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Philosophy of Math:
The Beginnings of Mathematical Deduction by Induction
Christy Ailman
Azusa Pacific University
Abstract
In attempt to provide an answer to the question of origin of deductive proofs, I argue that Aristotles philosophy of math is more accurate opposed to a Platonic philosophy of math, given the evidence of how mathematics began. Aristotle says that mathematical knowledge is a posteriori, known through induction; but once knowledge has become unqualified it can grow into deduction. Two pieces of recent scholarship on Greek mathematics propose new ways of thinking about how mathematics began in the Greek culture. Both claimed there was a close relationship between the culture and mathematicians; mathematics was understood through imaginative processes, experiencing the proofs in tangible ways, and establishing a consistent unified form of argumentation. These pieces of evidence provide the context in which Aristotle worked and their contributions lend support to the argument that mathematical premises as inductively available is a better way of understanding the origins of deductive practices, opposed to the Platonic tradition.
The Beginnings of Mathematical Deduction by Induction
The origin of deductive proof is a central question in the philosophy of mathematics; this problem has been commonly answered in one of two ways. The first, taking a Platonic understanding of knowledge as a priori, says deductive premises arise from within the intellect thus in this way are self-evident. T T The opposing response claims knowledge is a posteriori as derived from Aristotelian thought, mathematical truths built up through an n an inductive process, using the senses to firmly establish premises that can be applied toward deductive reasoning. In this paper I will argue for the Aristotelian position, and consequently that it is possible for deductive proofs to have begun inductively as a posteriori knowledge and to have then developed into deductive reasoning. I will begin by explaining Aristotles philosophy of mathematics. Of primary importance is he argues that mathematical objects exist, in a qualified sense, within the physical object. There is, however, a contemporary debate over the exact relation between the math object and the physical object, which I will also examine. I will look back to the Greeks who were the first to capitalize the deductive thought process in mathematics to see if deductive proofs actually developed through induction as Aristotle believed. The first piece of evidence I will present comes from contemporary scholarship on recently recovered texts of Archimedes showing the unique way in which diagrams and indirect proofs were used in his work; his proofs were focused on the images rather than the words. Next, I will utilize recent scholarship on the origins of deductive proofs that argues that their origins lie not in mathematics, but in Greek literary conventions. In conclusion, I will describe how these insights to Greek mathematics provide evidence for thinking that Aristotle's, rather than Plato's, overall approach to the philosophy of mathematics is the more accurate one.
Aristotles basic epistemology provides a foundation for how math reasoning is a form of a posterior reasoning. He claims the forms are in the matter; in that, the logos of the cosmos is in nature rather than transcendent. He says that our reason is limited, qualified, until we have investigated nature for its material and formal substance to then give unqualified knowledge. The sensible world is intelligible because it contains the formal principles within it. Following his belief that form is in object, he thinks mathematical objects are in a way in the objects themselves. For example, he might say that a round table contains within it circularity. Circle does not have its own completely independent being separate from all instances of circularity, as Plato would want to say. The form is found in the physical.
A significant difficulty in Aristotles metaphysics is explaining the exact mode of the forms existence in the object. On one end of the spectrum, Lear proposes an interpretation of Aristotle saying that the table actually contains the mathematical circle in it. The form is a substantive part of the object and is found by abstracting the matter away to find the specific mathematical object. This method of abstracting toward specific characteristics of objects is often done by examining what the object is qua its specific characteristic. For example, if we wanted to examine a golden triangles triangle-ness, we would consider the golden triangle qua triangle to find out more about what makes it so triangle-ly such as the sum of its interior angles is 180 degrees. In this, mathematicians study the physical world but not as physical because they study the mathematical objects qua abstract thought (Lear 247). The metaphysical location of Aristotles mathematical objects according to Lear is actually in the physical, as substantive matter.
To properly understand a position opposing Lears, it would be helpful to consider the distinction between potential and actual existence. As White explains it, it is certainly not possible for them to exist separately. But since they could not exist in sensible either, it is clear they either do not exist at all or they exist in a certain mannerthey do not exist unqualifiedly (White 166). Mathematical objects always exist in the object, but specifically they exist in a qualified, potential sense. Due to certain math absurdities that arise from assuming that math objects have substantive physical existence, they must exist potentially. Rather, they exist potentially so that if a division is made the point is still indefinitely divisible, thus refuting the problem of division. To exist actually means to bring out the object through identification and abstraction (Pettigrew 248). White defines abstraction as a means of focusing ones attention, as it were, on those features by eliminating from consideration other figures not germane to ones present mathematical investigations (White 176). This interpretation of Aristotles mathematical objects says they exist in a particular way: potentially and actually in the object as abstract, nonmaterial aspects (White 182).
