Incompleteness, Undecidability, and Uncertainty

[SelfSim]

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."

I concur that there must be some philosophical position which permits science to exist (and do its thing - unimpeded) - it is after all, a human endeavour.
That philosophical position must be consistent with science's goals, ie: such that it doesn't interfere with its conclusions.
Science, by its very nature draws on many parts of philosophies for its utility purposes, eg: logic/axiomatic math. In fact, math logic is widely used in science, specifically for tracking dependencies on any assumptions made throughout its process. The usefulness of doing this is to keep visible the implications of these assumptions in any inference based conclusions formed. (Same goes for philsophical Realism).
However, making use of the assumed 'truth' of math axioms (or Realism) is of no concern to science's conclusions because it tracks and deliberately neutralises/'ejects' any residual philosophical implications of this, and resists incorporation of them into future iterations.

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.

Ok then .. (perhaps I misunderstand?) .. what 'philosophical implications' and 'philosophical assumptions' are you referring to?
What process would you recommend for dealing with them?

My disagreement with most of your conclusions is only further evidence that science raises meta-scientific questions which are inherently philosophical.

I'd say that different minds demonstrably think differently on the same topics .. and that's all you're seeing there.

However, I don't have a problem with you choosing to see or dwell on 'philosophical implications/assumptions' and handle these philosophically, but this is then not part of science's domain. They have been 'ejected' by the scientific process as being unusable for science's purposes.

One might say that your concerns (or 'philosophical implications/assumptions/roots) are of little/no impact to science, as it continues with objective testing.

 

[FrumiousBandersnatch]

Friend, if you're interested, I've written about this in my essay On Axioms.

It seems to me that the philosophical presuppositions you mention are models - the result of fruitful hypotheses.

 

[0r0r]

It seems to me that the philosophical presuppositions you mention are models - the result of fruitful hypotheses.

I've used this as an opportunity to write down my thoughts on the matter: Why Philosophy Is Necessary.

 

[0r0r]

I concur that there must be some philosophical position which permits science to exist (and do its thing - unimpeded) - it is after all, a human endeavour.
That philosophical position must be consistent with science's goals, ie: such that it doesn't interfere with its conclusions.
Science, by its very nature draws on many parts of philosophies for its utility purposes, eg: logic/axiomatic math. In fact, math logic is widely used in science, specifically for tracking dependencies on any assumptions made throughout its process. The usefulness of doing this is to keep visible the implications of these assumptions in any inference based conclusions formed. (Same goes for philsophical Realism).
However, making use of the assumed 'truth' of math axioms (or Realism) is of no concern to science's conclusions because it tracks and deliberately neutralises/'ejects' any residual philosophical implications of this, and resists incorporation of them into future iterations.
Ok then .. (perhaps I misunderstand?) .. what 'philosophical implications' and 'philosophical assumptions' are you referring to?
What process would you recommend for dealing with them?
I'd say that different minds demonstrably think differently on the same topics .. and that's all you're seeing there.

However, I don't have a problem with you choosing to see or dwell on 'philosophical implications/assumptions' and handle these philosophically, but this is then not part of science's domain. They have been 'ejected' by the scientific process as being unusable for science's purposes.

One might say that your concerns (or 'philosophical implications/assumptions/roots) are of little/no impact to science, as it continues with objective testing.

Many thanks for the reply. I've used this as an opportunity to write down my thoughts on the matter: Why Philosophy Is Necessary.

 

[FrumiousBandersnatch]

Many thanks for the reply. I've used this as an opportunity to write down my thoughts on the matter: Why Philosophy Is Necessary.

The Principle of Sufficient Reason implies causality, and, arguably, determinism and the lack of 'true' free will. Spinoza used this and the infinite attributes of God to argue for a panentheist determinsim, where God is nature and its immutable laws - which, by Christian standards, is no God at all...

 

[0r0r]

The Principle of Sufficient Reason implies causality, and, arguably, determinism and the lack of 'true' free will. Spinoza used this and the infinite attributes of God to argue for a panentheist determinsim, where God is nature and its immutable laws - which, by Christian standards, is no God at all...

