Stephen Wolfram, Author of Mathematica and 'A New Kind of Science' was invited to speak by my employer today. Here are my running notes of his talk:
How do structures form?
1st assumption - easy to do; didn't work.
Approach fundamentally wrong - mathematical analysis worked for simpler phenomena, what can you do differently?
Previously based on existing mathematics; wanted a more general construct, which led him to programs.
Consider space of all possible programs, and consider what kinds of programs are found in nature. Mathematica is a new tool like telescope or microscope - reveals the computational world.
256 1-d Cellular Automata is a simple program space to explore. (If you're running OS X, try my program that draws them)
Patterns in nature look more complex than man-made things. The simple rules can give rise to natural complexity - mollusc shells have very similar patterns.
Without computational tools it is hard to see these patterns; people looked for regularity not irregularities.
Principle of Computational Equivalence - complex systems are of equivalent computation. Not linear - phase transitions between kinds of sophistication.
Computational limit is achieved often.
Is rule 30 computationally universal? No, but 110 is. You can make a Turing machine from rule 100 - it is computationally universal.
If the observer is computationally more complex than the system, they can see the patterns; in practice the observer is equivalent in complexity, so systems can look complex.
For a system that is linear or repetitive, you can find out an arbitrary future state; for a computationally complex system you can't. Simulation is necessary, not just convenient.
Mathematics has chosen a subset of possible systems that are tractable, rather than complete.
Simple underlying equations of physics - a 14 bn year program. CA's not appropriate. Do not distinguish matter and space, just define space. Think of space as like water - looks continuous, but is made of interacting particles. Underlying space is a network of nodes with connections between them. Look at dimensionality d dimension r^d nodes.
Are space and time different or not? He thinks yes. Global sync like a CA? Not needed. Consider a single active cell that moves change around. Can only follow the causal network, so you only know others have changed when you get updated too.
Can derive special relativity from the low level network model, and move on to General Relativity.
Not yet derived Quantum Mechanics, but can model particle interactions.
Ongoing reduction of 'specialness' - every time we get less special, science gets more general. Simple abstract systems are as computationally complex as we are.
Process of paradigm shifts - see notes in back of book.
Mathematica - algorithm resource base. Mathematica notebooks as publication model. Symbolic language under Mathematica - symbolic structures.
Schr�dinger's equation relating to these systems - do they apply to PDE's? Yes - start with a Gaussian distribution, get the same kinds of complexity. Can one make PDE's from discrete systems? Yes can do it with fluid mechanics.
Can't derive Schr�dinger's equation at the moment. Rigid body mechanics is harder than fluid mechanics.
Bells inequality - atoms can be joined through the 'space' network - only approximately 3-dimensional.
PCE vs awareness & consciousness? Hierarchy - life, intelligence, consciousness.
Life definition gets harder as we get machines and programs. Life as we know it has a common ancestor, so there is a historical definition. PCE says there is not a general notion of intelligence or consciousness - they are connected through our 'computational history' through pre-existing evolution. Self-awareness 'Real-time philosophy is hard'. Neural net approach to brain modelling is static - machinery is continuum dynamics.
Penrose's discrete and continuous equations - discrete space 'spin networks' fit with the network model of Wolfram. Penrose thinks computers are different from humans. Penrose's view of computing is that brains are different - he considers computers do predicate logic, and is wrong. Continuum vs discrete mathematics - had thought that continuum could transcend Turing, but can't prove, but can't find examples.
I asked him about the connection between his categories and the applicability of the Central Limit Theorem, and he pointed out that while it would break down for his complex systems, it will also break down for long-tailed distributions too.