In either case, the process of abstracting mathematical objects must be based upon inductive processes. The material substance must be investigated scientifically and critically considered until the form is found. The intellect begins with qualified understanding, very basic knowledge-for example, that a round table is circular. To gain unqualified knowledge, the table must be closely examined by taking measurements, sketching the basic shape, and such. After the material properties of the table have been abstracted away, and the roundness of the table has been considered, circularity can be identified to then distinguish properties of a geometrical circle. This unqualified knowledge was found through the process of investigating the physical objects to come to a greater understanding of the formal properties. Nothing is in the intellect that wasnt first in the senses; in that, knowledge such as this comes a posteriori. The inductive process hinges on the idea that the sensible world is intelligible and thus rationally ordered. By observing this order, one can discover patterns and principles. Once principles have been established, they can then be used to reveal necessary entailments and absurdities through the process of deduction. Aristotle describes this process of using deduction in his definition of syllogism, certain things having been supposed, something different from those supposed results of necessity because of their being soX results from Y and Z if it would be impossible for X to be false when Y and Z are true (Smith 1.2).
Aristotles philosophy of math begins inductively because his epistemology is inductive and because one finds the math objects within the existence of physical, empirical objects. I will now demonstrate how the actual development of deduction in mathematics in the Greek culture proves to be compatible with Aristotles claims. Historian of ancient Greek mathematics, Reviel Netz has made a recent discovery in early Greek mathematics, documented in his book The Archimedes Codex that describes how Archimedes mathematics was a visual science. Netz uncovered some of Archimedes proofs hidden behind layers of other writings in the Palimpsest document. Within these pages, the secret of how Archimedes did math was revealed. Netz explains how our mathematics is concentrated on the words in proofs, whereas Archimedes proofs concentrated on the images (Netz 31). With that, he sees the importance of diagrams and indirect proofs in mathematics as visual experiences of uncovering truths in the physical world (Netz 36).
In understanding Archimedes mathematics, it is necessary to know what he was trying to do. The holy grail of Greek math was to find the area of a curved object. Archimedes succeeds in this, but through unconventional ways. The first way being through indirect proofs which are defined by an initial false assumption; it assumes the opposite of the truth (Netz 47).
Archimedes starts out promising to make some incredible measurement, and you expect him to fudge it out somehow, to cut cornersAnd then he begins to surprise you. He accumulates results of no obvious relevancesome proportions between this and that line, some special constructions of no direct connection to the problem at hand. And then, about midway through the treatise, he lets you see how all the results build together (Netz 44).
In this, he is deceiving the reader by beginning with what they would not expect; but about midway he reveals his ways and ties it all back together. His answer to measuring curved objects follows this form of proof as it does not begin with known shapes. Rather, he invents his own curved objects and then finds unexpected ways to measure them. This method of indirect proof was a hallmark in Greek mathematics (Netz 47).
The second component of Archimedes mathematical process was the way he used diagrams. In deductive proofs today, diagrams are illustrative in attempts of making things easier to comprehend: they represent particulars (Netz 91). Diagrams for the Greeks, were general demonstrations to provide us with the most basic information in the proof while retaining veracity (94). Mistakes or false conclusions did not result because the same diagrams were used over and over again. Imagination is necessary to understanding the diagrams because they only suggest an object. Greek art shows the capability of drawing with precision, but mathematicians chose not to. Instead, they copied each others diagrams as a means to represent the broader, topological features of a geometrical object (Netz 105). Like a green triangle is no different than a red triangle, a precise angle is no different than an estimated one. These Greek diagrams are better called schematic representations, where precision is not a factor (101).