The Principle of Sufficient Reason and the Principle of Causality are inequivalent. I will not offer a demonstration here, but I may write one soon.

On the question of determinism, please see my essay A Solution to Modal Collapse. I also have a forthcoming paper which uses calculus to disprove determinism.

Leibniz's theory of contingency was aimed at Spinoza, as you may know. Have you read The Strong Free Will Theorem? Just curious what your thoughts on it are.

Many, many thanks for your comments. God bless!

 

[SelfSim]

@Catholic Philosophy: I Just read your 'Why Philosophy is Necessary' paper.

One comment I have is that I think you trivialize and skim over how we determine reality. It seems to me you assume a 'true objective reality' which simply exists independently from the mind which conceives it(?) This is demonstrably nothing more than a pure belief .. and being such a fundamental, it sets the basis of reasoning for everything you're writing there.

Its not surprising I suppose, when your basic tenet is the assumption of the existence of God.

 

[FrumiousBandersnatch]

Have you read The Strong Free Will Theorem? Just curious what your thoughts on it are.

I just read it - I didn't follow all the details, but it seems to come down to the idea that if experimenters can 'freely' choose the kind of quantum observation they make, the quantum system in question (e.g. a particle) also can be said to have the same class of free will. They define an experimenter's 'free will' as the capacity to make a choice that is "not a function of the past".

I'm wondering why we should assume that capacity, and if we do, then in what sense is 'will' involved?

As I understand it, the 'folk' sense of exercising free will to make a choice involves evaluating the available options according to some criteria and selecting the option that best satisfies the criteria. The authors explicitly reject such formulations, in favour of what appears to me to be true randomness (i.e. acausal), but which they suggest is not "accounted for by mere randomness".

The observed outcome of QM measurements is stochastic, with the probability of a possible outcome given by the amplitude for that outcome squared. This is true whatever means the experimenter uses to choose which measurement to make.

It may be that if one considers the experimenter to be a quantum system in a superposition of measurement choice outcomes, then their choice could also be considered stochastic, but the brain of an experimenter, by any normal understanding of the term, is not an isolated QM system, and could not achieve or maintain such a superposition.

So, yeah, it's not clear to me what they think they've achieved, why they think it's important, or how they justify the capacity to make a 'choice' that is "not a function of the past" (what does that mean and how could it happen?), or how they justify calling 'free will' and deny that it implies 'mere randomness'.

I suspect much of it (apart from the experimenter's 'free will') would be straight-forward under a fully deterministic formulation of QM (e.g. Everettian 'Many Worlds'), but I don't know.

 

[0r0r]

@Catholic Philosophy: I Just read your 'Why Philosophy is Necessary' paper.

One comment I have is that I think you trivialize and skim over how we determine reality. It seems to me you assume a 'true objective reality' which simply exists independently from the mind which conceives it(?) This is demonstrably nothing more than a pure belief .. and being such a fundamental, it sets the basis of reasoning for everything you're writing there.

Its not surprising I suppose, when your basic tenet is the assumption of the existence of God.

Many thanks for the insightful comments.

Gödel did prove the existence of true yet (axiomatically) unprovable statements. We are not on shaky ground here: the Incompleteness Theorems are generally accepted. I offer a sketch of the proof in my article Gödelian Incompleteness, Platonism, & the Mind of God.

But if the truth of a mathematical statement does not inhere in its axiomatic provability, then wherein does it lie? This seems like strong evidence for an objective mathematical reality that is independent of our axiomatic theories. Don't you think so?

As for the existence of God, my belief does not hinge on any one argument, but I have worked rather extensively on Gödel's proof of the existence of God.

Merely believing in God, however, is not all that much of an accomplishment (though it is a requirement for salvation); for as it is written, "Thou believest that there is one God. Thou dost well: the devils also believe and tremble. But wilt thou know, O vain man, that faith without works is dead?"

God bless!

