Symbolic, Discreet, Fuzzy and other Kinds of Truth

Why do we need adjectives. What are we going to do with them? It is indeed a complex question. I am a meteorologist, but I deal a considerable amount with linguistics. I developed a number of languages for my own work with computers. However, so far, I considered a vocabulary exclusively based on objects, describable concepts such as properties, attributes, and verbs that combine these. An adjective can be considered an attribute. Then a further attribute can be assigned on the previous attribute and ad infinitum. We like this. In this system, there is no adjective. So how do I introduce the idea of an adjective? An adjective is a first order attribute to an object. So we have : \( rule : Object \boxplus Attribute \implies intermediate \: result \) \( rule : intermediate \: result \implies final \: result\)

See, for simple systems, we can already apply the first rule, and use an intermediate result as parts of our vocabulary. That will be our adverb. For systems one order complexer systems, we can take and attribute of attribute, use the intermediate second result as an adverb.

Gerog Cantor has had been an interesting man.

That’s Georg Cantor, at the age of 25 (i am 26). A father of 6, devout Christian, and a great mathematician who ran into a lifelong contradictions and conflict, finally dying from malnutrition.

His first conflict arises from his work on transfinite numbers. The very same discovery which he claimed to have been given to him by god himself in fact challenges the infinteness of the same god. Rest of his life revolves around many more conflict with other contemporary stars, e.g. Kroencker, Dedekind, and others, that systematically destroyed his self confidence. Finally he succumbed to depression, leading to a shift of his focus from mathematics to philosophy. After the first world war, he fell in poverty, and malnutrition, finally dying in a sanetorium, almost hidden from everyones eyes. Germany lost a great jewel. But then again, at that time, Germany had too many jewels. So, George Cantor has had proposed a new kind of infinite numbers - a number that is strictly larger than every finite number. Nobody really knows how numbers came into being (theories exist, yes, but no definite knowledge). The so called Natural numbers start at zero. The earliest use of negative number is around 200 BC in China - the exact dating of comparable Indian manuscripts remain controversial. From their, via Arabian mathematicians, the concept arrived to Europe. Either way, it is easy to intuitively understand that the numbers start at zero, and at each successive addition of a certain standardized object (called the unity) a new number comes out. You can compose multiple such additions with a multiplication, and then you are walking towards the creation of groups. For negative numbers, you just walk the other direction. Now, you can add an unity with zero, creating one. You can add one to it, creating two and so on. However, the operation one addition unity makes two is independant of the definition of three or four, which will appear somwhere later in the chain. This also makes causal sense, i.e. works trivially with causality. You need to start at some (depending on context, arbitrary-ish) zero, and continue, based solely upon the last number handled. Of course, this means that the following axiom is granted : zero is a number. This makes sense, if we take numbers to be a description of an amount. 

So we get a few ideas, what numbers can / shall do :

They can describe an amount, what ever the ontological definition of “amount” might be.

Some numbers are successor to some others. The general implication (not really a direct result of the last sentence) is that the total set of numbers is totally ordered.

There is an arbitrary-ish starting point, and then the other numbers can be constructed by a successive train similar (standard) operations applied repeatatively (This one’s a bitxh). These operations can be scaled up or down, to describe fractions, and tens and hundreds and so on. The standard operation is also arbitrary-ish. It is the addition of unity, with unity defined somewhat arbitrarily. Another ontological issue, if you will.

The construction of a number does not depend on a number that comes after it in an ordered sequence, nor is another extra arbitray-ish point like zero necessary. This holds ad infinitum.

Number 4 is where Herr Cantor comes in. His idea is the existance of another, ab initio existing, number, that is not constructed via repeatative similar standard operations - the Omega, the first transfinite number. It is impossible to creat this number by repeatatively adding unity to zero (or to any other finite number, for that matter). Still, we have two different classes of numbers: Finite ones, created from zero, and transfinite ones, created from Omega, resp. minus Omega. Even then, all numbers still admit total ordering.

