Divine Mind

The science of God consists in the pursuit of the perfection of our nature. The most perfect of all sciences is that which is sought for its own sake, and which is the proper end of man. Since the pursuit or attainment of this end is the greatest possible fulfillment that man is capable of. And it is attained by virtue, which results in accuracy in choice. For to hit the proper mark in each situation is requisite for that particular circumstance. For choice is concerned with the measure or proportionality towards its end and is known in terms of the end that is sought. And the end of man is the perfection of his own life and the shape of his environment in accordance with God’s supreme plans. 

Now the most perfect of all sciences is that of Salvation or Divine Economy, where in one is known to prefigure the Glory of being face to face with his own source and creator. And to attain this, means only follow the Eternal Word of Jesus Christ. It means only to have the fullness of Faith and the infusion of sanctifying grace by submission to the Holy Church. As a consequence we discover the inherent harmony of created things as they are situated inside of the Uncreated Splendor in its operations of divine grace outpouring upon all of creation.

The intellectual science of Salvation concerns the reasonable justification of this Science that is pursued for its own sake and does not require any kind of intellectual defense for it rests on the self sufficiency of God’s Revelation to mankind and His Eternal decrees which have validity and Authority simply, and of their own accord, as being given by God.

But to men who know not God or doubt his Majesty they must be convinced by proper and beautiful reasons emanating from that divine Splendor though in accord with the solemn reasons of the human mind, in accordance with geometric reasons and deductions from premises. The Eternal Word must be given valid and reasonable justification through logical accuracy and choice that hits the proper mark in each circumstance particular to that circumstance.

Now we can consider every word or natural expression as essentially variable, within a limited range of associations, we can replace it with a like term or another term that is similar and equivalent. Since human reason relies on accurate representation of reality. And human speech as it assigns and evaluates words as application to things, is a kind of convention indicated by the fact that words for the same thing differ among different peoples. But the relation between the representation and what it represents is an underlying or implicit structural constant, or universal form, or is like an underlying constant proportionality in the order of things. And our natural expressions in some way converge or are approximate upon it. Since each expression is itself and also interacts with each other. The way that we derive understanding is by the same substitution of like terms, or transposition of analogs.

Now the way that we prove this is as follows: If we consider every word or natural expression as essentially similar and equivalent, we can replace it with a like term or another term that is a variable. Since human reason relies on accurate assignation and evaluation of reality. And human speech as it represents words as application to things, is a kind of relation indicated by the fact that words for the same thing differ among different peoples. But the conventions between the representation and what is represents is an underlying or implicit structural proportionality, or like an underlying universal constant in the order of things. And our natural expressions in some way interact or are approximate upon it. Since each expression is itself and also converges on each other. The way that we derive understanding is by the transposition of analog terms, or substitution of like terms. 

What is revealed herein is the constant of proportionality or the object of proportionality that is the interior form or inherent structural order of intellect itself, in its relation to tangible experience. For the transposition of like terms is how motions operates, by substitution of like choices one for another in selection of pathways through space towards the proper ends, but transposition of terms is also how our representations of reality operate in such a way that the best summation of the whole is always preserved and rotated to its best aspect. This object of proportionality is revealed by the ratios of similarity and equivalence in like terms or analogs of each other, which also shows the proportionality of uniquely situated identity and the consonance of the plurality of such uniquenesses. This is the precise object of universal import revealed by such transpositions and rotations of underlying shape.








Science of God

The science of God is also the only true science of human happiness and the most practical science of human social relations. Since reality is always identical with itself it can be taken as mere chance or change, because each particular place is unique or identical with itself, the relation between places or circumstances point to universal order that places them in relation to each other. So these circumstances reveal a kind of alphabet or divine economy of human thought, which is the proper object of study in this science of human relations.


We take each human experience as implying the whole of reality. And so each natural expression is summary of all of experience in a partial way. And the best reorganization of the parts reveals new knowledge about how the whole is preserved. And so the proper method of study both preserves the whole and reveals new information.

This means that the science of God is also the only true science since reality is always identical with itself, so these circumstances reveal a kind of alphabet or divine economy of human thought.  Human happiness can be taken as mere chance or change because each place is unique or identical with itself which is the proper object of study. The most practical science of human social relations is the relations between places or circumstances that points to universal order that places them in relation to each other in this science of human relations.

This demonstrates the diagonal transposition 

ABC.DEF.GHI. ->= ADG.BEH.CFI.

And therefore, we take each human experience as summary of all of experience in a partial way. And the best reorganization of the parts both preserves the whole and reveals new information. And so each natural expression reveals new knowledge about how the whole is organized. And so the proper method of study implies the whole of reality.

AB.CD.EF.GH. —>= AD.CF.EH.GB





Understanding

What we mean by understanding in speeches or written representations is a vertical or analog substitution of like terms. This means that when we speak or understand something the mind replaces one specific term or reference with one that has already been used. This is a kind of analogical displacement that reflects the structure of reality itself, or the way that motion itself works. For example, when we speak or understand something the mind replaces one specific substitution or term with one that has already been used. This is the opposite of a statistical conception of speech and understanding, since substitution of terms involves symbolic capacity. And the choice of how to move through space is fundamentally about symbolic capacity, or how man achieve his proper ends.

Now the way that we can show this is that what we mean by understanding in speeches or written representations is a vertical or analog replacement of like terms. This means that when we speak or understand something that the mind substitutes one specific term or reference with one that has already been used. This is a kind analogical reflection that represents the structure of reality itself, or the way that motion works. For example, when we speak or understand something the mind displaces one specific substitution or term with one that has already been used. This is the opposite of a statistical conception of speech and understanding since the space of substitutions involves symbolic capacity. And the choice of how to move through terms is fundamentally about symbolic capacity, or how man achieves his proper ends.
















Coordination Without Algorithms

If there is a way to coordinate digitally without algorithms, then it would be possible to discover a new science of human experience. There is a certain kind of geometric proof that relies only on the shapes of natural language expressions, which allows us to recombine these arrangements into shapes that reveal new information. 

This means that if there is a way to coordinate digitally without algorithms then there is a certain kind of geometric proof that relies only on natural expressions. It would be possible to discover a new science of human experience which allows us to recombine these arrangements into shapes that reveal new information.

This is an example of transposition: 
AB.CD-> AC.BD

And this means that we can also partition the initial arrangement in a different way that is also self-consistent: if there is a way to coordinate digitally it would be possible to discover a new science. Without algorithms there is a kind of human experience which allows us to recombine these arrangements. Geometric proof relies only on natural expressions in shapes that reveal new information.

This is an example of tripartite partition of the original and its transposition:

ABC.DEF -> AD.BE.DF

Why does this work? Because the realm of thought is consistent with the realm of experience. The two realms are harmonious and translate in a self-consistent manner. The initial condition is where everything is arranged a certain way that action is the transposition of those terms into forms that understand and interpret reality.

Which means:

The realm of thought is consistent with two realms that are harmonious and translate the initial condition where everything is arranged in a certain way. The realm of experiences is in a self-consistent manner of action, the transposition of those terms into forms that understand and interpret reality.

AB.CD.EF -> ACE.BDF
Demonstration of Commutativity

Let’s say we want to demonstrate the property of commutativity which Penrose holds as paradigmatic of a truth that is so self evident that it suggests the non-computational nature of consciousness. We would have to show that equivalent and analogous terms substitute with one another and don’t change their meaning, which is so clear that we don’t realize it is how we already understand things.

As a test of the definitions of terms we replace like terms or clauses with one another, and so therefore show how the meaning is preserved through transposition or rotations of fixed relations of points. In the same way that the laws of nature expressed in mathematics are algebraically recombined to reveal formulas that are conserved throughout all changes, these transpositions or rearrangements of like terms are the best stable representations of motion that we observe in things themselves.

Let’s say we want to demonstrate the property of commutativity which Penrose holds as paradigmatic of what is so clear that we don’t realize it is how we already understand things. We would have to show that equivalent and analogous terms substitute with one another and don’t change their meaning which is a truth that is so self evident that it suggests the non-computational nature of consciousness.

