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**Gemma De las Cuevas**: *on* 6/25/20 at 11:47am UTC, wrote Hi Rob, Thanks so much for reading the essay, and I'm really glad you...

**Robert Spekkens**: *on* 6/24/20 at 16:09pm UTC, wrote Hi Gemma, I really enjoyed your essay! I particularly appreciated...

**Gemma De las Cuevas**: *on* 5/14/20 at 13:58pm UTC, wrote Dear Markus, thank you so much for your comments! I'm so glad you enjoyed...

**Markus Mueller**: *on* 5/14/20 at 12:56pm UTC, wrote Dear Gemma, I enjoyed reading your essay immensely! I feel like I have...

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**Alyssa Adams**: *on* 5/11/20 at 1:37am UTC, wrote Hi Gemma! This was a very fun read! I really love your point...

**Gemma De las Cuevas**: *on* 5/7/20 at 16:06pm UTC, wrote Dear Branko, Thanks for your comment, but I disagree. I don't think it's a...

**Branko Zivlak**: *on* 5/6/20 at 15:24pm UTC, wrote Dear Gemma De las Cuevas When I read the essays of young scientists a look...

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Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020)
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TOPIC: Universality Everywhere implies Undecidability Everywhere by Gemma De las Cuevas [refresh]

TOPIC: Universality Everywhere implies Undecidability Everywhere by Gemma De las Cuevas [refresh]

Why is it so easy to generate complexity? Because essentially every non-trivial system is universal, that is, capable of exploring all complexity in its domain. In this sense there is Universality Everywhere. Automata, spin models and neural networks are the best understood examples of systems that jump to universality, and I also discuss some more speculative ones. One crucial consequence of universality is that it allows for self-reference and negation, which gives rise to paradoxes such as ‘I am a liar’. The truth of this statement cannot be established — it is undecidable. The liar paradox is at the core of the undecidability results in automata, provability theory, set theory, truth theory and epistemology. It is a very powerful paradox which cannot be fixed in a finite way. Undecidability is thus an inescapable consequence of the expressive power of the system, that is, of universality. In this sense, Universality Everywhere implies Undecidability Everywhere. I discuss some conceptual and practical implications of undecidability, as well as perspectives of overcoming these limitations. I also argue that universality is a statement about the power of simulation, that is, of how much a system can reach with the help of additional variables, and I discuss the relation of this notion of universality with emergence.

Gemma De las Cuevas is an Assistant Professor at the Institute for Theoretical Physics of the University of Innsbruck (Austria). Her research is centered around mathematical physics and quantum many-body physics. She is also interested in the universality and undecidability notions presented in this essay.

Dear Gemma De las Cuevas,

"The liar paradox is at the core of the undecidability results in automata, provability theory, set theory, truth theory and epistemology."

If the above assertion is at the heart of undecidability, the problem can be easily solved. Take for instance: 'All Cretes are liars said the Crete', which is a problem only in formal logic, i.e. when the Crete is considered an element of the set of Cretes. The Cretes as a people, however, belong to a different category than individual Cretes. So, the liar paradox speaks of a trait of the people of Crete expressed by an individual Crete, which is perfectly fine. In Tarski's words: the individual Crete speaks in meta-language about an object-language concerning the traits of the people of Crete. The same holds for "This sentence is false" or "I'm lying". In all those examples categorically separated instances play a role (which logicians and computers can not deal with for principled reasons).

The 'undecidability' illness was basically cured

a) by Gödel's (1931) incompleteness theorem, which states that Logic has no domain of applicability other than itself

b) by Tarski's (1935) theory of truth, which states that no object-language can discuss it's truths.

Some illnesses though seem to be fairly attractive...

Heinz

report post as inappropriate

"The liar paradox is at the core of the undecidability results in automata, provability theory, set theory, truth theory and epistemology."

