goedel and more
Nadin, Mihai
Discussing Penrose is justified. BUT: the focus should be on Goedel. Allow me to repeat my advice: read the paper, i.e. his proof. Read also Turing, i.e. his proof for the Hilbert challenge.
Against my better judgment, I am suggesting one of my papers:
(in my book https://link.springer.com/book/10.1007/978-3-031-43957-5 this is discussed in detail)
But I am not reacting to recent opinions only for the sake of pointing to my work.
The thing that concerns me regarding the discussion is expressed as
…and perhaps some day someone will somehow prove things that are unprovable according to GT/C
Let us agree that science is not only about what is possible, but also about what is not possible. Squaring the circle—anyone? Just to give an example. Within the number construct
we are using, there is PI
)—no
need to redefine it here. As long as we stick to the numbers used currently in order to represent our measuring (quantitative aspects) of reality, the doubling of the cube and trisecting the
angle will remain impossible because of how we defined 
. NOT even the not yet invented new forms of computation will do it.
Goedel discovered for a well-defined aspect of mathematics that consistency and completeness are not possible. Those using the outcome of his proof owe it to him to understand what the meaning of this is. Doubt is good (Dubito ergo sum), on the assumption that is grounded in knowledge. Get if from the source.
Mihai Nadin
https://www.nadin.ws
https://www.anteinstitute.org
Alex Shkotin
Just a few ideas inspired by your letter - an exchange of ideas.
We use arithmetic to get some knowledge about natural numbers. Gödel invented a way to say something about arithmetic theory itself using natural numbers. For example his so-called Second theorem shows that if we express the hypothesis of arithmetic consistency as an arithmetic formula we can't prove it in arithmetic. But this is not a problem for those who study natural numbers, not a usage of arithmetic to study theories features.
I am sure that we need not to read the initial KG publication. Any textbook for Math-Logic is a good source. On my e-bookshelf there is [1]. Let me cite it [rus]==>(google-trans)
"This result is called Gödel's second theorem (see Gödel [1931]). Roughly speaking, this theorem states that if a theory S is consistent, then a proof of the consistency of the theory cannot be carried out by means of the theory S itself, i.e., any such proof must necessarily use ideas or methods that are inexpressible in the theory S. Examples of this are the proofs of the consistency of S proposed by Gentzen [1936], [1938b] and Schütte [1951] (see Appendix), which use concepts and methods (for example, a fragment of the theory of countable ordinal numbers) that are obviously not formalizable by means of the theory S."
My hypothesis is that the language of science, technology, and mind is HOL.
And when we have a framework for theories of mathematical logic, it will be the primary source for referencing any definitions, theorems, and proofs. We have the beginnings of a framework for undirected graph theory.
Incidentally, we also need a framework for tasks, problems.
Alex
[1] Introduction to Mathematical Logic by Elliott Mendelson Associate Professor of Mathematics Queens College Flushing, New York ~1964(1971)
D. VAN NOSTRAND COMPANY, INC. PRINCETON, NEW JERSEY. TORONTO NEW YORK
[rus] p.165 "Этот результат носит название второй теоремы Гёделя (см. Гёдель [1931]). Грубо говоря, эта теорема утверждает, что если теория S непротиворечива, то доказательство непротиворечивости теории не может быть проведено средствами самой теории S, т. е. всякое такое доказательство обязательно должно использовать невыразимые в теории S идеи или методы. Примерами тому могут служить доказательства непротиворечивости теории S, предложенные Генценом [1936], [1938b] и Шютте [1951] (см. Дополнение), в которых применяются понятия и методы (например, один фрагмент теории счетных порядковых чисел), очевидно, не формализуемые средствами теории S."
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Chuck Woolery
Alex, Thank you for this paragraph...
My hypothesis is that the language of science, technology, and mind is HOL. And when we have a framework for theories of mathematical logic, it will be the primary source for referencing any definitions, theorems, and proofs. We have the beginnings of a framework for undirected graph theory. Incidentally, we also need a framework for tasks, problems. Alex
With your permission I will be sharing it with other in my blog posts attempting to bring sanity to a world where religious, economic, and political words can mean damn near anything...
