Conceptual overview of the understanding of the workings of nature...
Illustrations to the following text can be found in the techWiki: Slides: From Fundamental to Complex
Fundamental Level
Ideas
What is science?
- Science is the quest to capture the processes of nature in formal mathematical representations
So "math is the blueprint of reality" in the sense that formal systems are the foundation of science.
In a nutshell:
- Natural systems are a subset of reality, i.e., the observable universe
- Guided by thought, observation and measurement natural systems are "encoded" into formal systems
- Using logic (rules of inference) in the formal system, predictions about the natural system can be made (decoding)
- Checking the predictions with the experimental outcome gives the validity of the formal system as a model for the natural system
Physics can be viewed as dealing with the fundamental interactions of inanimate matter.
For a technical overview, go to the techWiki section. See also techWiki: Slide 1.
Paradigm
- Mathematical models of reality are independent of their formal representation
This leads to the notions of symmetry and invariance. Basically, this requirement gives rise to nearly all of physics.
Classical Mechanics
Symmetry, understood as the invariance of the equations under temporal and spacial transformations, gives rise to the conservation laws of energy, momentum and angular momentum.
In layman terms this means that the outcome of an experiment is unchanged by the time and location of the experiment and the motion of the experimental apparatus. Just common sense...
Mathematics of Symmetry
The intuitive notion of symmetry has been rigorously defined in the mathematical terms of group theory.
Physics of Non-Gravitational Forces
The three non-gravitational forces are described in terms of quantum field theories. These in turn can be expressed as gauge theories, where the parameters of the gauge transformations are local, i.e., differ from point to point in space-time.
The Standard Model of elementary particle physics unites the quantum field theories describing the fundamental interactions of particles in terms of their (gauge) symmetries.
Physics of Gravity
Gravity is the only force that can't be expressed as a quantum field theory.
Its symmetry principle is called covariance, meaning that in the geometric language of the theory describing gravity (general relativity) the physical content of the equations is unchanged by the choice of the coordinate system used to represent the geometrical entities.
To illustrate, imagine an arrow located in space. It has a length and an orientation. In geometric terms this is a vector, lets call it a. If I want to compute the length of this arrow, I need to choose a coordinate system, which gives me the x-, y- and z-axes components of the vector, e.g., a = (3, 5, 1). So starting from the origin of my coordinate system (0, 0, 0), if I move 3 units in the x direction (left-right), 5 units in the y-direction (forwards-backwards) and 1 unit in the z direction (up-down), I reach the end of my arrow. The problem is now, that depending on the choice of coordinate system - meaning the orientation and the size of the units - the same arrow can look very different: a = (3, 5, 1) = (0, 23.34, -17). However, everytime I compute the length of the arrow in meters, I get the same number independent of the chosen representation.
In general relativity the vectors are somewhat like multidimensional equivalents called tensors and the commonsense requirement, that the calculations involving tensor do not depend on how I represent the tensors in space-time, is covariance.
It is quite amazing, but there is only one more ingredient needed in order to construct one of the most estethic and accurate theories in physics. It is called the equivalence principle and states that the gravitational force is equivalent to the forces experienced during acceleration. This may sound trivial, has however very deep implications.
See techWiki: Slide 2.
Physics of Condensed Matter
This branch of physics, also called solid-state physics, deals with the macroscopic physical properties of matter. It is one of physics first ventures into many-body problems in quantum theory. Although the employed notions of symmetry do not act at such a fundamental level as in the above mentioned theories, they are a cornerstone of the theory. Namely the complexity of the problems can be reduced using symmetry in order for analytical solutions to be found. Technically, the symmetry groups are boundary conditions of the Schrödinger equation. This leads to the theoretical framework describing, for example, semiconductors and quasi-crystals (interestingly, they have fractal properties!). In the superconducting phase, the wave function becomes symmetric.
Conclusion
The Success
It is somewhat of a miracle, that the formal systems the human brain discovers/devises find their match in the workings of nature. In fact, there is no reason for this to be the case, other than that it is the way things are.
The following two examples should underline the power of this fact, where new features of reality where discovered solely on the requirements of the mathematical model:
- In order to unify electromagnetism with the weak force (two of the three non-gravitational forces), the theory postulated two new elementary particles: the W and Z bosons. Needless to say, these particles where hitherto unknown and it took 10 years for technology to advance sufficiently in order to allow their discovery.
- The fusion of quantum mechanics and special relativity lead to the Dirac equation which demands the existence of an, up to then, unknown flavor of matter: antimatter. Four years after the formulation of the theory, antimatter was experimentally discovered.
The Future...
Albeit the success, modern physics is still far from being a unified, paradox-free formalism describing all of the observable universe. Perhaps the biggest obstacles lies in the last missing step to unification. In a series of successes, forces appearing as being independent phenomena, turned out to be facets of the same formalism: electricity and magnetism was united in the four Maxwell equations; as mentioned above, electromagnetism and the weak force were merged into the electroweak force; and finally, the electroweak and strong force were united in the framework of the standard model of particle physics. These four forces are all expressed as quantum (field) theories. There is only one observable force left: gravity.
The efforts to quantize gravity and devise a unified theory, have taken a strange turn in the last 20 years. The problem is still unsolved, however, the mathematical formalisms engineered for this quest - namely string/M-theory and loop quantum gravity - have had a twofold impact:
- A new level in the application of formal systems is reached. Whereas before, physics relied on mathematical branches that where developed independently from any physical application (e.g., differential geometry, group theory), string/M-theory is actually spawning new fields of mathematics (namely in topology).
