Need to know the Topology on the Space of all Theories?

In summary: then we have to at least know to topology on the space of all possible theories to know if convergence possible in the first place.
  • #1
FallenApple
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So according to Dr. Frederic Schuller, we need to at least know the topology on the space of all theories in order to know that we are getting closer to the truth. I take that this is because we need to know the topology to establish that convergence is possible in the first place. How does this stack against the idea that some theories predict better numerically? Is it because it could be that numerical precision is only convergence in numbers but not necessarily in truth? Then how does one even prefer any theory over another? Or is it that among equally predictive theories, no further notions of convergence can be established?

He says that if we cannot establish a transitive quality function to know if we are getting closer to the truth, then we have to at least know to topology on the space of all possible theories to know if convergence possible in the first place, but that we cannot hope to do so because we do have such a space to work with.

Thoughts?



Discussion peaks roughly at 1:21:45
 
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  • #2
The first thing that occurs to me is that the discussion presupposes that there is such a thing as a Grand Unified Theory that explains everything and nowhere contradicts itself. I see no reason to suppose that is the case. We have lots of beautiful, useful theories, each of which work well within certain restricted domains. These could be like the charts of local coordinates on a manifold. The fact that such local charts exist in no way implies that there exists a global chart and, for many manifolds, it is possible to show that there cannot be a global chart.

The second thing is that, in order to define convergence, I imagine we need not only a topology but also a metric*. We need to be able to put a real number on the distance of one theory from another. It's easy to put a metric on theories. For instance we can find the shortest expression of each in a given language and alphabet and then convert those expressions into integers by regarding them as numbers in a base-n system where n is the number of symbols in the alphabet. But that metric bears no relation to the accuracy of a theory. I think it would be hard and likely impossible to devise a metric that measured accuracy. I didn't have time to watch the video, but I expect he may be pointing out similar problems.

* However I found the following about an analog to the notion of convergence, for non-metric topologies. https://math.stackexchange.com/questions/2012020/cauchy-sequence-in-non-metric-space
 
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  • #3
andrewkirk said:
The first thing that occurs to me is that the discussion presupposes that there is such a thing as a Grand Unified Theory that explains everything and nowhere contradicts itself. I see no reason to suppose that is the case. We have lots of beautiful, useful theories, each of which work well within certain restricted domains. These could be like the charts of local coordinates on a manifold. The fact that such local charts exist in no way implies that there exists a global chart and, for many manifolds, it is possible to show that there cannot be a global chart.

The second thing is that, in order to define convergence, I imagine we need not only a topology but also a metric*. We need to be able to put a real number on the distance of one theory from another. It's easy to put a metric on theories. For instance we can find the shortest expression of each in a given language and alphabet and then convert those expressions into integers by regarding them as numbers in a base-n system where n is the number of symbols in the alphabet. But that metric bears no relation to the accuracy of a theory. I think it would be hard and likely impossible to devise a metric that measured accuracy. I didn't have time to watch the video, but I expect he may be pointing out similar problems.

* However I found the following about an analog to the notion of convergence, for non-metric topologies. https://math.stackexchange.com/questions/2012020/cauchy-sequence-in-non-metric-space
That makes a lot of sense. There might not be one grand truth that exists for everything simultaneously. So we have to approximate with charts that together do not build up to a whole. It's a bit similar in flavor to Godel incompleteness, or the ontic status of a particle prior to measurement, and some other phenomena that I'm not thinking about. So the idea is that truth is more of a local phenomenon can be generalized.

What's also interesting is figuring out what's the dividing line between what makes sense in a chart and when it stops making sense when it stops making sense when zoomed out.I think his point is that if we cannot find a metric, then the next best thing is neighborhoods, which gives a very loose view on the relation between points, and that it is needed to find a mapping based on the natural numbers to the topological space where in the long run, the neighborhoods containing the "truth" would eventually capture every output. It seems to be a much weaker notion than convergence using a metric since the neighborhoods that contain the "truth" could have many points with no way to order them. I think this might give "quasi-truth" if there is such a thing. So the truth searching function eventually gets "closer" to the truth, in a way, without actually reaching it. Then he said that not even this is possible because there is no topological space of all possible theories describing a phenomenon. Which is similar to the chart thing you mentioned, but focused on one domain and the logic inverted, where there is apparently one truth out of many hypothesis( ultimately, all but one has to be false).

