are.
I cite here Gershom Scholem in his 'major trends in jewish mysticism', the second lecture, on merkabah mysticism (very important subject if you want to get some sense of the symbology of space travel, in fact I recommend this whole book as essential reading).
'As is well known, the anti-Jewish gnostics of the second and third centuries drew a sharp distinction between the unknown 'strange' good God, and the Creator, whom they identified with the God of Israel'
(There is more to be said about the role of things like 'empiricism' (in BIG quote marks), figures like David Hume and the scottish enlightenment clusterfuck, Rudolf Carnap and Logical Positivism, the 'falsifiablity' principle of Karl Popper and the way that such things as 'rationalism vs. empiricism' are portrayed as historically necessary dialectical movements rather than complete psyops set in motion and controlled from the middle in creating this situation. I dealt with this a little in the 'hegelian dialectics' thread, along with the sokal hoax, which along with the 'johann hari psyop' deserves its own thread. I would write and publish a book about my ideas but I am just a skint, angry, scottish, state-educated, down-on-his luck layman in his late twenties who can type fast and has read too many books but possesses no real motivation or agenda other than feeling it to be a major fucking inconvenience to have been obliged to incarnate into such a nightmarish fake world. My own mother even called me a racist the other day for attempting to apply the Kantian categorical imperative to the relatively inocuous question of free speech. I did not even articulate any specific ideas! Racist indeed! I would not know how to advance my ethnic interests even if I wanted to. Which I don't, I barely even want to be alive. But while I am, I speak for myself and myself alone.)
The thing about, say, christian gnosticism, neoplatonism or any of these transcendental world views is that they have always regarded 'the demiurge' as the adversary. It is not for nothing that true christians are accused of 'hating the world' and wishing for the next one. real
Muslims also. They believe that they are being lied to in every way by the deceiver (whether he be called Iblis, Satan, Ahriman or the demiurge), that the combination of their senses and reason is not enough to get them to a transcendental truth about the nature of the 'world' (I have said it before, but if you ever feel like you are getting drawn into a stupid, go-nowhere flat vs.round dialectic, READ THE CRITIQUE OF PURE REASON BY IMMANUEL KANT! Or The World As Will and Idea by Schopenhauer.) True Buddhists (unlike our plastic Dalai Lama) just dismissed the material world as maya and did not, for a long time, make massive technical or material progress. The christian west, and indeed the muslim world (and others, this is not an ethnocentric matter, Japan's departure from Shintoism and Emperor-worship for example, into the world of Honda motorcycles, robotics and pokemon), cut a kind of deal with the devil, took the material world at a sort of 'empirical face value', as it were, and set about changing it, getting to know it and in due time, inventing everything from musical instruments to sanitation to recipes for foods to transport to, yes, computers, they gave the world nearly everything that they have now in terms of prosperity and progress but thereby lost their soul.
And you are never allowed to say 'ENOUGH'!
This is what Goethe's Faust is about and we are all living through it and this is also why Goethean Science is one of the answers to our problems.http://wn.rsarchive.org/Books/GA001/Eng ... index.html
(As far as I can tell this is being put freely online by the publishers, although it is copyright. Mods feel free to remove it if you have a problem with it.)