As a test of the definitions of terms we replace like terms or clauses with one another, these transpositions or rearrangements of like terms are the best stable representations the motion that we observe in things themselves. In the same way that the laws of nature expressed in mathematics are algebraically recombined to reveal formulas that are conserved throughout all changes, we therefore show how the meaning is preserved through transposition or rotations of fixed relations of points.











Transposition of Proportional Analogs

We can consider all natural or mathematical expressions as shapes or representations of reality or things in reality. The arrangement of representations is in a linear series. But the way that we form meaning is actually through a vertical association of more precise analogies of the same thing. 

And the way that we represent things is actually inherently ordered geometrically and algebraically by an underlying constant even though we don’t realize it at first. The way that mathematics works is through transpositions of analogous terms. For example when we say that 3x + 5 = 17, then by substitution we get 3x = 12 and then x = 4. 

If we scale this up to more and more technical complexity then arguably we can see more and more prediction and anticipation of the structure of reality. Mathematics is effective because the transpositions of terms preserves the underlying causal order of what we are representing by it. What Leibniz figured out is how to do this through the geometry of natural expressions to essentially prove vertical causality and solve the measurement problem.

I can demonstrate this to you directly by the following geometric proof, which requires careful reading, in relation to what we have just said:

We can consider all natural or mathematical expressions as shapes or the arrangement of representations, but the way we form meaning is actually through a vertical association.  Representations of reality or things in reality is in a linear series of more precise analogies of the same thing.

And the way we represent things is actually ordered geometrically and arithmetically by an underlying constant in the way that mathematics works. For example when we say that 3x + 5 = 17, even though we don’t realize it at first, through the transposition of analogous terms, then by substitution we get 3x = 12 and then x = 4.

If we scale this up to more and more technical complexity then arguably mathematics is effective because the transpositions of terms preserves what Leibniz figured out, which is how to do this through the geometry of natural expressions. We can see more and more prediction and anticipation of the structure of reality, the underlying causal order of what we are representing by it, to essentially prove vertical causality and solve the measurement problem.

Now we have to consider how to understand this:

[AB. CD. EF] -> [ACE. BDF], but does [AB. CD. EF] = [ACE. BDF]?

What it means to “mean” something is to assign a proportional analogy through slices of time. We can call this a “vertical analogy” or “vertical causation”, since it does not take place in the horizontal time series. This is essentially to demonstrate matrix transposition, or rotation, through natural language. The reason that this is important is because it demonstrates the exact relation between free will and necessity in mathematical reasoning. The free will is what chooses how to partition the series of moments or grammatical clauses to form the vertical analogy that makes the most sense. There may be multiple ways to partition the series so as to preserve meaning in vertical analogy. And those alternative partitions may reveals slightly different emphases.





The Shapes Implied by Representations That Ground Motion

If there is a certain way to represent the changes or contingency in things, that nevertheless rests upon a kind eternal and divine consistency, then this method would be a structural constant that orders the way in which motion unfolds. Since motion is always to be consistent with itself or in accordance with unchanging laws like gravity or the mathematical formulas by which we represent eternal objects. Now the way in which motion happens or the laws that guide or constrain how motion unfolds, are alike to the way that we communicate or represent reality to ourselves and others, by various terms that signify conventional relations of things. Since the way that we communicate or represent reality to ourselves and others to accord with the nature of reality itself, as it actually is, rather than how it appears, relies on the understanding. And the understanding is grounded in a kind of transposition and substitution of proportional and equivalent terms that are constraints on the way that we convey things. And since the whole of experience is always a simple summary of the whole of things themselves, our representations of understanding are always commutative, or are merely self-consistent rotations of the same things said in analogous and equivalent ways.

Therefore if there is a method to represent the motion or contingency in things, that nevertheless rests upon a kind of eternal and divine constant, then this representation would be a structural consistency that orders the way in which changes unfold. Since motion is always in accordance with itself or is consistent with eternal laws like gravity or the mathematical formulas by which we represent unchanging objects. Now the way in which communication happens or the laws that guide or constrain how representations unfold, are like to the way in which we signify reality to ourselves and others, by various motions that are the terms for relations of things. Since the way that we communicate or represent reality to ourselves and others to understand the nature of reality itself, as it actually is, rather than how it appears, relies on accord. And the accordance is grounded in a kind of transposition and substitution of proportional and equivalent understanding of the constraints on the way we convey things. And since the whole of things themselves is always a simple representation of the whole of experience, our summary of understanding is always self-consistent rotations, or are merely commutative of the analogous and equivalent things said in the same ways.

From this is clear that conventional representations are always a kind of shaped analogy. Whereby each term is a kind of point and the relations between points form various geometrical shapes. And understanding or meaning or the purpose of the human being in reasoned inquiry convergent upon its own self-sufficent ends, is to transpose these shapes of analogy in precise ways. This makes clear that representations of conventions are always analog shapes. Whereby the point is a term and the points of relations form geometrical shapes. And understanding or meaning or purpose of the human being in convergence upon its own reasoned inquiry is to transpose these precisions of analogy towards self-sufficent ends.

Since the human soul is a summary of the whole of reality, then motion will always be consistent with itself or will always be commutative. The complete set of terms is always available to the understanding, and in the conventional terms we see the relations of these terms to the things themselves. Thereby the relation between thought and experience is separated and also at once harmonious. Since each is its own set of terms, but their translation one into the other is a kind of self-consistent rotation. 

This is because the the human soul is a summary of the whole of reality in the complete set of terms always available to the understanding whereby the relation between thought and experience is each its own set of terms. Then motion will always be consistent with itself or will always be commutative in the conventional terms that we see relations of terms to things themselves as separated and also at once harmonious, but their translation one into the other is a kind of self-consistent rotation.

Or AB.CD.EF.GH = ACEG.BDFH
















The Christian Political Science & Exoteric Writing 

In 20th century thought, there was a rediscovery of the tradition of political science or political philosophy, of authors who wrote exoterically, or popularly. And in so writing publications for popular audience, would appear as to conceal their true views "between the lines" as it were. Their motives for doing so, it was argued, were to avoid political persecution and to communicate with other careful readers or philosophers, who could discern their true meaning. 

Now, in the Christian tradition of political philosophy, this form of exoteric writing must take on a slightly different meaning. Since avoidance of political persecution could not be a primary motive of the Christian political philosopher, since, in the Christian tradition, the exemplary or highest calling of the Saint is to die for one's Faith which confers immediate entrance into the Heavenly Kingdom. That is, the highest calling of the Christian is to die for the Truth about reality itself, Jesus Christ, in the same way that Christ Himself died in order to save mankind from their folly. Therefore it follows that to avoid political persecution is not the primary motive for exoteric writing in Christian political philosophy. 

And so we must look to another motive for the exoteric writing in Christian political philosophical tradition. 

In the pre-Christian political philosophy the essential question is of the relation between appearances and reality, or conventions and nature, or opinions and truth. And the political philosopher is the one who understands to a greater degree than others, the reality, nature, and truth of things themselves. Whereas the others who do not understand, or are not compelled to learn, are inevitably offended by such things. 

And therefore the motivation of the exoteric writing in pre-Christian political philosophy is to conceal his view from those who would be offended and, at once communicate those views to other political philosophers who either already understand or are apt to learn to understand the reality, nature, and truth of things, while also flattering those who are ignorant of learning and understanding such things. 

However, the Incarnation of God resolves this dilemma of the ancient political philosophy, or it Reveals the answer to the question of Socrates and Plato, concerning the relation between appearances and reality, conventions and nature, opinions and truth. Or it answers the rebuttal of the sophists and the pharisees alike, that there is no truth as such, or that there is no god but Caesar. 