These features of mathematics, diagrams as schematic representations and indirect proofs, were not unique to just Archimedes; these are thought to have been prevalent practices among all of Greek mathematics. There were not many mathematicians at the time of Archimedes; around fifth century BC poetry was the dominant vocation so there were very few doing pure mathematics (Netz 39). Indirect proofs and diagrams demonstrate how Greek mathematics was an imaginative and visual process similar to the more popular vocation of poetry. The works of Archimedes are described as creative and playful; he was a poetic mathematician and consequently his entire science was based on a sense of play and beauty, on hidden meanings (Netz 55, 58). This shows how early forms of mathematical works were not done in a clear cut, deductive method; rather it appears to have been a visual process that required imaginative ways of finding the potential mathematical objects in nature. For example, diagrams as schematic representations necessitated a mathematician to look at figures in new, in depth ways to find what was not seen before. If the initial deductive premises were this closely related with visual and imaginative practices, how then did they develop into logically precise deductive proofs?
According to recent scholarship on the interplay of narrative and mathematics in Circles Disturbed, Apostolos Doxiadis writes on the origins of deductive thought in A Streetcar Named Proof. He says culture-specific factors played a crucial role in the development of mathematical deduction; there were many people in fifth and fourth century BC who combined the skills of the craftsman and the thinker (Doxiadis 328). In looking specifically at the Greek culture and what was unique to it to originate deductive proofs, he asserts that their frame of mind to provide logical evidence in the practice of forensic rhetoric along with the common literary structure of ring composition were the keys (325). He believes deduction was developed through a process rather than a sudden event. This process began with narrative, then rhetoric, and then finally came to mathematical proof. Greek narratives then will be the starting point to understand the development of deductive proofs.
Showing how deductive proofs developed as a result of a literary structure demonstrates that deductive thought originated through a process correlating with Aristotelian induction rather than an immediate Platonic intellectual intuition. Doxiadis argues that narrative was used in a particular way; its aim was good mimesis. In order to create a believable narrative about reality, the Greeks used the structure called ring composition. Ring Composition is a symmetrical alignment of phrases: A,B,A*. There are seven conventions of ring composition according to a modern anthropologist Mary Douglas as described in her book, Thinking in Circles. These are not requirements, nor are they exact; but they are consistent guidelines to help find and support ring composition in narratives. The first convention is a prologue which lays out the dilemma, command, or doubt in anticipation of the pivot. Second, the narrative will split in that it will make a distinction between the prologue and the coming middle section. The third convention is alignment of parallel sections, reflecting a pattern across the central dividing line. Next, indicators are used to clearly mark the various sections either with key words, a switch in genre, or repeated lines. The fifth convention is called central loading, which refers to the dependence on or importance of the turning point. Next, a feature often used but not necessary is a ring contained within another ring. The seventh convention is closure at two levels: completing the response provoked in the prologue through repetition of the key words and completing the prologue thematically (Douglas 36-37). Overall, the attention is drawn to the place of transition from A to A*, this is the pivot, B, ring compositions structure draws all eyes to the center in which everything changes.
Homers Iliad, a very widely known epic for the Greeks, shows great promise for Douglas theory in that it meets all seven conventions. Douglas does not only find one, but two rings in this epic. The epic has a prologue, convention one, in the first book where the dilemma of how to give Achilles kleos is proposed to Zeus and the theme of rage is introduced. Convention seven is met in the final book with closure thematically, transforming rage into grace and showing Achilles eternal kleos; and closure in repetition of the key phrases. Convention four, the indicating lines are seen when a new day is introduced; most often done in a form of personification of the Red Dawn (Fagles 563). Structural and thematic parallels are made between the first and final books and the days surrounding the pivot point, thus meeting convention three. From the macro-ring it is also evident that the epic loads the center of the narrative with importance, convention five; in this it actually contains another ring, convention six. There is a change in pattern from the macro ring to the micro ring, an indication of a division in the narrative going from the prologue into the center, which aligns with the only convention yet to surface, convention two (Douglas 36-37).
The use of ring composition in Homers epic indicates it was a recognizable structure to the Greeks. As Douglas says, [Rules] are not imposed from outside of the literary work. They are not there first. They emerge from the first completed works (Douglas 17). It is not known if the Iliad was the first piece of Greek literature or not so either ring composition was already a common structure and Homer ascribed to, since it would be recognized; or he was the first to use it in literature and due to the epics widely held use, the structure emerged from it. Either way, this form must have at least been prevalent in thought because one of the special literary merits of a ring is to anticipate its own form of closure from the beginning (Douglas 17). Homer would not want to use a literary form if the audience was not going to recognize it, the Iliad is intended to be heard or read as a whole, and delivered to an audience who are familiar with the story, or at least with the style (Douglas 31). So this thought process of building parallels around a central turning point was at least recognizable due to previous literature or previous cognitive patterns. They [the audience] will expect symmetry and balance, and they will judge how well the ending slots on to the start. To bring the preordained ending elegantly back to the beginning is not so easy as it may sound (Douglas 17).