Well. Now we have the permission of of having multiple ab-initio numbers, we might as well use more than one of them to construct the successor number. Some or other variants of this idea existed for quite some time.

Imagine a set of points together defining a polytope - say, an open polygon.  So we end up with a metric set. Take any two neighboring points of this set, and introduce new points in between them, such that a scaled version of the original polytope appear between these two points. Yes, we are talking about self similar sturctures with an Haussdorf dimension.

Fast forward about a century. We meet a gentleman called John Horton Conway. Yes, that Conway. Yes, Donald Knuth was also interested in this business. Now, we consider the set of numbers to be a metric set, an one dimensional line. We take two neighboring points (points in this set are numbers) in it, and instead of creating another set, similar to the entire one, in between these selected points, we require the existance of just another point (which is then automatically another number). Then we can take the newly created number, and one of the previous one, and create yet a new number in between, and ad infinitum.

The set of all such created numbers is also a totally ordered set.

I guess there are two aspects in such numbers that immediately become obvious.

We want to create numbers, such that, starting at an arbitrary initial point, by iteratively adding something to it (stepping towards a direction), we can continue towards infinity, even if we can’t reach it (ah, Herr Cantor). At the same time, we want to repeatatively, reach finer and finer resolution, deeper and deeper, anywhere between any two points. The anywhere part is interesting. We can go further, or we can go deeper, finer.

I already wrote how I feel about categories and types. I am not interested in empirically expecting an error free program - I am interested in applying rules on categories. 

How do you apply rules? 

There are several ways. 

You can grab the D compiler and modify it - but I don’t want to - but remember that D actually permits you to write inline assembler code - but that’s where I stop.

You can write a metacompiler - which you can do in several ways. This is where I am investing my sanity in. I want to write genetically modifiable code. So this becomes interesting. 

The metacompiler can be written in D itself, or it can be written in, say Perl, which will analyze the target script, modify it, and throw it to the D code. But I don’t like this. 

Remember that the internal representation of typenames are random but unique strings in D. So you don’t really have much choice about it. So you can write everything with perfect typing, and then use a perl script to create types that won’t make much linguistic sense. This is not very appealing, is it?

The second option is to write everything in D, and then create functions which work on categories. Here you can predict the behavior of the function - i.e. a function that is supposed to compute the sea level pressure duirng an El Nino period will truly do it. You generally can’t predict the outcome of a general purpose of a computer program without running it (maybe this is why we observe strange things in the context of software … forget it). 

So, let us start. The basic idea of category theory is given by a rather small set of theorems (if you compare that with axiomatic set theory) . 

Theorem: \( Z \rightarrow X \) and \( Z \rightarrow Y \), along with \( W \rightarrow X \) and \( W \rightarrow Y \) implies \( W \rightarrow Z \)

I wouldn’t prove it - has had been prooven many times till now. However, the implication is, if there are two things, both of which you can reduce to two other things, then they are reducable between one another. Reducibility here comes with a problem. Reducibility here simply means that a function can take the input of a certain and reduce it to, i.e. produce an output of another category - each of the outputs will have an unique input (to decrease the complexity). 

However, objects of different categories contain different types of information, and a function, according to some rules (a calculus, in effect), manipulates those informations. While globally, information may be a conserved quantity, locally it is possiblee to destroy information  but conversely, in general, it is not possible to create information, 

Now, if the said \(W\) and \(Z\) are such, that \(Z\) pertains a category which makes it to contain information that \(W\) does not, then we have a problem. In certain cases, however, it might be possible, to introduce default values of missing information. For this, we may want to exploit default values as encompassed by a natural language. This is one practical reason why I am interested in programming a storm class type as a “storm” - because then every time we have the occurance of the word storm in the type descriptor, we do not have to go through a dictionary to find out the default properties of strom. We know them. 