And so the Christian exoteric writing does not rest on the same question, since reality, nature, and truth have been decisively Revealed in the Son of God, the question of Socrates and Plato is answered and not completed. Rather, the Christian exoteric writing rather elevates the whole dilemma of Socrates and Plato to a higher plane: that of the Hidden and the Revealed. 

The Christian is motivated by the Truth about reality, which has been decisively Revealed, and therefore he has no reason to conceal his views or to communicate them only to others who would understand him, since the Truth has been Revealed to all strata of society alike, both political philosophers and the ignorant. Rather, the Christian exoteric writing is concerned with communication of the intrinsic relation between the Hidden and the Revealed. And the Hidden is not understood in the sense of "occult" but rather in the sense of what is internal, interior, intrinsic to Revelation itself. Or what is inherently implied or consequent of Revelation itself. 

In other words, the Christian political philosopher is concerned with the interior and exterior reality in themselves, and in the relation of these, since the interior life is the life of sanctifying Grace or the infusion of God's Own Life in our own, whereby we participate in a prefiguration of God's Glory and Salvation. And thereby the Christian political philosopher is motivated in exoteric writing by translation of What is Hidden to others into What is Revealed in general or for everyone simply, which is the perfectly Eternal Word of God. Since the Christian religion inevitably concerns the conversion of the whole world to the Kingdom of God: "Go and make disciples of all nations". That Jesus Christ is the Truth about reality itself is acknowledged implicitly every time anyone says what year it is, at once we recognize what all of time is oriented around - the Incarnation of God.

Its Transformation in the Early Modern 

Now this relation of the Hidden versus the Revealed takes on a new dimension in the early modern period. Since the Christian and pre-Christian political philosophy or political science are both inherently oriented towards the pursuit of Virtue, or the pursuit of Good itself.

In other words, when asking about the appearances versus reality, or convention versus the natures, or the opinions versus truth, the pre-Christian political philosopher was asking about the reality, nature, or truth about the Good itself. Or how to live a good or virtuous life. And similarly, the Christian political philosophy in asking about the relation between the Hidden versus the Revealed, asks about the Revelation of the Good itself, or how to live a virtuous, holy, sanctified life. 

And so on this point, the pre-Christian and the Christian political philosophy are in complete agreement: that political philosophy is concerned with the good itself, or the good life, or the life of virtue, or the best ends of the polity as a whole in the common good.

And yet in the early modern period, the pursuit of the Good itself is translated as it were, into another aspect. The question of the Hidden versus the Revealed takes on another dimension when the natural law, or the moral truth, is translated in laws of nature, or mathematical truth. The pursuit of virtue or the good life, is translated into a pursuit of mathematical reason. Or precise mathematical reason is taken to be virtue itself, since mathematical reason is effective in Revealing the truth about God's designs in the creation, and seems to confer Godly powers on man. And in this way, the pagan idea of the Cosmos is reinstantiated in the search for mathematical truth. And the Heavenly Spheres become material and inert, and the Cosmology is that of being specks of dust floating through an infinitely empty vacuum.

And nowhere is this debate of mathematical versus political virtue more clear than in the contemporary debate over classical liberalism and nationalism. Where the classical liberalism would like goverance to be reduced to technocracy and pursuit of wealth, whereby the organization of human things themselves is reduced to a kind of utilitarian calculation or computation, and the polity is a kind of economical competition with other nation states. On the side of nationalism, the question is whether virtue is to be understood in terms of political honors or rather as the pursuit of the common good, and these are not necessarily in conflict.

The Harmony of Classical and Ancient Liberalism

Now the resolution of ancient and classical liberalism would appear to be found in the political philosophy of Gottfried Wilhelm Leibniz. Throughout his life, Leibniz pursued the project of reunifying the Christian churches throughout Europe. And this theological project took on the dimension of a geometrical inquiry, in the analysis situs, also known as the geometric characteristic. In this, the true political philosophy is to be found in the relation of interior and exterior reality, which bears forth the relation between the different strata of society. Since the interior pursuit of virtue is a plenitude of the divine outpouring, and its translation to the exterior reality is only summary and indirect in its capacity to convey the divine splendor, this serves as the image of the polity as a whole, whereby the lower strata approximate the magnanimity exemplified in the just statesmen.

Since the science of God, is also the science of human happiness and virtue, geometrical reasoning is in no way separated from the pursuit of the common good. And the way that Leibniz preserves the philosophy the ancients, and defends against the rationalism, materialism, and atomism of the moderns is to defend teleology in physical matters, by showing the geometric relation of interior to exterior reality. The method is substitution of like terms, which is the analogy of how motion itself happens, or how reality as a whole is always preserved in summary within each human soul. Since the soul of the human being is approximate or convergent upon the whole of reality, the change or contingency that we witness as readily apparent is merely a kind of recombination, and reordering of the parts already present in such a way that they reconstitute or converge upon the whole.

In this way it is clear that, the science of God is the way that Leibniz preserves the philosophy of the ancients in the method of substitution of like terms, since the soul of the human being is approximate or convergent upon the whole of reality. The science of human happiness and virtue defends against the rationalism, materialism, and atomism of the modern and defends teleology in physical matters, by the analogy of how motion itself happens, whereby the change or contingency that we witness as readily apparent is merely a kind of recombination. Geometrical reasoning is in no way separated from the pursuit of the common good by showing the geometric relation of interior to exterior reality, or how reality as a whole is always preserved in summary within each human soul by reordering of the parts already present in such a way that they reconstitute or converge upon the whole.


Or by transposition, w have shown that: ABC.DEF.GHI.JKL. = ADGJ.BEHK.CFIL










The Science of Salvation

The perfect science of human happiness, consists in first of all, pursuit of virtue and nobility and just actions, and second of all, in expectation that this science admits of the degree of precision appropriate to its activity. And we ought not expect more precision than this. Now the highest happiness comes from participation in God’s Own Life, by sanctifying grace which is the infusion of superabundant love. And such participation and adherence to God’s Own Perfections, gives insight to the order of reality itself, and the designs of the Creator. 

Since this conformity to the order of reality is what the human being is made for, his very purpose and essence, to find total fulfillment in loving his Creator and source, and the divine operations sufficient unto themselves for all eternity, and furthermore then being the vehicle for this outpouring of love from the Creator, to flow to other human beings, thereby  allowing them also to come to know God intimately. And in this approach comes of the greatest humility, for proximity to God always remains at an infinite distance, however intimate and close one comes to clear vision on this side of life.

Now this science consists in combinations of representations of reality. And these come by way of conventional signs, attached to the designation of proper things, or tangible objects. And the order of the combinations adheres to structures and rules called grammatical. And the order of combinations encompasses a range of thoughtful considerations of general scope of things, at the same time each representation in itself is unique and particular to itself and its own context. When we seek to represent the arrangement or the order of things themselves, we would look to the order and arrangement of our representations of those things as exemplary of the way things themselves are ordered and arranged, or the converse in looking to arrange our representations in the best order we would first look to the proper arrangements of the things themselves that we would represent. By study of our representations of reality, or by our study of reality itself, we come to knowledge of the reality of representations or representations of reality, and furthermore of the generalized structure of correspondence between the two. That is, by a kind of implicit interaction of each particular we discover new resonant and unstated particulars that then take on structure in our natural articulations.

Now the way of motion, and the operations of motion according to general or universal laws, is that motion is always the particular expression of the whole, or the person’s perspective is always a summarization of the whole, and the whole is greater than its parts, and the summary is partial, but approximates the whole. And so the particular perspective is some combination of parts. And our represenations of reality are partial combinations. And the way that the partial combinations summarize the whole is by interaction of relations of conventional designations. Since the interactions prove an overlap of partial signfications that points towards something implicit between them, which reveals a kind of structural association that is beyond natural expression itself, but relies on the relations of the combinations of habits. And in general, motion here once again reiterated in a similar way, operates on the same basis, that change from moment to moment is based on some measure or criterion, which is found in choice, and choice selects things on the basis of its ends, or its supreme purposes. And as we have seen the highest end, or the most fruitful of all ends, is the science of salvation or participation in God’s Own Life, and the operations of His Grace sufficient to express all of Infinity without closure or finality but infinitely convergent upon what is most perfect of all. And so motion itself is based on a relation of temporal choices that converge in series, and the relations point towards an association that links them together which is ever recedent above the human reason’s ability to grasp it. Since the connection of moments is grouped into clauses, or limit-boundaries of memory’s capacity to cognize wholes. Thus we may consider a fifteen minute block as a kind of moment, or perhaps an hour, but at some point there are bounds to the memory, and the thought is separated into clauses of some kind.

