9781909287723 Topics in Programming Languages:A Philosophical Analysis Through the Case of Prolog

8/22/2019 9781909287723 Topics in Programming Languages:A Philosophical Analysis Through the Case of Prolog

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LUS HOMEM

TOPICS INPROGRAMMINGLANGUAGES

A PHILOSOPHICAL ANALYSISTHROUGH THE CASE OF PROLOG


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Topics in Programming Languages


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Topics in Programming

LanguagesA philosophical analysis through the

case of Prolog

LUSHOMEM


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Chartridge Books OxfordHexagon HouseAvenue 4Station LaneWitneyOxford OX28 4BN, UKTel: +44 (0) 1865 598888Email: [email protected]: www.chartridgebooksoxford.com

Published in 2013 by Chartridge Books Oxford

ISBN print: 978-1-909287-72-3ISBN digital (pdf): 978-1-909287-73-0ISBN digital book (epub): 978-1-909287-74-7ISBN digital book (mobi): 978-1-909287-75-4

L. Homem 2013

The right of L. Homem to be identified as author of this work has been asserted in accordance withsections 77 and 78 of the Copyright, Designs and Patents Act 1988.

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Typeset by Domex, IndiaPrinted in the UK and USA


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Dedication

Para o Afonso, meu filho


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Contents

Dedication v

Abstract ix

Acknowledgements xi

1 Section I 1

Arguments 7

) The phonetics and philosophical argument 7

) The symbolic or rational argument 8

) The difficulty argument 8

) The content-and-form artificial intelligence argument 9) The efficient cause argument 9

) The model theory argument 10

Notes 11

2 Section II 13

Arguments 26

) The endogenous to exogenous language argument 26

) The efficient cause continuance argument 27

) The reviewing incommensurability argument 28

) The functional and declarative programming languages argument 30

Notes 33

3 Section III 35

Arguments 39

) The -calculus argument 39


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) The Prolog argument 41

Notes 50

4 Section IV 51

Topics in programming languages: a philosophical analysis

through the case of prolog 51

Summary 51

State of the art 51

Goal 53

Detailed description 54

Bibliography 59


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Abstract

As opposed to the rhapsodically recounted myth of the Tower of Babelit is commonly held that Programming Languages have superseded Natural

Languages in concentrating on two fundamental aspects that have arisen

from Recursion Theory. Not that Natural Languages did not cleave to

similar cases in point, but the status of one Universal Language in which

all problems could be declared, and one decision method to solve them

were, as literature specifies, and respectively, a sort of Declarative

demand and Procedural request of the Philosophy of Language and

Computation that most constituted last centurys prominent boundary.

We search for the Classical Antiquity of the functional Paradigm,

wherefrom Programming Languages were born, willingly inviting the

Philosophical starting place as a Science and Language, since the

existence of the modern Alphabet, after which we wish to account more

consistently Programming Languages Topics. The last question entails

the well-known Church-Turing Thesis, and the status of effectively

calculable functions, to which we will offer one Philosophical route of

Understanding, convoking what we have decided to address as the

Calculus and the Computus traditions and their sharp distinctions. The

first question addresses a much deeper problem, especially because,

inasmuch as only partial functions pervade, Universality is a restricted

concept in what relates to possible constituted Languages.

This is why the premise of this Thesis is the immiscibility of Natural and

Programming Languages, not restricted to the Syntax and Semantics

partaking from each to the other, as is the habit, but, more imperatively,astride both in a conjoined ascent from one Sound of Speech Analysis. In

such a way, the status of the Programming Language Prolog will, by the

end, be elevated, not through typical Computer Science jargon, but,

inversely, through an Analysis of the conceptual intricacy ofArguments

that such Artificial Intelligence privileged historical Language seizes upon.

These Arguments being exposed in isolation, our aim is to offer a

consistent Philosophical listthat can best take on, inproblematisingone


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such unbreakable knot as the balance between Natural and Programming

Languages, from the core of Prolog, an original purposed Natural-

Language Analysis tool.

Notwithstanding being prevented from adventurous conclusions, this

book holds responsively its finale, having clearly ascertained that the

Philosophy of Computation is to find its place.


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Acknowledgements

I wish to express my sincere gratitude to Mara Manzano Arjona PhD,adviser for this book, not only for having served as a hard-working

reader and critic, but overall, for having, since the very first moment,

welcomed me in the warmest and most humbling manner, accordant

with her profound wisdom and selfless concern, for having assisted so

much through her motivation and encouragement, through her prompt

and cogent replies, as much as in content, and first and foremost for

having influentially guided this project. The author would like also to

convey thanks to all involved in the design of the Epimenides Official

Postgraduate Program in Logic and Philosophy of Science of all

Universities and Departments, and particularly thosewith whom I tooklessons. The interdisciplinary focus was very useful, the coordination andmediation was excellent, and the richness of depth invoked was truly

worth the Quality Grant, something that, very much appreciable to the

Students, must have been seen to have called for no small investment and

true dedication.

I would like also to sincerely thank Olga Pombo PhD, presently Head

of the Executive Committee of the Centre for Philosophy of Science at

the University of Lisbon, for having invited me to be a member in the

summer of 2010, and for having assisted me with gifts of understanding,

direction and continuous support.

To all, my deepest sense of gratitude.


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1

Section I

In the transcription of Ancient Greek characters and words, we have

chosen to use the conventional Middle Ages form [which includes both

majuscule and minuscule letters; in graphical terms, equivalent to the

use of uppercase and lowercase letters], as opposed to the original

exclusive use of the capital form of Classical times. It was also precluded

the use from the original Ancient Classical form the absence of any space

between words or paragraphs, with only one string of capital letters in

the case of expressions for reasons of comprehension.

Following the text, what might come up first is the Greek word, or the

synonym in English, in terms of words or expressions, until the end of

the best-conveyed occurrence by the author. Often, in such a context,when absolutely required, it shows the proper transliteration in English

inside brackets. Greek words forgo the use of brackets.

Aristotle used the term (endoxa) in alliance with the word

(topos) with the aim of addressing the title The Topics of one of the

six works on Logic, known as the Organon.

The sense ofcommon-placedassumptions was, thus, as if Arguments

in Logic had found the same locus as Space itself, that is, presumably

backed by the Euclidean Geometrizing rigour, binding together Logic

with Physics, not only because (topos) was close to the sense of

Elements () (stoichea) precisely the title of Euclides GeometryBook ( ) but also due to the existence of prior notes

credited by Aristotle himself, as when he echoed Zeno of Elea:

We say that a thing is in the world, in the sense of in place, because

it is in the air, and the air is in the world; and when we say it is in

the air, we do not mean it is in every part of the air, but that it is in

the air because of the outer surface of the air which surrounds it;


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for if all the air were its place, the place of a thing would not be

equal to the thing-which it is supposed to be, and which the primary

place in which a thing is actually is. (Aristotle, Phys. IV, 209a)

According to the Eleatic School Representative, whom Aristotle called

Dialectic in person, each things place (topos) is sculpted by the closure of

air, as if things were luminescent intervals of one Monism, and one

content-and-form paradox. Movement, (kinesis) thought to be a

spelling illusion of the one and only principle by Parmenides and Zeno

both chief and long-lasting references of the Eleatic School was, too, a

considerably difficult concept for Aristotle, which denotes, moreover,striking affinities with sensory experience. This is supremely important,

as it is in the same work that Aristotle defines place (topos) in the

following manner, Now if place is what primarily contains each body, it

would be a limit, so that the place would be the form or shape of each

body by which the magnitude or the matter of the magnitude is defined:

for this is the limit of each body. (Aristotle, Phys. IV 212a).