 

[0r0r]

I just read it - I didn't follow all the details, but it seems to come down to the idea that if experimenters can 'freely' choose the kind of quantum observation they make, the quantum system in question (e.g. a particle) also can be said to have the same class of free will. They define an experimenter's 'free will' as the capacity to make a choice that is "not a function of the past".

I'm wondering why we should assume that capacity, and if we do, then in what sense is 'will' involved?

As I understand it, the 'folk' sense of exercising free will to make a choice involves evaluating the available options according to some criteria and selecting the option that best satisfies the criteria. The authors explicitly reject such formulations, in favour of what appears to me to be true randomness (i.e. acausal), but which they suggest is not "accounted for by mere randomness".

The observed outcome of QM measurements is stochastic, with the probability of a possible outcome given by the amplitude for that outcome squared. This is true whatever means the experimenter uses to choose which measurement to make.

It may be that if one considers the experimenter to be a quantum system in a superposition of measurement choice outcomes, then their choice could also be considered stochastic, but the brain of an experimenter, by any normal understanding of the term, is not an isolated QM system, and could not achieve or maintain such a superposition.

So, yeah, it's not clear to me what they think they've achieved, why they think it's important, or how they justify the capacity to make a 'choice' that is "not a function of the past" (what does that mean and how could it happen?), or how they justify calling 'free will' and deny that it implies 'mere randomness'.

I suspect much of it (apart from the experimenter's 'free will') would be straight-forward under a fully deterministic formulation of QM (e.g. Everettian 'Many Worlds'), but I don't know.

I hadn't realized John Conway died! He was (and is) a truly marvelous mathematician. I'd like to read On Quaternions and Octonions. I remember reading one of his books on the surreal numbers, which I believe he discovered.

 

[SelfSim]

Gödel did prove the existence of true yet (axiomatically) unprovable statements. We are not on shaky ground here: the Incompleteness Theorems are generally accepted. I offer a sketch of the proof in my article Gödelian Incompleteness, Platonism, & the Mind of God.

Will read, and try to provide some feedback shortly .. (I, for one appreciate these essays as a way of understanding how you're thinking about these matters .. but not as being 'true' themselves).

But if the truth of a mathematical statement does not inhere in its axiomatic provability, then wherein does it lie? This seems like strong evidence for an objective mathematical reality that is independent of our axiomatic theories. Don't you think so?

No .. it just suggests that maths is perpetually incomplete.
Eg: our minds are capable of exploring our own perceptions and conceptualisations. The concept of 'truth' is one such conceptualisation, as is: how do we know what's real? None of that is independent of the way we devise axiomatic theories, or how we go about figuring out what's real. Its our minds doing all of it!

As for the existence of God, my belief does not hinge on any one argument, but I have worked rather extensively on Gödel's proof of the existence of God.

Thanks for the link .. will refresh some faded memories ..

Merely believing in God, however, is not all that much of an accomplishment (though it is a requirement for salvation); for as it is written, "Thou believest that there is one God. Thou dost well: the devils also believe and tremble. But wilt thou know, O vain man, that faith without works is dead?"

Hmm 'written' in a way that mysteriously we are able to comprehend .. Sounds like the simplest explanation leading to understanding there, is that: 'twas written by human minds?

 

[FrumiousBandersnatch]

I hadn't realized John Conway died! He was (and is) a truly marvelous mathematician. I'd like to read On Quaternions and Octonions. I remember reading one of his books on the surreal numbers, which I believe he discovered.

Yes; I find his 'Game of Life' cellular automaton one of the most compelling ways to understand emergence.

 

[FrumiousBandersnatch]

But if the truth of a mathematical statement does not inhere in its axiomatic provability, then wherein does it lie? This seems like strong evidence for an objective mathematical reality that is independent of our axiomatic theories. Don't you think so?

Mathematics is the exploration of the implications of a particular set of axioms. We understandably base the standard axioms on mappings of our experience of the world, so it's no surprise that there's a good correspondence when we interpret the world mathematically.

But we can choose different axioms and get a different kind of mathematics.