Now it is requisite that to truly explore the structure of reality, and the operations of the Divine Designs, that our mind be entered into a kind of fluidity and constant choice of the freedom conducive to skilled practice and good habit. For the habit of perfection lies in mastery of the craft to which are dedicated. And the craft is always some kind of operations on reality, or on the materials of reality, to produce artifacts. And to explore the structure of reality is to consider each thing in it, and all of the spatial organization and each particular as a kind of represenation or mark in a series, and the mind as estimation of the series of things in anticipation of their unfolding. And so the series of things or the series of representations of things, is the best approximation of how things unfold themselves, and each kind of natural expression or mark is like a designation of a point in time or in the series, or is like an abstract spatial variable arranged in the whole summary of the order of representations. And so meaning or purpose of the ends of man, are represented by a kind of arrangement of points which is always an approximate or convergent upon the summary of the whole. And so the whole is always plainly and simply present or at least represented in the series, and thereby the unfolding of moments or the estimation of how moments transpire, is seen in the transposition of points, like rotations of a shape in the mind. Where the points in the shape are complex, and fold into each other in different and varied ways. For example, each point in the series is a kind of analogy, that fits partially and overlaps proportionally with certain other like terms in the series and can thereby in the mind already substitute for it. The intellect makes such proportional associations implicitly and already, in the same way that it anticipates structures of unfolding event trajectories, in consideration of myriad possibiles, and the interactions of both known and unknown circumstances contingent upon what actually happens. And so we see the way in which this kind of abstract substitution or transposition transpires when we say that it is requisite that to truly explore the structure of designs, and the operations of the Divine Reality, we have to posit a kind of analogical equivalence or that the mind be entered into a kind of constancy and freedom of fluidity conducive to good practice and skilled habits. And we see by this that transposition is proportional and approximate though by this reveals precise ratios of differentiation between modulations to the purpose. Since the purpose is divinely one and inexhaustible and in its own self-sufficiency, is Perfectively Infinite. The replacements or subsitutions of like terms reiterates proportionalities or ratios of different designations to one another in order to unveil the best and most proper designations of particular things in series.

If we had a way to rank representations or to assign to each a number or a probability, based on some cannon frequency, and then we saw the series as sequence of pure quantities, and then performed some kind of operations on those, such that we found the pattern inherent to their order, what we would find is that the series, though convergent upon this pattern, would always recede into a kind state just beyond the grasp of human understanding, since its object is above human understanding, yet perfectly accords with the grounds of reason itself, it’s object being the divinity and perfections of God’s designs. And so, if the whole is perfectly summarized, then what we find in the series, and in things themselves, is perfect rotations or transpositions of shapes, or matrices, or otherwise patterns of folding across the matrix, that can in no wise be concisely summarized or perfectly compressed, since their pattern is to much identical with itself, its definitions always recede just beyond the horizon of the mind’s grasp, just outside the range of intelligibility. Yet it is precisely this boundary that we are in exploration of, since it is this boundary that constitutes the correspondence between interior and exterior, or allows for the mind to see clearly the relations of unfolding. And so we always find a pattern herein, not in the kind of way that by a random configuration of points we, as it were, see a kind of face or a design that is accidental, or in some ways, when we look at a cloud, and see a certain kind of animal or the figure of person, accidentally conveyed, but the mind’s habitual association marks out its shape in the random configuration of the contours, not in this way is the pattern herein to be found, but even furthmore, that there is a non-accidental or substantial association that inheres in the familiarity that is so close and obvious and self-evident that its intimacy of presence cannot be fully grasped, or can be reasoned as only implict and indirect, such is its proximity. Since being is itself, or is equivalent to itself, or is self-identical, it cannot not be itself, nor in any way be its opposite, or composed of its opposite, but rather in contingent things, or changing things that unfold, such things do admit or appear to admit of their opposites. But being or what is self-identical in no way contradicts or opposed itself, and thereby what is changing must admit of what is eternally consistent with itself, and must admit of being proven self-consistent in an underlying law or pattern.






The Non-Technological Science

What we are searching for in this is a kind of non-technological science, or a science that takes as its instruments not extrinsic tools, but conventions or representations themselves, as convergent upon being or upon what is only itself, since being in its own perfection is infinitely perfective or is just beyond the grasping of the intellect in a self-sufficient activity of God’s own operations. Since the understanding of our representations of things does not construct reality in a way which it is not but reveals the way in which we understand reality itself. So the combinations of representations as grounded in perfective being is convergent upon the true structure of interior reality and gives a kind of freedom of exploration in pursuit of just and noble action.








Organic Exploration & the Test of Organic Experience

What would it be to have a science of organic experience, as was emblematic of the Aristotelian and Thomistic sacred science? That is, a science not reliant on extrinsic instruments, but requiring only the human intellect and organic experience of reality. What is the precise measure of such a science? Or what is the criterion or standard for evaluation of things? We would have to know the goal or the intended outcome of such a science.

Now the goal that we posit is salvation, or entrance into the Kingdom of God, in accord with the teachings of the Saints and the Universal Church, who propound the Science of the Saints. In other words, the result of intended outcome of sacred science is what is known as the beatific vision, or the sight of the blessed. Now the question is how are we to arrive at such a measure? Or how are we to present out inquiry in the first place?

For the vast run of men in our day know nothing of Christ or His Church, notwithstanding his Salvation or the beatific vision.

What we would say is that the science of the Saints, is the science of happiness, or the science of the best life. Now the measure or criterion of such a science cannot be that we already possess the best life or that we are already in full possession of truth. We can only posit that such a thing exists, and that we are capable of grasping it, or prefiguring it in all its fullness, by a kind of integration.

And so to practice the science of the Saints, is to practice the science of virtue, or to cultivate the habits of soul that are conducive to love of God, and charity to our fellow men. Now Aristotle says that the political science consists in just and noble actions, and furthermore, that we must not expect greater precision that the subject matter admits of. That is, the foundations of the political science is virtue or just and noble action. But this science admits of degrees of perfection. That is, it speaks of the good in itself, the self-sufficient good, in two ways at once. It speaks of the good in itself or the self-sufficient good as eternal and pre-existent in things themselves and as already obtainable inherently. But it also, at once, speaks of the good in accordance with the common notions which do not admit of precision. For the vast run of men know nothing of Jesus Christ and His Universal Church which is the very doorway to the Heavenly Kingdom. And so what they know as the good is pleasure, honors, wealth, indulgence. And so to have charity to the vast run of men, or to take them as our audience, we would do well to speak to them in the degree of precision that is adequate to their tastes and dispositions.

Now the science of the Saints does admit of a high degree of precision in the case of the self-sufficient good, or the good that is pursued for its own sake. For the end of man is to live inside of the operations and activities that are sufficent of themselves. This is known as maturity in the spiritual and material sense. To have sufficiency of one’s own accord. Now the operations and activities that are sufficient unto themselves are to dwell inside of the divine operations of sanctifying grace, or the operations of Providence, which show forth the Harmony of God’s works within creation, in a way that is indirect or implicit yet clearly known. For it cannot be said all at once what such things are, only that they are, and how to grow in such virtue, so that the soul is adequate vessel to receive this santifying grace which is of greater goodness than all of the good in the created universe and in all of the heavenly Angels.