Cw
Chuck Woolery, Former Chair
United Nations Association, Council of Organizations
315 Dean Dr., Rockville, MD 20851
Cell:240-997-2209 ch...@igc.org
Blogs: 435 Campaign: www.435globaljustice.blogspot.com (May 2017 through today)
Dothefreakinmath http://dothefreakinmath.blogspot.com (June 2006 to Nov 2016)
The Trilemma http://trilemma.blogspot.com/ (Oct 2011 to Nov 2013)
“Today the most important thing, in my view, is to study the reasons why humankind does nothing to avert the threats about which it knows so much, and why it allows itself to be carried onward by some kind of perpetual motion. It cannot suffice to invent new machines, new regulations, new institutions. It is necessary to change and improve our understanding of the true purpose of what we are and what we do in the world. Only such an understanding will allow us to develop new models of behavior, new scales of values and goals, and thereby invest the global regulations, treaties, and institutions with a new spirit and meaning.” President Vaclav Havel, Czech Republic.
Here’s a video of optimism if you dare watch it https://www.rethinkx.com/videos
"A human being is part of the whole, called by us 'Universe'; a part limited in time and space. He experiences himself, his thoughts and feelings, as something separated from the rest - a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and affection for a few persons nearest us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole nature in its beauty. Nobody is able to achieve this completely, but striving for such achievement is, in itself, a part of the liberation, and a foundation for inner security." -Albert Einstein. As quoted in Quantum Reality, Beyond the New Physics, p. 250.
“The sad truth...is that most evil is done by people who never made up their minds to be or do either evil or good.” Hannah Arendt quoted in The Bulwork.
What are you doing to ensure the funding and achievement of the 17 Sustainable Development Goals by or before the year 2030? Connect the dots! See the web of life! Achieve ‘justice for all’. Or, prepare for the catastrophic consequences. Because...
“Everything is connected, everything is interdependent, so everything is vulnerable.... And that’s why this has to be a more than whole of government, a more than whole of nation [effort]. It really has to be a global effort....” Jen Easterly. Cyber and Infrastructure Security Agency’s director in a speech Oct. 29, 2021. [CISA is our nation’s newest federal agency established by the Trump Administration in 2018] I’m guessing Ms. Easterly understands that our environment is our most vital infrastructure! And human security everywhere is inherently and irreversibly connected to it everywhere
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Alex Shkotin
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John F Sowa
Alex, Thank you for this paragraph...
My hypothesis is that the language of science, technology, and mind is HOL. And when we have a framework for theories of mathematical logic, it will be the primary source for referencing any definitions, theorems, and proofs. We have the beginnings of a framework for undirected graph theory. Incidentally, we also need a framework for tasks, problems. Alex
With your permission I will be sharing it with other in my blog posts attempting to bring sanity to a world where religious, economic, and political words can mean damn near anything.
Cw
John F Sowa
Sent: 9/20/24 5:39 PM
To: "ontolo...@googlegroups.com" <ontolo...@googlegroups.com>
Subject: [ontolog-forum] goedel and more
Mike Bergman
Hi John,
OK; I'll bite, and give you two. See below.
I agree with the note below. But nobody but a logician who has studied advanced issues in logic knows how to state an undecidable proposition,
I would challenge anybody to find a single undecidable proposition in any branch of science or engineering or economics or politics or any field other than advanced mathematics.
- "The laws of nature are consistent across time" (the counter was a favorite of Peirce)
- "The laws of nature are universally applicable across all regions of the universe."
Thanks, Mike
I won't deny that a professional logician might contrive an undecidable proposition about some subject matter in any of those fields. But I will predict with almost absolute certainty that no professional in that field would understand it. And even if the logician could explain that proposition to somebody in that field, I predict with almost absolute certainty that the professional in that field would consider it irrelevant or even ridiculous.
And by the way, if anybody could contrive such a proposition, I would be delighted to see it. Please send it to the list, state the proposition in English, translate it to some version of logic, prove that it is indeed undecidable, and find a professional in that field who would consider it significant.
I seriously doubt that anybody could do that, but I'd be happy to be proven wrong.
John
From: "Nadin, Mihai" <na...@utdallas.edu>
Sent: 9/20/24 5:39 PM
To: "ontolo...@googlegroups.com" <ontolo...@googlegroups.com>
Subject: [ontolog-forum] goedel and more
Discussing Penrose is justified. BUT: the focus should be on Goedel. Allow me to repeat my advice: read the paper, i.e. his proof. Read also Turing, i.e. his proof for the Hilbert challenge.