His definition of convergence is:
A sequence ##q## ( i.e a map ##q:\mathbb{N}->M##) on a topological space ##(M,\mathcal{O})## is said to converge against a limit point ##a\in M## if

## \forall \mathcal{U} \subseteq \mathcal{O} : a\in \mathcal{U}, \exists N\in \mathbb{N} : \forall n>N, q(n) \in \mathcal{U}##
I think in statistics there are some ways of measuring. Things like the Kullback Leiber divergence, which measure the distance between two distributions, or the Fisher info metric, which I believe works on a manifold. And then we can look at the distances between distributions on a hypothesis space of all hypothesis which can form a manifold. Not sure if statistical theory currently robust enough to arrive at the truth this way. This seems to go way beyond regular hypothesis testing, where there is usually a single pair of null and alternative.
 
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  • #4
I think it is not likely we will find a metric since metric implies Hausdorff , meaning we would be able to somehow separate any two theories with an open set, which seems to imply theories do not have too much overlap -- or we may have to collapse theories with significant overlap into a single point/object/theory. Convergence in non-metric topologies is done through nets: you have "chains" of comparable elements ( the directed sets) that somehow come together. This is the type of convergence used, e.g., in Riemann integration. You have a net assigned to the partitions , defined by inclusion, so you end up with chains of inclusion, then your Riemann integral converges . Sorry, they're closing the coffee shop down, I will elaborate on this tmw if you or anyone is interested.
 
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  • #5
WWGD said:
I think it is not likely we will find a metric since metric implies Hausdorff , meaning we would be able to somehow separate any two theories with an open set, which seems to imply theories do not have too much overlap -- or we may have to collapse theories with significant overlap into a single point/object/theory. Convergence in non-metric topologies is done through nets: you have "chains" of comparable elements ( the directed sets) that somehow come together. This is the type of convergence used, e.g., in Riemann integration. You have a net assigned to the partitions , defined by inclusion, so you end up with chains of inclusion, then your Riemann integral converges . Sorry, they're closing the coffee shop down, I will elaborate on this tmw if you or anyone is interested.

What about if the amount of overlap between the theories are somehow encoded in the structure of the propositions? For example, Relativity is different from Newtonian physics but it's also similar in the idea that concepts such as time, momentum, space, etc are used in both, but concepts such as absolute time and space is deleted in the latter and used in the former. So there are new propositions, deleted propositions, etc. Maybe we can build a topology and nets based on this. New theories are not completely new, it uses preexisting notions, deletes things and add things, but usually wouldn't replace everything. But when it does, it can be disjoint.

Ok, sure, thanks. This is a very interesting topic.
 
  • #6
FallenApple said:
What about if the amount of overlap between the theories are somehow encoded in the structure of the propositions? For example, Relativity is different from Newtonian physics but it's also similar in the idea that concepts such as time, momentum, space, etc are used in both, but concepts such as absolute time and space is deleted in the latter and used in the former. So there are new propositions, deleted propositions, etc. Maybe we can build a topology and nets based on this. New theories are not completely new, it uses preexisting notions, deletes things and add things, but usually wouldn't replace everything. But when it does, it can be disjoint.