And so to prepare our soul to receive Heavenly Grace and ascend the Heavenly Orders, it is requisite to have tests such as appropriate to the intellectual virtues and to those skeptical minds who would demand a kind of geometrical rigor in their doubts against the faith. Now, all admit that reasoning is based on certain presuppositions. The question is as to the nature of our presuppositions, whether there are certain natural presuppositions, or certain presuppositions that are true by nature. Since it is commonly understood that presuppositions cannot be demonstrated, but that demonstrations proceed from them.

We, on the other hand, hold that presuppositions, or beliefs, or axioms, of the tenets of the Faith, can in a certain way be demonstrated. Holding that the presuppositions are self-consistent and complete. This means that every true axiom is compatible with each other. Now the measure or the test of such a thing, is that our representations of created reality itself, are compatible with one another, or work upon one another in certain definite ways.

The measure or test of our representations is a kind of proportionality that is intrinsic to their arrangemenet. For all representations are shapes and all shapes are arranged in a certain manner. And the manner of their arrangement, if it is the best arrangement of all, corresponds most precisely or summarily to the arrangement of things themselves. And if such is the case, then the representations touch upon one another in such a way that they enhance each other’s claim or force upon the understanding.
Since these are four sentences or presuppositions or representations of shapes that are arranged in a certain way, we may thereby test its claim upon things themselves in the following way:

The measure or test for all representations and the manner of their arrangment if such is the case [is as follows]: Our representations are shapes if they are in the best arrangement of all, then the representations touch upon one another [therefore it follows that]: A kind of proportionality that is intrinsic of all shapes corresponds most precisely or summarily in such a way that they enhance each other’s claim [whereby it follows further that]: Intrinsic to their arrangmeent is being arranged in a certain manner that is the arrangement of things themselves as a force upon the understanding.

And so we see that 
ABCD.EFGH.IJKL.MNOP ->=AEIM.BFJN.CGKO.DHLP






Geometric Representation

If we wanted to posit a test of natural expressions and their geometry of meaning, we would have to say that certain shapes are related to one another as analogous and equivalent, and we would have to decide what is the criterion for deciding such a relationship. If we assign each unique word a random unique variable.

AR BM BL BI AX AA BE AV AU AN AC BH AP AV AT, BM BP AQ BI BB BF AH BC AE AZ BI AW AD AF AB AC AM, AC BM BP AQ BI AJ BN AS BG AI AO AK BD AA BA. AR BM AG AL BJ BO AA AY BJ BK.

If a human being tries to reduce a series of representations to random variables, or to a random order of representations, it isn’t possible for someone to produce a truly random series, because the closer that you approximate to true equality of representations, the more the intellect inherently associates the equality of things with other similar things, since each particular thing is purely unique to itself in that specific position or place or situation. 

But when we take pure computations or calculations, such as 3x + 4x + 7 = 42, we consider them as abstract intellectual puzzles, and then we have to solve for x, the characters transpose in certain definite ways in order that we arrive at x = 5. First we subtract the 7 from 42, to get 35, then we combine 3x + 4x into 7x. Then we divide 7x by 7 and 35 by 7. And so we get x = 5. We basically have an input and output of a function, or cause and effect, and the two are always perfectly equal. 

But the transpositions of characters are only similar, not equivalent, because the variable is still unknown before the unknown is solved for, so the representations cannot be totally equivalent. The solution of the unknown variable is a kind of approximation of total certainty. When things are perfectly equal then there is perfect randomness or disorder which isn’t actually possible in reality. This kind of abstract intellectual puzzle is just a reflection of the certainty or determination of the will to accomplish its aims. But it does also in a sense reflect the nature of independent objects, or our ability to coordinate through pure intellectual abstraction. Since the just is the equal or fair. The human will in being convicted or certain about the nature of reality seeks to equalize everything order to control it.

How would we posit it were possible to do that with a natural expression?





Form in Natural Expression

We take all natural expressions or representations of reality, to be indications of some thing. No matter in what order or grammaticality or in what soundness of relation, all natural expression is indicative of an underlying relation, an invisible object of meaning to which it relates. This is shown by Aristotle when he takes as the first object of all first philosophy to be being. Indeed, all philosophers treat of being as such, as the first principle of philosophical and scientific inquiry, since being refers to what is and excludes what is not. Therefore, all expressions no matter their order, refert to being as to some underlying or implicit object of perfection.

Now of all natural expressions, those are well ordered who series accords with grammatical convention and nature. For grammar is both conventional, since it varies according to particular languages, and also natural, since there are certain universals of grammar, such as noun / verb, subject / predicate, is / and, etc. 

In all natural expresssions, we have those which are grammatical or non-grammatical, and those which are meaningful and non-meaningful. For example, we may have a grammatical and meaningful statement, a grammatical and non-meaningful statement, a non-grammatical and meaningful statement, and a non-grammatical and non-meaningful statement. Of these, we are only concerned with those that are grammatical and meaningful. For these both accord with the rules of nature and convention in the order of terms, and also, the terms are substantial or they accurately imply the object of their indications.

Now, in the geometrical form of natural expressions, we take each term as a kind of point, or unit. And at the same time, as the implication of a line or a variable, when it is combined in an ordered pair. Since the combinations of terms applies to a kind of interaction between two given points, which points to a third beyond them. And those implications themselves also interact in the same way, ultimately pointing to a single unknown unit that unites them all. And this is what we are searching for as the form of the natural expressions that organizes them.

Now, it seems that there are three ways in which we can prove this geometrical form of natural expressions: 1.) rotation, 2.) transposition, 3.) substitution.

For example, with a rotation, then we take into account the meaning of the statement/s as a whole and we undertake the same mutual rotation on all parts simultaneously. We choose how to partition the terms, or how to apply the variable indications to the partitions of the statement.

For example, with a rotation, then we take into account the meaning of the statement/s as a whole or we choose how to partition the terms. We undertake the same mutual rotation on all parts simultaneously or how to apply the variable indications to the partitions of the statement.

Here we applied the rotation ABCD ->= ACBD

Rotation can also be applied at the level of individual terms:

Rotation of terms is symmetrical
Symmetry is in rotation of terms.
Terms are symmetrical in rotation.

Now, with transpositions we can replace a single term with its analog in another statement. The means that the two terms are sufficiently equivalent and similar to one another that the replacement is indistinguishable.

For example, with replacement we show a single term with its analogy in another statement. This means that the two terms are sufficiently equivalent and similar to one another that the transposition is indistinguishable.

Or with transpositions we can replace a single term with its similar and equivalent one in another statement. Which means that the two terms are sufficiently analogous to one another that the replacement is indistinguishable.

Now there is a certain way of constructing such transpositions in order that the series of terms cascades into the indication of its unit of interaction. In order to accomplish this we have to be able to define the central concept or operation of this kind of mathematics. And we take this is as equilibration. Now the operations of equilibration we take as the cascade of mutual implications of suggestion, such that we immediately see before the mind’s eye the specific object of perfection which organizes the terms into the best series.

And so we take equilibration as an exercise in reordering of terms such that they imply each other in a self-consistent whole where each mutually defines the others. And the idea of this is given in perfect transpositions of definitions. Wherein we would show the perfect equilibration of freedom with constraint, for freedom ought to be constrained towards virtue and the pursuit of God’s Glory alone, towards the ends of Salvation and Sanctity. For freedom other than this end, is license and violates the nature of true pursuit of freedom, which is free within the bounds of perfect beauty of the good. For each terms is a kind of constraint since it is a convention arranged according to conventional grammatical rules. But the interactions of the grammatical and significantly implicational terms, signifies interaction of higher variables of arrangement, which are the higher order constraints, and when these are properly aligned then we see the interaction of the unit itself.

Now this requires a certain grace from on high, given by God alone, who pours out his Glory on those Whom It Pleases Him To Do So. For the order of complexity is made null by the Godhead Who Is Ultimate Simplicity and Perfection, and whose abundant graces are never lacking, and even the smalles of grace is worth more than the goodness of the whole created universe, since grace is the germ of Salvific Perfection, and this is the end of all rational creatures to dwell inside of God.

