Axioms of the cosmological model in Timaeus
The cosmological model proposed by Plato is a “scientific model” in today’s strict sense of the word. As a result, this construction, this theoretical model of the universe, must be accepted as a formal axiomatic system: a collection of primordial propositions, axioms, which must first be established, and all observable manifestations of the universe must in principle be derived as theorems from these axioms. Of course, it is possible that Plato was not fully aware of the fact that he was shaping a model based on a list of axioms and that these axioms had no other justification than their epistemological consistency. But he accepted this idea: whoever discovers a different set of axioms, whoever can advance another model or try to explain it differently, has an equal right to be heard; and only the model that best corresponds to the data, the model that is “most acceptable” (Timaeus 54a-b, Timaeus 55d).Cosmological axioms1. Reality is divided into two domains: intelligible forms (eidos, idea), pure, eternal, unchanging, and simple; and complex objects of sensory perception, always changing (kinetos) in time.This division of reality into two domains corresponds to the difference between existence and becoming, between that which remains the same forever and that which never ceases to change, between that to which the predicate “true” can be attributed and that to which it cannot.Only knowledge of intelligible forms can be said to be “true”. However, finite, transient, and limited human beings cannot attain such knowledge in this world. Knowledge of these divine entities is the exclusive privilege of the gods and a small number of their friends (Phaedrus 278d). Knowledge directed towards the world of objects of sensory perception, which Plato classifies as “opinion” (doxa), is a lower form of knowledge that, at best, only reaches similarity. 1. Truth holds a special position. It is an intelligible form among others, such as Justice, Unity, Man, Animal, etc., but it plays a crucial role within the Platonic system, especially in Timaeus. This form transmits to other intelligible forms its distinct characteristics: beauty, harmony, order, simplicity. According to Plato’s proposed cosmological model, the other intelligible forms then transmit these characteristics to sensory objects.
2. In the sensory world, everything that comes into being does so as a result of a cause. Causality is only applicable to the sensory world because in the reality of intelligible forms, which, according to Axiom 1, are eternal and unchanging, there is no change, and therefore there are no causal relationships in the intelligible world. In the sensory world, change implies relationships among elementary components (eternal and unchanging), but all changes are subject to causality and can be mathematically expressed as much as possible. The word “cause” (aitia) is borrowed, and From the legal dictionary, where it denotes responsibility, here it denotes this chain of relationships. 4. The sensory world is the result of divine introduction of order. One of the causes of the sensory world is God, also known as “father”, “creator”, and “demiurge”, the most commonly used expression. This God does not create the world; his action is limited to the partial ordering of the primordial chaotic substrate (ref. axiom 7). 5. The demiurge is good (agathos). The goodness of the demiurge dictates a certain way of acting (ref. axiom 10). 6. The demiurge is not omnipotent. This God is not omnipotent, for two reasons set out in the form of axioms. Intelligible Forms of ichor, primordial matter or “medium of space”, exist independently of him (axioms 1 and 7). And the demiurge must face the shenanigans (axiom 8), which always resist his effort to introduce order. 7. The demiurge organizes the primordial matter, khora. Khora is simultaneously what the sensory forms are located in (space or place) and what they are made of (something like matter). Khora (“medium of space”) is “Khoraija is a mysterious entity. It is eternal, existing even before the demiurge introduced order into it. However, everything that is within the Khoraija, everything that originated from it (the sensory world), is constantly changing. Khoraija is eternal and knows no destruction; it provides conditions for everything that arises, but we perceive it through a kind of “mixed reasoning,” rather than with our senses or convictions. Khoraija is the recipient and protector of all creation (Timaeus 49a). Cause, Ananke, constantly resists the order that the demiurge tries to establish in the world. Ananke is