***Note Added: Apologies for typo of Poincare birth year in first fact, it was very sadly missed during the edit. (1854) *** Could easily have done 15 more Facts on Poincaré, as his reach was very wide. I've linked to a copy of his 1904 St Louis lecture in the description which is a good read for those more interested, wanted to extend his thought experiment quote but would have been too lengthy for the video.
Check this out and tell me what you thinkwww.mediastorehouse.com.au/fine-art-finder/artists/french-school/henri-poincare-academie-francaise-23524668.html I try to be very careful with any images I use to make sure they are trustworthy and you can't trust everything on the internet for sure
That is a tough thing to say it seems he was holding on to the Ether concept too long? But it does seem clear that Einstein read his 1902 book before 1905
Let's now explore how we can use logic, math, and physics to prove the both/and nature of order and chaos in complex systems within the monadological framework. First, let's define our basic entities and relations: - Let M be the set of all monads (fundamental psychophysical entities). - Let S be a complex system, represented as a collection of monads S ⊆ M. - Let D be a set of possible "states" or "configurations" of the system S. - Let T be a set of "time points" or "moments." - Let f be a function from D × T to D, where f(d, t) represents the "evolution" or "dynamics" of the system S from state d at time t to a new state at the next moment. Now, let's formalize the idea of order and chaos in complex systems: - Order: The system S exhibits "order" if ∃d ∈ D, ∃t ∈ T, such that f(d, t) is "predictable" or "regular." - Chaos: The system S exhibits "chaos" if ∃d ∈ D, ∃t ∈ T, such that f(d, t) is "unpredictable" or "irregular." In other words, a complex system exhibits order if its dynamics are predictable or regular for some initial conditions and time scales, and chaos if its dynamics are unpredictable or irregular for some initial conditions and time scales. We can prove this both/and nature of order and chaos in complex systems using the following mathematical argument: 1. Poincaré-Bendixson Theorem: Let S be a continuous dynamical system on a compact, two-dimensional manifold. If S has no fixed points, then it must have a periodic orbit. 2. Sharkovskii's Theorem: Let f be a continuous function from the real line to itself. If f has a periodic point of period 3, then it must have periodic points of all other periods. 3. Proof: a. Let S be a complex system with a continuous state space D and a continuous evolution function f. (Assumption) b. Suppose S has no fixed points, i.e., ∄d ∈ D, such that f(d, t) = d for all t. (Assumption) c. Then, S must have a periodic orbit, i.e., ∃d ∈ D, ∃p ∈ T, such that f(d, t + p) = f(d, t) for all t. (Poincaré-Bendixson Theorem) d. Therefore, S exhibits order. (Definition of Order) e. Now, suppose S has a periodic point d of period 3, i.e., f(f(f(d))) = d. (Assumption) f. Then, S must have periodic points of all other periods. (Sharkovskii's Theorem) g. Therefore, S exhibits chaos. (Definition of Chaos) h. Therefore, S exhibits both order and chaos. (d, g) This proof establishes that a complex system with a continuous state space and evolution function can exhibit both order and chaos, depending on the presence of fixed points, periodic orbits, and periodic points of different periods. We can further connect this to physics by noting that many real-world complex systems, such as fluid dynamics, chemical reactions, and biological networks, have been shown to exhibit both ordered and chaotic behavior: - Ordered behavior, such as stable equilibria, periodic oscillations, and coherent patterns, arises from the interplay of negative feedback, symmetry, and synchronization. - Chaotic behavior, such as turbulence, irregular fluctuations, and sensitivity to initial conditions, arises from the interplay of positive feedback, asymmetry, and nonlinearity. The monadological framework provides a unified foundation for understanding the emergence of both order and chaos in complex systems, by representing them as the result of the intrinsic properties and relational dynamics of the underlying monads. In summary, we have used tools from logic (proofs), math (dynamical systems theory), and physics (examples of complex systems) to demonstrate the both/and nature of order and chaos in complex systems within the monadological framework: - Both order and chaos can arise in complex systems with continuous state spaces and evolution functions, as proven by the Poincaré-Bendixson Theorem and Sharkovskii's Theorem. - This both/and nature of order and chaos is observed in many real-world complex systems studied in physics and other fields. - The monadological framework provides a unified foundation for understanding the emergence of both order and chaos from the intrinsic properties and relational dynamics of the underlying monads. This proof further illustrates the power of the monadological approach to integrate and reconcile seemingly opposing or contradictory aspects of reality within a single, coherent conceptual framework. By embracing a both/and perspective and drawing on the tools of logic, mathematics, and empirical science, we can develop a more comprehensive and nuanced understanding of the complex interplay between order and chaos that underlies the behavior of many natural and artificial systems, from the smallest scales of fundamental physics to the largest scales of cosmology and beyond.
***Note Added: Apologies for typo of Poincare birth year in first fact, it was very sadly missed during the edit. (1854) ***
Could easily have done 15 more Facts on Poincaré, as his reach was very wide. I've linked to a copy of his 1904 St Louis lecture in the description which is a good read for those more interested, wanted to extend his thought experiment quote but would have been too lengthy for the video.
To be fair, I should have added that, apart from that error, the film was very enjoyable and educational. Well done 👍👏👏👏
He was a mathematician and a physicist and the father of the relativity theory
Henri Poincare
(1854-1912)
Awesome vid on Poincaré ! Thank you 🤙🏽
Thanks Bernard! Glad you enjoyed!