We also have a problem, if there is a necessity to destroy some information, and doing so would fundamentally change the properties of the object. In general, we would do such an action, when, for example, we want to trade off, e.g. precision for speed. There are certain fundamental changes, which we overlook, when we destroy high precision information. However, it becomes complicated, if, for example, we have a complex number. Nonetheless, there exists certain function, which drops the real (resp. imaginary) part of it ( = destroyes the information) and retruns the imaginary (resp. real) part of it. 

[copyright : Gunnar Staalesen - I think]

My Norwegian is not that good - but I believe that means that In darkness, all wolves are grey. It happens to be the title of a detective novel, and a film of the same name. The hero is called Varg Veum, whose name is also supposed to mean the wolf (from Skriften på veggen, writings on the wall -another novel with the same hero - also, Tip : Okey novels, Okey movies - but not worth 6+/10 rating in IMDB - but watch them.). But how grey is a grey wolf?

In other words : Given a wolf is grey, how likely is it that it is actually grey?

MEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEH. I have not slept for a while, and I am not particularly feeling well at this moment - but i want to speak my heart now.

I can’t remember the book, but it was about Bayesian Reasoning. The credit of Bayesian reasoning lies in answering the question, knowing \( a \implies b \) and having observed \( b \), what is the probability of \( a \). Given that a wolf seen in night looked grey, and knowing that a grey one would appear such, how likely is that the wolf was actually grey? Or even, how likely is the wolf brown instead? The answer to the particular question may be interesting to Norwegian wolf biologists, but similar questions appear in many fields of science. Repeatedly.

I am a meteorologist. I am always dealing with this problem - knowing \( CO_2 \) driven absorption of long wave radiation triggers increase of atmospheric temperature, and observing an increase of temperature in atmosphere, what is the probability that the \( CO_2 \) is actually absorption is happening. (Remark : it is happening - but put it aside for now.)

What I am very interested in is the application of the same in SLAM. SLAM stands for simultaneous localization and mapping. While the application is very much focused in the field of autonomous robotics, e.g. driverless cars, one can also use it on atmospheric sondes. I want to use it on a - no, I won’t say it here before actually doing it. But point is, we are stuck with the probabilistic view of things.

This may have to do smething with the way we are raised - out education implants the idea of probability firmly in our minds. It might have to do something about our dichtomous way of perception - either is the world deterministic, or it is not, in which case people talk about probability.

Probability is a very strange thing. The first rise of the concept of probability is well beyond my knowledge, but in school one starts with randomness and probability. The classical idea of probability is the one of Gauß :

gauß kam auf den zufall zu sprechen, den feind allen wissens, den er immer habe besiegen wollen. aus der nähe betrachtet,sehe man hinter jedem ereignis die unendliche feinheit des kausalgewebes.trete man weit genug zurück, offenbarten sich die grossen muster, freiheit und zufall seien eine frage der mittleren entfernung, eine sache des abstands. ob er verstehe?

- Die Vermessung der Welt, Daniel Kehlmann

I don’t read fiction, and this is not a very great fiction - it has some poetic licences, but this is an exception in the sense I read it.

I won’t translate this. This is one of the reasons Gauß is that he handled the properties of randomness as a outcome of something whose outcome is not definite. He tamed the beast of randomness. And that made him great. To quote again the same book,

seltsam sei es und ungerecht,sagte gauss, so recht ein beispiel für erbärmliche zufälligkeit der existenz, dass man in einer bestimmten zeit geboren und in ihr verhaftet sei,ob man wollte oder nicht. es verschaffe einem einen unziemlichen vorteil vor der vergangenheit und mache einen zum clown der zukunft.

Those of you can understand this without translation, forgive my rough translation. Roughly, Existance is random, that one is born in a particular time, [...] and may either profit from the past or may become the clown of future.