However, Aristotle, by choosing the title The Topics for one of the six

Books of the Organon, was doing so under the overall dominance of

Dialectic or Argumentative Studies. Under these circumstances

(topos) was taken as a commonly held argument of determined instances

laid down to favour the approval of the several propositions that entailed

the same argument. This had much more to do with Persuasion thanDemonstration, and much more to do with the intermittent Dialectic

Studies than with the scrupulous rigour of Geometry.

Therefore, it seems that (topos) was close to the sense of the void,

bereft of anything, and the content-and-form paradox of every being, to the

extent that Logical Arguments, themselves bereft of the ideal Geometrising

locus of syllogism and demonstration, would fall under this judgement.

The sense that something is intermittently absent is evident not only in the

claim by Aristotle that (topos) was a heading under which truncated

syllogisms would be revealed, such as enthymemes, but also in the fact that

one critical reader would find it more appropriate referring to it as proper

places scientific syllogisms, instead of rhetoric riddles.Dialectic () is, thus, a touchstone concept that serves to

unveil properly the use of (topos).

Plato envisaged the concept, rescuing it, as well, from the Eleatic

School, as a path to Truthful sources, and acknowledgeable Truth

firsthand. Inasmuch Plato promoted the concept from the Eleatic sense

of one inflamed wrangling altercation, eristic in essence (Plato, Soph.

224e-226a, Rep. 499a, Phaedrus, 216c), to the pathway of the Good, so


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Section I

too Aristotle crafted scientifically the concept of (Dialectic),

expurgating it from alien symbolism.

Hereafter, and since Aristotle, (Dialectic) has been relative,

not to the Good in substance under one holistic Theory of Forms in

Platonic fashion, but merely to the truth of the dialogue, with reasonable

arguments being paramount.

This aspect, by gravitation, shifted also the profound semantics of the

elenchus of all other terms, and, most notably, the proper meaning of

Elements (stoichea), extensively perdurable in Europes History

and Philosophy of Science as to have retained in Latinised form its lexical

provision in the title Philosophiae Naturalis Principia Mathematica byIsaac Newton in the seventeenth century, emulating Euclid.

Elements (stoichea), as an ancient term, that is, prior to

Euclids and Aristotles time and intellectual ambience, denoted and

connoted various things, to the degree that it meant each of the atomised

twenty-four letters of the Greek alphabet. This is true not only of this,

but, usually also of other properties and relations, other than the

common letterform and respective utterance.

These genres of other properties and relations were affiliated with the

idiosyncratic Hellenistic Atomist tradition on one side, and on the other

side, to others, most especially Pythagoreanism. Pythagoreanism was

very receptive to the domains of Gymnasium exercising Sapience as

Astronomy or Music, not excluding hermetical investigations, and wasfirst entrenched-synchronically in Hellenistic Egypt.

For Aristotle and Euclid, Elements (stoichea), were not taken

as having risen out of these conjuring spirits, nor did they have a magical

nature anymore, but were strictly scientific. Maybe all that lasted with

abiding force was the irreplaceable vital metaphor, distributed among

various traditions of the world, being composed not quite by letters or

atoms but, instead, by their generated (taxis) (order) and

(thesis) (position), which have led, as well, by Philosophical consentaneity,

to the introduction, coherently, of one linear interpretation from indivisible

(atoms) to wholly ordered (harmony or cosmos).

The arch between both was nothing but (Logos).This is in accordance with the on-going course of our dissertation that

will eventually guide us to better understand how (topos) suitably

goes as the head of the content-and-form paradox, pursuing the sense of

this undetermined evanescent limit the content-and-form paradox

ascribed most predominantly to (dynamics), otherwise known

as potentia through the Latin literary legacy, and, in contrast with the

expected, not that much conditioned by the force of Geometry, the


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primary interested field that would presumably welcome, precisely by

reasonable assumption, the concept of (topos) in the first place.

This approach will hopefully stage its abundantly evident kinship with

the concept of (Logos), to serve as a medium for our prior

desideratum, that of further discussing what it means, then, to be

speaking about Topics in any Language. Continuing to reason about

what it means to be speaking about Topics in Programming Languages

will be, henceforth, a complementary problem. None of the latter can be

accomplished if we cannot grasp the fundamentals behind Aristotles

choice of (topos) for a Work on Logic, once it fairly constitutes the

preliminarygroundof our thesis.One fundamental assertion is recognising that the emancipation of the

Alphabet as one naturalPhilosophical Language, neutrally or scientifically

constricted to mere formality, additionally of a more constringent

connotation to one being or abstract, predominantly functional, provided

in Aristotles and Euclids time, was, thus, verifiably concurrent with the

Euclidean Geometrising notion of ruling Principia, and also Aristotles

envisagement of transforming the archaic exterior principle from the

Pre-Socratic (arch), to the Parmidean (all that is one), comprising

almost four centuries of Greek Philosophy into the inherent, autonomously

intrinsic elementat the core of Physics by asserting as follows in

, (Lectures on Nature), (or simply, The Physics):

Of things that exist, some exist by nature, some from other causes.

By nature the animals and their parts exist, and the plants and the

simple bodies (earth, fire, air, water) for we say that these and the

like exist by nature. All the things mentioned present a feature in

which they differ from things which are not constituted by nature.

Each of them has within itself a principle of motion and of

stationariness (in respect of place, or of growth and decrease, or by

way of alteration). (Aristotle, Phys. II, 192b).

I claim, hence, both to be equivalent:

The former Symbolical and Semantic exteriorisation in (Logos)

(here as Language)placedinstead by one scientifically and philosophically

interiorised Alphabet permitted, in similitude, in the all-encompassing realm

of Nature and Physics, to interiorise the verge of one principle that had since

the pre-Socratics seen several and plentiful intellectual ownerships, in all

forms of externalised principles, basically contending to dissert on (the

origins), was now redirected to a principle by nature and within itself.


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Section I

Thales of Miletus water, Anaximanders of Miletus unlimited

(apeiron), Anaximenes of Miletus air, Xenophanes of Colophons earth,

Heraclitus of Ephesus fire, Pythagoras of Samos number, Empedocles

of Agrigentums four elements, Anaxagoras of Clazomenaes (nous),

(intellect, mind, common sense), Democritus of Abderas (atoms),

are all stepping stones to further achieving a sufficiently from within all

elements of the Nature principle, thus catapulting (the origins) to

the meaning of (stoichea) (elements).

This shift pointed sharply to the need Aristotle had to respond to the

problem already mentioned of (dynamics). The vulnerable and

volatile character of change, movement and sense experience was, insuch a way, brought about to be self-absorbed, and introspectively

naturalised. This is what propelled Aristotle to proportion and come up

with the entirely original concept of(hyle) (matter).

In effect, (hyle) (matter) was just the right analogy with the

AB(Alpha, Beta), (Alphabet), once, apart from having been

materialised, the Alphabet retained the idea that it was not anything else

other than what it is, by the iterative nature of its lexicon.

The shift in debate is an Olympic remodelling.

For Plato, recognisably, the Theory of Forms was not only, so to

speak, carved in the air in one general plausible embodiment of

Arithmetic, but, furthest from this original position, it was, beyond that,

respective to (the entirety, the total). Its origins were, likeMathematics, very worldly, though, of accountants dealings and agrarian

measurements. These manifestations were hospitable to the context of

two-line-segment, instead of the two numbers abstract, which is

clearly in detriment to the non-geometrised views of Nature.

The intricacy of quantities to objects was mainly operated by means

of what was called anthyphairesis1 from (anti) (closer to the sense

ofinstead of, as a substitute for, than, over against, or opposite to) and

(a taking away, an abstraction of), now commonly known as

a continued fraction, or as that which continuously subtracts the smaller

from the larger, in an essentially geometrised way.

This operation was performed by (analogy) in all possibleconfigurations of space and (topos), allowing algebraic expressions

to be animated in a geometrised style and vogue.

Anthyphairesis consisted, thus, of passing on from dividing the larger

by the smaller (finding, thus, a quotient and a remainder) to dividing the

smaller by the remainder.