Ok, sure, thanks. This is a very interesting topic.
Just to followup, re Riemann's integral and Net Convergence, we have the directed sets d in D are finite sets of partition points in the interval of integration [a,b] of f , i.e.,
## a\leq x_1, <x_2,...,<x_n \leq b ## and R(f)_d converges netwise to the number R if the net R_d (f):= Riemann integral using partition choice d in D is eventually in every neighborhood of R, ordered by inclusion , i.e., ## d_i < d_j ##(with < as a generalization of ordering) if ## d_i \subset d_j ##. Here, R is a Real number, so its neighborhoods are of the form ##(R-a,R-b) ##, but we can use ##(R- \epsilon, R+ \epsilon) ##. Then the net ##R_d(f) \rightarrow R ## Netwise ,iff(Def.) for any ## \epsilon >0 ##
##|R(f)_{d_i} -R | < \epsilon ## whenever ##d_i <d_j ##, where this last means the net ##R(f)_d ## is eventually in every 'hood of R.
 
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  • #7
WWGD said:
Just to followup, re Riemann's integral and Net Convergence, we have the directed sets d in D are finite sets of partition points in the interval of integration [a,b] of f , i.e.,
## a\leq x_1, <x_2,...,<x_n \leq b ## and R(f)_d converges netwise to the number R if the net R_d (f):= Riemann integral using partition choice d in D is eventually in every neighborhood of R, ordered by inclusion , i.e., ## d_i < d_j ##(with < as a generalization of ordering) if ## d_i \subset d_j ##. Here, R is a Real number, so its neighborhoods are of the form ##(R-a,R-b) ##, but we can use ##(R- \epsilon, R+ \epsilon) ##. Then the net ##R_d(f) \rightarrow R ## Netwise ,iff(Def.) for any ## \epsilon >0 ##
##|R(f)_{d_i} -R | < \epsilon ## whenever ##d_i <d_j ##, where this last means the net ##R(f)_d ## is eventually in every 'hood of R.
Apple, there is an actual topology define on the collection of theories. I did a small presentation for class, but that was a while back. It is Stone Spaces:
See here : https://en.wikipedia.org/wiki/Type_(model_theory) Go to the part on Stone Spaces for a topology defined on a collection of theories.
 
  • #8
WWGD said:
Apple, there is an actual topology define on the collection of theories. I did a small presentation for class, but that was a while back. It is Stone Spaces:
See here : https://en.wikipedia.org/wiki/Type_(model_theory) Go to the part on Stone Spaces for a topology defined on a collection of theories.
Ah ok got it. So the spaces have to be Hasudorff and totally disconnected. Does this imply the theories are independent? Because some theories are just modifications of others and so might not follow the Hausdorff condition.
 
  • #9
FallenApple said:
Ah ok got it. So the spaces have to be Hasudorff and totally disconnected. Does this imply the theories are independent? Because some theories are just modifications of others and so might not follow the Hausdorff condition.
I think it depends on the underlying Boolean algebra. Isomorphic Boolean algebras give rise to homeomorphic Stone spaces, which can then be collapsed to a single theory space, but let me double check on that. But you're right that the Stone spaces are totally -disconnected , Hausdorff and Compact. I am thinking of something alone the lines of the standard Cantor Set.
 

1. What is topology?

Topology is a branch of mathematics that studies the properties of geometric shapes and spaces that are preserved under continuous deformations, such as stretching, twisting, and bending.

2. What is the Space of all Theories?

The Space of all Theories refers to the collection of all possible scientific theories or models that can explain a particular phenomenon.

3. Why is it important to know the topology on the Space of all Theories?

Knowing the topology on the Space of all Theories allows scientists to understand how different theories or models are related to each other and how they can be connected or transformed. This can help in evaluating the validity and accuracy of different theories and identifying possible gaps or overlaps between them.

4. How do scientists determine the topology on the Space of all Theories?

Determining the topology on the Space of all Theories involves analyzing the structure and relationships between different theories using mathematical tools and techniques. This can include studying the properties of the theories themselves, as well as the connections between them.

5. Can the topology on the Space of all Theories change over time?

Yes, the topology on the Space of all Theories can change as new evidence or data is discovered, leading to the development of new or modified theories. The understanding of the relationships between theories can also evolve as more research is conducted and new connections are discovered.

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