As described, this is a complex structure so it was not used lightly; it was a difficult task to create a narrative fitting to ring composition. Douglas explored the possible reasons for its use, the Iliad, being a lyrical poem, would have needed to have a structure designed to help the memory of those who relayed the poem. By using ring composition they would only have to know the storyline leading up to the turning point and then simply reverse or redirect all that came prior. Learning by route is a characteristic of literate society from the Greeks back to the Egyptians (Douglas 13).
Another explanation is merely that ring composition makes the poem clear and orderly, which would have appealed to the Greek mind. Especially since the Greek mind admired unity, harmony, and clarity, this structure might have appealed to them visually. Regardless of its logic, it was aesthetically pleasing. None of these explanations explain, however, why ring structure worked so well and continued on into written narrative. Douglas goes on to say, the ring convention does something to fill the interpretative gap by virtue of its symmetry, its completeness, and its patterned cross-referencing (Douglas 13-14). Arguably, it continued to be used because of its efficaciousness to create clarity and unity in communicating a narrative. The parallelism of the structure is designed in a way to frame the mindset of the reader. It gets rid of ambiguity by creating a formulated structure and process by which the reader can know what will happen after reaching the turning point. Ring structure creates an argument; its clear structure persuades the reader.
This third explanation for why ring composition was used fits with what Doxiadis has argued for. From narrative, rhetoric was developed using this same ring structure in attempts to make a strong, convincing argument. The first kind of rhetoric in the Greek culture was public-occasion rhetoric. This was used to praise or blame a person by appealing to general opinions or rules. These general rules were derived from the narrative use of analogies (or otherwise called parallels), where two things not thought to have been put together are considered analogous to stress a point. The second type of rhetoric was oratory, which was used in politics or courts to convince a jury by probable cause. This was seen in the probable or necessary connections of narrative, specifically in ring composition where the prologue is anticipating the pivotal action and reversal in the rest of the narrative. The third and most commonly use was forensic rhetoric. This was used for argumentation in any area with the pure intent of persuasion by proof. In this, Doxiadis believes logic began as a method for comparing contesting narrative accounts of events and as a result people became aware that it was not the one and only narrative but a narrative that was presented. This form of argumentation was very similar to the Greek narratives structure of ring composition (Douglas 295-303). The aim of the orators importing poetic techniques into rhetoric could well have been originally aesthetic [but] those same techniques soon became the basis of the budding method of logical argumentation (Doxiadis 301).
Following Doxiadis train of thought, rhetoric then turned into mathematical proofs. It is argued that rhetoric provided the context and model for mathematical, deductive proofs (Doxiadis 302). The Greek mind had developed the need for purely logical proofs as a result of mimesis, the court system, and other cultural factors. As literacy and written documents increased, mathematical proofs became the written model of the verbal rhetoric: let us think of [deductive proofs] not as abstract thinking but as the symbolic representation of the action of proving something (Doxiadis 331). A mathematical proof contains several parts: enunciation, setting out, specification, construction, proof, and conclusion. The prologue sets out the claim, the setting out specification and construction make these claims concrete, the proof makes the claims stronger with counter arguments, and then the conclusion repeats the claim in a general form. From this, it is seen how ring composition matches up very easily: both have prologues to lay out the dilemma, both have a crucial central pivotal section which answer the dilemma, and then conclusions to reiterate the claims in light of the new insight (Doxiadis 335).