 

[0r0r]

Mathematics is the exploration of the implications of a particular set of axioms. We understandably base the standard axioms on mappings of our experience of the world, so it's no surprise that there's a good correspondence when we interpret the world mathematically.

But we can choose different axioms and get a different kind of mathematics.

It seems to me that Gödel's incompleteness results constitute strong evidence against your view of mathematics. You can, of course, choose different axioms, but you would only change how you are describing an underlying reality. Moreover, Gödel proved that axiomatic provability and mathematical truth are not the same by producing true but unprovable mathematical statements.

By the way, when you read, for example, a book like Robin Hartshorne's Algebraic Geometry, do you get the impression that axioms are based on "ordinary" experience of the world? After all, Gödel's results -- along with my own experience in mathematics -- suggest to me that mathematical objects are as much a part of the world as anything implied by physics.

 

[0r0r]

Will read, and try to provide some feedback shortly .. (I, for one appreciate these essays as a way of understanding how you're thinking about these matters .. but not as being 'true' themselves).

No .. it just suggests that maths is perpetually incomplete.
Eg: our minds are capable of exploring our own perceptions and conceptualisations. The concept of 'truth' is one such conceptualisation, as is: how do we know what's real? None of that is independent of the way we devise axiomatic theories, or how we go about figuring out what's real. Its our minds doing all of it!

Thanks for the link .. will refresh some faded memories ..

Hmm 'written' in a way that mysteriously we are able to comprehend .. Sounds like the simplest explanation leading to understanding there, is that: 'twas written by human minds?

While I agree that the mind is always involved with human experience (trivial), it seems wrong to say that Gödel's result is psychological. It is, in fact, a logical result. Or did I misunderstand you? Please advise.

 

[FrumiousBandersnatch]

It seems to me that Gödel's incompleteness results constitute strong evidence against your view of mathematics.

Are you saying that mathematics is not the exploration of the implications of a particular set of axioms?

You can, of course, choose different axioms, but you would only change how you are describing an underlying reality.

Are you suggesting that any combination of arbitrary axioms necessarily describes some reality? Do you subscribe to Max Tegmark's 'Mathematical Universe' hypothesis, that our universe is a mathematical structure - and that every mathematical structure is a universe?

Moreover, Gödel proved that axiomatic provability and mathematical truth are not the same by producing true but unprovable mathematical statements.

Sure - and?

... when you read, for example, a book like Robin Hartshorne's Algebraic Geometry, do you get the impression that axioms are based on "ordinary" experience of the world?

I don't think I've read a book like Robin Hartshorne's ¯\_(ツ)_/¯. But I do think axioms are based on our experience of the world. What you intend by '"ordinary" experience, I don't know. I suppose it's possible that some axioms are based on 'extraordinary' experience, but I had in mind refinements of the pre-arithmetical experiences of collating, categorising, comparing, accumulating, sharing, etc.

Gödel's results -- along with my own experience in mathematics -- suggest to me that mathematical objects are as much a part of the world as anything implied by physics.

Sounds reminiscent of Platonic idealism; objects in the world approximate their mathematical models.

 

[SelfSim]

While I agree that the mind is always involved with human experience (trivial), it seems wrong to say that Gödel's result is psychological. It is, in fact, a logical result. Or did I misunderstand you? Please advise.

Logic (and math, science, God etc) is a product of our minds. It is a way of making sense via its explanations, to our own minds (for our own minds).

Gödel was human. He used hid mind. All this may seem trivial .. but it is always completely overlooked and disregarded! Its like a blind spot we seem to have developed over time(?)

There is no objective evidence that we perceive (or conceive) anything which escapes having 'the fingerprints' of the mind all over it .. and thus there is abundant objective evidence for this .. but one has to actually look at it, to see that!

 

[0r0r]

Logic (and math, science, God etc) is a product of our minds. It is a way of making sense via its explanations, to our own minds (for our own minds).

Gödel was human. He used hid mind. All this may seem trivial .. but it is always completely overlooked and disregarded! Its like a blind spot we seem to have developed over time(?)