- These theories tell us very strange things about reality:
- Time does not exist on a fundamental level
- Space and time per se become quantized
- Space has more than three dimensions
- Another breed of fundamental particles is needed: supersymmetric matter
Unfortunately no one knowns if these theories are hinting at a greater reality behind the observable world, or if they are "just" math. The main problem being the fact that any kind of experiment to verify the claims appears to be out of reach of our technology...
Real Complex Systems
Outline
While physics has had an amazing success in describing most of the observable universe in the last 300 years, the formalism appears to be restricted to the fundamental workings of nature. Only solid-state physics attempts to deal with collective systems. And only thanks to the magic of symmetry one is able to deduce fundamental analytical solutions.
In order to approach real life complex phenomena, one needs to adopt a more systems oriented focus. This also means that the interactions of entities becomes an integral part of the formalism.
Some ideas should illustrate the situation:
- Most calculations in physics are idealizations and neglect dissipative effects like friction
- Most calculations in physics deal with linear effect, as non-linearity is hard to tackle and is associated with chaos; however, most physical systems in nature are inherently non-linear
- The analytical solution of three gravitating bodies in classical mechanics, given their initial positions, masses, and velocities, cannot be found; it turns out to be a chaotic system which can only be simulated in a computer; however, there are an estimated hundred billion of galaxies in the universe
Systems Thinking
Systems theory is an interdisciplinary field which studies relationships of systems as a whole. The goal is to explain complex systems which consist of a large number of mutually interacting and interwoven parts in terms of those interactions.
A timeline:
- Cybernetics (50s): Study of communication and control, typically involving regulatory feedback, in living organisms and machines
- Catastrophe theory (70s): Phenomena characterized by sudden shifts in behavior arising from small changes in circumstances
- Chaos theory (80s): Describes the behavior of non-linear dynamical systems that under certain conditions exhibit a phenomenon known as chaos (sensitivity to initial conditions, regimes of chaotic and deterministic behavior, fractals, self-similarity)
- Complex adaptive systems (90s): The "new" science of complexity which describes emergence, adaptation and self-organization; employing tools such as agent-based computer simulations
In systems theory one can distinguish between three major hierarchies:
- Suborganic: Fundamental reality, space and time, matter, ...
- Organic: Life, evolution, ...
- Metaorganic: Consciousness, group dynamical behavior, financial markets, ...
However, it is not understood how one can traverse the following chain: bosons and fermions -> atoms -> molecules -> DNA -> cells -> organisms -> brains. I.e., how to understand phenomena like consciousness and life within the context of inanimate matter and fundamental theories.
Illustrations
See techWiki: Slide 3.
Category Theory
The mathematical theory called category theory is a result of the "unification of mathematics" in the 40s. A category is the most basic structure in mathematics and is a set of objects and a set of morphisms (maps). A functor is a structure-preserving map between categories (see Slide 3).
This dynamical systems picture can be linked to the notion of formal systems mentioned above: physical observables are functors, independent of a chosen representation or reference frame, i.e., invariant and covariant.
Object-Oriented Programming
This paradigm of programming can be viewed in a systems framework, where the objects are implementations of classes (collections of properties and functions) interacting via functions (public methods). A programming problem is analyzed in terms of objects and the nature of communication between them. When a program is executed, objects interact with each other by sending messages. The whole system obeys certain rules (encapsulation, inheritance, polymorphism, ...).
Some advantages of this integral approach to software development:
- Easier to tackle complex problems
- Allows natural evolution towards complexity and better modeling of the real world
- Reusability of concepts (design patterns) and easy modifications and maintenance of existing code
- Object-oriented design has more in common with natural languages than other (i.e., procedural) approaches
Algorithmic vs. Analytical
Perhaps the shift of focus in this new weltbild can be understood best when one considers the paradigm of complex system theory:
- The interaction of entities (agents) in a system according to simple rules gives rise to complex behavior: Emergence, structure-formation, self-organization, adaptive behavior (learning), ...
This allows a departure from the equation-based description to models of dynamical processes simulated in computers. This is perhaps the second miracle involving the human mind and the understanding of nature. Not only does nature work on a fundamental level akin to formal systems devised by our brains, the hallmark of complexity appears to be coded in simplicity ("simple sets of rules give complexity") allowing computational machines to emulate its behavior. See Slide 4.
It is very interesting to note, that in this paradigm the focus is on the interaction, i.e., the complexity of the agent can be ignored. That is why the formalism works for chemicals in a reaction, ants in an anthill, humans in social or economical organizations, ... In addition, one should also note, that simple rules - the epitome of deterministic behavior - can also give rise to chaotic behavior.
The emerging field of network theory (an extension of graph theory, yielding results such as scale-free topologies, small-worlds phenomena, etc. observed in a stunning veriety of complex networks) is also located at this end of the spectrum of the formal descriptions of the workings of nature.
Finally, to revisit the analytical approach to reality, note that in the loop quantum gravity approach, space-time is perceived as a causal network arising from graph updating rules (spin networks, which are graphs associated with group theoretic properties), where particles are envisaged as 'topological defects' and geometric properties of reality, such as dimensionality, are defined solely in terms of the network's connectivity pattern.
Go to the techWiki for a list of open questions in complexity theory.