generally translated as “necessity” in ancient Greece (the prevailing law of the material universe, an essential characteristic of Khoraija). Plato considers Ananke as a “cause,” but a negative one, labeled as a “wandering cause,” since it represents the non-rational element that permanently opposes the demiurge’s effort to establish order. Before the intervention of the demiurge, the actions of Ananke in Khora were such that the four elements, believed to create the entire sensory world, “behave without order and measure” (alogos).” andvariable.Random'”,time}) (Timaj 53a).Necessitycontinuesto manifest in the sensory world as the “wandering cause” after the demiurge withdraws from the world. As a result, there is always a factor of complexity and disorder in the universe.Objects of sensory perception, including celestial bodies, are composed of only four elements: fire, air, water, and earth.Here, Plato follows a tradition that was likely introduced by Empedocles and widely accepted until the birth of modern chemistry in the 18th century.Whatever the benevolent demiurge (axiom 5) can do, seeking to introduce order (axiom 4), is to use a “perfect paradigm” as his model and try to achieve that the result of his efforts is the best possible copy (eikon) of that model (axiom 6).From the postulated axioms so far, it follows that the demiurge’s effort consists of transforming the chaotic spatial-material substrate into a construction ordered according to symmetry, which is an operation that in itself will enable the designation of sensory and of the world. In order to achieve this, the demiurge takes as a model what Plato calls the “perfect paradigm”, the intelligible reality governed by the Good. Thus, the modeled universe represents the following qualities in their highest sense: beauty, symmetry, order, harmony, simplicity, etc. 11. As a copy of the perfect paradigm, the sensory world created by the demiurge can be nothing other than a living being whose body is composed of four elements (ref. axiom 9), and whose soul (psukhe) is endowed with reason (nous) (ref. axiom 12). The most prominent feature of a living being is the autonomous principle of proper change and movement (kinesis). Plato called this autonomous source of order the “soul” (psukhe) and thus gave a new meaning to an ancient word. And it is precisely this soul endowed with reason (nous) that directly explains the proper, orderly, and permanent, so to speak, “rational” movement of celestial bodies… the world soul places itself in the role of “mistress and ruler” (Timaeus 34c). 12. As the transmitter of reason, the world soul is mathematics. can be understood only if we consider the symmetries in Plato’s cosmology. The word “summetros,” derived from the Greek word “metron,” meaning a unit of measure, means “with a common measure.” If things can be measured using a standard measurement, we call them “measurable by the same standard,” otherwise they are not. Proportions appear in standard measurement. Therefore, it can be said that things are in proper proportion (analogoi); and if the proportion is always the same, then these things are also symmetrical. By accepting such ideas, “symmetrical” can become synonymous with “harmonious,” and more importantly, with “ordered” (kosmetos).
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The mathematical forms in which we express elementary particles today are much more complex than the geometric shapes established by the Greeks. In both cases, these forms arise from certain simple mathematical foundations… Ultimately, in both cases, the concept of matter is essentially a mathematical concept. The fundamental essence of everything material, both for us and for the ancient Greeks, lies in the mathematical structure. And the same goes for Plato, it is a mathematical form, not just some material content. Heisenberg, of course, saw an extraordinary similarity between Plato’s theory of matter and its contemporary counterpart, at a fundamental axiomatic level. The theory of matter is indeed based on the following axioms:
13. The entire universe can be reduced to separate, elementary components.
14. These elementary components are infinitely small and therefore “invisible.”
15. Essentially, these components are mathematical entities.
16. These separate components are numerically small, simple, imperceptible, and indestructible.
17. These entities are the fundamental components of all sensory objects; everything in the universe is derived from them.
18. In the universe, all visible phenomena, which Plato calls “change,” can be reduced to interactions between elementary components.