In geometrical terms, this corresponded to, in a Pythagorean square,

transporting any side to the interior diagonal, finding, thus, one point


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(the rest of the segment being the remainder). From this point of the

interior diagonal to any vertex, we find the side of a new square,

repeating the process of transporting the side to the interior of the new

and ever shorter diagonal, thus agitating observers by showing magnitudes

to be incommensurable.

This method shows coherence, for instance, with the usual search for

the least common factor, based on the pattern of subtractions, and with

the Greek general refusal to use numbers (fractions or real) for the

standard description of graduate shifting quantities.

Curiously, Plato vindicated Nature as a shadowy cave presence, but

almost impossibly so, except metaphorically, that is, by transferringsimile or (metaphor) as objects would instead emanate light,

and therefore emit their concrete architectural forms, Nature being not

exactly a shadow, but a copy of light, something that is close to the

actual sense of photography or, fundamentally, a Model.

These notes are also intended to throw light on the congruence with the

Ancient Greek Emission Theory of Vision supported by Plato and how far

Geometry was integrated into Platonic Philosophy and the Theory of forms,

to the degree that it was not just all previous (the origins) that were

ascribed one specific Geometrical solid the Tetrahedron (four faces) to fire,

the Hexahedron (six faces) to earth, the Octahedron (eight faces) to air, the

Dodecahedron (twelve faces) to the excluding ether, and Icosahedron

(twenty faces) to water as, for example, water, which was one such elementthat was unable to hold any vertex, of which there is no better example in

Nature of change and movement, and which was itself a placeholder for one

very complex solid of Geometry (of twenty triangle sides).

In Plato, Geometry was thoroughly comprehended from within

(dynamics).

Aristotle exempted, therefore, in a way, the world from Geometry, in

spite of being a contemporary of Euclid and having a thirst for knowledge

of the Platonist fountain.

Maybe there isnt, consequently, a more extremist demonstration of

the aforesaid than the fact that the founder of the Academy in Athens

defended Time, the vessel for sensory experience and movement, as animage of eternity (Plato, Timaeus, On Physics, 37c-e), while, in contrast,

his student form Stagira merged eternity with time, through the idea of

one substratum, thus, stating Time itself to be one of a kind, of everlasting

existence and sempiternity.

This sort of characterless , (hypokeimenon) or substratumcharacterisation conjoins excellently with the idea of the exposed

content-and-form paradox.


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Section I

In agreement with the exposed was the Aristotelian new Theory of the

Four (causes, emergence), a sort of Philosophical Analytical

Cosmogony that was simultaneously a new Methodology, in lieu of the

former four (origins).

Most prominent, and that which paves determinately the way for our

Topic of Discussion, is precisely the graded movement between the

efficient cause and the final cause.

The efficient (cause) (Aristotle, Parts of Animals 641b24-25;

Physics 194a29-30, 199a8-9) was described as (the origins, the

commencement) of (motion), and the final cause as (the

purpose), both causes much more conducive to movement and (dynamics) than the material (hyle) and formal (ousia).

Notwithstanding this, what is revealed is that not only do these two

separated pairs form the border of the division between Nature and

Artificiality for Posterity, but more importantly, they are synthetically

brought together, to the effect that out of each of the four causes, there

is found to be one content-and-form paradox, a limit, as mentioned

before, to have held the harmony of one irrepressible flux, of such

alterations, as in quantity, quality, several instances of being acted upon,

time and, of course, space or (topos).

Nature and Artificiality are, by this effect, and in Philosophical terms,

irreconcilably reconciled to being forthcoming, eventually extending to

such areas as the Philosophy of Computation and Artificial Intelligence.We defend the fact that, more important than setting the border

between Nature and Artificiality, it was the evanescent characterisation

of the content-and-form paradox that silently recorded for posterity the

essence of the Natural Philosophy Debate.

Now, to condense as much as possible our first draft conclusions, we

outline them below in paragraphs, with considerations put forward in a

declarative style, as far as possible:

Arguments) The phonetics and philosophical argument

After the Phoenician Alphabet, the first ever non-pictographic Phonetic

Alphabet (in which the script denoted that one sound, a phoneme of the

spoken language, was represented by one symbol, even though it

consisted of consonants throughout, lacking vowels), thus producing a


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sort of Linguistic Universal Algebra Method we find as being

complementary to the consequent Method by constraintof the derived

Greek Alphabet (a proper modification of the Phoenician Alphabet),

which was Semantically and Philosophically inspired by Aristotles time

and Aristotle himself, and which was that of ascribing little or less

meaning to each word and letter form except its phonetics or sound. This

brings up again Derridas not too trifling claim that Philosophy would

not be possible without the Alphabet.

) The symbolic or rational argument

Following the above-mentioned breakthrough, we recognise that this

licensed another Symbolical axial distinguisher between two views, not

by exactly diminishing one in favour of the other, but by acknowledging

first and foremost the same as that which is perceivable hereafter as a

boundary demarcation in the Philosophy of Language.

According to this view the World and its Natural constituents are seen

as (material) or (ethereal) signifiers (almost as of one

absconded meaning, purely of a Philosophical Quest), transforming any

Language into one epiphenomenon, as if the Language and Alphabet

were primarily the (cosmos).

According to this other view Language is (Logos).It remains to say that it is possible to have passed the first view, having,

therefore, one scientifically Alphabetised Philosophy, and still be a strong

proponent of the first. Likewise, yet less convergent due to the orderliness

of History, it is possible to hold the second view and lack any Alphabet.

) The difficulty argument

The united prevalence of the idea of one limit, already mentioned as the

content-and-form paradox, out of which arises the doubt about which

excludes the other, at what time, by which actions are taken, to which

space, and out of which causes of emergence, with (topos)characterised here as one fugitive limit, having absconded to Geometry,

prevails in the exact same way as the holistic coeval emergence of the

four different causes propounded by Aristotle, after his having grasped

this apperception with a similar sensitivity. Similarly, an Analysis of

Topics, as is the intention of this Thesis from the Title, can bear affiliated

paradoxes, as if we have written a frontispiece with the intention of

writing an Analysis of Dialectics, with one maximum aporia.


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Section I

) The content-and-form artificial intelligenceargument

In exchange for the view by Aristotle, borrowed from Zeno of Elea, of

limits therein present, like air surrounding beings, we could thereon

satisfy the same model, by extending it to appropriately meet the

emergence of the efficientcause in the given example of the art of casting

a statue (Aristotle, Phys. 195 a 6-8; Cf. Metaph.1013 b 69).

As a continuation, we can extend further the content-and-form

Philosophising Paradox, to a view similar to that of Plotinus of man

sculpting its own statue (Plotinus, Enneads, I,6,9).I argue that this categorically finds and superbly orchestrates the

deepest insightful and distinctive mark that we encounter in Artificial

Intelligence (AI).

Though not the place for the enunciation of prolific examples, we

argue, wholly and declaratively, that Greek Culture, Mythology and

Philosophy, conserving this content-and-form paradox, in both the

Symbolic and Scientific poles, can confidently be appointed as the

genuine precursors of AI, forbearing signs of such intellectual portent, of

which the gear Antikythera of Archimedes was the pinnacle, but in the

group of other impeccable demonstrations, such as the Elements of

Greek Mythology in Homers Iliadand Odyssey, Hesiods Works and

Days and the Theogony, and many other Historic Classical accounts. It

has also demarcated sharply the distinctive line between Nature and

Artificiality by formal means, wherefore we conclude that it seems more

just to substitute the view of Artificial Intelligence as a Discipline of

Computer Science, by the exact opposite: Computer Science seems to be

one very late survey of the age of Gods, semi-Gods and mortals, in which

Artificial Intelligence was experimented.

