Reconsidering of Wittgenstein Philosophy of Mathematics

These are some notes of mine on Matteo Plebani doctorate theses, Universit`a Ca’ Foscari Venezia.

 

Hossein M. Bojnordi' s note : (Page 48) Controversy I think, lies here: Reduction of all semantics relations to just one denotation is an useful, practical then legitimate course of action for filtration and reduction in size of information, because there are practically infinite extensions for anything under considerations, in the world. It's easy to drown under avalanche of data, very soon, if the strategy is to collect as much as data and relations that thought to be relevant, and they are, actually, relevant but having them, or better said, (just before this "measure" stage is one fundamental pre-condition, which is "the first and second act of intuition" according to Brouwer. I'll elaborate on this at the next paragraph), measure, calculate relations, preserve and all of that is done utilizing some formal language or breed of languages (built and structured on top of an appropriate logico-mathematical language like intuitionistic type theory which is a constructive mathematics formalization. In an intuitionistic type theory language which is a standard logical formalization and on a par with the classic logic, computation and instructions (or Programming) of the process of computation is equal to the writing of the notation of the formal language framework under consideration. It means you have computation power (relative to the available material in the universe and bounding with the laws of physics) to model and represent the re-construction of the world (remember models are always constructed with less data than the original universe) it means we omit, factor and filter information and use generalizations. If we are not going to filter most of data then our computation time bill will exceed far beyond the life-time of the universe.

 

Hossein M. Bojnordi' s note : (Page 48) I have got some aggregation of my thoughts at this point after reading this paragraph:

It was Lord Berkeley or Breuwer (I'm in doubt!) who said once that "Not any conscious moment can be represented in terms of a formal notation or language". And this Wittgenstein idea that every word in language points to many things at once. And, also, his rejection of ontological side of Platonism, they are all pointing to the same direction: The complexity and vague nature of a conscious moment and instance of an intuition. A conscious moment represents, seemingly, from point of view of an agent like the author of these lines, a very unclear boundaries, its core content isn't much crystalline though but boundaries are (feel!) so much thicker than the core, almost infinite, one semantical extension after another attached to many more in a messy bosh that hopelessly unmanageable. Do we capture and represent that event fully and flawlessly in a formal notation or language? Are all the ingredients of an event of intuition also involved in the understanding process? Or some of them may only be some form of infrastructure for that process? There's a plausible possibility that there are things extra to needs of intuition process, in a conscious agent's brain. So is it permittable to skip them in our formal notations? Are they even be able to be formulated in any notation? We are far from preceision (reminding myself of Nima Arkani-Hamed thought on practical limitations of precision experiment, there are ten to the power ten to the power 140, possible configuration of matter in the universe, means write 1 with 10 to the 140, zeros in front of it!

 

Some more speculative thoughts: the form, feeling, function and essence of an intuition or a conscious moment, may well be a result or partially a result of physio-chemical processes in a thinking agent's brain. What should we do about them, to represent them as a notation? What is this state of separation we feel to everything else (a state of being in super-position)? Could un-necessary elements (well seemingly), noises and even unrelated evolutionary rubbish, in a intuition process, have vital roles? In what ways?

 

Answers to these questions could potentially soften (a little bit!) the hardness of the question.

 

Of-course one more plausible idea (in a very fundamental fashion), is Roger Penrose involvement of quantum mechanical phenomena in the process.

 

Hossein M. Bojnordi' s note : (Page 49) I am thinking, some concept (formations) like Two, Twoness, One-more, addition of exactly one reality to my previous one, a divide in my grasp of reality--"meanings of number two". Are they not different in kind, and for shaping of our function of mentality etc. , from word perhaps, love. Can we differentiate between basic, mind forming concept words (like basic rules of logic or in Wittgenstein words an operational role) from higher-order, more complex concept words?

 

Hossein M. Bojnordi' s note : (Page 59) I think there are, in the philosophy of mathematics community, a great amount of obsessive behaviour towards ontology of mathematical objects. The answer to Quine's question, I believe, is, in form of an Information-Theoretic Formalism. More about this later. But:

 

First, there is a reasonable chance that the information in the universe be the most fundamental level of reality, even more fundamental than matter and energy and space-time (space-time is already out of game) itself. Then it follows that a given mathematical structure that tries to explain or predict some aspect of the physical world, is a (always a smaller) model of reality. Second, even if information is not fundamental, but still our mathematical models can give a very good approximations to explain the functions in the universe. Third, the very notion of existence, in a materialistic or in whatever standard contemporary scientists language, is a very fuzzy notion. Numbers exist in the sense that, from Brouwer intuitionistic point of view, an agent's conscious moment (in his/her brain) split into two distinct pictures and a two-ity is born. From there, the agent's one giant tree of two-ities events will evolve. Mathematical structures, not inhibit, but are the very same events in the, and from the agent's point of view, evolution of these trees. The important point is, the structures (Or the Super-Structure) is there (probably always been, or in the instance of the time be an emergent, non-fundamental faculty of the universe, the question itself can be ignored.) and of course the structure of the universe is there and real. The formal language of mathematics is a tool (in our human-agent hand and from our point and position in the structure of reality) to model our mathematical structures that captures the properties of the original reality or making approximations to them. So here, I am suggesting an augmented-reality, which is our constructed mathematical modelling of the universe. Here, It is important to understand that a modelling (the information of the augmented-properties of the universe) can't be equal or larger than the original. It is impossible, impractical, not useful and nonsensical.

 

To conclude:

Numbers and mathematical entities do exist, but in an augmented world. Augmented information realities are as real as the original ones. They inhibit somewhere, in the world, whether be in an animal-brain (making and utilizing models, to live, evolve and navigate in the world), in an artificially intelligent agent's mind, in a computer chips for instance or in an university text-book, chalk's powder sticking on a blackboard.

 

So there is not a separate, Platonistic place for them to live. They are here!