19. These interactions can only be described mathematically.
20. There is no complexity factor associated with a specific scale of magnitude. For every level, from the scale of microscopic realities, invisibility From the smallest particles visible to the naked eye to the gigantic astronomical objects, the explanation of visible phenomena can only be found in the elementary components that follow the same mathematical laws. Just like in the case of many fundamental ideas presented by Plato in Timaeus, this theory of matter was neither accepted nor understood for centuries. These ideas were rediscovered only in the 19th century and modern physics has developed models of matter based on them. Plato’s assumptions reduce all objects of sensory perception to “phenomena” that result from the combination of elementary particles, which are few and simple, and the mathematical rules that govern their correlations… Timaeus relates the geometrical properties, such as those of regular polyhedra corresponding to the four elements, to the physical characteristics that these elements represent when perceived by the senses. The universe is not uniform. The visible motion in the universe, its constant change, is a result of the unequal size of the elementary polyhedra… the motion will not It can never exist in a state of homogeneity or uniformity (Timaeus 57e). The sizes of the elementary polyhedra that create all sensory objects must also be different. The absence of uniformity creates a cause of change that always acts on the sensory world, a change that the world soul tries to put in order, but only where it rules as its mistress and ruler. In the sensory world, there is no vacuum (Timaeus 58a, ref. 79c). The sphere of the world encompasses everything that is physical. Within this sphere, the four elements extend in four concentric layers (Timaeus 33b, 53a, 48a-b). These four concentric layers are pulled by circular motion that moves the entire sphere. Since there is no vacuum (axiom 22), the elementary polyhedra cannot extend beyond the sphere. Within it, they can only move in perpetually filled intermediary spaces that arise due to the absence of polyhedra homogeneity (axiom 21). Hence the chain reaction that results from the condensing process, it pushes small bodies into the intermediary spaces between large ones. h (Timej 58b, ref. 76c and Laws X 849c). Modern cosmologyWhat makes modern cosmology a scientific inquiry is the mathematical description of the universe, which became possible only after the achievements of Kepler and Galileo during the renaissance period, a period that coincides with the rediscovery and translations of all Plato’s manuscripts. Modern cosmology can be defined as the pursuit of a mathematical formulation that encompasses natural phenomena as a whole. However, cosmology is not a science of everything that exists, but a science that collects and organizes natural phenomena into a unity called “the universe”.We, the people of today, have become entangled, due to the strange dominance of modern science, in a bizarre misunderstanding that science represents the source of knowledge and that reasoning must submit to scientific judgment. But whatever a thinker can express, it can never be logically or empirically confirmed or refuted. And it is not a matter of belief either. It can only be approached through questioning and contemplation. And what The divine is observed in such a way, it is always worth exploring… The text by M. Heidegger, here in a free translation, argues that “true” knowledge (episteme) can never arise from the natural sciences as they are understood today, and that the only remaining option is the relentless reformulation of questions, the continuous repetition of questions worth asking. Scientific research is also based on a system of assumptions, temporarily accepted as the foundation of a theoretical model of the physical world, or on axioms. Axiomatic systems on which the cosmology of Plato’s Timaeus and the standard Big Bang model are based sufficiently illustrate this important question. It follows that the analysis of knowledge revealed by the scientific method must focus on the system of axioms permitted in each particular case. But the questioning of these axioms is equivalent to adding additional axioms to the system. And therefore, the initial question can be reformulated. Instead of asking “What knowledge does science convey?” we now ask: What knowledge is transmitted by science?
The cosmological model that Plato provided is a “scientific model” in today’s strict sense of the word. From this, it follows that this construction, this theoretical model of the universe, must be accepted as a formal axiomatic system: a collection of primordial propositions, axioms, first of all, must be established, and all observable manifestations of the universe must then in principle be derivable as theorems from these axioms.
Of course, it is possible that Plato was not fully aware of the fact that he was shaping a model based on a list of axioms and that these axioms have no other justification than their epistemological consistency. But he accepted this idea: whoever discovers a different set of axioms, whoever can improve another model and If he tries to explain it differently, he has an equal right to be heard; and only the model that best suits the data, the “most acceptable” model, will be chosen (Timaeus 54a-b, Timaeus 55d).
Cosmological axioms
1. Reality is divided into two domains: intelligible forms (eidos, idea), pure, eternal, unchanging, and simple; and complex objects of sensory perception, always changing (kinetos) in time.
This division of reality into two domains corresponds to the difference between existence and becoming, between what remains the same forever and what never ceases to change, between what can be attributed the predicate “true” and what cannot.
Only for the knowledge of intelligible forms can we say that it is “true”. However, finite, transient, and limited human beings cannot attain such knowledge in this world. Knowledge of these divine entities is the exclusive privilege of the gods and a small number of their friends. (Fedar 278d). Knowledge directed towards the world of objects of sensory perception, which Plato classifies as “opinion” (doxa), is a lower form of knowledge that at best only achieves similarity to the truth.
2. Good occupies a special position among forms.
Good is an intelligible form among others: Justice, Unity, Man, Animal, etc., but it plays a crucial role within the platonic system, especially in Timaeus. This form transfers these particular characteristics to other intelligible forms: beauty, harmony, order, simplicity. According to Plato’s proposed cosmological model, the other intelligible forms then transfer these characteristics to sensory objects.
3. In the sensory world, everything that comes into being is the result of a cause.
Causality is only applicable to the sensory world since in the reality of the intelligible forms, which are eternal and unchangeable according to axiom 1, there is no change, and therefore there are no causal relationships in the intelligible world.