If the above assertion is at the heart of undecidability, the problem can be easily solved. Take for instance: 'All Cretes are liars said the Crete', which is a problem only in formal logic, i.e. when the Crete is considered an element of the set of Cretes. The Cretes as a people, however, belong to a different category than individual Cretes. So, the liar paradox speaks of a trait of the people of Crete expressed by an individual Crete, which is perfectly fine. In Tarski's words: the individual Crete speaks in meta-language about an object-language concerning the traits of the people of Crete. The same holds for "This sentence is false" or "I'm lying". In all those examples categorically separated instances play a role (which logicians and computers can not deal with for principled reasons).

The 'undecidability' illness was basically cured

a) by Gödel's (1931) incompleteness theorem, which states that Logic has no domain of applicability other than itself

b) by Tarski's (1935) theory of truth, which states that no object-language can discuss it's truths.

Some illnesses though seem to be fairly attractive...

Heinz

report post as inappropriate

Dear Heinz,

Thank you for reading the essay and for your feedback.

As I understand, the solution you are proposing to the liar paradox is an example of (the first step in the construction) of a hierarchy to solve the paradox. Namely, you suggest that Cretes as people belong to a different category to individual Cretes, thereby establishing "two levels" which avoid self-reference. With these two levels, one can devise a new liar paradox (which will sound involved). Building hierarchies to escape the paradox is indeed the commonly adopted solution, as mentioned in my essay.

Regarding your two final comments, I am not exactly sure with what you mean in 1). Regarding 2) I agree that Tarski's theory of truth is a way of circumventing the paradox, but I don't think I agree with the statement that this cures the illness - for one reason, it is not a finite solution. (I recommend reading T. Bolander's entry on Self-reference in the Stanford Encyclopedia of Philosophy to learn about approaches to solve the liar paradox).

Thanks for your input again!

Best,

Gemma

Thank you for reading the essay and for your feedback.

As I understand, the solution you are proposing to the liar paradox is an example of (the first step in the construction) of a hierarchy to solve the paradox. Namely, you suggest that Cretes as people belong to a different category to individual Cretes, thereby establishing "two levels" which avoid self-reference. With these two levels, one can devise a new liar paradox (which will sound involved). Building hierarchies to escape the paradox is indeed the commonly adopted solution, as mentioned in my essay.

Regarding your two final comments, I am not exactly sure with what you mean in 1). Regarding 2) I agree that Tarski's theory of truth is a way of circumventing the paradox, but I don't think I agree with the statement that this cures the illness - for one reason, it is not a finite solution. (I recommend reading T. Bolander's entry on Self-reference in the Stanford Encyclopedia of Philosophy to learn about approaches to solve the liar paradox).

Thanks for your input again!

Best,

Gemma

Dear Gemma,

thanks for your reply, there seems to be some common ground...

As regards a) another way to express it is the generally accepted claim that logic has not solved its GROUNDING problem. When logic demands to be universally applicable (to any problem) it is equivalent to saying that it has no domain of applicability.

In my FQXI essays as of 2015 I have argued that legitimate domains, hierarchies or whatever one wants to call these categories are separated by orthogonality and thus rest on Absolut non-contradiction and hence imply nescience. That means, every legitimate object-language is not as per Tarski member of the meta-language, but categorically separated from it. The trouble with logic is in its very positivity (affirmativity)! Whatever we say can not be better as 'not-false' in the context of natural- and legitimate object-languages. That is, the optimum way to speak about the world is the metaphor.

Heinz

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thanks for your reply, there seems to be some common ground...

As regards a) another way to express it is the generally accepted claim that logic has not solved its GROUNDING problem. When logic demands to be universally applicable (to any problem) it is equivalent to saying that it has no domain of applicability.

In my FQXI essays as of 2015 I have argued that legitimate domains, hierarchies or whatever one wants to call these categories are separated by orthogonality and thus rest on Absolut non-contradiction and hence imply nescience. That means, every legitimate object-language is not as per Tarski member of the meta-language, but categorically separated from it. The trouble with logic is in its very positivity (affirmativity)! Whatever we say can not be better as 'not-false' in the context of natural- and legitimate object-languages. That is, the optimum way to speak about the world is the metaphor.