In addition to the larger scale ring composition, deductive proofs also have rings at the smaller level. Doxiadis begins this exposition by arguing that the basic syllogisms of logic that underlie so much of mathematical thinkingare in RC form (345). Ring composition exists in three ways at this micro level: rule centered, binary X, and substitution RC form. Rule centered rings begin with a concrete situation, a general rule is applied, and then the answer is the rule applied to the situation. An example is: 1. CA and CB are equal to AB. 2. But things equal to the same thing are also equal to one another 3. Thus, CA is also equal to CB (Doxiadis 349). The binary X form is also known as reduction ad absurdum. This essentially is an if-then statement, an extremely significant form of deductive proof. An example is: 1. If AC is equal to AB 2. Angle ABC is equal to ACB. 3. Angle ABC is no equal to ACB. 4. Thus AC is not equal to AB (Doxiadis 350). The final way ring composition exists in deductive proofs is in substitution rings. These rings are equivalent to what Aristotle calls syllogism. They contain a middle which unites two opposing statements. An example is: 1. A has property a. 2. All xs have property a have property b. 3. A has property b (Doxiadis 356). As seen in these three types, ring composition exists in mathematical deductive proofs on the macro level and micro level. This gives an account for how deductive proofs began, not as a sudden explosion of new thought; but rather a process of fine graining the form of argumentation through ring composition (Doxiadis 347-356).
In summary, Netz provided an explanation for how deductive premises could have arisen inductively and Doxiadis provided an account for how deductive thought grew. In both of these views, the activity of the mathematician can be described as experiential; deduction was not a matter of coming upon a revelation, rather it was a process of interacting with the material and discovering what worked. Ring composition showed how the most convincing way to represent a narrative was through bringing the audience into the dilemma, presenting a solution, and seeing it worked out. The answer could not simply be laid out from the beginning; it is not as convincing that way. Rather it must be good mimesis, necessary connections leading one action into the other, drawing the audience into persuasion as Aristotle describes in Poetics (Janko 4.4). It is the experience of persuasion, just as indirect proofs did not simply give the answer from the start. It too drew the reader in with surprise or confusion to then later be enlightened. Aristotle supports this idea as well saying, Homer above all has taught the other poets to tell untruths in the right way, that is, by false inference (Janko 5.3.2.2). Diagrams too are a matter of experience as they are only schematic representations. Netz states that mathematics is some sensual packaging, in the sounds of language, and in the artifices of vision (115). They did not use equations as we use today to make logic visible, rather they used diagrams as necessary components to their logical thought process (Doxiadis 88). The Greek mathematician draws a picture of a polygon, not the actual polygon. Words are conceptual, but drawings are physical and need to be taken, played with, and then conceptually understood rather than simply taken at face value (Doxiadis 100). In light of these descriptions of how math began as a process of experiences forming unity with creativity, I conclude that math began inductively.
Now having two foundational pieces of evidence from early Greek mathematics, the Greek Miracle I argue is actually closer to a process of induction rather than an explosion of deduction. Netz showed how the work of early mathematicians was focused on the images, the diagrams in proofs as schematic representations rather than precise illustrations. They also used the convention of indirect proof which began with a false assumption to surprise and redirect the reader. Inductive thought is evident in both of these as they use the senses to attain knowledge and the experience of getting it wrong before realizing the correct answer. Doxiadis then proposed the development of deduction to have arisen through a process of ring structure being adopted by various forms of communication. The process began with narrative, then refined its intent of persuasion in rhetoric, and further developed in persuading the more abstract ideas of mathematics. We now have two accounts of Greek math, one gives a basic understanding of mathematical truths as arriving from the senses; the other gives an explanation for how mathematical deduction arrived from a process beginning in narrative.
Furthermore, with the evidence gathered from Doxiadis and Netz, the development of Greek mathematics aligns with Aristotles philosophy of mathematics. Doxiadis showed how deductive proofs were derived from rhetoric, and rhetoric from narrative. The common thread was the use of ring composition. Aristotle claimed deduction to have arisen as a result of induction first, and this shows that deduction was not self-evident but began from a gradual development. Ring composition showed a unified thought process at work within each of the disciplines, opposed to a creation of a brand way of thinking. Indirect proofs do a very similar thing as they bring the reader into a state of vertigo realizing that the initial assumption is wrong. It is about beginning in a state of assumption, as Aristotle would call qualified knowledge, to being abstracted away, experiencing a formal understanding, and then come back with a more thorough, unqualified understanding. Diagrams too, aside from being visual, are tools used to realize mathematical truths through the senses by actualizing them. They are left open for interpretation or alteration to see new possibilities. As Aristotle would say, the potential is there within the diagrams, there are infinite potentials, but only do they become actualized when abstracted. Inductive mathematics for the Greeks meant mathematical truths were inherent in nature, the cosmos contained intelligible matter as Aristotle would say. The truth is just waiting to be pulled out and seen in the right way. Within the nature of Archimedes diagrams and indirect proofs and the account of ring composition, Aristotles philosophy of how mathematics began proves to be supported by this evidence.