HA! Well! I am ****ing JEALOUS ! Gauß was born in a time when randomness was not tamed. I don’t know if the idea of randomness and probability was formulated and known to Gauß or not. If it was, then at least he had had a direction to pursue.

Either way, there is still the example of Schrödinger. He realized the concept of existance of multiple states to coexist, and probability of each to manifest themselves at an outcome. This seem to be an idea more original than that of Gauß -but that is my feeling.

Somewhere in between is Bayes, and his interpretation of probability as a description of a continuing reality. While Gauß’s probability is the prediction of reality as an outcome of a non-predictable event - in a way, Bayes’s probability is a description of continuous reality, from \( a \) to \( b \) - a confidence of asserting a particular train of events.

It is a capital mistake to theorize before one has data.

... said Sherlock Holmes , as it happenes in “A Scandal in Bohemia” - by Aurther Conan Doyle, as mentioned in the book “GIS : A Computer Science Perspective“ by Michael F.  Worboys.

It is?

Gödel has said quite a long time ago that you can’t have a logical system which is both consistant and complete. I will not go in the exact meaning of this statement, but is suffices to say, that generally, a logical system. i.e. a system of statements, where you deduct some statement from others, and there are no conflicts, if consistant, i.e. does not lead to paradoxes, can not be complete- i.e. there will always be some statements which, although you can express in the system, you won’t be able to prove. Let that sink in for a bit. Example : The theory of relativity is a rather well known theory of physics, and while using the theory, you can deduct a vareity of effects observed in Mercury’s orbit, you can’t prove that the light’s speed is a natural limit / a constant. You can formulate the statement, that the speed of light is the maximum and a natural constant everywhere in the observable universe, but you can’t prove it. (In fact therein lies the strength - starting form a very simple assumption, we can actually predict - using theoritical calculation and logical reasoning - without having to depend on observation, which is actually reflected in the observation !! ) So what do we learn ? We learn that we often theorize without Data. In fact, in science, itis more important to be consistant than complete. I found this image in the net :

Strictly speaking, I am possibly doing a semantic argument (not forbidden, but not the strongest form of the argument) - but that does nto matter - because I am not interested in knowing absolute “allness“ of the qualifier “all” here - I will speak of the tendency. It is very clear that the word “all” here means “tedencially all”, and that is not up to debate. Right? See while the intention in Bible is to be as complete as possible, the important part in science is to be consistant. We know we are not complete. That is okey. We are more interested in being consistant. Now that science is consistant, it is necessarily incomplete. There is always a limit of what science can deduce. Thus, logically, science has (at least) a start and an end. In practice, science is not a single prose with a start and end (with each intermediate sentence being logically deduced from the previous one), but a collection of such proses. As such it has several beginnings. None of these beginnings can be deduced from anything else. These are called postulates or axioms (actually, there is a difference between them, but that is not important here). In the previous example, the statement about the speed of light was a postulate. In Fermi - Dirac statistics, the postulate is that the Pauli Exlcusion Principle is obeyed. In case of Bosons, they don’t obey the Pauli Exclusion Principle, and an unlimited number of particles can condensate to the same energy state. This, in fact, where the vision of the theory lies, that Bose allowed the particles to break PEP. Bose did not have the data, but that did not stop him from theorizing. 

Moral : The Vision of a scientist is the ability to understand where a particular formulated theory describing the events of nature ends and what comes beyond the same.

Global warming naysayers, take note : Just because you claim otherwise, that the theories governing the warming world does not break. Moreover, if they do break in an exotic setup, present earth does not belong to that setup. So for the case of Earth, the status quo remains as the theories of warming predict. Moreover, vision is not what Steve Jobs had, but what Plank had, what Bose had, what Mandelbrot had ... These days, just because Elon Musk thinks of a fast transport system, we assign him the title of visionary, while he is simply digging up / tinkering around existing ideas - he does not / did not come up with a fundamental conflict with a theory describing the universe - nor did he offer anything fundamentally new. I do understand, that to the mass of the populace, the concepts of Bose condensate is out of reach, so they can hardly realize where the wonder of the same lies, whereas Musk’s new transportation system lies out of their box all right, and as such it suddenly becomes a ‘visionary’ one.

