Philosophy of Science

Pythagorists-world rational, math is perfect

Epistemic Mediation

Planck, Heisenberg,Schrödinger (and the cat) and the uncertainty principle.

 Gödel 1931

Starting with the cracks in the wall of the platonic heaven

Incommensurability: The diagonal of a square can't be measured by the same units used to measure the sides.

Infinitesimals: If you go half way every time you will never get there.

The set of all sets contains more subsets than it contains sets. This is impossible.

"The Gödel Debacle"

Gödel's Theorem

All consistent axiomatic formulations of number theory include undecidable propositions

A Simple Proof

Chaos Theory

 Desireable Features of Science

Internal Consistency

Beauty

Simplicity

Parsimony

Power

What distinguishes science from religion?????

It must be..!!!

What is Ecology? "the body of knowledge concerning the economy of nature-the investigation of the total relationships of the animal both to its inorganic and its organic environment; including, above all, its friendly and inimical relations with those animals and plants with which it comes directly or indirectly into contact-in a word ecology is the study of all those complex interrelations referred to by Darwin as the conditions for the struggle for existence."

(Ernst Haeckel 1866)

Greek Oikos = household

"the study of the structure and function of nature"

(Odum 1971:3)

"the scientific study of the relationship between organisms and their environments."

(Mc Naughton and Wolfe 1979)

"the study of the relationship between organisms and the totality of the physical and biological factors affecting them or influenced by them" (Pianka 1988:4)

"the study of the adaptation of organisms to their environment"

(Emlen 1973:1).

"the study of the relationship between organisms and their physical and biological environments"

(Ehrlich and Roughgarden 1987:3)

"the science of the universe"

(Hutchinson)

"the study of how the world works"

(Colinvaux)

What is the environment?

Dictionary Definition of Environment

(Merriam-Webster 1974)

1. The circumstances, objects or conditions by which one is surrounded;

2. The complex of climatic, edaphic and biotic factors that act upon an organism or an ecological community and ultimately determine its form and survival;

3. The aggregate of social and cultural conditions that influence the life of an individual or community;

4. An artistic or theatrical work that involves or encompases the spectator.

This is a diagram of environment. What is wrong?

Environment as circular causal nexus

(von Uexkull 1926)

The entire function circle formed from inner world and surrounding world constitutes a whole.

Continuity of the complete whole must never be lost sight of.

 Causation in Environments

Everything is lawfully produced ...and lawfully

produces something else.

"Every object (H) defines two environments: an input environment and its associated causal nexus (H'), and an output environment and its associated causal nexus (H''). The prerogative of environment definition is that of the object" (Patten 1988).

What is a system? (White, Mottershead and Harrison1992)

a set of elements with a set of properties:

1. All systems have structure or organization

2. All function in some way

3. There are functional and structural relationships between parts

4. Function implies that flows and transfers occurr

5. Function requires a driving force or energy source

6. All systems have some degree of integration

Systems possess boundaries, surroundings, elements, states

The state of the system is defined when each of its properties has a definite value.

Systems can be isolated,closed or open.

In any closed system, the final state is determined by the initial condition.

In open systems the final state may be acheived with different initial conditions and in different ways.

Because systems contain elements systems are decomposable.

Everything is a Vector (Whitehead)

Horn 1981

A STATE VECTOR

1= Gray Birch	5
2= Blackgum	36
3= Red Maple	50
4= American Beech	9

TRANSITION MATRIX (50 year tree-by-tree transition matrix).

	1	2	3	4
1	.05	.36	.50	.09
2	.01	.57	.25	.17
3	0	.14	.55	.31
4	0	.01	.03	.96

Transition Matrix * State Vector= New State Vector

NEW STATE VECTOR (Year 50)

1= Gray Birch	1
2= Blackgum	29
3=Red Maple	39
4= American Beech	31

NEW STATE VECTOR (Year 100)

1= Gray Birch	1
2= Blackgum	29
3= Red Maple	39
4= American Beech	31

Example of matrix addition:

A=	1 3		4 1		1+4	3+1	5 4
	2 5	+	3 7	=	2+3	5+7 =	5 12

Examples of Matrix and vector Multiplication

Rows of first * collumns of 2nd

A=	3 1	1		3*1 + 1*2		5
	2 4 *	2	=	1*2 + 4*2	=	10

So you can multiply a 2*2 * 2*1

BUT YOU CAN'T MULTIPLY 2*1 * 2*2

Result- dimensions of result=dimemsions of the 2^nd matrix

Why bother? It can be used to map flows, map changes, calculate utilities, and evaluate direct and indirect effects.

Patten 1985

Adjacency Matrix (A_ij) Matrix denotes numbers of paths from column compartments j to row compartments I.

