Incompleteness, Undecidability, and Uncertainty

sfs

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Ok, but still we proceed under the axiomatic assumption that the laws of physics are consistent and build the whole edifice of cosmology starting with that assumption. Science treats this consistency as an axiom basically. Perhaps it could be another way. But thats not whats happening.
I don't think that's true. There is a small but active branch of physics that tests whether the laws of physics are consistent across space and time. You don't test axioms. Now I do think it's true that if there were no consistency to physical phenomena, physics wouldn't have been invented, i.e. some degree of consistency is a precondition for physics to have developed as a field, but that's a different claim.
 
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durangodawood

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But observing the consistency of the redshifted spectra compared with local star spectra can establish that the laws of physics are the same...
Only if you assume youre looking at the same type of objects - for which we rely on the consistency-of-nature. Theres no way to properly compare if the laws of nature may vary in some unknown fashion from one data point to the next.
 
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durangodawood

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I don't think that's true. There is a small but active branch of physics that tests whether the laws of physics are consistent across space and time. You don't test axioms....
Interesting. If they can do this then consistency can be retired as an axiom. But for sure its served as a fundamental assumption behind the interpretation of data for a long long time..... and still does for the most part.
 
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durangodawood

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I gotta wonder how much of the resistance to the idea of axioms underpinning scientific endeavor is wrapped up with the atheism/theism debate where theist's like to beat atheists over the head with their (toothless) "axioms/assumptions = faith!" argument all the time.
 
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FrumiousBandersnatch

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Only if you assume youre looking at the same type of objects - for which we rely on the consistency-of-nature. Theres no way to properly compare if the laws of nature may vary in some unknown fashion from one data point to the next.
It seems to me that you can discover that the laws appear to be consistent; you make observations of tiny bright points of light, discover they have a spectrum similar to local stars but redshifted, so you can then test the hypothesis that they are distant stars. In a similar fashion, what were thought to be local nebulae were discovered to be distant galaxies when Hubble found sources with cepheid light curves in them; further observations confirmed this.
 
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FrumiousBandersnatch

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I gotta wonder how much of the resistance to the idea of axioms underpinning scientific endeavor is wrapped up with the atheism/theism debate where theist's like to beat atheists over the head with their (toothless) "axioms/assumptions = faith!" argument all the time.
I'm comfortable with the concept of axioms, both for formal systems and potentially for science; but I've become sceptical that the axioms science is said to rest on are really necessary. It seems to me that they can equally well be testable hypotheses ¯\_(ツ)_/¯
 
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SelfSim

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I'm comfortable with the concept of axioms, both for formal systems and potentially for science; but I've become sceptical that the axioms science is said to rest on are really necessary. It seems to me that they can equally well be testable hypotheses ¯\_(ツ)_/¯
Nope .. a testable hypothesis is never assumed to have any 'truth' value before testing, whereas axioms are so assumed.
This is because 'what is true' is objectively untestable and also because science starts with no going-in assumptions.

The 'formal system' of mathematics is not science and so 'axiom' acquires logical truth value in all math contexts.
 
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FrumiousBandersnatch

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Nope .. a testable hypothesis is never assumed to have any 'truth' value before testing, whereas axioms are so assumed.
This is because 'what is true' is objectively untestable and also because science starts with no going-in assumptions.

The 'formal system' of mathematics is not science and so 'axiom' acquires logical truth value in all math contexts.
I'm not suggesting that hypotheses have truth value. I'm suggesting that, in science, the assumptions often referred to as axioms are unnecessary, and can be replaced with testable hypotheses. So, for example, rather than assuming that the laws of physics remain the same beyond the local domain, we can instead hypothesise that they do and make observations to test that hypothesis.
 