) The efficient cause argument

Of all the four Aristotelian causes, we designate a leading and prominent

importance to the efficient cause. This is because the efficient cause

sublimates into abstraction every inquiry into Nature by contemplating

its initiating motion that leads to change. Passing over the naturally

imperfect disambiguated term (cause) in the confrontation with

Aristotelian and Greek fonts, and even the explanatory priority in return

for one axiomatic listing by the Author from Stagira, we can, though, see

from an Aetiological perspective crucial semantic turnovers.


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On the one hand this form of abstractness in the efficientcause (Phys.

195 a 6-8. Cf. Metaph. 1013 b 6-9) that ranks more highly a principle

of action interiorised in the human mind as an agent (e.g., the art of

sculpting), allowed the passage of the notion of self-autonomous efficient

cause that has, ultimately, been driven to the notion first put forward by

Leibniz of function f(x) and, of course, mixing along the way with the

notion of artefact, the model found to parse between one agents

intention and an object or machine, of which there is no better example

than the so-called Universal Turing Machine.

The determinant topic under the attention of Programming Languages

is more sharply foreseen by this assessment.On the other hand, the emergence of the efficient cause was the

keystone to the first off space apart from Teleology or the finalcause

from the more static natural inaugural causes. One such envisagement

(Phys. 198 b, 19-27) that was first ever proposed scientifically by

Aristotle, and which was barely distinguishable from atomist and

mythological bases, though, was in full comprehension of the principles

that would cause the ruin of the Teleological Principle by Darwin in the

nineteenth century, and permit the advance to Computation and

Computationalist views.

) The model theory argumentAs Maria Manzano points out in the preface to the book Model Theory,

(1999) Oxford Logic Guides, the Discipline born fundamentally from

Alfred Tarskis insights has rooted in itself an inestimable Epistemological

integrity, by convoking, as a result of Mathematical Logic, the

representation as a model, this outcome being the result of the

establishment of one Language Land a class of objects M, which are

structures with the notion of truth bringing them together.

Although aware that what is being depicted is one Mathematical

Theory between mathematical structures by means of the apparatus of

formal language, we wish, additionally, to save from confinement and

the still reliable, broader and faithful historical view of Model Theory,

that is, (...) the study of the interpretation of any language, formal or

natural (...).2

In this fashion, it stands to reason as well, that the interpretation of

any language, formal or natural, towards any other, testing both Natural

and Programming Languages through the notion of a truthful full

interpretation, in which, again, any Language L, natural or programming,


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Section I

can be seen as playing the role of a metalanguage to the other, granting,

fundamentally, in a panoramic view, such a one that allows Languages

as Structures to face each other.

We wish, meanwhile, to contemplate more steadily the basic notions

of Universal Algebra and, in agreement, the basic notions of Model

Theory.

Universal Algebra endorses topologically enriching concepts such as

groups, rings, fields and orders, which assert consistently the dissertations

goal, and, consequently, set the right scenario and help the progress of

Model Theorys notes about Natural and Programming Languages.

On this basis, inarguably, lies the symbiosis we found betweenSymbolical and Philosophical views, as contemplated in Platos Philosophy

as well as in the Western tradition. The class of objects M and the

Universal Algebra method of the anthyphairesis bond have been so

strongly forced into one interpretation, as to have been foreseen from the

core of the Platonist Philosophy and obelised against contrary points of

view, an excellent way of allowing its isomorphism from the mind to

mathematical structures.

To reinforce this assertion, we shall, in conclusion, appeal to the

corroboration of the previously mentioned perspective with the fact that

we have the continent-and-form paradox on one side, and the carved in

stone (metaphor) on the other, as though it were the sculpting

of structures to find one interpretation, which inevitably joins securely,in terms of Mathematical and formal objects such as quantities,

magnitudes or letters of the Alphabet, the method of continuous

subtraction or anthyphairesis.

Notes

1. After the Work by Fowler, D., The Mathematics of Platos Academy: A NewReconstruction, (1987) Oxford, Clarendon Press, we acknowledge that thisterm rescued to stand out by the cited author anthyphairesis was derived

from the Greek verb anthuphairein used in Elements VII, 1 and 2, and X, 2 and3, by Euclid. The meaning of reciprocal subtraction has been promoted to othermore ancient sources by David Fowler, to such an extent that the author comesto argue that the original meaning of anti-hypohairesis, reciprocal sub-traction, was one throughtout not silent and assumed Greeck Mathematicaland Logocratic Method. (Vide, p.62).

David Fowler came to give prominence to one passage by Aristotle in The

Topics (158b29ff) in which Ratio Studies were brought to be disscused by

means of the above-mentioned term. He said of himself as agreeing with


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Alexander of Aphrodisias Comments on Aristotle when he stated that the

Philosopher from Stagira understood by that process the same as anthyphairesis,

corresponding, contrary to division, to the method of repeated subtractions.

2. Hodges, W. (2009) Model Theory, The Stanford Encyclopedia of Philosophy,

Zalta, E.N. (ed.). http://plato.stanford.edu/archives/fall2009/entries/model-theory/


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2

Section II

We shall now proceed by highlighting the Mathematical and Philosophical

role offunctions to further come up with the resource of-Calculus the

smallest universal programming language in the world1 in the words of

Ral Rojas and Prolog, one of the first Logic Programming Languages,

to assess properly the aim of discussing Topics in Programming

Languages.

-Calculus was created by Alonzo Church in the 1930s as a sequel to

the Investigation for the Foundations of Mathematics, a field of study so

closely adjoined to Philosophy, that it set up and provided a debate with

the notion of the distinct assorted Philosophies of Mathematics of

Platonism, Formalism, Intuitionism and Logicism acting together.

This intellectual vibrancy obeyed the urgency of avoiding the pitfall ofparadoxes and tides of axiomatic misconceptions that characterised the

beginning of the twentieth century search for the Foundations of

Mathematics, following the heritage of Hilberts thorough

Entscheidungsproblem. This was only one among twenty-three open

questions or collections of problems presented by Hilbert and often

described as one Krisis.

Alonzo Church attempted, thus, by inventing -Calculus, to formalise

Mathematics through the notion offunctions, more noticeably, effective

procedures, contrary to typical Cantors and Dedekinds Set Theory

(or, preferably, at the time its limpid Zermelo-Fraenkel version

contemplating the non-existence of some sets and the famous Axiom ofChoice).


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The idea at its core was to enhance in simplicity the realm offunctions,

by drawing a sharp line between Consistency, Independence and

Universality, in an approximate accomplished style of Set Theory,

conjecturing, based on the Theories of Recursion, Type, Logic, Axiomatic

and Proof.

In other words, Alonzo Church attempted to substitute theplace of Set

Theory into that of a functional language in the well-ordered pair of

Foundational Mathematics that corresponds to the twofold Set Theory

and Propositional Calculus.

It renders in the affirmative the postulation that such an idea never

expanded, but on the contrary, it helped towards the progress ofComputability Theory.

Indeed, in this respect, the 1930s were an absolutely seminal decade:

while Gdel destroyed the aspirations for the wholly Complete resolution

of one of the problems posed by Hilbert, namely theEntscheidungsproblem,

(with much more at stake, coming as if from the depths of Mathematics,

rather than being one mere problem), through the presentation of his

two Incompleteness Theorems published in 1931, Alonzo Churchs

interest in the theme, and Turings work on Computability and Artificial

Intelligence (McCarthys 1955 original term) prevailed.

This made Church and Gdel each publish independently in the years

19361937 papers in which, similarly, the case against any solution to

the Entscheidungsproblem was defended, having plunged Mathematicsinto ignoramibus. However, some have speculated that Hilberts original

stand was such that a negative answer was still an answer, and thereafter

a position maintained by Hilbert to the beginning of the decade in

question.