Heinz

report post as inappropriate

Dear Gemma,

I enjoyed your essay tremendously, for not only is well-written and your thesis crystal clear, but I think that it brings forward some deep insights too. In particular, I never thought of universality (as power of simulation) in terms of additional variables. Indeed, it makes a nice connection with my (and Gisin's) proposal, and it would be nice to explore this further.

I found insightful your analysis of (a certain class) of paradoxex as arising from the self-reference problem and negation. And how this issues is implied by universality, in your words: "since a universal system can be fed the description of any other system, it can also be fed its own description".

I also have two naive questions:

1) The whole starting point of your argument, namely the jump to universality, relies on the concept of complexity, which, however, I don't think it is defined in your essay besides an intuitive level (which perhaps is enough!). I was wondering if the universality claim is based on some particula, formal notion of complexity. And since you "define universailty as the ability to explore all complexity in a given domain", is complexity bounded from above?

2) You mention Wolfram's Principle of Computational Equivalence ("all processes which are not obviously simple can be viewed as computations of equivalent

sophistication"). Do you have any intutions on how this is justified?

I wish you the best of luck for the contest!

Cheers,

Flavio

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I enjoyed your essay tremendously, for not only is well-written and your thesis crystal clear, but I think that it brings forward some deep insights too. In particular, I never thought of universality (as power of simulation) in terms of additional variables. Indeed, it makes a nice connection with my (and Gisin's) proposal, and it would be nice to explore this further.

I found insightful your analysis of (a certain class) of paradoxex as arising from the self-reference problem and negation. And how this issues is implied by universality, in your words: "since a universal system can be fed the description of any other system, it can also be fed its own description".

I also have two naive questions:

1) The whole starting point of your argument, namely the jump to universality, relies on the concept of complexity, which, however, I don't think it is defined in your essay besides an intuitive level (which perhaps is enough!). I was wondering if the universality claim is based on some particula, formal notion of complexity. And since you "define universailty as the ability to explore all complexity in a given domain", is complexity bounded from above?

2) You mention Wolfram's Principle of Computational Equivalence ("all processes which are not obviously simple can be viewed as computations of equivalent

sophistication"). Do you have any intutions on how this is justified?

I wish you the best of luck for the contest!

Cheers,

Flavio

report post as inappropriate

Dear Flavio,

Thanks a lot for reading the essay and for your feedback. I am very glad we found some points in common and would very much like to explore them.

Regarding your questions:

1) A priori I don't think one can characterise the jump to universality in terms of some complexity measure, since complexity measures are defined for specific fields (for formal languages, spin...

view entire post

Thanks a lot for reading the essay and for your feedback. I am very glad we found some points in common and would very much like to explore them.

Regarding your questions:

1) A priori I don't think one can characterise the jump to universality in terms of some complexity measure, since complexity measures are defined for specific fields (for formal languages, spin...

view entire post

Dear Gemma,

thanks for a highly readable and insightful essay! I love the phrase 'the long reach of undecidability'; it is perfectly evocative of the often subtle ways in which the phenomenon of undecidability is relevant to often seemingly quite remote areas, such as determining whether a certain Hamiltonian has an energy gap in the ground state.

You present a few first steps...

view entire post

thanks for a highly readable and insightful essay! I love the phrase 'the long reach of undecidability'; it is perfectly evocative of the often subtle ways in which the phenomenon of undecidability is relevant to often seemingly quite remote areas, such as determining whether a certain Hamiltonian has an energy gap in the ground state.

You present a few first steps...

view entire post

report post as inappropriate

Dear Jochen,

Thank you so much for your comments! I believe they are very useful and will help me formalise some of the ideas of the essay - on universality, the jump to universality, and their application to various domains. I will keep you posted if I do any progress in these fronts :) Let's stay in touch.