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sfs

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Interesting. If they can do this then consistency can be retired as an axiom. But for sure its served as a fundamental assumption behind the interpretation of data for a long long time..... and still does for the most part.
I'd say it's largely been an assumption for a couple of centuries, maybe, but it was originally a conclusion, and in fact a conclusion that contradicted the prevailing assumptions at the time. The heavens were not assumed to operate in the same ways as the sublunary realm, and uniformitarianism was not assumed.
 
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durangodawood

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I'd say it's largely been an assumption for a couple of centuries, maybe, but it was originally a conclusion, and in fact a conclusion that contradicted the prevailing assumptions at the time. The heavens were not assumed to operate in the same ways as the sublunary realm, and uniformitarianism was not assumed.
If the heavens werent even accessible to natural science, then I'm not sure what difference that makes to the discussion.

I would bet a lot of money that natural investigators of any era wouldnt even question whether their discoveries would hold true in the next town over, or the next country, or anywhere they could conceivably move their investigation.
 
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SelfSim

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I would bet a lot of money that natural investigators of any era wouldnt even question whether their discoveries would hold true in the next town over, or the next country, or anywhere they could conceivably move their investigation.
Then go and put your suspicion to the test.

I would question discoveries made in different contexts.

Consistencies are noticeable when one also notices what's been discarded and labelled as inconsistent.
 
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durangodawood

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Then go and put your suspicion to the test.

I would question discoveries made in different contexts.

Consistencies are noticeable when one also notices what's been discarded and labelled as inconsistent.
How might I put my suspicion to the test?

I suppose I could read up on history of science and see if various scientists thought natural principles varied across time and place. But Im going to plead too-busy for that right now.
 
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SelfSim

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I suppose I could read up on history of science and see if various scientists thought natural principles varied across time and place. But Im going to plead too-busy for that right now.
But not too busy to make endless posts based on misconceptions centered around what distinguishes science from mathematics, from logic, and from philosophy eh?
 
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durangodawood

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But not too busy to make endless posts based on misconceptions centered around what distinguishes science from mathematics, from logic, and from philosophy eh?
You seem rather prickly about this topic. I'm not immune from saying something wrong or dumb. But I'm not going to go down some giant rabbit hole to fix it if theres a more economical way at hand.
 
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0r0r

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I made an offhand comment here, but now it's going to bug me. I might as well get it out of my system.

It seems to me that if science were necessarily tied to mathematics in an existential way, a scientific incompleteness theorem would simply be an extension of the mathematical one (i.e. Godel). However, that seems impossible to establish.

Maybe something based on measurement, as in physics' uncertainty principle, would be a more realistic possibility. That is, maybe some measurement in biology (etc.) would imply a biology uncertainty principle.

What do you think? Is there, maybe, an incompleteness, undecidability, or uncertainty principle for sciences other than physics?

Friend, it has often been asked, "Can one get a Gödelian incompleteness result for physics?" Under the aegis of twentieth-century physics, the answer appears to be "yes." What is accordingly at issue is the undecidability of propositions with physical content, e.g., the location of a billiard ball in a universal Turing machine that is "billiard ball computer." For another example, the Kochen-Specker Theorem states that, given plausible restrictions, it is not possible to assign definite values simultaneously to all quantum absurdities. If you would like to investigate this, a nice resource is "Incompleteness, Nonlocality, and Realism" by Michael Redhead (Oxford: Clarendon Press, 1987).

You may be interested in my article Gödelian Incompleteness, Platonism, and the Mind of God: Gödelian Incompleteness, Platonism, & the Mind of God

I've also written about Gödel's proof of the existence of God: An Introduction to Gödel’s Proof of the Existence of God

God bless!
 
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0r0r

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I'm comfortable with the concept of axioms, both for formal systems and potentially for science; but I've become sceptical that the axioms science is said to rest on are really necessary. It seems to me that they can equally well be testable hypotheses ¯\_(ツ)_/¯

Note. This post initially linked to an essay entitled On Axioms, but I have replaced that writing with Why Philosophy Is Necessary.