Gdels First Incompleteness Theorem attempted successfully to

instantiate that no metamathematical proof of consistency was possible

within a system comprehensive enough to contain Arithmetic per se, as

the one described in Principia Mathematica, Bertrand Russells work on

Elementary Logic. Gdels Second Incompleteness Theorem sought to go

further, and advocated that Axiomatically, in its plentiful sense of

deriving from (topos) to construct Theorems from all or some ofthe ones previously postulated, and again, one such as the Principia

Mathematica, was essentially Incomplete.

An Axiomatic System of Number Theory such as Principia Mathematica

or any other, holding a consistent formalisation of Arithmetic could not

necessarily derive number-theoretical statements from within the system.

The shock that followed was so great that it was if it had been discovered


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Section II

that Mathematics was not Mathematical, and, in essence as if it had not

derived from Inference, or Deduction had not been its underlying method

since Euclid in its standard form.

Therefore, the rendition of proof was sought in just about the same

(topos) where it had disintegrated and surrendered to

Incompleteness, that is, the proper place of Mathematical Proof and

Deduction.

Just as deceptive as the recognition of Propositional Logic as being just

one truly weak conceptualisation of Formalised Logic, and, in addition,

not even borrowing First Order Sentencing, it was capable, though, of

achieving expression through its formal apparatus to describe ElementaryArithmetic, so too did Set Theory find through the Axiomatic Collection

of Sets compromising Paradoxes.

But none was such a blow as Gdels Second Incompleteness Theorem,

once it relegated Proof to a balance between, on the one side, Completeness

and, on the other, Compactness of the Axiomatic inside Number Theory

proper. The astonishment was so great it was as if it has been asked what

exactly the Philosophical birthright of Mathematics would be ifwe had

at our disposal all its theoretical statements. Would that collection be

more correctly said to be non-derivable or, else, a paradox of such

amplification that Number Theory was most inappropriate for

Mathematics, as Mathematics was appropriately just a shadow to

another sort ofMathematicalRealism, as Gdel himself envisaged?It is worth stressing that Mathematical Realism or Platonism is, out of

the quartet of Platonism, Logicism, Formalism and Intuitionism, the

most inexpugnable of them all by the principle ofnecessary reason only

due to Theoretical and Methodologically naturally non-disprovable and

unsolvable assumptions.

For Gdel, though, this conjecture was a principle ofsufficient reason,

and most importantly, what would be taken under this sufficient

reason principle was the absolute order of things, from (atoms,

elements) to wholly ordered (harmony or cosmos), in Greek

conceptual terms.

Alan Turings testimony on the problem follows smartly from the articleOn Computable numbers, with an application to the Entscheidungsproblem

(1936). This article, in my opinion, is a summula of the traditions I have

chosen to address as being ofCalculus and Computus.

To make a point, we shall try to understand how much the rise of

Programming Languages, and the evolution from Aristotelian efficient

cause to the coadunate Mathematical function, under the timeline landscape

from Calculus to Computus, was completed, to which is compelled, in


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addressing the task, to convoke the progression of Philosophia Naturalis,

the work of authors of repute, and research with an exceptionally

outstanding methodology, either at the forefront or in the matrix of

progress, to which end the manifestation of Machinery into Computus is

rendered easier to understand, au pair the well-ordered entanglement from

Natural Languages to the first Artificial Programming Languages.

In more detail, our digression will identify Calculus and Computus

landmarks, without which the proper rise of Computation and -Calculus

cannot be evaluated, through the propagated Church-Turing Thesis, and,

likewise, without which proceeding to study closely Topics in Programming

Languages would be flawed. Conjecturally, any passage or attempt, tolearn the Declarative status and Philosophical Impingement of the

Programming Language Prolog originally PROgrammation en

LOGique would be equally bound to fail. It is virtually impossible to

understand the full rise of Prolog as a Programming Language without

setting forth its most undeviating forerunning influences.

Traditionally, Computation is considered to mean any part ofCalculus

expressing algorithms precisely, and following some kind of architectural

model, by the use of Computer technology. But, in the same way

Programming Languages predate Computation, likewise, Machinery

expertise, besides predating Computer Science, was also eximious and

competent beforehand in the technique ofComputus.

There no author so correctly associated with the immediacy of thegoal of our Philosophical Investigation, as regards Historical Timelines,

as well as Philosophical significance and magnitude, as Kant.

Prior to Kants death (1804) at the dawn of the nineteenth century,

looking back at the Computata and Automaton History, from the very

probable Archimedean Antikythera, to Frederick II (The Great) of

Prussia (17121786), whose patronage of the Arts, Science and Religious

tolerance made possible the building of the Prussian (Berliner) Academy

of Sciences, and so we can, without any hesitation, say that Kant, and

Kants moment in History was, though an advocate of Criticism apposite

to the new Copernican Revolution, very orientated still, for the most

part, to the paradigm of the Anthropocentric Humanist view of theseventeenth century of Pascal, Leibniz, Descartes, Bacon and Newton.

Kants idea ofFuture Metaphysics was still captive within the medieval

conceptual-frame that admitted such figures as angels (one Bio-Theo

hybrid) enclosed in a strong Ontological Hierarchy; also, fundamentally, to

outline its biggest distinguishable mark, was considering one of the

Axioms of Intuition Totally Humanist. That is to say, if we imagined


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Section II

Humanity as a domain, in accordance with Kant, there had to be an

irrevocable ascent, with a strong perception of knowledge developing as an

increasing, all-pervading function from a planar region, whose arbitrarily

considered units were to be summed up as the ideal of Mankind.

Even after the proper inception ofPhilosophia Naturalis into Natural

Philosophy with Darwins On the Origin of Species by means of natural

selection, or the Preservation of Favoured Races in the Struggle of Life

(1859), and chiefly because of its spurious interpretations from the field

of Social Sciences afterwards, the anthropocentric preceding idea has

prevailed. Furthermore, it has not by any means, lost such a sense of

Moral Ontological Law or Anthropological Hierarchy in theContemporaneous Era of Computation.

Yet in essence, what we want to overcome through possible minor

issues, is the critical disparity that has to do with the fact that in Kant s

time, and to Kant, such an irrevocable Humanist function of Knowledge,

prefiguring monolithic absolutist Philosophies of History, was never

considered of such a quality that would have to have an outer-

empowerment of man from non-sentient, unperceptive, automated

means, and least of all, say, with the decisive help of machines.

Having said this, however, ideas must not be misconstrued. The

seventeenth century gave rise to all the foundations of countless axes to

the forthcoming driving forces of Computus. The Leibnizian binary

code, machinery enterprises such as Le Pascaline, the idea of the ModernAge disembodiment of the soul and body by Descartes (a sort of inner

hiatus between Greek Mythological constructs and the Contemporaneous

Bio-Machine Hybrid Problem), and the strongly Scientific Inductive and

Experimental Baconian Method, under the umbrella concept of

commanding Nature in action, are all fitting examples.

In reality, for the Topic under discussion, fusing Language with

Computation, and calling for an awareness of the immiscibility of Natural

and Artificial Language, we should be attentive to the fact that the

resurrection of the Encyclopaedia, namely the Encyclopdie, ou

dictionnaire raisonn des sciences, des arts et des mtiers, (17511772),

edited by Diderot and dAlembert, both members of the Prussian Academyof Sciences, was coeval with the fortuitous rise of Mechanics and

Machinery expertise, illustrated eloquently by the fact that men such as

La Mettrie, author ofLhomme machine, have ingeniously advanced such

themes at the forefront of contemporary Natural Science Academies.

Thus, it is legitimate to declare that (Encyclopedia,

Universal Education) and (craftsmanship) have joined together

again for the Epistemological test in the Modern Age, especially from the


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end of the seventeenth century onwards, which would prepare the

ground for functions to enter the scene in the Calculus moment.

What, then, is the right time to effect the passage to Computus? The

answer willl not be rendered in the form of one narrow convergence, but,

rather, in the form of precisely the original sense behind the first use of

functions, that is, one curve of different values. Kant is, to this effect, as

said before, a superlative example that reflects this curve from Calculus

to Computus.