Thanks again, really.

All the best,

Gemma

Thank you so much for your comments! I believe they are very useful and will help me formalise some of the ideas of the essay - on universality, the jump to universality, and their application to various domains. I will keep you posted if I do any progress in these fronts :) Let's stay in touch.

Thanks again, really.

All the best,

Gemma

Dear Gemma De las Cuevas

When I read the essays of young scientists a look at reference to see whether they are on the right track. To be on the right track, you need to penetrate the minds of the greats of science of the past who have unrivaled achievements (Newton, Boskovic, Maxwell,…, Planck, Einstein, Euler, Ramanujan). Everything else leads to paradoxes, singularities… and a waste of time.

Regarda,

Branko

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When I read the essays of young scientists a look at reference to see whether they are on the right track. To be on the right track, you need to penetrate the minds of the greats of science of the past who have unrivaled achievements (Newton, Boskovic, Maxwell,…, Planck, Einstein, Euler, Ramanujan). Everything else leads to paradoxes, singularities… and a waste of time.

Regarda,

Branko

report post as inappropriate

Hi Gemma!

This was a very fun read! I really love your point "Undecidability is thus an inescapable consequence of the expressive power of a system — it is the other side of the coin of universality. Universality Everywhere, thus, implies Undecidability Everywhere." This is a very good point. but also, it seems that physical systems like biology don't seem to mind.

It's interesting to think that we, as humans, can hold these paradoxes in our heads without any issue. I don't explode (like computers do in movies) when presented with a paradox. We just kinda go "Huh" and then move about our daily lives. it's strange to think that maybe paradoxes don't affect us that much, yet have such a profound effect on mathematics. I personally think it's due to the disconnect between our current mathematical theories and physical reality. We have much more to model!

Cheers!

Alyssa

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This was a very fun read! I really love your point "Undecidability is thus an inescapable consequence of the expressive power of a system — it is the other side of the coin of universality. Universality Everywhere, thus, implies Undecidability Everywhere." This is a very good point. but also, it seems that physical systems like biology don't seem to mind.

It's interesting to think that we, as humans, can hold these paradoxes in our heads without any issue. I don't explode (like computers do in movies) when presented with a paradox. We just kinda go "Huh" and then move about our daily lives. it's strange to think that maybe paradoxes don't affect us that much, yet have such a profound effect on mathematics. I personally think it's due to the disconnect between our current mathematical theories and physical reality. We have much more to model!

Cheers!

Alyssa

report post as inappropriate

Hi Alyssa,

thanks for reading my essay and I'm glad you enjoyed it!

>> This is a very good point. but also, it seems that physical systems like biology don't seem to mind.

I agree with you. It seems that the paradoxes cannot be "implemented" (in biological sytems or in physical computers). Only things that "make sense" can be implemented. Make sense is here defined with respect to the usual logic.

>>It's interesting to think that we, as humans, can hold these paradoxes in our heads without any issue. I don't explode (like computers do in movies) when presented with a paradox.

I don't explode (obviously) but I am left in a blocked state: I don't know how to solve it, and I cannot conclude that 'I am liar' is true nor false. So in this sense I am in a dead end - maybe that's the cognitive version of exploding :)

>>I personally think it's due to the disconnect between our current mathematical theories and physical reality.

I am also very interested in the question of whether one could define some mathematical models that circumvent some of these paradoxes. Or, alternatively, that model better the way we think. Or the two wishes together. The first wish has been addressed in alternative models of truth, such as Kripke's theory of truth and developments thereof. People investigating models of consciousness may be have addressed (in a perhaps oblique way) the second wish.

Thanks again for your input :)

All the best,

Gemma

thanks for reading my essay and I'm glad you enjoyed it!

>> This is a very good point. but also, it seems that physical systems like biology don't seem to mind.