And so when we arrange the representations of the created order in their proper combinations, then we ought not see the shapes themselves only, but the shapes that are implied by the transformations of their interactions. For each of the terms in its series is a kind of interaction of shapes. For these shapes are but plain designations or conventions upon a surface, and are arranged according to the order of convention and nature, but their deeper implication is that their interaction implies an order of perfect objects that converge. For each of the terms in its series is a kind of interaction that implies an order of perfect objects that converge.












Representations of Being

All natural expressions, speeches, actions, intuitions are representations convergent on being. And being we define as the object of perfection. Thus all of our representations are like terms which specify aspects of this one object. And the object itself is what makes all of speech, action, and intuition, compatible or harmonious with one another. In the same way, two mathematicians independenly of one another can discover the circumference of circle, because the circle exists independently of subjective opinions, it is a perfect object in the intellect. So if we abstract all of the perfections of mathematical objects, then we have the object of Salvation, or the supra-rational object of Grace, beyond mere human intellection, can only be known by Revelation and Tradition, as a kind of Revealed axiom. This is the outpouring of God’s Salvific Glory upon mankind, attained by the pursuit of truth in itself, as Jesus promises us that if we seek the Truth, we shall also thereby find it, in Himself, who is the object of perfection, and also the subject of all history.

Thus, all terms are either simple or are reducible to simple terms. Since being is perfect, it is also simple. And so we can either move from simple terms to more complex definitions, which afford greater accuracy into the nature of reality, or a higher resolve as to the composition of the continuum. Or we can move from definitions to simple terms, which is the movement towards being. Now the object of perfection is being, and all representations are convergent upon the simple object of perfection. The movement of the first kind, from simple terms to definitions, is the inductive movement from common notions to what is better known in itself or by nature. The movement of the second kind, from definitions to simple terms, is the deductive movement from what is better known by nature, towards what is better known to all.

Now what is the way that both of these movements can be achieved within one method? It is that our terms are common notions, and at the same time are accurate definitions which hit the mark of accuracy in the natures of things. This is because the object of being is the nature of the intellect itself. Thereby all particular intellects are compatible in God, or the object of perfection.







Organic Science & Geometric Equilibrium of Motion

Motion itself consists in transpositions of similar terms. For the criterion of decision is always some kind of measure. And the measure assume a unit, or an original identity of a thing with itself. Now, we decide according to similarities or proportionality, which thing to choose is the best of all possibles. Because all of our similarity and proportionalities are proportional to the good itself, because we choose for the sake of the good, or for the sake of our Best End, which is God Himself, or the Perfection of Reality itself. And this is identical with our own perfection or salvation. And this is a kind of science, whereby reality is fulfilled by pursuit of the truth of itself, in us. 

Now this sort of motion or task, is oriented to the highest object of perfection that we can conceptualize in the mind, even beyond mere conceptualization, this object of perfection is an experience of reality, that is tangible and real, moreso than an abstraction, though it is also the summation of all abstractions of perfect ideas.These perfections nevertheless correspond to actual things in their formation or movement. Since it is in this way that things unfold or move, according to transpositions. The perfections of things organize them in their unfolding in relation to one another. Like a kind of universal substance of implication, each particular or unique things is related to each other by nature of the fact that their uniqueness points towards the unit of measure in particularity. And what unites all particularities is participation in some unique universal that applies to all of them. And the object of being or of simplicity and perfection is what harmonizes these particulars through motion. And so we look for the laws of motion in a kind of convergence. By which we associate certain particular with each other in a precise way. This is both identical and analogous, or the analogy is causal and literal. Only literal analogies can be causal. For example, when we say that human action always seeks its own perfection, the perfection is only known by a kind of analogy, since we do not already possess it in its fullness. Nevertheless the literal truth of this analogy makes it causal, or makes efficacious in its fruitful application to reality.









Science of God & the Test of Reality

Is there in the tradition sufficient basis for us to say that there is a political science that is at once theologically and geometrically sound?

Aristotle says in the Nichomachean Ethics (1094b.15) the nature of political science is that it a.) investigates just and noble actions, and also that it b.) does not expect more precision than the subject matter admits of:

 

Leibniz reiterates this point that the political science does not of complete precision, in his Preface to the General Science.  

 

If there is some end of action that we seek in itself, this is the end of political science, 

 

Then if there is one thing that is chosen because it is complete or perfect in itself, rather than being chosen for the sake of something else, this is more complete or perfect than things that are chosen for the sake of something else.

 

Now, we can see that thing that is sought for its own sake, is identical with the object of Salvation, or the perfection of reality, or the supra-celestial science.

 

And the reason that this method does not admit of more precision than is to be expected of it, is that it moves in two directions simultaneously: it treats of what is first in the order of intention, is last in the order of execution, and simultaneously, of what is first in the order of execution, which is last in the order of intention.

 

In other words, the political science must treat of the common notions of the good, at the same time that it treats of the good in itself, or the self-sufficient good.

Now, in what sense can we say, in addition to being politically and theologically sound, that that this science is geometrically sound? In the sense that this science contains its own test of reality within itself:

 

The test is not made on the object itself but on our representations that we have substituted in place of the object:

 

And furthermore, the test of reality, or the science of reality is not performed on a perfect whole number, but on irrational numbers, or to say it another way, the composition of reality is not a whole number ratio, but an infinite decimal expansion of an underlying constant.
 

And therefore, this means that our reasons always incline rather than necessitate. And we can always give a reason why things are thus and not otherwise. Thus, our reasons always incline towards what is good in itself, or self-sufficient.

 












Perfected Symmetry in Rotational Language Matrices

Let us posit that this exposition as a whole is a kind of matrix. Let each word be represented as a point or a variable [{x}, {y}, {z}...].Let each equivalent or repeated word be indexed to itself in every other location. Let each word be indexed to the position of every other word in the order. 

As an example, let us take the first word in the series:

“Let” = [A, B, C] = N

A = {x}
B = {[x], [1.2], [1.3]…., 
          [x], [2.2], [2.3]…}
C = {[1.1], [y], [z]…., 
          [2.1], [h], [k]…}

Vector A is the identity variable or a variable that represents the word itself.
Vector B is the positional distribution of the word, where [1.1] means [sentence 1, word 1], and [2.1] means [sentence 2, word 1], etc.. 
Vector C is the meaning distribution of the word.

The positional distribution and the meaning distribution are inverse of each other. In the position distribution, the variable is indexed to all other positions. In the meaning distribution, the position is indexed to all other variables.

Now, we can derive a few insights from this: first, that the grammatical words are the most commonly used words, such as “is” “of” “and” “the.” Thus, these words will have the most positional distribution.

The most specialized words are the least commonly used, such as “indexed” “positional” “matrix.” These words have very specific usage and so will be least commonly used.

But there should also be meaningful words that are in between grammatical and specialized words, such as “location” or “word.” These words are not very common nor are they very uncommon.




Science of Reality

The central unstated assumption of number theory is that numbers are abstract objects in the intellect. But even mathematicians do not take this assumption seriously. It is an artifact of mathematical Platonism which was basically abandoned in the 20th century. The assumption that numbers are abstract objects is a pragmatic assumption rather than proven. Abstract symbols are easier to manipulate than things themselves. The manipulation of abstract symbols proves to be a powerful explanatory method for discovery of previously unknown truths about reality. 

But it also does not explain the reason that mathematics is a powerful tool for discovery of previously unknown truths about reality. If we can show the reason why mathematics is effective for discovery of previously unknown truths about reality, then we would evidently, discover far greater truths about reality.

Again, the central unstated assumption of number theory is that numbers are abstract objects. But we have seen that even mathematicians do not take this assumption seriously. And by dropping this assumption we can very simply show the reason why mathematics is effective for discovery of previously unknown truths about reality. The alternative is to assume that numbers are always an example of things we see. 