We learn that theorization is a bit complex. I would like to first show this :

Courtesy: Spiked Math. Mathematics, in its complete entirity, is mostly an intricate interplay of axioms and postulates - practically assumptions. Precisely as such arises the question 10, 13 and 14 this list. This is from Wikipedia, and in case a future version delets the list, here is a local copy :

What is the role of Mankind in developing mathematics?

What are the sources of mathematical subject matter?

What is the ontological status of mathematical entities?

What does it mean to refer to a mathematical object?

What is the character of a mathematical proposition?

What is the relation between logic and mathematics?

What is the role of hermeneutics in mathematics?

What kinds of inquiry play a role in mathematics?

What are the objectives of mathematical inquiry?

What gives mathematics its hold on experience?

What are the human traits behind mathematics?

What is mathematical beauty?

What is the source and nature of mathematical truth?

What is the relationship between the abstract world of mathematics and the material universe?

Honestly, I don’t even know how to approach this. I tried to read on the topic, but it is a very very slippery slope. The authors pretty soon start appealing to properties of human nature - such as search for beauty, truth, perfectness ... That does not answer the questions of the list.

If a sufficiently inteligent machine can create axioms, and feed them to an appropriate neural network, very possibly the same mathematics will rise as we know it.

May be not. Since we do not really know much about the ontological status (heck, do we even know the meaning of “being” ? ) mathematical objects or the truth they represent, we don’t even know the status quo of their existance. Thus we can’t take the uniqueness thereof as a universal truth, and can’t claim all mathematics, regardless of source is the same.

Then comes the question of the Frame Problem . As a meteorologist, I am interested in pushing the envelops of radiosondes, and that is why I am interested in the frame problem. While this is often studied in the frame of logic, it has an extremely profoud impact in robot based cartography. A robot does not necessarily take the same set of statements as axioms as a human mathematician is taking. Thus, their  perception of world is different than that of a human mathematician. Wait a minute! A human mathematician does not base his axioms necessarily on the observable world.

I do not know how far these two points drive / modify each other.

Nonetheless, theorization does appear to be more complex than Holmes is saying.

Moral : Theorization is a process that can start without Data, and often does so. That is how science progresses.

Is Sherlock then completely out of his minds? 

Very possibly not.

Let us take a closer look at theorization.

Bose was thinking of the limits of Pauli Exclusion Principle (PEP). He thus theorizes the existance of a world without PEP, and data was found to support his claim. Notice, that the process that started with an assumption, completes the cycle at the finding of the necessary data.

Bose has the vision (probably not an epiphany, but unimportant) of breaking the PEP. That is his axiom (ehhh, postulate). He then deduces a set of statements using the Axiom and other proven statements. The new set of statements, along with his vision / axiom is now a Hypothesis. The hypothesis is a perfectly valiid theory in the sense of the word as in mathematics. In mathematics, one does not need a set of confirmation form the natural world.

Now once we have data to verify this, the hypothesis becomes a valid physical theory.

The entire process is the one of theorization. Moral : You can very well start theorization without data, but you cant finish it with a guaranteed validity in the physical world without data.