In attempt to provide an answer to the question of origin and veracity of deductive proofs, I argued that Aristotles philosophy of math was more accurate opposed to a Platonic philosophy of math, given the evidence of how mathematics developed. Aristotle says that mathematical knowledge is a posteriori, known through induction; but once knowledge has become unqualified it can begin deduction. Through Netz and Doxiadis, we have two pieces of evidence which reveal that math began gradually and inductively in the Greek culture. Both claimed there was a close relationship between the culture and the mathematics; many mathematicians were artists or craftsman as well. Netz described the way Greek math was done in diagrams and indirect proof which speaks to sensible nature of mathematics at that time, it was understood with imagination and experiencing the proof in a tangible way. The origin of deductive thought then was explored by Doxiadis who found a correlation between narrative and deductive proofs. He claims deductive reasoning was a result of using the common literary ring composition to make a clear, unified argument. Aristotles claims are supported in the Greek mathematical assumption that intelligence could be found in nature and once principles were established, they could be argued in a persuasive manner of deduction. The arguments from Netz and Doxiadis do not establish a definitive interpretation of how Aristotle thought that mathematical objects could exist in a way that makes them inductively available. Their arguments do, however, establish that Aristotles thinking dovetails nicely with Greek mathematical practice and literary conventions. The question of how mathematical truths might exist in the sensible world still remains, but Netz and Doxiadis help us to see the context in which Aristotle worked and their contributions lend support to my argument that mathematical premises as inductively available is a better way of understanding the origins of deductive practices.
Works Cited
Curd, P., & McKirahan, R. D. (2011). Heraclitus of Ephesus. A Presocratics Reader Selected Fragments
and Testimonia. (2nd ed., pp. 39-54). Indianapolis: Hackett Pub. Co..
Douglas, M. (2007). Thinking in circles: an essay on ring composition. New Haven: Yale University Press.
Doxiadis, A. (2012). Circles Disturbed: The Interplay of Mathematics and nNarrative. New Jersey:
Princeton.
Fagles, R., & Knox, B. (1990). The Iliad. New York, N.Y., U.S.A.: Viking.
Janko, R. (1987). Poetics I. Indianapolis: Hackett Pub. Co..
Lear, J. (1988). Aristotle's philosophy of mathematics. Aristotle: the desire to understand (1. publ. ed.,
pp. 231-246). Cambridge u.a.: Cambridge Univ. Pr..
Netz, R., & Noel, W. (2007). The Archimedes codex: how a medieval prayer book is revealing the true
genius of antiquity's greatest scientist. Philadelphia, PA: Da Capo Press.
Pettigrew, R. (2009). Aristotle on the Subject Matter of Geometry. Phronesis: A Journal Of Ancient
Philosophy, 54(3), 239-260.
Smith, R. (n.d.). Aristotle's Logic (Stanford Encyclopedia of Philosophy/Spring 2012 Edition). Stanford
Encyclopedia of Philosophy. Retrieved December 10, 2012, from HYPERLINK "http://plato.stanford.edu/archives/spr2012/entries/aristotle-logic/" http://plato.stanford.edu/archives/spr2012/entries/aristotle-logic/
White, M. (1993). The Metaphysical Location of Aristotle's 'Mathematika'. Phronesis: A Journal Of
Ancient Philosophy, 38(2), 166-182.
Logos, as I use it, means account or explanation for reality. Nature is intelligible because it contains the logos within it.
This major Aristotelian concept is called hylomorphism: all objects are composed of both matter and form.
Aristotles Physics 190b11
For example, the problem of Division as described by White in The Metaphysical Location of Aristotle's 'Mathematika'.
Mimesis as I use it means a representation of reality, a copy, not reality. The Greeks sought to mimic reality in a believable way, to represent reality as even better than it was.
23.129
Douglas offers another proposal that it is a part of the human cognition. Ring structure could be a psychological understanding of how our minds work universally.
Doxiadis specifies narrative from lyrical poetry and says that poetry was derived from narrative, but I have just combined the two for the sake of my argument.
RC being ring composition
I think indirect proofs are essentially rings compositions.
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