First we have to state the argument more informally and then we will give the initial statement more and more precision. Instead of assuming that numbers are abstract objects, we posit is that number always depends on the measure or unit assumed. We never see “3” “6” or “9” in themselves, as abstract objects. What we see are only examples of “3” “6 “9”, such as 3 boxes, 6 inches, 9 liters. 

Thus, we say that number always depends on the measure or unit assumed. Because we can assume different measures for numbers themselves, for example, rational numbers, real numbers, whole numbers, irrational numbers, each assume different measures or units. Thus, the first pass at an answer to the question will seem trivial, though it is not: the reason that mathematics is effective for discovery of truths about reality is that it allows us to order things under a given measure. 

Now one reason we can give as to why this is not trivial is that it appears to show clearly why the different approaches to foundations of mathematics arose in the 20th century: the approaches to foundations were an attempt to answer the question of what measure or unit is assumed. In set theory, numbers are assume to be sets of things. In type theory, numbers are assumed to be types of things. In category theory, numbers are assumed to be categories of things. 

They each assume a different measure, or different collection of axioms and rules. But all of them still adhere to central unstated assumption of modern mathematics: that numbers are abstract objects, rather than tangible examples of things.

And now we can also see why Godel’s proof is limited in its application. Recall that Godel’s first theorem (roughly stated) proves: in any consistent formal system that is capable of expressing basic arithmetic, there exist true statements that cannot be proven within the system. And Gödel's second theorem (also roughly stated) proves: No consistent formal system that is capable of expressing basic arithmetic can prove its own consistency.

Godel’s theorems apply only to formal or abstract systems. They do not claim that it is impossible to prove the completeness and consistency of informal systems. Because it is assumed that informal systems do not follow the law of non-contradiction while formal systems do. 

Though there are many informal ways of speaking which are contradictory, it also seems evident that there are certain informal ways of speaking which are consistent or non-contradictory or non-tautological, e.g. when I say I am going to do something and then I do it. Or when I say I am going to define something and then I proceed to do so.

And furthermore, it seems evident that all mathematics, from Archimedes, to Euclid, to Descartes, to Peano arithmetic, to Grothendique, in some way depends on some kind of informal expressions to serve as scaffolding to more complex and technical and formal ideas. 

And if the completeness and consistency of a certain informal system were proven, then would not also the completeness and consistency of a formal scaffolding built on top of it, also be proven?

This leads us to ask how the completeness and consistency of an informal system would be proven. Since certainly not all informal systems are non-contradictory or non-tautological. What would be the initial conditions in which an informal system could derive a rigorous proof of their own consistency and completeness? Or what are the initial conditions for an informal system which would allow it to seamlessly scaffold into reasoning in a consistent and complete way using abstract symbols in a formal system? 

If a certain informal system can demonstrate its own consistency and completeness, and also seamlessly scaffold into a formal system, then it reasonably follows that the informal and formal system in some way share an underlying grammar or rules of syntax.  If that is true, then it seems that the formal system would specify the grammar that is intuited or already implicit in the informal systems. 

The specifications by abstract symbols allow us to isolate and reduce very specific forms of meaning, and to guard against the generalized analogical or polysemous meanings that natural expressions often convey. However, if there is a kind of scale-invariance, or computational irreducibility , or incompressibility  in grammatical expressions, then informal and formal systems could be shown to be different degrees of formality.

And if they are merely different degrees of formality, then the possibility arises that the consistency and completeness of an informal system would be able to precisely demonstrate how both consistency and completeness arise and can then be generalized to a formal system. We would have to precisely prove the process by which formal notions are formalized, and show the steps or degrees of ascent and descent.

Now we are in a position to make some of these arguments more precise. We posited before that we do not see numbers in themselves or abstract objects. Rather what we see are numbers always as examples of particular things, i.e. 5 shirts, 2 miles, 7 continents. 

And this led us to to the position that number depends on the measure or unit assumed. The reason mathematics is effective is that it allows us to order things under a given measure. And we asked, what is the measure of unit that number universally assumes? Now the prior foundations of mathematics attempt to answer this question via sets, types, and categories. But all of these still treat of mathematical objects as abstract objects rather than examples of specific tangible things within immediate experience. In other words, sets, types and categories are formal things, they are not immediately applicable to real sets of things, types of things, or categories of things, that we observe.

What we will demonstrate here is the way in which formal systems are applicable to real things that we observe right now: Each statement we make is composed of words in an order. And so we index each word to its corresponding number in the order. For example, if we use the first sentence of this paragraph, we can say that ‘What’ = 1 ‘we’ = 2 ‘will’ = 3 ‘demonstrate’ = 4 ‘here’ = 5, ‘is’ = 6 ‘the’ = 7, ‘way’ = 8, etc.

The informal system is the words (e.g. “What we will show here is the way…”), and the formal system is their mapping to the order of occurrence in the series (1. 2. 3. 4. 5. 6. 7. 8). In this way, we show how mathematics works if we assume that numbers are an example of things, rather than abstract objects. Or we show how informal systems may acquire a certain formalization. What we have to prove is how the grammar of the informal system can match up with the grammar of the formal system, in order to show they have the same scale-invariant grammar.

In other words, if we consider the formal system of numbers as abstract objects, we can say that 1 + 2 + 3 + 4 + 5 = 15 and equally say that 1 + 5 + 2 + 4 + 3 = 15. But we cannot apply this same logic to the corresponding informal system, if each number maps to its correspondent word in order. 

In their mapping to the informal system, we can say “What we will demonstrate here…” [1 + 2 + 3 + 4 + 5 ] but we cannot say  “What here we demonstrate will…” [1 + 5 + 2 + 4 + 3] because the latter informal sentence would not be grammatical, even though formal sentence is grammatical.

This shows what we mean by grammatical in both formal and informal systems: the rules that allow us to rearrange and compose new sentences. The rules for informal systems are things like subject-predicate, noun-verb, syntax structures. The rules of sytax or grammar in formal systems are things like axioms of commutativity, transitivity, identity, etc. 

We conjecture that these can be proven to be analogous and equivalent to one another. They simply represent different degrees of formality. But the informal and formal grammars are mutually dependent on one another for accuracy and precision.

In other words, we posit that the reason mathematics is effective for discovery of truths about reality is that the grammar for transpositions in informal expressions is in some way equivalent or analogous to the grammar of transposition in formal systems. Or that formal systems in some sense correspond to informal systems, in a irreducible proportionality.

They only represent different degrees of formality. It is the same grammar of algebraic transposition at different scales. When the grammar is informal, it allows us to state new sentences. When we abstract the same grammar to greater and greater complexity, it allows us to discover new things about reality through technical formulas. Because we can derive terms that were not obvious before, even though they already existed in some timeless sense.

The scaffolding from informal expressions into complex formal systems, will always in some way translate back down to informal terms, expressible in common grammars. The accuracy of the formal system always depends on the integrity of the informal system that it scaffolds from. And likewise the integrity of the formal system can fine-tune and expand the accuracy of informal foundations.

Let us give an example of a clear demonstration: We demonstrate how numbers are not the fundamental unit of measure, but depend on the consistency and completeness of the underlying informal system. The underlying informal system is shown to be consistent and complete if we can choose partitions or its statements that can be recombined in a grammatical and non-contradictory way. This is analogous to the practice of algebraic transposition or substitution that balances two sides of the equation.

Now we take the above four sentences and partition them into two parts each, and we represent it as: (AB. CD. EF. GH), and then we show that it can be diagonally reflected in a grammatical way (ACEG. BDFH), where the meaning is unchanged.

Let us give an example of demonstration of how numbers are not the fundamental unit of measure and the informal system is shown to be consistent and complete, as analogous to the practice of algebraic transposition: A clear demonstration that depends on consistency and completeness of the underlying formal system, is a choice of partitions or cuts that can be recomined in a grammatical and non-contradictory way, as a substitution that balances the two side of the equation.