It has been about an year, since I said I would write about abstractness. I programmed a skeleton of a climate simulation in D. It’s not how things work, I know that. But I really like the way D lets you handle types. Okey I don’t like it that much. It is not as sophisticated as Python, but it allowes me a “little” bit of abstraction. It will be soon clear, what I mean by a little bit of abstraction. The feel of D

Programming D will be mostly smooth to most programmers of the C genre. It is highly inspired by C++, but claims to be better. So I decided to play with it. D claims to be a system programming language by engineers, for engineers, to address real problems, etc. I couldn’t care less. What an engineer thinks is relatively unimportant for me. There are two and maybe a half things that are currently keeping me to D. First of all, I like to do things differently. The scientific programming scene, after Fortran, is dominated by C++/ C, which is closely tailed by python. So D is a natural choice to do the job of C++/C. However, D still can’t do everything C can. At the time I started, D could not load libraries dynamically at runtime - that is, you counld not write plugin-s in D. But still D was alluring. Then the second reason to feel attracted to D is its typing. You can use the alias command to redefine types. Read the part on Flexibility. In the earlier versions of D, you could use the alias keyword to create a new native type. So alias newtype oldtype would create a new type called newtype, which will be any exact copy of oldtype - Since the second version of D, you can’t do this any more, now newtype is merely a new idetifier of oldtype. (These engineers ... nevermind, I know their arguments). The 2.5-th reason is speed. I am not completely convinced, but D can be allegdly faster than C. the lack of convincing makes it half of a reason to stick to D. But never mind, I want to talk about reason #2. Types - category for programmers?

There is a considerable amount of debate on whether category theory helps you or not. Please also see this. In practice, I am less worried about type safety. Being a scientist, I really need to know each letter of my programming and its function, and not just use some method to solve some thing, and as such (obviously metaphorically) I typecheck every byte of my program manually anyway, and I do not whine about the additional work - i.e. I dont need to put this task on the compiler / interpreter. Since I am using D anyway, the type checking is not really my decision anyway, but I want to mention that I am not appealed by / concerned witth the type checking argument at this stage, and thus I will analyze the feature from my POV.

Die Bücher

So, in the quest of learning about type , category theory, and programming in a more detailed manner, I wanted to study a bit. Two books helped me a lot, but, considering I actually was thorough in the investigation of only two books, (everything else I just skimmed) my understanding is horribly partial. But that is what I want to talk about.

The first book is given by ISBN 978-0262660716, and the other one is given by 963 05 6332 0. The second book can justify the learning of hungarian language. This gave me a very good guidance. Beyond Computation

Not everything we do is calculation. When I started to consider these things,  thought that the holy grail for learning something mathematical would be its application to make predictions - and the kind of predictions that you can immediately identify as predictions, e.g. if you add two with two, you will find four [ Easter egg: I am not defining two or four ]. However, that is not all what we do. We may also need to organize our thoughts. The organization of thoughts is rather a set of regulations, than predictions, but all regulations can trivially be reduced to predictions. If you apply the regulation, then the world will be in a state where the regulation has had been applied. That is, a trivial kind of prediction as \(a \vdash a \) can be easily made. However, the purpose is not to make those predictions, but coherent descriptions. The coherence necessarily implies predictions, that coherence would be achieved from something ( \( something \vdash coherence \) ) . But since the target of the prediction, the coherence by itself, is set, we are mostly interested in the said \( something \). As such, most of the literature in this area I am familiar with, does not really compute anything, but lead to the conclusion, that certain regulation of thoughts will ensure the entire chain coherent. Often is the guarantee of coherence is not enough. We want to categorize. We want to see that after certain manipulation of reality as a machine reads it, the coherence is still guaranteed. This is where one comes into the discussion of logic and category theory. Logic We can talk about the purpose of logic. The second book mentions Carnap, and his concept of logic as method of philosophizing. Genau darüber will ich jetzt reden - über das Philosophieren.

Think about it. I want to materialize my thoughts in a program. I have mentioned that detecting / avoiding errors by implementing strict types in not important to me. However, such type checking effectively categorizes objects, i.e. assigns some properties to it as postulates, and you can check those properties. The actual mathematical stream categorization will be dealt later. So you can take the categorization as a rule of thought. 

In fact, you have the follwoing rule :

Categorize the representation of a real object as a representation of a real category.