I agree with you. It seems that the paradoxes cannot be "implemented" (in biological sytems or in physical computers). Only things that "make sense" can be implemented. Make sense is here defined with respect to the usual logic.

>>It's interesting to think that we, as humans, can hold these paradoxes in our heads without any issue. I don't explode (like computers do in movies) when presented with a paradox.

I don't explode (obviously) but I am left in a blocked state: I don't know how to solve it, and I cannot conclude that 'I am liar' is true nor false. So in this sense I am in a dead end - maybe that's the cognitive version of exploding :)

>>I personally think it's due to the disconnect between our current mathematical theories and physical reality.

I am also very interested in the question of whether one could define some mathematical models that circumvent some of these paradoxes. Or, alternatively, that model better the way we think. Or the two wishes together. The first wish has been addressed in alternative models of truth, such as Kripke's theory of truth and developments thereof. People investigating models of consciousness may be have addressed (in a perhaps oblique way) the second wish.

Thanks again for your input :)

All the best,

Gemma

Dear Gemma,

I enjoyed reading your essay immensely! I feel like I have learnt a lot — I was not aware before of the Boltzmann-machine and neural-network-types of universality, and it was nice to see the different examples put side-by-side. As universality is everywhere, undecidability must be too. And this raises several fascinating questions: does knowing transcend proving? Can the limitations be overcome by some other type of logic? What about the relation (as pointed out by Deutsch) to physics? Your essay gives a fascinating exploration of those ideas.

One thing that suggests itself to be explored further, I think, is your statement that (Turing-type) universality only happens in digital systems. It feels like there must be some grain of truth to it. But it is probably not that easy to make it fully rigorous. For example, quantum theory has a continuous (non-digital) set of wave functions, and yet, it admits error correction (and contains a kind of “hidden” form of digitality)… So what kind of “digital” we need for reliable self-representation (for the liar paradox, as you pointed out) is perhaps a deeper question?

All the best,

Markus

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I enjoyed reading your essay immensely! I feel like I have learnt a lot — I was not aware before of the Boltzmann-machine and neural-network-types of universality, and it was nice to see the different examples put side-by-side. As universality is everywhere, undecidability must be too. And this raises several fascinating questions: does knowing transcend proving? Can the limitations be overcome by some other type of logic? What about the relation (as pointed out by Deutsch) to physics? Your essay gives a fascinating exploration of those ideas.

One thing that suggests itself to be explored further, I think, is your statement that (Turing-type) universality only happens in digital systems. It feels like there must be some grain of truth to it. But it is probably not that easy to make it fully rigorous. For example, quantum theory has a continuous (non-digital) set of wave functions, and yet, it admits error correction (and contains a kind of “hidden” form of digitality)… So what kind of “digital” we need for reliable self-representation (for the liar paradox, as you pointed out) is perhaps a deeper question?

All the best,

Markus

report post as inappropriate

Dear Markus,

thank you so much for your comments! I'm so glad you enjoyed the essay!

>> quantum theory has a continuous (non-digital) set of wave functions, and yet, it admits error correction (and contains a kind of “hidden” form of digitality)… So what kind of “digital” we need for reliable self-representation (for the liar paradox, as you pointed out) is perhaps a deeper question?

This is a very interesting question. But I don't know if the digitality needs to be "hidden" or mysterious. Consider natural language. At the spoken level it is continuous; yet, as a code, it is digital. The latter is reflected in the fact that it admits a representation with finitely many symbols. Listening to someone speak a language with their accent is doing error correction, i.e. mapping the sounds to a finite set of objects that give rise to a word and thereby an element of the language.

I'd love to discuss this next time we meet :)

Thanks again for reading the essay and for your very encouraging comments.

All the best,

Gemma

thank you so much for your comments! I'm so glad you enjoyed the essay!