Now we take the prior four sentences partitioned into two parts each (AB. CD. EF. GH), and then we show that it can be diagonally reflected in a grammatical way (ACEG. BDFH), where the meaning is unchanged and we represent it as (AB. CD. EF. GH) →= (ACEG. BDFH)

And now if we translate the partitioned variables (AB.CD.EF.GH)  into numbers, we have:

(AB. CD. EF. GH) →= (ACEG. BDFH)
= (1 + 2. 3 + 4. 5 + 6. 7 +8) →= (1 + 3 + 5 + 7. 2 + 4 + 6 + 8)
= (3. 7. 11. 15) →= (16. 20)
= (36) →= (36)

This demonstrates that there are certain informal systems which are both consistent and complete, i.e. both non-contradictory and proven to be non-contradictory. The reason that the informal system is non-contradictory is because the underlying informal meaning is essentially identical in both cases 
(AB. CD. EF. GH) →= (ACEG. BDFH). The reason it is complete is that we have proven that the transposition (AB. CD. EF. GH) →= (ACEG. BDFH) can correspond to formalization in arithmetic which is proven to be non-contradictory.

The important point here is that while the informal underlying meaning of (AB. CD. EF. GH) is essentially the same as (ACEG. BDFH), the transposition (AB. CD. EF. GH) →= (ACEG. BDFH) is both the same, and also may reveal some new underlying meaning without changing the original. This shows precisely how mathematics discovers truths that were not as obvious before but were always there.

Now it will be argued that this depends on an informal or common sense agreement about when two statements logically contradict each other. And it seems that the whole reason that formal systems of mathematics and logic arose was because of the ambiguity or polysemy in informal expressions, which allows them to contradict each other with different individual interpretation, and there is no clear way to judge of different individual interpretations.

But we have just given a clear criterion for consistency and completeness in informal expressions, namely that 1.) they can be partitioned, 2.) the partitions can be transposed or substituted in place of one another, 3.) such a transposition is grammatical, 4.) such that underlying meaning is unchanged by the transposition.

The only dimensions in which individual interpretation comes in is (4.) whether the underlying meaning is unchanged. Only rarely is there a question of (3.) whether the transposition is grammatical. However, in this case, the reason that there will be a question of whether the underlying meaning is unchanged is not because of the difference between individual interpretations, but the proportionality between individual interpretations. The transposition (AB. CD. EF. GH) →= (ACEG. BDFH) can reveal new implicit information about (AB. CD. EF. GH) without changing its underlying meaning. In the same way, a proportion can state the same thing in a different way, such as 25 / 100 = 10 / 40. 

When we are talking about specific tangible things, the difference between 25 / 100 and 10 / 40 may matter in the same way the difference between 223/71 and 22/7. 

Since before we knew that 2[π][r] was equal to the circumference of a circle, we would have had to derive that truth from a more informal algebraic transposition of terms. This is in fact what Archimedes did when he approximated π by inscribing and circumscribing polygons around a circle.  By increasing the number of sides of the polygons, the perimeters approached the circumference of the circle. Starting with a hexagon and doubling the number of sides repeatedly, Archimedes calculated that π lies between 223/71 (approximately 3.1408) and 22/7 (approximately 3.1429). In an informal and formal way, Archimedes approximated the value of π in showing that: π ≈ [Diameter of circle] / [Perimeter of polygon]
Formal or rigorously abstract systems alone do not give us a sufficient basis to know the foundations of mathematics. Because the foundations of mathematics would give us a way to prove that arithmetic is consistent, i.e. that it does not contain contradictions. And this would give us a way to prove why mathemtics is effective in discovery of previously unknown truths about reality. 

We know that calculation with numbers is effective. And we know that with numbers we can calculate almost anything. And it seems we know that arithmetic is consistent or non-contradictory. But we have not been able to prove why calculation with numbers is effective, why numbers can calculate almost anything, why arithmetic is consistent or non-contradictory. In other words, with modern mathematics we know only the effects without knowing the cause.

Presumably the reason that we calculate with numbers is to pursue the truth about reality. However, numbers can and are used to deceive. And so the answer cannot be that mathematics is effective and useful because it allows us to calculate anything at all, if we do not qualify this with “can be used to calculate almost anything.” Because if numbers could calculate anything at all, then they could calculate a contradiction, in which case it would follow, that they also could not calculate anything.

And so we can begin see how these considerations are related: in order to understand why mathematics is effective and useful, we would have to know the foundations of mathematics. In order to know the foundations of mathematics we would have to demonstrate that arithemetic is consistent and complete. And in order to demonstrate that arithemetic is consistent and complete, we would have to know precisely the way in which mathematics calculates, not anything at all, but almost anything. 

And therefore, by knowing precisely where the limit of calculation or mathematics is, we would have a certain and implicit knowledge of why mathematics is useful and what its foundations are, without necessarily having uncertain and explicit knowledge.







To demonstrate the purpose of numbers would be to demonstrate what thing it is that numbers are an example of, what measure or unit they assume, universally. Numbers cannot be an example of sets, types, or categories because these are not sufficient foundation to establish both the consistency and completeness of arithmetic.

This would be to explain how grammar or definition arises. Or, it would be to give definition or grammar to meaning itself. Or, to put it another way, it would be to show that arithmetic is consistent with itself. 

This would mean that we can treat the words as geometrical objects and simultaneously as definitions or grammar. So that the representations operate on two levels simultaneously without diluting either level.





If there is an order or words that can be recombined in such a way, then we will have demonstrated the consistency and completeness of arithmetic because we make the assumption that words themselves are the measure or unit of arithmetic.

This would be to demonstrate that arithmetic or rational numbers are self-consistent and complete, meaning that rational numbers have no contradictions and that we can prove they have no contradictions using only rational numbers themselves. This is based on the supposition that numbers are always specific things. We never have 5 as such or 7 in itself, we always have 5 or 7 apples or tables or inches or liters. It always depends on the measure or unit that is assumed. The question to be answered is how is it demonstrated that the order of arithmetical rational numbers is both consistent and complete? Or why is the order both non-contradictory and proven to as non-contradictory within itself. Because this shows the ends or purposes or meaning of arithmetic and rational numbers.

For example, we have simple reality and we have representations of ways of combining those. ABCD -> CABD





Here we show how principle of excluded middle (A v ~A)  implies principle of analogy (A → B → C) → (A → C):

A: We posit that: (A = A) → (A v ~A) → (A ≠ ~A)
B: Therefore: (A = A) → (A ≠ ~A)
C: Therefore: (A = A) ≠ (A ≠ ~A)

A’: Now, we reduce these three steps to: A → B → C
B’: Therefore: A → C
C’: Therefore: A ≠ C

A’’: And therefore we know that: (A ≠ C) ≠ (A → B → C) → (A → C)
B’’: Therefore: (A ≠ C) ≠ (A → C)
C’’: Therefore: (A → C) → (A ≠ C)



This leads us into the central paradox or even contradiction in the foundations of modern mathematics, stated by Vladamir Voevodsky, in his 2010 talk, “What if the Foundations of Mathematics are Inconsistent”? 

 

Von Neumann recognized that Godel’s first incompleteness theorem implied a second incompleteness theorem. His statement could not be more clear: “there is no rigorous justification for classical mathematics.”

This leads to what Voevodsky calls Godel’s paradox:

 

Voevodsky then states the three choices open to mathematicians, given this paradox:

 

He says that 90% of mathematicians have gone with the first option, that the second incompleteness theorem is false as stated. But Voevodsky, instead of attempting to either refute Godel’s second theorem or to find a statement of it that is not false, instead pursues the third option, that first order arithmetic is inconsistent.


This would thereby demonstrate the completeness and consistency of arithmetic, because it would prove both that rational numbers have no contradictions and also that we can prove that they have no contradictions. And this goes back to Hilbert’s second problem and Godel’s theorems.

The way that we demonstrate the completeness and consistency of arithmetic is that we assume that numbers are not abstract objects, but rather that numbers are always an example of the measure or unit that is assumed. In this case, the measure or unit that is assumed is words or the qualities indicated by words. Every mathematical theory in some way relies on common words, or on the things themselves in experience.