Effect of measurement:
Effect of measurement:
Taking whatever happens in nature and transducing it to the point it is accessible to human senses
[or to the “senses” of some device that’s going “take action” based on what’s happening]
[or to the “senses” of some device that’s going “take action” based on what’s happening]
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EntityList["PhysicalQuantity"]
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QuantityVariable["Pressure"]
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Pressure
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,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
E.g. pressure
E.g. pressure
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What is the destination for the measurement?
What is the destination for the measurement?
E.g. direct human senses
E.g. direct human senses
Measurement: turn things into numbers?
Measurement: turn things into numbers?
Another definition of measurement: filling in parameters from the world into a predefined model
Another definition of measurement: filling in parameters from the world into a predefined model
For “mathematical models” the parameters tend to be numbers
Measurement is evolving to some “attractor” that represents a “thing we understand”
Measurement is evolving to some “attractor” that represents a “thing we understand”
The world has lots of details; we want to extract a symbolic description that we “understand”
The world has lots of details; we want to extract a symbolic description that we “understand”
Another operational definition: it leads to nerve firings
Another operational definition: it leads to nerve firings
“Happening” is related to building up a entailment cone of consequences
“Happening” is related to building up a entailment cone of consequences
The measured vs unmeasured case:
The measured vs unmeasured case:
Without “measurement” you have some system and all its details matter to the future
With “measurement” the only causal consequences are ones in the “measured attractor”
With “measurement” the only causal consequences are ones in the “measured attractor”
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ArrayPlot[CellularAutomaton[184,RandomInteger[1,50],20]]
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ArrayPlot[CellularAutomaton[90,Last[CellularAutomaton[184,RandomInteger[1,50],20]],20]]
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BlockRandom[SeedRandom[24425];ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0},7]/.{l3_,_,l1_,c_,r1_,_,r3_}:>If[If[c==0,r1+r3,l1+l3]+c>=2,1,0],2],2,3},RandomChoice[{.5,.5}->{1,0},1000],500],ColorRules->{0->Hue[0.15,0.72,1],1->Hue[0.98,1,0.8200000000000001]},Frame->False]]
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BlockRandom[SeedRandom[24425];ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0},7]/.{l3_,_,l1_,c_,r1_,_,r3_}:>If[If[c==0,r1+r3,l1+l3]+c>=2,1,0],2],2,3},RandomChoice[{.6,.4}->{1,0},1000],500],ColorRules->{0->Hue[0.15,0.72,1],1->Hue[0.98,1,0.8200000000000001]},Frame->False]]
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BlockRandom[SeedRandom[24425];ArrayPlot[CellularAutomaton[{FromDigits[Tuples[{1,0},7]/.{l3_,_,l1_,c_,r1_,_,r3_}:>If[If[c==0,r1+r3,l1+l3]+c>=2,1,0],2],2,3},RandomChoice[{.4,.6}->{1,0},1000],500],ColorRules->{0->Hue[0.15,0.72,1],1->Hue[0.98,1,0.8200000000000001]},Frame->False]]
Claim: statistical mechanical “measurements” might be of things that are basically additive
Claim: statistical mechanical “measurements” might be of things that are basically additive
The “measurement” is implementing a contractive mapping
The idea of number is a similar form of data summarization
Contractive mapping: like identifying equivalence classes
Contractive mapping: like identifying equivalence classes
Relation to foliations:
Relation to foliations:
A foliation is defining what events are “somehow equivalent”
Symbolic dynamics-like approach: take a state space, and partition it into equivalence buckets [AKA lossy compression]
Symbolic dynamics-like approach: take a state space, and partition it into equivalence buckets [AKA lossy compression]
Input: detailed raw data; output: which bucket / AKA which numerical / ....
Dynamics of measurement: involves time and/or involves “extent of the observer”
Dynamics of measurement: involves time and/or involves “extent of the observer”
Purpose of observation/measurement/perception
Purpose of observation/measurement/perception
Take the details of some part of the world and equivalence/compress/attractorize them to some model which has a predefined structure “known to the observer”
How do measurements work?
How do measurements work?
If the state of the world is the same, the measurement will be the same [though consider QM]
If the state of the world is the same, the measurement will be the same [though consider QM]
There is a measuring device ; and it is reused
There is a measuring device ; and it is reused
System is interacting somewhat weakly with the measuring device
System is interacting somewhat weakly with the measuring device
Consider a manometer
Consider a manometer
Two fluids made of molecules; density of events is much higher in the “measuring device” fluid than the in the measured gas; viscosity of liquid higher than gas
Human eye
Human eye
Several photons hit the photoreceptor [ultimately only one photon matters]
Some photons never make it to the photoreceptor
Measure: is it red or green?
Imagine we wanted to make a universal measuring device ... out of certain components
Imagine we wanted to make a universal measuring device ... out of certain components
In WPP, if we could extract the causal graph we could measure everything
Given the causal graph, we know what energy is ; we know what length is
If we know MLT etc. then we can reconstruct everything.
If we know MLT etc. then we can reconstruct everything.
Imagine we have a region of causal graph .... can we evaluate its elementary physical dimensions
current ; luminosity ; amount ; temperature
[ number of electric charges ] ; [ number of photons ] ; [ number of atoms ]
[ ? temperature ]
[ number of electric charges ] ; [ number of photons ] ; [ number of atoms ]
[ ? temperature ]
A-to-D converter
A-to-D converter
What is measurable given a basis set of measurements?
What is measurable given a basis set of measurements?
p , p^2 [ momentum ]
{x , p } angular momentum
Given a certain set of measurements at a fixed time, what can we derive?
Given a certain set of measurements at a fixed time, what can we derive?
If we measure at multiple times, what else can we deduce?
If we measure at multiple times, what else can we deduce?
[ L^2 often associated with flow ]
Phase transitions as examples of continuous to discrete
Phase transitions as examples of continuous to discrete
Measurable features of gas vs non-measurable....
Measurable features of gas vs non-measurable....
Software radio vs. XXXX
Software radio vs. XXXX
Lumped models are another example of equivalencing....
Lumped models are another example of equivalencing....
Measurable quantities vs . all quantities [ cf computable reals ]
Measurable quantities vs . all quantities [ cf computable reals ]
Measurability complexity....
Measurability complexity....
Locality is easier....
Consider molecules hitting a sensor : can one think of looking at properties of the gas as decoding a code...
Consider molecules hitting a sensor : can one think of looking at properties of the gas as decoding a code...
Afterthoughts.....
Afterthoughts.....
Consider a membrane being hit by molecules...
Consider a membrane being hit by molecules...
It could be bizarrely distorted, or it could have a reasonable shape that “all moves together”