My original messages follows: Friend, if you're interested, I've written about this in my essay On Axioms (Now Why Philosophy Is Necessary). I had begun this essay in connection with my research/book on Gödel's proof of the existence of God, but it grew into something more general.

God bless!
 
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SelfSim

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Friend, if you're interested, I've written about this in my essay On Axioms. I had begun this essay in connection with my research/book on Gödel's proof of the existence of God, but it grew into something more general.

God bless!
The scientific method does not require any of the assumptions you list .. they can all be simply ignored throughout the process. None of them appear as prerequisite steps in the widely published, and taught, scientific method.

The supposed philosophical implications you mention, are in the mind of the philosopher and will invariably be debated following science's conclusions. Such debates then are philosophical and are not called for by the scientific method.

PS: Since when is statics 'reduced' to dynamics, optics 'reduced' to electromagnetism and thermodynamics 'reduced' to statistical mechanics? There are some big misconceptions (just plain errors) going on there, methinks!?
 
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0r0r

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The scientific method does not require any of the assumptions you list .. they can all be simply ignored throughout the process. None of them appear as prerequisite steps in the widely published, and taught, scientific method.

The supposed philosophical implications you mention, are in the mind of the philosopher and will invariably be debated following science's conclusions. Such debates then are philosophical and are not called for by the scientific method.

PS: Since when is statics 'reduced' to dynamics, optics 'reduced' to electromagnetism and thermodynamics 'reduced' to statistical mechanics? There are some big misconceptions (just plain errors) going on there, methinks!?

Friend, the existence of capacitors seemed to refute J.C. Maxwell's assumption that an electric current is closed. Maxwell sought to save his hypothesis by postulating that a variable current is propagated by dielectric as a displacement current. Others took a harsh view of Maxwell's maneuver for
  1. Introducing an unobservable; and
  2. Being ad hoc.
These criticisms are -- by their very nature -- fundamentally philosophical, and Maxwell's hypothesis was vindicated by H. Hertz after Maxwell died. The overall result only raises further philosophical and methodological questions about ad hoc hypotheses in science.

Indeed, it is so easy to show that science roots in philosophical assumptions and has philosophical implications that it is hardly worth pointing out. Then again, it is often the painfully obvious that gets glossed over.

Try an exercise. Are the "empirical curves" wholly empirical? If they are, what does that mean? Could such a graph be considered a scientific hypothesis? If it is a hypothesis, what would that mean? If not, why not? These questions -- whose answers stand coordinate with different conceptions of what a scientific method looks like -- bring us into the domain of philosophy.

The sort of textbook description (if there is just one) of science may do little more than impede scientific progress. A mature scientific theory is a hypothetico-deductive system, and the matter of empirical testability is anything but straightforward. One could say that a scientific hypothesis is empirically testable if it implies testable statements or if it is implied by higher-level hypotheses with testable consequences. In no case are hypotheses directly confronted with data sets, since even seemingly naked observations are generally theory-laden.

Moreover, theory selection often comes down to philosophical considerations like economy, elegance, and aesthetics. For example, Einstein knew either classical mechanics had to go or else classical electrodynamics, and he chose to keep classical electrodynamics out of a philosophical preference for field theories. Or take another example: Given two theories of anything like equal explanatory value, the more economical theory is bound to prevail.

Take another example of great importance, namely: Given the standard operationist interpretation of relativity, where does the concept of observer occur in the union of all Lorentz-covariant theories?
 
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SelfSim

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Indeed, it is so easy to show that science roots in philosophical assumptions and has philosophical implications that it is hardly worth pointing out. Then again, it is often the painfully obvious that gets glossed over.
Science is a human pursuit .. philosophy helps us humans in understanding how thinking varies across the set of 'healthy' (normal) thinking human minds.