Innovative Machinery such as the Spinning Jenny in the Wool Industry,

the Cotton Gin and Jacquards Loom, the Water Frame and moving

factory cogs, the Steam Engine, and even the Electrical batteries made byVolta, were all contemporary to Kant, wherefrom it is evident that we

should concentrate on the fact that Kants pre-Industrialist Weltanschauung

(world view), was effectively directed, but delayed, towards the critical

irrevocable entry of the Controlled Electricity Current that would facilitate

Computation through a constant flow of electricity, as is still the case.

In truth, Textile Machinery and Electricity were to together lead to the

first glimpse of Computation with something close to a Turing Machine.

Jacquards loom machinery was, lest it be forgotten, of a type using punched

cards, just the same as the early twentieth century digital computers.

Kants Philosophical evaluation of the impending Historical

phenomenon, would have been laid down, most surely, in the sort of

Machinery coming from the Textile Industry, as opposed to themechanised type akin to Watts improved steam engine, largely isolated

among the prevalent wooden-parts Machinery of the time. His perception

was well-inured miles away from England, the arena not of Metaphysical

debate, but of the debut of Machinery.

This was to prefigure the entry onto the scene of Programming

Languages, with the description of the Analytical Engine by Charles

Babbage in 1837, and the consequent algorithm to be performed on it by

Ada Lovelace (184243), which corresponds in substance with the first

accomplished Historical Theoretical instance of Hardware and Software

together, out of which a pair of Programming Languages stand out as the

function between the two. It was, additionally, a sort of empiricalpostulate that was unveiled to give proper rise to the Computus family,

but Computus as akin to the ideal Turing Machine.

It not only helps to consistently determine in History when the

restructuring from Calculus to Computus was achieved, but it really

expresses the affiliation of Natural to Programming Languages along the

same lines as Lovelaces direct influence from the poet Lord Byron in

symbolism.


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Section II

The Aufklrung age heralded, thus, Industrialisation and

Industrialisation Computation. Yet, in terms of establishing the Model of

Computation parsed with Language on one side, and Artificial Intelligence

principles on the other, none was so spectacular as both Churchs

-Calculus (1936) and Alan Turings paper On computable numbers

with an Application to the Entscheidungsproblem (1936). The idea of

calculable by finite means unfolded a drastic transformation, profound

enough to have unveiled a new Aesthetic Perception, in Kantian terms.

This new Aesthetic Perception is more profound than mere Cultural

views, and corresponds largely to the hereafter even more impossible

unravelling ofComputus from Bios, to which we shall lend some moreconsideration later on.

Bearing in mind the fact that all intuitions are of great magnitude,

Kant, nevertheless, opposed the Axioms of Intuition (Kant, I., Critique

of Pure Reason [A162/B202]) (connected with the categories ofUnity,

PluralityandTotality) to the Axioms of Mathematics, and comprehensively

so, as these are, by definition, synthetic, a priori and valid according to

pure concepts. We could say that according to Kant, the Axioms of

Mathematics were necessarily related to Knowledge (not to experience,

but to intuition only) and the Axioms of Intuition were necessarily

related to Experience (and not necessarily to Knowledge).

The criticism cultivated by Kant allowed, therefore, the introspection

of Philosophy into its forms of Judgement. The source of the principlesin accordance with which everything comes in the first place as an object,

necessarily stands according to these rules. And this centrality in

formality was proposed and prepared in anticipation of the Symbolic

Gdelian Mathematical Arguments such as the Turing-Churchs Thesis.

Moreover, it conserved a heritage intrinsic to the Platonist, Formalist,

Logicist and Intuitionistic Logical debate.

Here we put forward the excerpts from and relative to Leibniz and

Turing that help to connect function with the age of Calculus and

Computus; the notion set out by Turing himself in his Princeton PhD.

Thesis that of functions being effectively calculable is here recalled

by Andrew Hodges:

In 1694 German mathematician Gottfried Wilhelm Leibniz,

co-discoverer of Calculus, coined the term function (Latin: functio)

to mean the slope of the curve, a definition that has very little in

common with our current use of the word. The great Swiss

mathematician Leonhard Euler (170783) recognised the need to

make the notion of a relationship between quantities explicit, and


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he defined the term function to mean variable quantity that is

dependent upon another quantity. Euler introduced the notation f

(x) for a function ofx, and promoted the idea of a function as a

formula. He based all his work in Calculus and Analysis on this

idea, which paved the way for mathematicians to view trigonometric

quantities and logarithms as functions. This notion of function

subsequently unified many branches of mathematics and physics.

(...) Advanced texts in mathematics today typically present all three

definitions of a function as a formula, as a set of ordered pairs,

and as a mapping and mathematicians will typically work with

all three approaches.2

And by Turing:

A function is said to be effectively calculable if its values can be

found by some purely mechanical process. Although it is fairly easy

to get an intuitive grasp of this idea, it is nevertheless desirable to

have some more definite, mathematically expressible definition. Such

a definition was first given by Gdel at Princeton in 1934... These

functions were described as general recursive by Gdel... Another

definition of effective calculability has been given by Church... who

identifies it with lambda-definability. (...)We may take this statement

literally, understanding by a purely mechanical process one whichcould be carried out by a machine...(Turing, A. (1939) Systems of

logic based on ordinals, Proc. Lond. Math. Soc 45 (2): 161228.

From the end of the seventeenth century to the early twentieth century,

in actual fact encompassing the scope in time of both the above quotes,

and remembering what Kant acknowledged as the Axioms of Philosophy

(Kant, I., Critique of Pure Reason, [A733/B761]), essentially a Mechanism

of Proof, a Universal Deduction in service for Enlightenment, limited

only and enough to Criticism, we can observe how, from this stance, we

can diverge in our analysis.

On the one hand, there is an immediate contrast between explicitness ofthe Second Incompleteness Theorem by Gdel, and the Churchs unsolvable

answer to determining the truth of arbitrary propositions inside Peanos

Arithmetic, the obvious Axiomatic Number Theory chosen to test on

-Calculus. However, on the other hand, there is a more systematic

understanding of the impressive achievement of having launched the

Theoretical basis for Computation and the effectiveness of Programming

Languages, from the Theoretical limitation of Incompleteness.


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Section II

It becomes really impressive when we contemplate how such an

outstanding Platonist as Gdel was to work out Computation and

Programming Languages generally by Mathematical Apagogic or

(Diaeresis) (breaking, division, successive division Method, and so on [in

written natural language, the diacritic mark is placed over a vowel]),

rather than Proof, even if what was proved was Incompleteness itself.

A shortcut to best understanding the rise of Programming and

Artificial Languages is, hence, the differentiation between Calculus and

Computus.

The idea of Computare by means of Artificial Intelligence, here

understood along the lines of and akin to Turings foreshadowinginvestigations, has somehow a distinct imprint, effect and influence

compared to that of Calculus as disclosed simultaneously by Newton

and Leibniz.

Very curiously, and still in general terms, it can be said, that at least

there is one continuum from Leibniz to Gdel with respect to the inner

expressive power of Mathematical Realism. This, after fading away, was

to give up its place to Computational Science. This is as if the slope of

the curve of Calculus was now Computus in the form of an effective

calculability within General Incompleteness.

We could also cite the great discoveries in the realm of Mathematics,

such as that of Oresme, responsible for the first graph or pictorial

function, or those of Napier and Briggs, who worked on tables oflogarithms and Machinery applications. It is really an all-embracing

subject. However, we should bear in mind that the sharp end of its

emancipation was very much an omnia relata est Renaissance

Cosmological Vision as early as the fifteenth and sixteenth centuries,

with systems of thought hospitable to relationalthought, without which

the notion offunctions could not have been born.