In the sensory world In the world, changes imply relationships among the elemental components (eternal and unchangeable), but all changes are subject to causality and can, to the extent possible, be mathematically expressed. The word “cause” (aitia) is borrowed from legal terminology, where it signifies responsibility, but here it denotes this chain of relationships.
4. The sensory world is the result of divine imposition of order.
One of the causes of the sensory world is the god, also known as “father,” “creator,” and “demiurge,” the most commonly used term. This god does not create the world; his actions are limited to partially organizing the primordial chaotic substrate (ref. axiom 7).
5. The demiurge is good (agathos).
The goodness of the demiurge dictates a certain way of acting (ref. axiom 10).
6. The demiurge is not all-powerful.
This god is not all-powerful, for two reasons stated in the form of axioms. The Intelligible Forms and khora, the primordial substance or “space medium,” exist independently of him (axioms 1 and 7). And the demiurge must to je uzrok koji se stalno opire demijurgovom naporu uvođenja reda. Ananke se može doživjeti kao sila koja postoji prije demijurga i određuje uvjete pod kojima se sve stvari formiraju i mijenjaju. Ova sila, iako je protiv demijurgove svrhe, također ima svoju važnu ulogu u stvaranju i održavanju svijeta. negative, marked as a “wandering cause” 4, as it represents a non-rational element that permanently resists the demiurge’s efforts to establish order.
Before the demiurge’s interventions, the action of ananke in khora is such that the four elements, believed to create the entire sensory world, “behave without order and measure (alogos kai ametros)” (Timaeus 53a). Ananke continues to manifest in the sensory world as a “wandering cause” after the demiurge withdraws from the world. As a result, there is always a factor of complexity and disorder in the universe.
9. Objects of sensory perception, including celestial bodies, are composed of only four elements: fire, air, water, and earth.
Here, Plato follows a tradition likely introduced by Empedocles and widely accepted until the emergence of modern chemistry in the 18th century.
10. All that a benevolent demiurge (axiom 5) can do in attempting to introduce order into khora (axiom 4) is to use a “perfect paradigm” as its The goal is to aim for the best possible replica (eikon) of that model (axiom 6).
From the previously postulated axioms, it follows that the effort of the demiurge consists in transforming the chaotic spatial-matter substrate into a construction organized according to symmetry, which is an operation that will enable the naming of things in the sensory world itself. In order to achieve this, the demiurge takes as a model what Plato calls the “perfect paradigm,” an intelligible reality governed by the Good. Thus, the modeled universe represents the following qualities in their highest sense: beauty, symmetry, order, harmony, simplicity, etc.
11. As a replica of the perfect paradigm, the sensory world created by the demiurge can be nothing else but a living being whose body is composed of four elements (ref. axiom 9), and whose soul (psukhe) is endowed with reason (nous) (ref. axiom 12).
The most distinctive characteristic of a living being is the autonomous principle of regular change, and movement (kinesis). Plato referred to this autonomous source of order as the “soul” (psukhe), giving a new meaning to an ancient word. And it is precisely this soul endowed with reason (nous) that directly explains the correct, organized, and enduring, so to speak, “rational” movement of celestial bodies… the world soul puts itself in the role of “mistress and ruler” (Timaeus 34c).
12. As a carrier of reason, the world soul is mathematically structured.
We can only understand the scope of axiom 12 if we consider the role of symmetry in Plato’s cosmology. The word summetros, derived in ancient Greece from the word metron, a unit of measurement, means “with a usual measure.” If things can be measured using standard measurement, we call them “measurable by the same standard”; otherwise, they are not. Proportions emerge in standard measurement. Therefore, it can be said that things are in the right proportion (analogoi); and if the proportion is always the same, then those things are also symmetrical. By accepting such ideas, “symmetrical” can become synonymous with “harmonious”. “harmoniously”, and what is even more important, for “orderly” (cosmetos).
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Mathematical forms with which we express elementary particles today are much more complex than the geometric shapes established by the Greeks. In both cases, these forms stem from certain simple mathematical foundations… In the final analysis, in both cases, the concept of matter is essentially a mathematical concept. The fundamental essence of all material, both for us and for Plato, is a mathematical form, and not just some material content.
Heisenberg, of course, saw an extraordinary similarity between Plato’s theory of matter and its modern counterpart, on a fundamental axiomatic level. The theory of matter is indeed based on the following axioms in Timaeus.
