science and sampling - dynamic graphics project
TRANSCRIPT
“Reality” in Science
• Empiricism: need for evidence.• Realism: that there is something “out there” that we
can apprehend “in here”.• Analysis: breaking things down into smaller
(presumably more easily understood) components.• Reductionism: the belief that all phenomena can be
explained through analysis.
If a tree falls in the forest...
If there’s no one tohear it,or see it,or feel it,did it happen?
“Truth” in Science
• Confirmation/Validation: ensure a theory matches experimental observations. Need evidence!
• Deduction: reason from facts known to be true through the use of logic and mathematics.
• Induction: draw conclusions from known observations
Theory in Science
Any scientific theory provides an explanation, often through mathematics, for the behaviour of a phenomenon. A theory should explain current observations and predict as yet unseen ones.
Theories are normally accretive: they connect to provide more comprehensive accounts of the world.
Falsifiability in ScienceAny scientific theory must be capable of being shown to be incorrect by conjecturing observations counter to the theory.
Finding such observations “falsifies” the theory (i.e., there has to be a way for you to be wrong).
A “falsified” theory may be abandoned, modified, or circumscribed.
Belief and Scientific Theory
All scientific theories are beliefs but not all beliefs are theories.
Evolution is a theory; Intelligent Design is not.
That a belief is not a science does not inherently devalue it. Science places constraints on what is admitted as science, but is silent on other ways of trying to understand or appreciate the world.
Belief and Scientific Theory
There is a deep connection to semiotics and linguistics:
What we cannot speak about we must pass over in silence.
-- Ludwig Wittgenstein,Tractatus Logico Philosophicus
Richard Dawkins on Evolution“We are going to die, and that makes us the lucky ones. Most people are never going to die because they are never going to be born. The potential people who could have been standing in my place but who will never see the light of day outnumber the sand grains of Sahara -- more, the atoms in the universe. Certainly those unborn ghosts include greater poets than Donne, greater scientists than Newton, greater composers than Beethoven. We know this because the set of possible people allowed by our DNA so massively outnumbers the set of actual people. In the teeth of these stupefying odds it is you and I that are privileged to be here, privileged with eyes to see where we are and brains to wonder why.”
This is from a beautiful BBC Lecture in 1996.http://www.edge.org/3rd_culture/dawkins/lecture_p1.html.
Science is “Social”
Fads, fashions, orthodoxies, conventions, prizes, deciding what is important and what is not.
Some non-measurable “theories” are give the benefit of the doubt (e.g., string theory) while others are not (e.g., the universe is a computer).
Social acceptance of data, observations, theories.
Science is a method.
Limitations of Science
There are limits to:• measurement.• sampling.• certainty.• scientific method.• formal reasoning.• computability and complexity.
Uncertainty
The 20th Century was in many ways a century of uncertainty.
There was tremendous progress in science (and technology), as well as a recognition of its limitations.
Sampling Theorem
Whittaker, Nyquist, Kotelnikov, Shannon, 1928-1946 :When sampling a band-limited signal, the sampling rate must be greater than twice the input signal bandwidth in order to be able to reconstruct the original perfectly from the sampled version.
This connects “knowing” a signal exactly to the frequency of probes you must at minimum take of it.
Uncertainty Principle
Heisenberg, 1927:One cannot exactly measure the values of the position and momentum of a particle at the same time.
This extends to a very wide variety of pairs of attributes that we might want to measure.
Incompleteness Theorem
Gödel, 1931:For any consistent formal theory including basic arithmetical truths, one can construct an arithmetical statement that is true but not included in the theory.
Thus there are limits to what we can formally discover logically and mathematically.
Halting Problem
Turing, 1936:Given a description of a program and its initial input, determine whether or not the program, when executed on this input, will finish computing.
Turing proved that no computer can solve this problem.
Thus there are things that cannot be computed.
Sampling Theorem
Whittaker, Nyquist, Kotelnikov, Shannon, 1928-1946 :When sampling a band-limited signal, the sampling rate must be greater than twice the input signal bandwidth in order to be able to reconstruct the original perfectly from the sampled version.
This connects “knowing” a signal exactly to the frequency of probes you must at minimum take of it.
Cook’s Complexity Theorem
Stephen Cook, 1971:There is an NP-complete problem, and thus there are many NP-complete problems.
This technical result demonstrates that most “interesting” problems are not feasible to compute exactly.
Back to sampling ...
How many samples of something do you need before you can be confident that you have a good “representation” for it?
This is fundamental to much of our technology today.
Sampling a sine wave: measure the sine wave at discrete points and connect by a curve.
Its bandwidth is 1 period every 360 degrees.
Three samples per period (360 degrees) gives a straight line.
Four samples per period gives a sloppy curve.
Five samples per period gives a good result.
Even more samples per period doesn’t help much.
The Chain of Events
input signal S Discrete sampling
discrete values
Reconstruction
reconstructed signal S’
The goal is to make the (output) reconstructed signal S’ as “good” an approximation as possible to input signal S, given the constraints of sampling and the reconstruction mechanisms available.
Much of your technology depends on this: e.g., CD, mp3, DVD, digital radio, TV, telephony, internet, disk drives, control systems, sensors … .
CD, DVD, mp3, DAT
The Upshot?
There are limits to:• measurement.• sampling.• certainty.• scientific method.• formal reasoning.• computability and complexity.
The Upshot?
Science is necessarily:• uncertain (at some scale).• accretive.• approximate.• social.• constrained• constructive.
What of “Reality”?
It depends on your beliefs about the relationship to “the world”.
It is safe to say that in Science, “reality” is that which can be measured, modelled or discussed in a way that permits validation, experimentation, measurement, falsification.
What of scientific theories that are not measurable?