Against my better judgment, I am suggesting one of my papers:
(in my book https://link.springer.com/book/10.1007/978-3-031-43957-5 this is discussed in detail)
But I am not reacting to recent opinions only for the sake of pointing to my work.
The thing that concerns me regarding the discussion is expressed as
…and perhaps some day someone will somehow prove things that are unprovable according to GT/C
Let us agree that science is not only about what is possible, but also about what is not possible. Squaring the circle—anyone? Just to give an example. Within the number construct we are using, there is PI
)—no need to redefine it here. As long as we stick to the numbers used currently in order to represent our measuring (quantitative aspects) of reality, the doubling of the cube and trisecting the angle will remain impossible because of how we defined
. NOT even the not yet invented new forms of computation will do it.
Goedel discovered for a well-defined aspect of mathematics that consistency and completeness are not possible. Those using the outcome of his proof owe it to him to understand what the meaning of this is. Doubt is good (Dubito ergo sum), on the assumption that is grounded in knowledge. Get if from the source.
Mihai Nadin
https://www.nadin.ws
https://www.anteinstitute.org
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John F Sowa
Mike Bergman
Hi John,
You are changing the premise of your challenge.
Mike,
I agree that those two questions are extremely difficult questions about physics, and nobody knows how to anwer them. But that is not a question about mathematical logic, which Mihai Nadin was discussing in his note below.
You are correct; no one knows how to decide these questions, let alone answer them. Look again at the statement of your challenge, which you stipulated was outside of advanced mathematics. And, the scientific domain I proposed was not merely physics, but I would suggest, the entirety of science.
Unlike your premise, I further suggest there are many, many
scientific propositions that are also based on undecidable
propositions, in contrast to your challenge. Yet, in science, we
can assume these propositions to be true within accepted
scientific models because no counter-factuals have (yet) been
shown to be true and they are practical for constructing coherent
and predictive theories.
The undecidability of these propositions arises from limitations
inherent to any formal system (such as a formalized version of
scientific theories) and our observational constraints.
Undecidability stems from the incompleteness of observational data
and the limits of formal systems to capture the entirety, in these
instances, of space and time. The same applies to any science
beyond the lightcones of our circumstances, and likely many other
areas related to fundamental assumptions underlying science.
When you put forward these categorical statements with assertions
of certainty and ubiquity, I find it off-putting. There is no
reason to state your arguments in such absolute terms. In essence,
such undecidability as the examples I provided is at the heart of
fallibility, which, as Peirce asserts so eloquently, is common to
all inquiry.
Thanks, Mike
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Alex Shkotin
Hi Mike,
Shortly 1. and 2. are axioms of physics.
And more precisely an abstraction from more modest pragmatics propositions:
1p. "The laws of nature are consistent across time one billion years before and after today. "
2p. "The laws of nature are universally applicable across all regions of the universe visible from Earth."
But axioms are decidable by introduction. 🎉
Alex
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Alex Shkotin
Hi Mike
First of all we need axiomatic theory for one or another scientific knowledge. For example, for Mechanics we have axiomatizations: Newton, Lagrange etc.
And I think JFS is sure that pragmatically useful propositions there would be decidable, but if natural numbers are used and we need full Peano axioms then there are some propositions there being undecidable.
But first and most interesting is formal axiomatic theory for Mechanics, or any other science.
I think the JFS's hypothesis is that any formal theory for science or technology is decidable.
But formal theories first.
We need formalization to feed our robots with robust knowledge.
Alex
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Alex Shkotin
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jsi...@measures.org
On Sep 21, 2024, at 8:34 PM, Mike Bergman <mi...@mkbergman.com> wrote:
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Mike Bergman
Hi Alex,
Hi Mike
First of all we need axiomatic theory for one or another scientific knowledge. For example, for Mechanics we have axiomatizations: Newton, Lagrange etc.
And I think JFS is sure that pragmatically useful propositions there would be decidable, but if natural numbers are used and we need full Peano axioms then there are some propositions there being undecidable.
But first and most interesting is formal axiomatic theory for Mechanics, or any other science.