13. The entire universe can be reduced to separate, elementary components.
14. Those elementary components are infinitely small and therefore “invisible.”
15. Essentially, these components are mathematical entities.
16. The number of these separate components is 16. They are small, simple, inconspicuous, and indestructible.
17. These entities are the basic components of all sensory objects; everything in the universe has originated from them.
18. In the universe, all visible phenomena, which Plato calls “change,” can be reduced to interactions between elementary components.
19. These interactions can only be described mathematically.
20. There is no complexity factor associated with a specific scale of sizes. For every level, from the microscopic scale of realities invisible to the naked eye to giant astronomical objects, the explanation of visible phenomena can only be found in elementary components that follow the same mathematical laws.
Just like with many fundamental ideas presented in Plato’s Timaeus, this theory of matter was neither accepted nor understood for centuries. These ideas were only rediscovered in the 19th century, and modern physics has led them to a model of matter. Plato’s assumptions reduce all sensory objects to their constituent elements. Observations on “phenomena” that result from the combination of elementary particles, few and simple, and mathematical rules that govern their correlations… Time relates the geometric properties, such as those of regular polyhedra corresponding to four elements, to the physical characteristics that these elements represent when perceived by the senses.
21. The universe is not uniform.
The visible motion in the universe, its constant change, is a consequence of the unequal size of the elementary polyhedra… motion will never exist in a state of homogeneity or uniformity (Timaeus 57e). The sizes of the elementary polyhedra that create all sensory objects must also be different. The absence of uniformity creates the cause of change that always affects the sensory world, a change that the world soul tries to bring to order, but only where it governs as its mistress and ruler.
22. There is no vacuum in the sensory world (Timaeus 58a, ref. 79c).
23. The sphere of the world. It encompasses everything that is physical. Within this sphere, four elements are spread across four concentric layers (Timaeus 33b, 53a, 48a-b).
These four concentric layers are driven by circular motion, which powers the entire sphere. Since there is no vacuum (axiom 22), elemental polyhedra cannot extend beyond the sphere. Inside it, they can only move in perpetually filled intermediary spaces that arise due to the lack of homogeneity of the polyhedra (axiom 21). Hence, the chain reaction resulting from the condensing process pushes smaller bodies into the gaps between larger ones (Timaeus 58b, ref. 76c and Laws X 849c).
Modern cosmology focuses on the scientific exploration of the universe, which became possible only after the achievements of Kepler and Galileo during the Renaissance, a period that coincided with the rediscovery and translations of all of Plato’s manuscripts. Modern cosmology can be defined as the search for a mathematical formulation that describes natural phenomena. taking into account all phenomena. However, cosmology is not a science of everything that exists, but a science that collects and organizes natural phenomena into a whole, into a unity called “universe.”
We, the people of today, have become entangled, due to the strange dominance of modern science, in a bizarre misconception that science represents the source of knowledge and that thinking must submit to the scientific judgment. But whatever a thinker can express, it can never be logically or empirically confirmed or refuted. And it is not a question of belief either. It can only be approached through questioning and contemplation. And whatever is observed in this way is always worth examining… This text by M. Heidegger, here in a free translation, argues that “true” knowledge (episteme) can never arise from natural sciences as they are understood today, and that the only remaining available option is an unrelenting reformulation of questions, a continuous repetition of questions that are worth asking.
Scientific research is also based on engagement The basic assumptions, temporarily accepted as the foundation of the theoretical model of the physical world, namely the axioms. Axiomatic systems on which Plato’s Timaeus cosmology and the standard Big Bang model are based effectively illustrate this important question. Consequently, the analysis of knowledge discovered through the scientific method must focus on the set of axioms allowed in each individual case. However, questioning these axioms is equivalent to adding additional axioms to the system. Therefore, the initial question can be reformulated. Instead of asking “What knowledge does science convey?”, we now ask: “What knowledge do axioms convey that scientists consider to be true and valid?” A complex procedure allows sensory data obtained through experiments to be replaced with measurements.
(Inventing the universe, Luc Brisson and F. Walter Meryerstein)
1 Axiom – a statement accepted as true without proof; an initial statement regarded as unquestionable, from which theorems are derived.
2 Intelligible – capable of being understood. Smart only with reason, not with feeling; intellectual, comprehensible.
3 Causality – the relationship between phenomena in which one (cause) is a necessary condition for the occurrence of another (effect); causality.
4 Causality that is not subordinate to the ultimate “purpose” – Good, but rather a series of unpredictable secondary causes, all that is the result of the action of ananke.