There is no objective evidence that we perceive (or conceive) anything which escapes having 'the fingerprints' of the mind all over it .. and thus there is abundant objective evidence for this .. but one has to actually look at it, to see that!

You got me searching for a criterion of real existence, so I have proposed one in Reality & Infinity. To me, it seems plausible, but I don't think it will convince you. My argument is not baseless; it is, I think, supported by both mathematics and science. I agree that all things involve mind -- especially the Divine Mind.

 

[0r0r]

Are you saying that mathematics is not the exploration of the implications of a particular set of axioms?

Are you suggesting that any combination of arbitrary axioms necessarily describes some reality? Do you subscribe to Max Tegmark's 'Mathematical Universe' hypothesis, that our universe is a mathematical structure - and that every mathematical structure is a universe?

Sure - and?

I don't think I've read a book like Robin Hartshorne's ¯\_(ツ)_/¯. But I do think axioms are based on our experience of the world. What you intend by '"ordinary" experience, I don't know. I suppose it's possible that some axioms are based on 'extraordinary' experience, but I had in mind refinements of the pre-arithmetical experiences of collating, categorising, comparing, accumulating, sharing, etc.

Sounds reminiscent of Platonic idealism; objects in the world approximate their mathematical models.

Are you saying that mathematics is not the exploration of the implications of a particular set of axioms?

Are you suggesting that any combination of arbitrary axioms necessarily describes some reality? Do you subscribe to Max Tegmark's 'Mathematical Universe' hypothesis, that our universe is a mathematical structure - and that every mathematical structure is a universe?

Sure - and?

I don't think I've read a book like Robin Hartshorne's ¯\_(ツ)_/¯. But I do think axioms are based on our experience of the world. What you intend by '"ordinary" experience, I don't know. I suppose it's possible that some axioms are based on 'extraordinary' experience, but I had in mind refinements of the pre-arithmetical experiences of collating, categorising, comparing, accumulating, sharing, etc.

Sounds reminiscent of Platonic idealism; objects in the world approximate their mathematical models.

The crux of the matter is as follows: If axiomatic provability and mathematical truth are not the same, then in what does mathematical truth inhere? It is natural to think that a mathematical statement is true because it is provable, but Gödel proved that the set of provable mathematical statements is a proper subset of the set of true mathematical statements. Therefore, mathematical truth cannot be tamed by axiomatics. The bottom line: axiomatics cannot capture all of mathematical reality. Thus the introduction of large cardinal axioms may have to proceed by way of non-deductive methods of harmony and systematic fitness, which indeed brings us back to Platonism. I've written a number of articles in favor of Platonism, including Gödelian Incompleteness, Platonism, & the Mind of God.

Note. A better name for large cardinal axioms may be large cardinal principles.

I never set out to be a Platonist (I don't think), but the evidence forced it on me.

 

[SelfSim]

You got me searching for a criterion of real existence, so I have proposed one in Reality & Infinity. To me, it seems plausible, but I don't think it will convince you. My argument is not baseless; it is, I think, supported by both mathematics and science. I agree that all things involve mind -- especially the Divine Mind.

It'll take me some time to consume what you've written there but, put as simply as I can, 'existence' and 'reality' are our human (English) words. We control their meanings. The way we go about assigning meanings, is all important.

There are two ways; either by belief or, via the scientific method. The word may be the same, but the meanings are vastly different because of the way we go about choosing meaning.

No matter which way one looks at it though, the meanings of 'existence' and 'reality' have the human mind's fingerprints all over them .. There is no reasoned way know of (yet) for concluding that these meanings came about independently from human minds .. other than by just believing that. The idea that things exist independently from any/all human minds is just another belief.
There is no way known yet, that we can escape from the 'prison' that everything we perceive, leaves behind objective evidence of some active human mind at play and none for the idea that 'things exist' independently from some human mind or other.

The outlined thinking above is entirely consistent with the scientific method and scientific thinking. It is in fact, demonstrably, the philosophy of Science, in a nutshell.