I think the JFS's hypothesis is that any formal theory for science or technology is decidable.
But formal theories first.
We need formalization to feed our robots with robust knowledge.
As you know, a formal system provides a framework within which
statements can be proven true or false based on a set of axioms
and logical rules. Let's take one of my examples, "The laws of
nature are consistent across time," and formalize it.
∀t∈T,L(t) asserts that for every time t, the laws of nature
(L) apply consistently at all times past, present, and future.
Scientific theories (e.g., general relativity or classical or
quantum mechanics) can be viewed as formal systems in which the
laws of nature are treated like theorems derived from a set of
axioms. For example, the axioms of classical mechanics include
Newton's laws, and from these, we can derive predictions about
physical phenomena like motion. The proposition "The laws of
nature are consistent across time" asserts that the same axioms
(or laws) apply uniformly across all time points, meaning the same
rules of inference should yield consistent outcomes.
The undecidability of this proposition arises from Gödel's first
incompleteness theorem, which demonstrates that in any
sufficiently complex formal system (such as those representing the
laws of nature), there will always be true statements that cannot
be proven within the system itself. Similarly, a formal system
representing the laws of nature cannot prove that these laws are
consistent across all time points, per Gödel's second
incompleteness theorem, which would amount to the system proving
its own consistency over time.
So, really, the issue is not one of formal systems and axioms per
se, but the reality that in science empirical observations and
measurements act as inputs to models that can only cover a finite
portion of time (the observable past and present). Thus, the
axioms of a scientific system based on these observations can not
encompass all time points. As a result, we are limited by the
finite set of observations (axioms), and we cannot conclusively
prove that the laws remain consistent for all time. (Analogous
arguments can be made regarding space in my other example.)
To be useful or pragmatic, it is not a precondition that the
underlying propositions be decidable, as my two examples showed.
We don't, however, as a community of scientists, accept falsified
propositions nor ones not widely accepted as likely (even though
undecidedly) true. Decidability is not the pragmatic criterion.
Thanks, Mike
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Philip Jackson
Sent: Saturday, September 21, 2024 6:09 PM
To: ontolo...@googlegroups.com <ontolo...@googlegroups.com>; CG <c...@lists.iccs-conference.org>
Subject: RE: [ontolog-forum] goedel and more
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Philip Jackson
Sent: Sunday, September 22, 2024 1:00 PM
To: ontolo...@googlegroups.com <ontolo...@googlegroups.com>
Subject: Re: [ontolog-forum] goedel and more
John F Sowa
John F Sowa
Philip Jackson
To: ontolo...@googlegroups.com <ontolo...@googlegroups.com>
Subject: Re: [ontolog-forum] goedel and more
Nadin, Mihai
The undecidable: generalization from Goedel’s mathematical proof as it relates to provability in formal axiomatic theories.
In my own take: a closed system (i.e. limited phase space) is decidable. An open system (of a variable phase space) is undecidable. Physics results in decidable descriptions. Indeed, subject to experiment. Biology descriptions (of the living) are undecidable. The living is an open system of an ever changing phase space.
Mihai Nadin
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Mike Bergman
Hi John,
I did a quick search of my personal reference manager and came up with 150 potential citations that mention 'undecidable' in one way or another. See below.
Thanks, Mike
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To view this discussion on the web visit https://groups.google.com/d/msgid/ontolog-forum/489a8342b5af442d8a25831c43b92b4f%40bestweb.net.
Mike Bergman
Hi Kingsley,
I think the addressing of this email got screwed up. If possible (and if it did not already come through), could you forward this message to the ontolog forum?
Thanks, and hope all is well,
Mike
-------- Forwarded Message --------
| Subject: | Re: [ontolog-forum] goedel and more |
|---|
Mike Bergman
Hi Mihai,
The undecidable: generalization from Goedel’s mathematical proof as it relates to provability in formal axiomatic theories.
In my own take: a closed system (i.e. limited phase space) is decidable. An open system (of a variable phase space) is undecidable. Physics results in decidable descriptions. Indeed, subject to experiment. Biology descriptions (of the living) are undecidable. The living is an open system of an ever changing phase space.
When it comes to fundamental premises, the consistency or
emergence of space and time are also undecidable in physics. Your
points about open systems and biology are well taken and (in my
view) correct.