		0	0	0	0	0	0
		1	0	0	1	1	1
Ao= A1	=	0	1	0	0	0	0
		0	1	1	0	0	0
		0	1	1	1	0	0
		1	0	0	0	1	0

The product matrix A^l gives the number

of paths of length l.

 The Matrix A² gives the number of paths of length 2.

	0	0	0	0	0	0
	1	2	2	1	1	0
A^2=	1	0	0	1	1	1
	1	1	0	1	1	1
	1	2	1	1	1	1
	0	1	1	1	0	0

More Examples of Matrix Multiplication

Online matrix calculator

What is an ecosystem?= An open system (exchanges energy and matter with its environment.

1st law of thermodynamics-

energy can neither be created nor destroyed.

2nd law-

Spontaneous transformation of energy is not 100% efficient.

Hierarchies of scale and control.

Within hierarchically structured environments, the behavior of one level is strongly influenced by the behavior of the two adjacent levels (O'Neill).

analysis and synthesis

Models, cognition, and epistemology,

Model=a nonunique, homomorphic mapping or representation

ho*mo*mor*phism (noun)=imperfect representations of reality

[International Scientific Vocabulary]

First appeared 1935

: a mapping of a mathematical set (as a group, ring, or vector space) into or onto another set or itself in such a way that the result obtained by applying the operations to elements of the first set is mapped onto the result obtained by applying the corresponding operations to their respective images in the second set

isomorphic (equal)

Conceptual Models

Statistical models y=a+bx

Analytical Models eg. dN/dt=rN(K-N)/K :the equation can be solved!

Simulation Models lack simple mathematical solutions

deterministic, probabilistic, stochastic

The method used to understand a system is a function of the relative levels of understanding and data available (Holling 1978, Starfield and Bleloch 1986, Grant et al. 1997).

System Description	Effective Approach
Systems with few, highly connected components.	Analytical Equations Eg. Physics and Mechanics
Systems with many, loosely connected components.	Statistics
Systems with many, tightly connected components. "organized complexity"	Simulation and Systems Analysis

Validation: Does new data give results similar to that acheived by the data initially used to frame the model. This helps us determine the confidence we have in the model.

A tradeoff exists between the ability to predict, the precision, and the generality of a model.

Hypotheses and Ecology

Inductive Method-specific observation to general conclusion

Deductive Method-general understanding yields specific prediction

Data Collection:direct observation or natural experiments, experimental approach(dependent and independent variables

Replicates=systems receiving treatments, replicates should be independent

Hurlbert(1984) warns against pseudo-replication

Testing the Hypothesis

null hypothesis and alternative hypothesis

Beauty, Simplicity, Parsimony, Variance accounted for.

Distributions

Normal Distribution

Mean

Mean=Xi/n

Variance=s²=S(xi-x)²/n-1

Standard Deviation= sqrt( s² )

Types of Error

Type 1 error Alpha=probabiliy of rejecting a correct null hypothesis

Type 2 error Beta= probabilty of failing to reject a correct null hypothesis

Choosing a Statistical Test, from Chapter 37 of Intuitive Biotatistics by Harvey Motulsky.

Interactive Statistics Web Site

Examples of Statistical Analysis Output

 














The TTEST Procedure
 Statistics
 Lower CL Upper CL Lower CL Upper CL
Variable N Mean Mean Mean Std Dev Std Dev Std Dev Std Err
write 200 51.453 52.775 54.097 8.6318 9.4786 10.511 0.6702
 
 T-Tests
Variable DF t Value Pr > |t|
write 199 4.14 <.0001












































The ANOVA Procedure
 
 Class Level Information
 
 Class Levels Values
 
 Group 3 1 2 3
 
 
 Number of observations 12
 Simple one-way anova 2
 
 The ANOVA Procedure
 
Dependent Variable: Response
 
 Sum of
 Source DF Squares Mean Square F Value Pr > F
 
 Model 2 127.1666667 63.5833333 7.76 0.0110
 
 Error 9 73.7500000 8.1944444
 
 Corrected Total 11 200.9166667





























 The REG Procedure




 Model: MODEL1




 Dependent Variable: weight









  Analysis of Variance









 Sum of  Mean




 Source DF Squares Square F Value Pr > F









 Model 1 526.39286 526.39286 14.55 0.0034




 Error 10 361.85714 36.18571




 Corrected Total 11 888.25000














 Root MSE 6.01546 R-Square 0.5926




 Dependent Mean 62.75000 Adj R-Sq 0.5519




 Coeff Var 9.58638














 Parameter Estimates









  Parameter Standard




 Variable DF Estimate Error t Value Pr > |t|









 Intercept 1 30.57143 8.61371 3.55 0.0053




  age 1 3.64286 0.95512 3.81 0.0034