>> quantum theory has a continuous (non-digital) set of wave functions, and yet, it admits error correction (and contains a kind of “hidden” form of digitality)… So what kind of “digital” we need for reliable self-representation (for the liar paradox, as you pointed out) is perhaps a deeper question?

This is a very interesting question. But I don't know if the digitality needs to be "hidden" or mysterious. Consider natural language. At the spoken level it is continuous; yet, as a code, it is digital. The latter is reflected in the fact that it admits a representation with finitely many symbols. Listening to someone speak a language with their accent is doing error correction, i.e. mapping the sounds to a finite set of objects that give rise to a word and thereby an element of the language.

I'd love to discuss this next time we meet :)

Thanks again for reading the essay and for your very encouraging comments.

All the best,

Gemma

Hi Gemma,

I really enjoyed your essay! I particularly appreciated all of the scholarship that went into your exposition of the topic. It seems to me that the role of universality is underrepresented in such discussions, so it was nice to see it centre stage in your essay. Concerning this: “Perhaps computable, provable, true, etc could be defined in a completely different yet-to-be-discovered way, which would rid them of the paradoxes.” Do you have any ideas of what this would look like? Also, have you thought at all about whether universality for quantum computation (as opposed to classical computation) comes with any novel consequences for self-reference? Thanks!

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I really enjoyed your essay! I particularly appreciated all of the scholarship that went into your exposition of the topic. It seems to me that the role of universality is underrepresented in such discussions, so it was nice to see it centre stage in your essay. Concerning this: “Perhaps computable, provable, true, etc could be defined in a completely different yet-to-be-discovered way, which would rid them of the paradoxes.” Do you have any ideas of what this would look like? Also, have you thought at all about whether universality for quantum computation (as opposed to classical computation) comes with any novel consequences for self-reference? Thanks!

report post as inappropriate

Hi Rob,

Thanks so much for reading the essay, and I'm really glad you enjoyed it!

>> Concerning this: “Perhaps computable, provable, true, etc could be defined in a completely different yet-to-be-discovered way, which would rid them of the paradoxes.” Do you have any ideas of what this would look like?

Not really. As mentioned in the essay, many people have tried to solve these paradoxes, mainly by building explicit or implicit hierarchies in the definition of truth, or computable, or knowable. (This is wonderfully explained in the Thomas Bolander's article on Self-reference - https://plato.stanford.edu/entries/self-reference/ ). But these hierarchies are infinite, so they only provide a solution in the limit, which I don't see as a solution. But perhaps we ought to take Markus P. Mueller's view and do not see this as a limitation, but rather as an indication that 'what is true / provable / knowable' is the wrong question.

>> Also, have you thought at all about whether universality for quantum computation (as opposed to classical computation) comes with any novel consequences for self-reference?

No, I haven't thought about that yet, but will definitely do. I would also love to discuss this with you :)

Thanks again for reading and commenting, and all the best,

Gemma

Thanks so much for reading the essay, and I'm really glad you enjoyed it!

>> Concerning this: “Perhaps computable, provable, true, etc could be defined in a completely different yet-to-be-discovered way, which would rid them of the paradoxes.” Do you have any ideas of what this would look like?

Not really. As mentioned in the essay, many people have tried to solve these paradoxes, mainly by building explicit or implicit hierarchies in the definition of truth, or computable, or knowable. (This is wonderfully explained in the Thomas Bolander's article on Self-reference - https://plato.stanford.edu/entries/self-reference/ ). But these hierarchies are infinite, so they only provide a solution in the limit, which I don't see as a solution. But perhaps we ought to take Markus P. Mueller's view and do not see this as a limitation, but rather as an indication that 'what is true / provable / knowable' is the wrong question.

>> Also, have you thought at all about whether universality for quantum computation (as opposed to classical computation) comes with any novel consequences for self-reference?

No, I haven't thought about that yet, but will definitely do. I would also love to discuss this with you :)

Thanks again for reading and commenting, and all the best,

Gemma

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