Thanks, Mike
To view this discussion on the web visit https://groups.google.com/d/msgid/ontolog-forum/BL3PR01MB6897DC7A479F507868BCBDE0DA6E2%40BL3PR01MB6897.prod.exchangelabs.com.
Alex Shkotin
John,
Let me correct this way
John,
What a nice day #2 to read this
"Chuck's sentence is true about science and technology: "My hypothesis is that the language of science, technology, and mind is HOL." "
There are a lot of details to discuss later.
But the main direction is to formalize theories we already have in "science, technology"
Thank you,
Alex
--
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Alex Shkotin
Hi Mike,
Just some points:
It's not interesting to formalize one particular proposition of any kind. But if formalized you should point out the real formal language you use.
And this is one idea of theory framework: one knowledge unit and many languages to express the same meaning.
Like here
The sentences of the kind "The laws of nature are consistent across time." is a philosophical one. And any philosophical doctrine may be formalized: we have formalization for Hegel's Science of Logic, E. Zalta issued the first part of formal Metaphysics, etc. Let me point again: English is a HOL, i.e. formal language 🎯
You wrote "Scientific theories (e.g., general relativity or classical or quantum mechanics) can be viewed as formal systems in which the laws of nature are treated like theorems derived from a set of axioms."
But the task is to create formal axiomatic theories for "general relativity or classical or quantum mechanics".
Let me mention that for Euclid's geometry we have maybe 10 axiomatic theories but 0 formal ones.
I am looking at how to formalize Mechanics where we have some axiomatic (more or less) theories.
After formalization it would be very interesting to discuss properties of this theory, like decidability etc.
But formal theory first 👏
I am talking about a very straightforward activity: take a theory and formalize it.
As this is a way to clarify, reorder and concentrate knowledge.
Thanks, Alex
alex.shkotin
IN ADDITION:
Formalization of one proposition is practically useless but very funny.
"The laws of nature are consistent across time." ⇒(English is a HOL)
(((The laws) of nature) are (consistent across time))
Where "the" is an unary operator, "of", "are", "across" - binary one in infix form.
In a slightly veiled form it is said that there are repeating processes. Processes that, having started in one specific state, end in another specific state and always the same one. The main idea of experiments.
By the way, to be "consistent across time" is one of the requirements to be the "law of nature". So we can say that "The laws of nature are consistent across time." BY DEFINITION. 🔬
Alex
Kingsley Idehen
Hi Mike,
Hi Kingsley,
I think the addressing of this email got screwed up. If possible (and if it did not already come through), could you forward this message to the ontolog forum?
Thanks, and hope all is well,
Mike
Yes, all is well :)
Kingsley
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Mike Bergman
Thanks, Kingsley.
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Alex Shkotin
The Concept of "Undecidable" in Experimental Science
The term "undecidable" is not commonly used in the context of experimental science, particularly when discussing observational data. This is because experimental science aims to gather and analyze data to draw conclusions and make predictions.
Here's a breakdown of why "undecidable" might not be a suitable term:
- Data Interpretation: Experimental scientists strive to interpret data within the framework of existing theories or to propose new ones. While data might be challenging to interpret or might lead to conflicting interpretations, it's rarely considered "undecidable."
- Scientific Method: The scientific method involves a systematic approach to observation, hypothesis testing, and data analysis. If data cannot be interpreted or reconciled with existing theories, it often leads to further investigation or the development of new hypotheses.
- Statistical Analysis: Statistical methods are used to analyze experimental data and determine the likelihood of certain outcomes. While statistical analysis can reveal uncertainties or limitations, it doesn't render data "undecidable."
However, there might be instances where the term "undecidable" could be used metaphorically or philosophically:
- Quantum Mechanics: Some interpretations of quantum mechanics, such as the Many-Worlds Interpretation, suggest that certain outcomes might be truly "undecidable" until observed, due to the nature of quantum superposition.
- Theoretical Limitations: In certain theoretical frameworks, there might be inherent limitations or paradoxes that make some questions or observations "undecidable" within that context.
If you can provide more context or specific examples, I might be able to offer a more tailored response. For instance, are you referring to a particular experiment, theory, or philosophical debate?
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