The background of my question relies on the following points.

  • Quine-Duhem thesis: it is difficult to test a theory with an experiment, because the test would rely on other theories.

  • The discussion of the previous point is difficult because we do not have a "theory of everything", so it is not clear if the chain of theories involved in an experiment ends somewhere.

  • It could be nice to make an attempt to reverse the logic: imagine a universe ruled by a given mathematical model (that we completely know), including entities able to perform experiments, and try to discuss to which extent they are able, by means of experiment, to discover the rules.

  • This approach does not require to mimic the real physical laws. In the past, it has been tried to imagine very unusual hypothetical universes and discus what the hypothetical inhabitants would see, e.g., "Flatland", or the non-Euclidean geometries. In some cases, scientists later discovered that the imagined universe had some relevance for the real life.

Now, the question is: given a mathematical model of a hypothetical universe, possibly including entities (animals, or robots) that are able to perform experiments, how is it possible to discuss the results of their experiments? And the models and theories that they can test and verify? Is there any philosophical attempt in this direction? Can anyone suggest some literature?

I do not think that these questions have a straightforward answer.

Just to avoid misunderstandings: I already deposited a pre-print on this topic, with some thoughts in this direction, but without references to literature. I do not put here the link because I do not want that my question looks as an advertisement for my manuscript.

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    Do you ask how the experimenting entities in your universe can discuss the results of these experiments, or how we, external observers can discuss these results? – Dirk Horsten Apr 24 at 11:01
  • It seems to me that the greatest obstacle for this thought experiment (like all others) to wield successful results is that it itself is unisolable from external (human) thought processes. We cannot but have a biased position. – Joachim Apr 24 at 12:12
  • @DirkHorsten : I'm asking about the second. I imagine to simulate the sytem, and then conclude that the robot did an experiment and got a given result, which would imply a given choice between theories (but without discussing the ability of the robot to design the experiment or to make logical reasonings on the theories). – Doriano Brogioli Apr 24 at 12:28
  • Got a cite for (link to) that preprint? – John Forkosh Apr 24 at 12:48
  • @Joachim : I partially agree with you! But, on one hand, most persons would think that the problem is trivial instead. On the other hand, your negative feeling could be correct, but maybe something interesting can be said anyway (I believe that I was able to partially takle the problem mathematically). Whatever is the outcome of the discussion (even negative) I think that it should deserve space in literature. – Doriano Brogioli Apr 24 at 12:51

The thought experiment is to build a simulated universe that is

ruled by a given mathematical model (that we completely know), including entities able to perform experiments, and try to discuss to which extent they are able, by means of experiment, to discover the rules.

One might imagine that the results of their experiments will lead right back to the mathematical model, that we already completely know, upon which the simulation was constructed.

The goal is to avoid the Quine-Duhem thesis: (Wikipedia)

...that it is impossible to test a scientific hypothesis in isolation, because an empirical test of the hypothesis requires one or more background assumptions....

However, one has embedded in the simulation itself the hypothesis that the mathematical model has something to do with our reality. This would be the background assumption that would make any results found by the isolated simulation problematic.

Here are the questions:

Now, the question is: given a mathematical model of a hypothetical universe, possibly including entities (animals, or robots) that are able to perform experiments, how is it possible to discuss the results of their experiments? And the models and theories that they can test and verify? Is there any philosophical attempt in this direction? Can anyone suggest some literature?

If the experiments inside the simulator are determined, with possible random inputs, by a mathematical model, then in order to get information about what is going on add additional software, that is, "instrument" the code, to print out a trace of interesting states that can be analyzed outside the simulation. This provides a window to see what is going on while the simulation is running.

Simulations are valuable, however, one should not assume that the programmed entities have any understanding nor can they make any but optimization "choices" on input values. One reference to help avoid this is John Searle's, "Minds, Brains and Programs". In particular he writes this about simulations and strong AI:

The idea that computer simulations could be the real thing ought to have seemed suspicious in the first place because the computer isn't confined to simulating mental operations, by any means. No one supposes that computer simulations of a five-alarm fire will burn the neighborhood down or that a computer simulation of a rainstorm will leave us all drenched. Why on earth would anyone suppose that a computer simulation of understanding actually understood anything?