Until the role of the scientist's thinking is objectively re-integrated into science, the influences of philosophical worldviews/ideologies held by them, is purposefully regarded neutrally, yet distinct from, the resulting outcomes.
Catholic Philosophy said:
Try an exercise. Are the "empirical curves" wholly empirical? If they are, what does that mean?
(Depends entirely on what your mean by 'empirical').
Catholic Philosophy said:
Could such a graph be considered a scientific hypothesis? If it is a hypothesis, what would that mean? If not, why not? These questions -- whose answers stand coordinate with different conceptions of what a scientific method looks like -- bring us into the domain of philosophy.
So?
That question is also mostly irrelevant when it comes to producing a graphical representation of operationally interdependent quantities.
Catholic Philosophy said:
The sort of textbook description (if there is just one) of science may do little more than impede scientific progress. A mature scientific theory is a hypothetico-deductive system, and the matter of empirical testability is anything but straightforward. One could say that a scientific hypothesis is empirically testable if it implies testable statements or if it is implied by higher-level hypotheses with testable consequences. In no case are hypotheses directly confronted with data sets, since even seemingly naked observations are generally theory-laden.
All of science's conclusions are inferences and a properly formed hypothesis' test results are accepted as being objectively real.
There's no 'ifs' (implying assumptions) called for.

Human perceptions are integrated with being human, but that doesn't make them real.

Objectively produced observations and their hypotheses, contain the necessary information needed to track back to explicit (deliberately made) assumptions made along the way.
Hidden, or 'undistinguised' assumptions, are a different (human) matter.
Catholic Philosophy said:
Moreover, theory selection often comes down to philosophical considerations like economy, elegance, and aesthetics. For example, Einstein knew either classical mechanics had to go or else classical electrodynamics, and he chose to keep classical electrodynamics out of a philosophical preference for field theories. Or take another example: Given two theories of anything like equal explanatory value, the more economical theory is bound to prevail.
Since the goal of theory is to understand, and the simplest theory that agrees with data is the best path to understanding, then that's clearly the best theory.
The idea that this all leads to 'how things actually work' is distinguishable as being just another belief .. as if the universe was a simulation made by a fairly inexpert programmer who therefore had to 'keep it simple'.
Catholic Philosophy said:
Take another example of great importance, namely: Given the standard operationist interpretation of relativity, where does the concept of observer occur in the union of all Lorentz-covariant theories?
In the mind of the operationalist, of course! .. Simple!
 
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0r0r

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Science is a human pursuit .. philosophy helps us humans in understanding how thinking varies across the set of 'healthy' (normal) thinking human minds.

Until the role of the scientist's thinking is objectively re-integrated into science, the influences of philosophical worldviews/ideologies held by them, is purposefully regarded neutrally, yet distinct from, the resulting outcomes.
(Depends entirely on what your mean by 'empirical').
So?
That question is also mostly irrelevant when it comes to producing a graphical representation of operationally interdependent quantities.
All of science's conclusions are inferences and a properly formed hypothesis' test results are accepted as being objectively real.
There's no 'ifs' (implying assumptions) called for.

Human perceptions are integrated with being human, but that doesn't make them real.

Objectively produced observations and their hypotheses, contain the necessary information needed to track back to explicit (deliberately made) assumptions made along the way.
Hidden, or 'undistinguised' assumptions, are a different (human) matter.
Since the goal of theory is to understand, and the simplest theory that agrees with data is the best path to understanding, then that's clearly the best theory.
The idea that this all leads to 'how things actually work' is distinguishable as being just another belief .. as if the universe was a simulation made by a fairly inexpert programmer who therefore had to 'keep it simple'.
In the mind of the operationalist, of course! .. Simple!

Friend, your conclusions about the nature of science are not what I'm after. I merely wish to point out that our entire conversation only confirms that the scientific enterprise is enmeshed with philosophy. For example, when you say things like my question hinges on what I mean by "empirical."

You are free to think as you wish about the nature of science, but it seems incontrovertible that science not only roots in philosophical assumptions but also has philosophical implications. My disagreement with most of your conclusions is only further evidence that science raises meta-scientific questions which are inherently philosophical.

God bless!
 
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