Dynamic systems and mathematics

Lecture notes

Phil/Psych 256
March 25, 1997

Dynamic systems:

	The Watt governor

	Chaotic systems

	Dynamic systems and connectionism

	Relationship to CRUM

Q: What is a Watt governor?

	1. A device for maintaining constant flywheel speed 
	despite fluctuations in steam pressure

	2. The flywheel is equipped with two weighted arms; 
	arm angle governs valve setting via connecting sponel

Q: How does this compare to a "representational" system 
(van Gelder)?

	Watt governor:

		1. Control by coupling arm angle with engine speed
		2. Action/reaction is analog and continuous
		3. Control occurs in real time via embodied system
		4. Control directs system towards stable state

	Representational governor:

		1. Control by measurement and computation
		2. Action/reaction by representation is discrete
		3. Control occurs via communication
		4. Control monitors sensors and effectors

Q: Is the mind a Watt governor?

	1. DSs naturally account for the development of a 
	system over time
		- evolution equation, e.g., 

			[ODE here!]
	 
		- parameters, e.g., ( , )
		- infant locomotion (Clark et al.)

	2. "State of mind" corresponds to location of DS in its 
	state space
		- attractors (point, periodic, strange)
		- olfaction (Skarda & Freeman)

	3. Changes in state correspond to trajectories
		- phase transition
		- chaos
		- infant moods (Wolff)

	4. DSs may be self-organizing
		- meteorology

Example: Infant locomotion (Clark et al.)

	Early theories of human locomotion: maturational vs. 
	perceptual-cognitive.  Both focussed on the acquisition 
	of control programs.  Neither accounts gracefully for 
	development sequence.

	The DS account proceeds with the infant settling into a 
	favourable attractor in the state space of its locomotor 
	system.

	Weighting a toddler's ankles produces a more stable, 
	adult-like walking pattern!

	Convergence on the proper walking pattern emerges from 
	development of state space.

Example: Rabbit olfaction (Skarda & Freeman)

	Representational models depended on (1) "labelled lines," 
	(2) selection of neurons active and (3) intensity of 
	stimulus

	The DS account relies on (1) time between stimulus and 
	response, (2) activity pattern in olfactory bulb and 
	(3) EEG.

	EEG study indicates all neurons participate in discrimination 
	through chaotic phase transitions.  This makes the DS 
	flexible, quick (no search), and robust.

	Attractors place the bulb in a stereotyped pattern of 
	activity corresponding to a smell

	Fresh ideas come from chaotic leaps!

Example: Infant moods (Wolff)

	Nativist/cognitive account of basic emotions as "hard wired" 
	(Darwin); moods emerge during cognitive development.

	Facial expression is sensitive to initial conditions, 
	e.g., object tracking in 3 month-olds.  Cf. greeting and 
	masked faces.

	Facial expression is a set of emergent properties of 
	dynamic sensory/motor system, e.g., crying (Wolff)

	wakeful --> fussy --> crying
	asleep --> pain --> crying

	These point to phase transitions with normal crying as 
	an attractor

Q: What is an emergent property?

	A1. A property of some system S that is not a property 
	of any of its parts

	A2. A property of S that can be predicted given the 
	properties of its parts

	A3. A property of S that cannot be predicted given the 
	properties of its parts

	Chaotic systems have unpredictable properties.  Is mind 
	one of them? Is consciousness?

Q: How are DSs and NNs related?

	A1. Some DSs are also NNs (van Gelder)

	A2. All NNs are DSs (Smolensky)

	A3. All DSs are NNs?

	Note that "all NNs" is a moving target, not GOFC.

Q: How are CRUM and DSs related?

	A1. They are completely distinct (van Gelder)

	DSs handle embodiment and real time naturally

	A2. NNs form an "unstable mixture" of DSs and CRUM 
	(van Gelder)

	A3. CRUM is an emergent property of DSs (?)

	It is not clear what kind of entity a very large 
	DS might be.

Thursday:

	- Penrose and the "Gšdel argument"


Phil/Psych 256
April Fools' Day, 1997

Penrose and the "Gšdel argument":

	Gšdel's theorems

	Lucas's argument

	Penrose's argument

	Relationship to CRUM

Outline of the "Gšdel argument":

	1. Computer algorithms are equivalent to formal 
	logic, e.g., predicate logic

	2. Algorithms therefore share any limitations of 
	formal logic

	3. Gšdel's theorems constitute limitations of formal 
	logic, and therefore on algorithms

	4. Because algorithms are part of CRUM, CRUM is 
	limited by Gšdel's theorems (half the truth)

	5. Such limitations do not apply to the mind, 
	therefore CRUM cannot be good enough to model the 
	mind (the "catch")

Q: What are Gšdel's theorems?

	For any formal logic L that is adequate for arithmetic

	I. If L is consistent, then there exists a formula G 
	in L such that neither G nor ~G is provable in L, 
	i.e., L is incomplete

	G: "I am not provable in L".

	II. The consistency of L cannot be proved within L.

	Since L proves (I) itself, L can show that (I) applies 
	to any L', such that L' is also adequate for arithmetic

Q: What is Lucas's argument?

	1. Any CRUM model is equivalent to a formal logic L

	2. If L is consistent and adequate for arithmetic, 
	then L is incomplete (Gšdel I)

	3. But the mind can see that G (in L) is true, so it sees 
	something L does not

	4. Therefore, the mind is not an L

Q: What's wrong with Lucas' argument?

	A1. For G to be true, L must be consistent.  Lucas does 
	not show this about his mind.

	A2. Lucas does not say what "seeing" truth is - it cannot 
	be provability 

	A3. Lucas cannot exclude the possibility that his mind 
	is an L' that can show the incompleteness of L.  If the 
	mind is an L, it may not be able to say which one (Bencerraf)

Q: What is Penrose's argument?

	1. Penrose discusses "knowably" sound algorithms A that 
	generate the theorems of L

	2. A mathematician is not knowably sound (Gšdel I), but 
	nevertheless correct because he or she is conscious, which 
	allows access to a Platonic realm of mathematical truths.

	Consciousness is produced by effects of quantum gravity 
	within the brain.

	3. Mathematicians as a community are correct because they 
	come to agree on the truth or falsity of any theorem.

	Agreement comes from mathematicians all having access to 
	the same realm of truths.

Q: What's wrong with Penrose' argument?

	A1. Are non-mathematicians not conscious?

	A2. What does "knowably" mean? (CAM)

	A3. Mathematicians do make mistakes, even Ramanujan.  
	Also, some disagreement persists, e.g., the axiom of choice.

	A4. Digital computers depend on quantum effects, but are 
	still equivalent to formal logics

	A5. Penrose does not consider inconsistent, but reliable 
	algorithms for CRUM (Dennett)

Q: What does Gšdel's theorem say about CRUM?

	A1. Nothing, it has been misapplied.

	A2. That mathematical thinking has yet to be adequately 
	modeled in CRUM

Further remarks on Essay 2:

	Be sure to evaluate your analysis of your task:

		1. What does CRUM suggest about the task?  Could 
		CRUM be used to improve how the task is performed?

		2. What does your analysis of the task suggest about 
		CRUM?  Is CRUM adequate to model the task?  How might 
		CRUM be improved?

Thursday:

	- review
	- remarks on final exam
	- course evaluations

Further materials

• Chaos at Maryland
• Kurt Gödel biography
• PSYCHE: Symposium on Roger Penrose's Shadows of the Mind