If the simulation that one has in mind with this thought experiment has anything to do with programming an entity so the entity understands one should address Searle's objections. His objections will likely be among the first philosophical objections raised against it.

Searle, J. R. (1980). Minds, brains, and programs. Behavioral and brain sciences, 3(3), 417-424.

Wikipedia contributors. (2018, January 13). Duhem–Quine thesis. In Wikipedia, The Free Encyclopedia. Retrieved 13:03, April 24, 2019, from https://en.wikipedia.org/w/index.php?title=Duhem%E2%80%93Quine_thesis&oldid=820226330

  • Interesting! But I have two comments to focus more. 1) My question explicitly says that the model does not need to be connected to the real universe. It can be as strange as you want. 2) I also explicitly clarified that I do not expect the robots (or animals) to be able to think (not even to analyze a theory) nor to design an experiment. The "intelligence" will be provided by us, who are providing the initial state of the simulation and looking at the evolution. – Doriano Brogioli Apr 24 at 14:01
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    @DorianoBrogioli If you don't expect the robots to think then you don't have to worry about Searle's objections to strong AI. However, the mathematical model upon which the simulation was built may be an hypothesis that would allow people to invoke the Quine-Duhem thesis. – Frank Hubeny Apr 24 at 14:58
  • the Q-D thesis can be invoked, of course, but it will be possible to mathematically prove if the Q-D thesis is true or not. Since the model is perfectly known, we will discover if (inside that particular model) the Q-D thesis is true. My opinion is that this would be interesting, also in the case in which the model is extremely different from our universe. – Doriano Brogioli Apr 24 at 15:10
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    @DorianoBrogioli One might be able to show that the Q-D thesis is false in that simulation since we know precisely what the answers would be in the simulation - they would agree with the mathematical model. We don't know if there even is a mathematical model that governs reality no matter how predictive some of our models are. – Frank Hubeny Apr 24 at 15:14
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    @DorianoBrogioli Yes, that is what the instrumentation would provide. – Frank Hubeny Apr 25 at 13:05

I do not think that these questions have a straightforward answer.

"approach" might be more appropriate than "answer". Let's say your universe's "entities" comprise a "collection" (set or whatever) E. And let's say your model also describes a "configuration space" accompanying E that we'll call its "state space", S(E). Then we could typically represent/define an experiment as an initial-state to final-state mapping S(E)-->S(E).

So now, you want your entities to be able to perform such experiments, i.e., to be such mappings. That is, your entity collection E must itself contain a subset of the operator space S(E)-->S(E). And that's a pretty typical operator algebra property, e.g., the bounded operators of a Hilbert space are again a Hilbert space.

The above is far from a model of what you're asking for, but maybe it suggests an approach/direction that might be followed to pursue such a model. And, as per my comment above, can you provide a link to your preprint, which might provide enough additional info to suggest a clearer direction. (And then, math.se or similar, might be a way better place for discussing any such model.) P.S. Thanks for the link.

Edit -- I see from your preprint (which I've only just briefly glanced through), you're developing a model based on your "robots". Maybe consider them instead as "operators", but also as computable operators, which can be represented as programs. And then a book like Computability in Analysis and Physics https://books.google.com/books/about/?id=t9FwDgAAQBAJ develops that in detail. But it's not immediately clear to me how you'd directly apply those ideas to your kind of model.

But it's likewise not immediately clear to me how your complexity discussion is barking up the right tree. For example, the Shannon or von Neumann entropy/information acquired from performing a measurement is not (I don't think, but am not immediately 100% sure) related to the complexity of the experimental apparatus. Indeed, consider a yes/no test for a pure state. Exactly the same apparatus answers both yes or no, but you acquire lots more information about the measured system if the answer happens to be "yes".