This resulted in the confluence and wide conceptual Philosophical

relatedness between the Mathematical function, the Philosophical rule

and the Computational algorithm (a list of procedures) presented here.

One piece of stronger evidence for the aforementioned is the fact that it

was won through the designation of the Church-Turing Thesis forposterity.

The Church-Turing Thesis combines, at its heart, -Calculus, the

Universal Turing Machine and Recursion or Computability Theory

(which saw developments in the 1930s from Turing, Church, Kleene,

Post, Gdel and so on). At its core, the issue was the targeting of the same

class of functions, and, therefore, contemplating in essence the equivalence

of Computability, the Universal Turing Machine and -Calculus.


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It is interesting to consider the separation between the branches of

Mathematical Logic with stress and attention on Formal Languages,

Semantics, Syntax and Proof on the one hand, and Recursion Theory,

primarily concerned with establishing definability in the subsets of

Natural and Real Numbers on the other hand according to the criterion

of Programming Languages. The reason is that Artificial Programming

Languages are born out ofcomputable functions in Boolean order in the

first place.

It is also worth mentioning that problems of unsolvability relative to

the nature of functions are effectively calculable, that is, algorithmically

computable, as seen in the so-called Turing degrees, and are a validparallel with Natural Languages. It could be asked in a most

non-technical manner: even if computable effectively calculable functions

were one set only, without the subsets of Turing degrees to maintain

unsolvability levels, was Natural Language Processing on any level

different, at the most linguistically computable and more competent than

it is now, even by the slightest degree?

Would the answer tell us more about the relationship between Natural

numbers decidability and Natural Language Processing? If the answer is

yes, it would mean that the reducibility of Natural Numbers to Natural

Languages through algorithms stands up well.

Is the mirage of a positive answer to this problem not a response to the

unsolvability of Natural Languages, and so worth mentioning here as aresult; and are the levels of unsolvability different from Language,

Syntax and Grammar studies typical of Mathematical Logical inspiration?

In other words, is it possible to rank in hierarchal form the

non-computability of Natural Languages in any formal method, like

Turing degrees? Can Computability Theory analyse in negatio the

expressiveness of Natural Languages?

Certainly, the study of computable functions or effective calculation is

stimulated to pivot on unsolvability and decidability separation;

independently, Turing and Church in 1936 were to confirm Gdels 1931

result about the non-existence of such an algorithm that would positively

test completeness for any restricted number of central First OrderAxioms, precisely the Entscheidungsproblem, otherwise known as the

decision problem. As for Recursion Theory, Natural and Real Numbers

played the role of testing assets for decidability, in the same way First

Order Logic had done previously.

I wish also to draw another parallel about the transferring method,

now in Number Theory, with repercussions in the Philosophy of

Mathematics. This will help us to understand the invention of-Calculus,


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Section II

which is supported by the Mathematical heritage offunctions, as well as

to realise how Computing Programming Languages were born, essentially

out of a functional paradigm capable of relating fields of different values

with various arguments.

By this means, if we look carefully, we are arriving at an understanding

of how Turing could, using a function approach, fully and clearly

articulate effective calculability, as one argument, by means of idealised

machinery, so that Computation and Calculability became as one. Turing

fused, just as how Jacquards or Babbages engines interweaved webs of

textiles, in intellectual Philosophical terms, the tradition ofCalculus with

that of the new emerging Machinery powered by electricity and, mostimportant and above all, Computus.

Calculus and Computus were, therefore, explicitly joined, as were the

concepts of the dynamics offunction, rule, and algorithm.

Church, on his part, foresaw in N, as well, the isolation of the

Recursive Numbers Set, identifying them with the Computable Class.

Reducibility notions, structure degrees and even relative computability

concepts all derive from the standard form of combining arguments with

values, to which list we could add all formalised languages, including

Programming Languages.

Algorithms were, thus, found to be the benchmark in the arbitrary

mathematical proposition search for solvalbility, once any function

capable of being computable by an algorithm was discovered to be acomputable function. And, fascinatingly, algorithms, as a list of

procedures to resolve a function, were to divide into effectively unsolvable

mathematical propositions, to the point of admitted Incompleteness in

the core of the simple formalised Language of Arithmetic, which failed

to find new Theorems through algorithms.

Another important point to note is how Arithmetic and its very latest

developments in Number Theory had sought improvements, and finished

eventually by putting forward for the Philosophy of Mathematics and of

Languages a much more succinct outlook, with revisionist stances

towards the existing body of Mathematical and Philosophical

knowledge.In relation to the Philosophy of Mathematics, the decade in question

(the 1930s) and pure developments, there have been really well-ordered

and almost regimented different background claims, but the most

exceptional following results were, maybe, the spectacular outburst of

revisionist stances and the positive incursion of Mathematical methods

bearing on typical Philosophical questions, some of them with late

precedence (Set Theory, Proof Theory), and some others which are


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subsequent as well as consequential arrivals (Model Theory, Computability

Theory). The Topic of Programming Languages was not, for this matter,

and in a straightforward analysis, a single rising field combining historical

traces of Language and Concepts, to come into contact with partial

functions and Computability, progressing to contend with Design,

Implementation, Syntax and Semantics, in one already very fruitful

overlapping of disciplines, views and methods. It has, unequivocally, too,

played the role of the scales of justice while denoting a particular Syntax

and a particular Semantics inside the framework, and has also been able

to, outside the framework, connote different Syntaxes as well as different

Semantics. It linked very well the high expressiveness of NaturalLanguages and the high consistency of Mathematical Proof Theory in

one progressive set of gradations that, while having started out with the

attempt, especially in Natural Languages Processing, to produce a special

effect of meta-theoretical expression in formal Language with the

phenomena of Language in its fullest sense, from acoustics to writing,

from Linguistics to Informal Semantics, it has also been capable of, as in

the Cartesian mapping of Geometry onto Algebra, or the Gdel

numbering technique of mapping meta-mathematics onto Arithmetic,

mapping Programming Languages into Formal Semantics, even if in

negatio.

The array of influences from Programming Languages on the core of

the Recursion Theory has also helped to normalise the debate in thePhilosophy of Mathematics, specifically in negating prima facie claims,

and at the same time being a clear example of implementation away

from the conundrum-like effects.

What is also very evident is the fact that, from the widespread debate

between Mathematical Realism, Formalism, Logicism and Intuitionism,

with its various historical tergiversations such as Predicativism, inflated

or deflated Platonism, Naturalism, Structuralism and Nominalism,

Programming Languages are recognisably one combined field in

anticipation of and with a keenness for with the latest relevant outcomes.

One clear example is how suitably Programming Languages, following

Shapiros distinction of algebraic and non-algebraic theories, parse withthe characteristics of algebraic theories, not intending, thus, to a unitary

model of the theory, but, in contrast, despite having enough analytical

flexibility to accommodate non-algebraic theories, have first and

foremost instantiated and modelled, again with extraordinary adaptability,

structural relations allegedly very typical ofante rem Structuralism.

Additionally, the epistemological challenge conquered by practice is,

fundamentally, far away from often pointless debates.


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Section II

Furthermore, for the purposes of this book, one word is important in

relation to the notion of place (topos), as clearly it is not just that

mapping as a concept has served fully adequately in Mathematics as a

substitute for function, or in Formal logic as a functionalsymbol, but at

its foundations, it reveals through further investigations in Mathematics

and Philosophy, such as in Model Theory, its intrinsic topological

argumentfrom a Euclidean inspiration. Discussing Topics in Programming

Languages might, thus, as said before, bring us to the abyss of, this time,

non-geometrised and impossible mapping, even if at the nucleus of

functional Programming. This fact is coadunative with the relational

status and structuralinstantiation action of Programming Languages, ofwhich the Declarative style is prominently the best example, and to

which belongs Prolog.