That is, if you are working with a storm’s mechanics, then instead of \( \mathtt{struct\:Storm} \), use \( \mathtt{storm\:Storm} \). This is nothing exceptional. Every object oriented system lets you make a class, and to this. But inside the class / struct, you might have \( \mathtt{string\:StormName} \). However, the rule requires you to write \( \mathtt{stormName\:StormName} \).

There are several ways of doing this. You can create a \( \mathtt{stormName} \) class, which will simply wrap a string. In D, you use the alias keyword, to mean that a \( \mathtt{stormName} \) is actually a string. Great. what does it allow us to do?

I will discuss concepts related to and ways to deal with Abstractness.

Reality and Model

Entities Exist. This is an Axiom. Entities can be related. This is also an Axiom. Lack of everything is a thing on its own right, we will call it void, or the null set. Entities can be grouped together. Relating an unique entity of percievable reality to an unique symbol (any generic entity), such that the rules regulating the reality are also evident in the symbolic construction, is a model.

Abstracness, Types, and Tokens

Hadmund Bußmann [Lexikon der Sprachwissenschaft], and Wolfgang Künne [Abstrakte Gegenstände: Semantik und Ontologie] has defined the concepts related to types and tokens, as they seem to be essential building blocks of philosophy. Tokens are individual appearences, Vorkommnisse, as Künne puts it, of any symbol in a model, and types are the number of symbols different from each other. What us interests, is the concept of "difference" from each other - how does one differentiates things, or brings things together? According to Künne, the Platonic view of the world is that properties of things exist as well. The properties are abstract. While Künne argues that their constantness, their lack of effect on human senses (as in contrast to thoughts), their lack of systematic precievability, makes them abstract, we will not follow him. Reality is but a set of measurable (by a machine) properties, with certain clustering - these clusterings are the corresponding objects or catagories, to which the properties of a certain cluster belong to. So all properties must be measurable, and measurability can be classified as perception. We will consider objects that can collide with each other and upon which the equation of continuity can be applied as concrete objects, and their properties, as well as the properties of the properties, or any higher order property to be abstract. One searches for certain similarities in these properties. A similarity between multiple properties is only then admisable, when those properties themselves have a (one level abstracter) property, which admits the some measure for both. Consider a blue object, and a green one. They have the "property of being blue", and "the property of being green" respectively. However, both of these properties can be classified as color, and then a measure can be assigned to them, respectively "blue" and "green", the values of measure need not come from a continious interval. The property color is more abstract than both "blueness" and "greenness". We note, that "color", as an abstract property", encompasses both "blue" and "green"; as well as the laters can be reduced from the former by assiging a measure to them, or stripping of certains measures from the laters reduces them to the former. Thus, we look for a possible measure on abstractness with the following properties : 1. The property with higher abstractness does not encompass lesser number of objects than the one with less abstractness. 2. The property with more abstractness can be formed by stripping off measures of a less abstract property.

Abstract Representation of Objects:

An object is but a set of properties, let there be two objects : \( \mathcal{O}_1 = \{\alpha_{1}, \alpha_{2}, \ldots , \alpha_{n}\} \) and \( \mathcal{O}_2 = \{\beta_{1}, \beta_{2}, \ldots , \beta_{n}\} \mid \alpha_i, \beta_j \forall 1 \leq i,j \geq n\) are properties. Without lose of generality, let there be properties \(alpha_k\) and \(\beta_k\), such that \(\alpha_k = \phi \psi_{k,1} , \beta_k = \phi \psi_{k,2} \). Now a third object may be \( \mathcal{O}_3 = \{\gamma_{1}, \gamma_{2}, \ldots , \gamma_{n}\} \), and one may be able to write, \(\gamma_k = \xi \psi_{k,3} \). If it is the case that \( \phi := zy_{k,1}, \xi := zy_{k,2}\) then we have: \( \mathcal{O}_1 = \{\alpha_{1}, \alpha_{2}, \ldots , \alpha_k := zy_{k,1} \psi_{k,1}, \ldots, \alpha_{n}\} \) \( \mathcal{O}_2 = \{\beta_{1}, \beta_{2}, \ldots , \beta_k := zy_{k,1} \psi_{k,2}, \ldots, \beta_{n}\} \) \( \mathcal{O}_3 = \{\gamma_{1}, \gamma_{2}, \ldots , \gamma_k := zy_{k,2} \psi_{k,3}, \ldots, \gamma_{n}\} \) This definitely is a more abstract representation than the former representation of \( \mathcal{O}_1,  \mathcal{O}_2 \) . One issue one has to keep in mind is avoiding conflicts. One can, for example, take "blueness" instead of "color" as a more abstract property, then assign the measure "blueness \(:=\) 0" to a green object. However, then describe the "greenness", one must take an additional property-measure pair "greenness \(:=\) 1", and for a red object, both the "blueness" and "greenness" must be set to zero, while a yet new property "redness" must be set to one. This is a scenario we wish to avoid. In this case, obviously, "redness" encompasses a lesser number of objects (it is assigned to the red object only) than "blueness", while color could have encompassed all three of them, with complete description of each (in contrast, while "blueness" does encmpass the green and the red object, it alone does not completely describe them). Thus, from the argument of greater number of encompassed objects, "color" is a better candidate. Notice that if "redness", "greenness" and "blueness" were attributed to all objects, then in this case at hand, it would have encompassed all three, but at a cost of longer code length.

Measure of Abstractness:

Mathematical Description of the Process of Simulation

A simulation can be described as followes :

Let \(i \in \mathbb{I}\) be a generic collection, which is our some initial data. We apply no further restriction upon this collection, not even the axioms of set theory. \( \exists \) functions \(f_{x} \mid x \in X := \{ 0, 1, 2, ... , n, \omega \} \) where \(X\) is an index set.

Also, let \[f_0 : i \mapsto Z_1\] \[ f_{k} : Z_k \mapsto Z_{k+1} \] \[ f_{\omega} : Z_{\omega} \mapsto e \mid e \in \mathbb{E} \] Here, \(e\) is the output, the result we look for. Thus, one can write, \(e = f_{\omega} \circ \ldots \circ f_1 \circ f_0 (i)\). The application of \(f_{\omega} \circ \ldots \circ f_1 \circ f_0 \) on \(i\) such that the result is \(e\), is called a simulation. Properties of \(i\)

\(i\) is the initial human input. This is likely to contain elements of abstractness. In order to deal with such, one must quantify it.

Quantifying Abstractness Before quantifying, one has to define Abstractness. Abstractness may indicate:

Fuzzyness : Concepts may not be categorizable in sets with a uniquely defined boundary.

Contextual Change : That is, same term may appear in multiple contexts, and imply different other terms. Assume, \(f:x \mapsto ax \mid x \in X\). Thus \( f(y) \mid y \not \in X\) is undefined. But, in a human language, it occurs that, either \(X\) is not defined not even fuzzily, or a new definition is supplied for \(f(y)\). Often, the new definition of \(f(y)\) bears some similarity to the definition of \(f(x)\mid x \in X\). This similarity may be evident in terms of the instructions needed in order to compute \(f(x)\) Thus the quantification of abstractness should reflect the similarity, as well as disimilarity.

Polymorphism : In a deterministic computing environment, the processor may need to know the type of data it is going to process. A human language does not have such requirements.

Thus one may quantify abstractness as the \(\frac{dissimilarity\: of\: context\: in\: case\: of\: a\: contextual\: change}{similarity\: thereof} + number\: of\: different\: morphs\). This is just a tentative quantification, which may change in course.

Conclusions In order to deal with such, we need a language that is, besides being reasonable fast, inherently supports contextual changes, when the mentioned \(y\) is known, and on the fly polymorphism. For the first, I choose D, and for the second Lua, as it has the following properties :

Embeddable in D (it is designed to be embedded).