  • Do you have literature references? Or you plan to write something along these lines yourself? To focus more: I was thinking of more fundamental models. E.g. a cellular automaton: we can build a computer inside them, or a robot (and even a self-replicating robot). What happens if we give an initial configuration which evolves as a robot performing an experiment? The mapping S(E) -> S(E) is the time-evolution in my case. – Doriano Brogioli Apr 24 at 13:40
  • it's better you add the text as a comment so that the others can follow the flow! Reply to the question about Shannon entropy/information: actually, it has nothing to do with my approach. I believe that my approach cannot be guessed from the title: unfortunately, it is not a straightforward application of well known concepts. I hope you will have time to read the pre-print. – Doriano Brogioli Apr 24 at 13:54
  • I added a textbook reference in the above Edit before seeing your comment. But it's not 100% related to your particular interest. Just generally discusses computability with respect to operator spaces. And, no, I wasn't planning on writing anything myself. Just read your post and found it interesting, which started me thinking along these lines. My background (see profile) is physics and cs (emphasize the "s" for my interests, though I've done lots of programming). And that combination immediately suggested the "space of operators" approach. I hadn't previously thought about it at all. – John Forkosh Apr 24 at 13:54
  • @DorianoBrogioli Yeah, I'll take a more careful look. Sorry about the mistaken entropy discussion -- just briefly saw what I (apparently mis-)interpreted as a development foundationally based on complexity considerations. – John Forkosh Apr 24 at 13:58

I browsed quickly your preprint , and I read your comments here. Forgive me if I am off mark, but you might also find these thoughts useful (or not, but you wouldn't waste a lot of time reading this answer anyway ). I would suggest to look into AIT, algorithmic information theory (not just complexity classes). This field has been initiated by Leibniz, and later rigorously developed by Kolmogorov and Chaitin. Note that there is a strong connection here with entropy and information theory, so I think Hubeny's suggestion above is actually quite relevant in the following. I will refer to the information content of an object, whether the mathematical framework of a whole universe or a robot performing experiments within this universe. You can easily translate this notion to a notion related to the algorithmic complexity of these objects. Your universes seem to be cellular automata (Fredkin's perspective ) . My guess is that the object "a robot performing an experiment " can find the mathematical "rules" of the universe in which it is embedded (seen as another object) iff the informational content of the robot is comparable with the informational content of the embedding universe. No surprise then, that there is a connection between transformations under a certain threshold of complexity, and experimentally equivalent robots. It would be interesting if your results could be extended to stochastic models , not just deterministic models, like cellular automata. Note that probability distributions (just an example) can be classified within the framework of AIT, so you could consider stochastic models also within an AIT framework. I did not have time to study your work and the validity of your results, but my first impression is "interesting perspective".

You might also find this paper interesting. I would also recommend Tipler's "Omega Point Theory", as well as the work of Teilhard de Chardin. Wigner' s "Unreasonable Effectiveness of Mathematics in Natural Sciences" is also interesting reading, as well as Tegmark's work..

As a final remark, I would suggest the following project. Let's assume that a satisfactory mathematical model of the mind is possible. Let's also assume that a satisfactory mathematical model of a TOE (theory of everything, or quantum gravity ) is also possible. Could there be a connection between these two mathematical theories? Unfortunately , a mathematical theory of the mind, as well as quantum gravity are both work in progress. Not quite there yet, but if such a strong connection exists, I think that would require some reflection.

Translated into the world that you describe in your paper, if the robot performs experiments and finds the rules governing its own existence/consciousness/functionally, then the robot performs experiments and finds the rules governing the universe into which it (and everything) is embedded, and if this robot finds strong connections between these two sets of rules, well, I think it/he would be quite an impressed robot.

  • Very interesting suggestions. All the three answers received up to now contain good ideas to make progresses. Your comment "No surprise ..." is encouraging also: you feel that my main result is correct. However, it seems that there is still no literature tackling the problem explicitly (although, from the literature, my results can be obtained relatively easily). Do you have suggestions on how to stimulate such a discussion? And: is there anyone available for working together on the subject? – Doriano Brogioli Apr 27 at 10:42

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