As an outline, it is also very noticeable how the new-born Philosophy

of Computation, oscillating between the tenets of Nominalism and

Structuralism of the twentieth centurys Philosophy of Language, was

capable of especially after Turings investigations into Mathematical

Biology on Morphogenesis and the study of the Argument from

Continuity in the Nervous System resuscitating one not seen since

before the Greeks Bios-to-Computus hypothesis, to which must have

concurred, very powerfully, as said before, one relationalframe of mind.

In Alan Turings case in particular, combining all the quoted elements,

it is superbly appealing how, passing over precise technicalities, the verysame originally Greek Bios-to-Computus paradigm is exemplified, and

how much the method of modelling understood as one assigning

function from one arena to another was closely scrutinised. Turings

genre of predication, for example in the paper On computable numbers,

with an application to the Entscheidungsproblem (1936), contains the

passage of Calculus to decimals, decimals to Computus and, thus, in

perspective, Calculus to Computus, replicating exactly what was

mentioned earlier.

Moreover, Turing discloses some important insights, as for example, in

the process of comprehending that numbers themselves, just like decimals,

lack visible chief predicates such as divisibility, primality, ideals,greatest common divisor or unique factorisation, then, in the same

fashion, although the class of computable numbers is enumerable, it does

not include all definable numbers. Furthermore, this is not a shortcoming,

but very much the contrary: they are proficient at resorting complex

functions, to industrialise primes in tables, with all sorts of pointwise

operations. Indeed, the Bios-to-Computus feature is best explained by the


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fact that, in the shift from Calculus to Computus, the multi-dimensional

tables ofCalculus were reduced to the one-dimensional array expressed

in the Turing Machine, as well as the digital Computing in the blueprint,

being the consequence, in point of fact, of Mathematical, Symbolic and

Language expressiveness having become incomparably bigger.

It remains to mention various features critical to the rise of

Programming Languages. Nevertheless, pinning down our best

efforts onto the preliminary decade of the 1930s, we are sufficiently

keeping to major lines, so as not to prorogue any more the most

pertinent conclusions of this chapter, while keeping the track of our

investigation on course.

Arguments

) The endogenous to exogenouslanguage argument

In Section I we came to the conclusion that the Semantic and mainly

Symbolical exteriorisation outburst in (Logos) (here known as

Language), that dragged, as explained before, naturalised essentials

referred to retrospectively as elements that came into play conceptually

as (the origins), such as Thales of Miletus water, Anaximenes of

Miletus air or Heraclitus of Ephesus fire, was Philosophically and

Scientifically interiorised towards formality. This was done, in essence,

through the Aristotelian breakthrough of considering a principle of

Nature as contained within itself, authorising, thus, the passage to the

consideration of the future Elements () (stoichea).

Through these facts in Section II it becomes clear, apparently, that, not

necessarily from the time the concept of Artificial Intelligence was

coined in 1955 by John McCarthy, but, instead, during all of Historical,

time that has mediated over the all-encompassing framework from

Greek Classicism to the Contemporary period and also, as well as morenarrowly, essentially throughout the time from Calculus to Computus

during which process, the first instance of Bios-to-Computus has been

shifting, radically altering its core, to its opposite; the Computus-to-Bios

case in point.

Behind all this are, of course, the two most imperative ideas in

Artificial Intelligence:


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Section II

The Absolute Transcendental possession of all Knowledge of the

Human condition and Environment, rendered uncomplicated if

analysed though Kants Epistemological tradition.

The possible fabric of the alike, that is, not just all objects, but Minds

too, in one structuralconvolution.

At this instant, something to be analysed further is, therefore, the reason

why it seems that the first-hand Phonetics and Philosophical interiorisation

of Semantics in Natural Languages sets out a very strong Bios-to-

Computus motto, and why it seems that with the entrance onto the scene

of Artificial Programming Languages, the cycle seems to have beenreversed, to the extent that it likely has been put forward in its entirety

a newer and reversed Computus-to-Bios conceptual journey.

For us, and in retrospect, the most significant arrival to have clearly

changed the direction of this was the work of Alan Turing, of which the

highly important (Church)-Turing Thesis is not a full indicator.

) The efficient cause continuance argument

Retrospectively, and holding on to the conclusions aired in the previous

paragraph, we have to start by giving the Aristotelian efficient cause its

fair historical counterpart, which would eventually lead to the inauguralfunctional paradigm in Programming Languages.

The inner passages that are most influential are, evidently, the

Mathematical function and the Computational algorithm, which have

helped to bring together Natural and Artificial Languages, Calculus and

Computus, but, in many ways, the Philosophical rule is oddly elapsed in

consideration, once it had played the role of connecting both, from the

very beginning, as seen early on in Euclids Geometrical interpretation.

It was never imagined, in Kantian terms, though, that the forceful

aspect of the rule, as the natural critical admonition of Human Reason

to Moral Law (without which we were to contemplate reality as strangely

as if the world as we experienced it had ceased, something impossible,

a priori pertaining to the commitment to experience binding Reason andreality, in the first place, according to Kantian views) was to have this

dilemmatic Antinomy revealed.

This sort of dilemmatic Antinomy was not only revealed but also

divulged from the very interior of Mathematics in its continuation after

Gdels astounding discovery.

1.

2.


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If the entrusted ontological credence of Mathematical Axioms as

taken by Kant already set apart from the Axioms of Intuition (Unity,

Plurality and Totality) was, nevertheless, very much inclined towards

the contemplation of Absolute Transcendental Knowledge, then after

Gdel we can say that the gulf between the two augmented to

unparalleled levels of distrust.

Besides, the Axioms of Mathematics in Kantian terms were no longer

contemplated after Gdel within the integrity of the representation of

Mathematics inside an Axiomatic System proper, recounted in N, neither

synthetic nor a priori, concurrent nor separate. On other side, the Axioms

of Philosophy were, after Gdel, au pair with the Axioms of Mathematics.The Axioms of Philosophy were, if we remember, contemplated by Kant

as dispossessed of a thorough deduction, which has now rejoined, if we

look carefully, with the Axioms of Mathematics.

More profoundly, we can observe how much havoc was wreaked on

the Axioms of Intuition, because, having been characterised as principles

of pure understanding, what followed was a revelation of their

monumental inadequacy with the examination of the Mathematics

Incompleteness Theorems to confront the Enlightened Kantian claim

that all intuitions are of great magnitude.

This fact conjoins, in essence, elegantly with the original already

quoted Greek concept of anthyphairesis. The main reason behind this

idea, is most probably, the incommensurability disclosed at the end. Thisis a strong reason that is sufficient to be recounted in one independent

following argument, which is put forward next.

) The reviewing incommensurability argument

There is no better example of the immiscibility in Greek Classicism,

though demarcated, between Philosophy and Geometry, intricate forms

of Compositionality Phenomenology, as it were, lacking in better

expression, than the fact that in Platos Dialogue Meno, it was both the

Euclidean and the Socratic methods that came together in harmony.

Also, very appropriately in the same Dialogue, as Socrates was

drawing figures in the sand, laying Geometry in its proper terrain

(topos), so to speak, what is more to consider is how much the Euclidean

notion of one segment-line being commensurable to another by one third

segment that can be laid lengthwise one whole number of times to the

first segment, and the same thing to the new last segment, otherwise

called incommensurable, is apposite to the concept of anthyphairesis.


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Section II

Furthermore, the stated idea of Mathematical congruence from one

line-segment to the other, as found in by the quotient of two non-zero

integers, corresponds, in inverse similarity, to the original meaning of

Algebra the bringing together of broken parts and to which end no

incongruity shall come about, whereby the Philosophical notion of

Rationality was also torn apart after Gdel by the non-congruence of

Mathematics and N as a representation for Mathematics.

In this fashion, the discovered impossibly retrievable medium between

the two otherwise called a fraction or quotient becomes for thi

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9781909287723 Topics in Programming Languages:A Philosophical Analysis Through the Case of Prolog