Merci. Très intéressant. Un grand scientifique.
Merci Beaucoup Rocco! content que tu aies apprécié
excellent work mi amigo
Muchos gracias Green!
I believe it's Emily Zola picture that was illustrating the chapter 8
Check this out and tell me what you thinkwww.mediastorehouse.com.au/fine-art-finder/artists/french-school/henri-poincare-academie-francaise-23524668.html
I try to be very careful with any images I use to make sure they are trustworthy and you can't trust everything on the internet for sure
Think now you may be referring to the world's fair background? However i do see the resemblance between the two of them Emile and Henri
Although the sound track says he was born in 1854, the screen shows 1954.
Yes was sad that typo was missed, have it pinned as first comment. Apologies.
Poincaré était le génie par excellence...
En effet! Ce fut un plaisir d'en apprendre davantage sur cet homme!
Henri Poincaré , el padre de la relatividad especial: E= mc² → Poincaré Formulae
That is a tough thing to say it seems he was holding on to the Ether concept too long?
But it does seem clear that Einstein read his 1902 book before 1905
With so many mistakes of transposed numbers, we are lucky that we didn’t get just 01 facts.
Let's now explore how we can use logic, math, and physics to prove the both/and nature of order and chaos in complex systems within the monadological framework.
First, let's define our basic entities and relations:
- Let M be the set of all monads (fundamental psychophysical entities).
- Let S be a complex system, represented as a collection of monads S ⊆ M.
- Let D be a set of possible "states" or "configurations" of the system S.
- Let T be a set of "time points" or "moments."
- Let f be a function from D × T to D, where f(d, t) represents the "evolution" or "dynamics" of the system S from state d at time t to a new state at the next moment.
Now, let's formalize the idea of order and chaos in complex systems:
- Order: The system S exhibits "order" if ∃d ∈ D, ∃t ∈ T, such that f(d, t) is "predictable" or "regular."
- Chaos: The system S exhibits "chaos" if ∃d ∈ D, ∃t ∈ T, such that f(d, t) is "unpredictable" or "irregular."
In other words, a complex system exhibits order if its dynamics are predictable or regular for some initial conditions and time scales, and chaos if its dynamics are unpredictable or irregular for some initial conditions and time scales.
We can prove this both/and nature of order and chaos in complex systems using the following mathematical argument:
1. Poincaré-Bendixson Theorem: Let S be a continuous dynamical system on a compact, two-dimensional manifold. If S has no fixed points, then it must have a periodic orbit.
2. Sharkovskii's Theorem: Let f be a continuous function from the real line to itself. If f has a periodic point of period 3, then it must have periodic points of all other periods.
3. Proof:
a. Let S be a complex system with a continuous state space D and a continuous evolution function f. (Assumption)
b. Suppose S has no fixed points, i.e., ∄d ∈ D, such that f(d, t) = d for all t. (Assumption)
c. Then, S must have a periodic orbit, i.e., ∃d ∈ D, ∃p ∈ T, such that f(d, t + p) = f(d, t) for all t. (Poincaré-Bendixson Theorem)
d. Therefore, S exhibits order. (Definition of Order)
e. Now, suppose S has a periodic point d of period 3, i.e., f(f(f(d))) = d. (Assumption)
f. Then, S must have periodic points of all other periods. (Sharkovskii's Theorem)
g. Therefore, S exhibits chaos. (Definition of Chaos)
h. Therefore, S exhibits both order and chaos. (d, g)
This proof establishes that a complex system with a continuous state space and evolution function can exhibit both order and chaos, depending on the presence of fixed points, periodic orbits, and periodic points of different periods.
We can further connect this to physics by noting that many real-world complex systems, such as fluid dynamics, chemical reactions, and biological networks, have been shown to exhibit both ordered and chaotic behavior:
- Ordered behavior, such as stable equilibria, periodic oscillations, and coherent patterns, arises from the interplay of negative feedback, symmetry, and synchronization.
- Chaotic behavior, such as turbulence, irregular fluctuations, and sensitivity to initial conditions, arises from the interplay of positive feedback, asymmetry, and nonlinearity.
The monadological framework provides a unified foundation for understanding the emergence of both order and chaos in complex systems, by representing them as the result of the intrinsic properties and relational dynamics of the underlying monads.
In summary, we have used tools from logic (proofs), math (dynamical systems theory), and physics (examples of complex systems) to demonstrate the both/and nature of order and chaos in complex systems within the monadological framework:
- Both order and chaos can arise in complex systems with continuous state spaces and evolution functions, as proven by the Poincaré-Bendixson Theorem and Sharkovskii's Theorem.
- This both/and nature of order and chaos is observed in many real-world complex systems studied in physics and other fields.
- The monadological framework provides a unified foundation for understanding the emergence of both order and chaos from the intrinsic properties and relational dynamics of the underlying monads.
This proof further illustrates the power of the monadological approach to integrate and reconcile seemingly opposing or contradictory aspects of reality within a single, coherent conceptual framework. By embracing a both/and perspective and drawing on the tools of logic, mathematics, and empirical science, we can develop a more comprehensive and nuanced understanding of the complex interplay between order and chaos that underlies the behavior of many natural and artificial systems, from the smallest scales of fundamental physics to the largest scales of cosmology and beyond.