Philosophy of Science

Pythagoras, mathematical Platonism, and Aristotle - the dominant ideology of science well into the 19^th century declared that mathematics could provide the complete and final truth. The world was rational and math was perfect.

Cracks in the wall of the Platonic heaven:

1. The first problem arose during the Renaissance with the discovery of imaginary numbers.

2. Infinity, Zeno’s paradoxes and infinitesimals

3. Set theory and Cantor’s infinite sets.

But then…

Frege and the quest for Foundations of Mathematics including the introduction of variables in logic

Bertrand Russell and the effort to found mathematics securely on logic - the Principles of Mathematics

Russell’s Paradox-If R is the set of all sets that are not members of themselves, and if R qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves, but if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox. . Russell’s paradox seems to be a subset of self-referential paradoxes that go back at least as far as Eubolides who said, “This statement I am now making is false.”

"Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion." - Frege translated by Jean van Heijenoortr 1967.

This problem can be avoided by a weaker axiomatic set theory that does not assume that, for every property, there is a set of all things that satisfy the property. It is based on the axiom that for any set X, any subset of X definable using first-order logic exists. The set R described above can’t exist as such a set. Therefore, no propositional function can be defined prior to specifying the function's scope of application. So the problem is solved by Russell’s Theory of types??? But Wittgenstein said the Theory of Types was a vain attempt to say the unsayable. For example, what does this mean: “’The class of men is a man’ is a piece of nonsense” (Kenny 2006)?

Hilbert's (1921) Program called for the rigorous formalization of all of mathematics in axiomatic form, and proof that the axiomatization was logically consistent.

"The Gödel Debacle" 1931

Gödel's Theorems:

1. Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250).

2. If such a system is also capable of proving certain basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency of the system itself.

A Simple Proof but Tarski’s undefineability theorem

Götterdämmerung: Gödel, Russell, Frege, Turing, Wittgenstein, von Neumann

Now for Aristotle and Science

Theory of Colours - Newton (1672), Goethe (1810)

Werner Heisenberg and Ludwig Wittgenstein, said, Goethe was right about colour! (Ribe & Steinle, 2002).

"There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" is reputed to be Lord Kelvin's remark made to the British Association for the Advancement of Science (1900).

Heisenberg (1927) and the (uncertainty principle), (matrix mechanical formulation), Schrödinger (wave mechanical formulation) (and the cat!)

It is impossible to determine simultaneously both the position and the momentum of an electron (wave packet) or any other particle with any great degree of accuracy or certainty. This system cannot have simultaneously singular values of these pairs of quantities. The product of the uncertainties in these properties is > 1/2 the reduced Planck constant (ħ = h/2π) (Werner Heisenberg 1927).

Schrödinger's Cat was designed to elucidate the quantum entanglement as described by Einstein, Podolsky and Rosen (1935).

Schrödinger wrote:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small that perhaps in the course of the hour, one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges, and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation.

You are the only contemporary physicist, beside Laue, who sees that one cannot get around the assumption of reality, if only one is honest. Most of them simply do not see what sort of risky game they are playing with reality—reality as something independent of what is experimentally established. Their interpretation is, however, refuted most elegantly by your system of radioactive atom + amplifier + charge of gunpowder + cat in a box, in which the psi-function of the system contains both the cat alive and blown to bits. Nobody really doubts that the presence or absence of the cat is something independent of the act of observation. - Albert Einstein to Schrödinger 1950

Desirable Features of Science

Internal Consistency

Beauty

Simplicity

Parsimony

Power

What distinguishes science from religion????? 

It must be..!!!

Popper rejected classical empiricism and observationalist-induction in favor of critical rationalism. He argued that scientific theories are abstract and can only be tested by their implications. Popper claims to have found a solution to the problem of induction. He finds that though there is no way to prove that the sun will rise every day, one can develop a theory that predicts the sun will rise every day, and the theory can be falsified and rejected the day that the sun does not rise.

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 encompasses 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 Harrison (1992):

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 occur

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 achieved 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=Birch	5
2=Blackgum	36
3=Red Maple	50
4= Beech	9

A 50-year tree-by-tree from column to row transition matrix). 

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

Transition Matrix * State Vector= New State Vector

NEW STATE VECTOR (Year 50)

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

NEW STATE VECTOR (Year 100)

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

Example of matrix addition:

Examples of matrix and vector multiplication

Rows of first * columns 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 = dimensions 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
A0 = A^1 =	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^x gives the number of paths of length x:

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 e.g., 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 e.g., 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 achieved 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

o Hurlbert (1984) warns against pseudo-replication

· Testing the Hypothesis

o null hypothesis and alternative hypothesis

· Beauty, Simplicity, Parsimony, Variance accounted for

· Distributions

· Types of Error

o Type 1 error, Alpha = probability of rejecting a correct null hypothesis

o Type 2 error, Beta = probability of failing to reject a false null hypothesis

Choosing a Statistical Test,from Chapter 37 of Intuitive Biostatistics 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 
 
 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