Einstein defines a theory (i. e. a “complete system of theoretical physics”) as a system that “consists of concepts and basic laws to interrelate those concepts and of consequences to be derived by logical deduction.” (Einstein, On the Method of Theoretical Physics)
Given this definition, can we assume that a successful theory is structure preserving (i. e. homomorphic) to, say, the causal structure of the world?
If the answer is yes, then we can infer that that there must be a homomorphism between the structure of the brain and the structure of the world. Consider a claim from Neuroscience: C. R. Gallistel and Adam Philip King define a representation of a system in analogy to the mathematical concept of a homomorphism. More precisely, they put force the proposal that there exists a homomorphism between the representing system, i. e. the brain and the represented system, the world (Gallistel and King, pp. 59-63).
If a theory is a representing system and if Gallistel and King are correct, then there must by a homomorphism between the structure of the brain and the actual structure of the world, insofar as the theory is successful.
It seems to be the case that this conclusion depends crucially on the assumption that successful theories are structure preserving. Are there arguments for or against this assumption?
Best,
Sven Beecken
- Albert Einstein, On the Method of Theoretical Physics, Philosophy of Science, 1934
- C. R. Gallistel and Adam Philip King, Memory and the Computational Brain: Why Cognitive Science Will Transform Neuroscience, Blackwell, 2010.
The] “preservation of certain concepts” shows that theories are homomorphic to the structure of reality. I'm sorry for the mistake, Sven, instead of 'behavior', it should be 'preservation'
The problem is that there are many different theories and which explain partial effects. For instance, Classical Mechanics and Relativistic Mechanics are two theories of the same reality, only change the level of energy that the particles have.
Dear Daniel Baldomir,
Thank your interesting answer! If I understand you correctly, then the fact that we have many different theories that deal with partial effects means that the answer is: no. Informally, a homomorphism between two structures preserves the operations of the structures. That is to say, each theory that deals with a partial effect of reality is a structure that preserves the operations responsible for a partial effect. So, one can say that Classical Mechanics and Relativistic Mechanics preserve the operations responsible for effects at different energy levels, hence, of partial effects, without denying that both theories deal with the same reality.
If this is correct, then one could still argue that there is a structure preserving relation between successful theories and parts of reality.
Best,
Sven Beecken
Dear Sven,
The homomorphisms preserve the operations which in the case of Classical Mechanics we can think that they are Galilean transformations and Lorentz transformations for Relativistic Mechanics. Thus we have one problem for identifying a homomorphism associated to even operations. I don't want to say that you cannot do it formally, but its usefulness would be doubful.
There is a form to overcome this difficulty just to take into account that Classical Mechanics might be obtained as a "subtheory" of the Relativistic Mechanics. But you have to define clearly every operation that you consider. In any case what forbits you is to have an isomorphism.
Dear Daniel,
You are certainly right, an actual argument that shows the desired relation would require a clear and detailed discussion. At this point, I would consider it a success if it is not impossible that theories are homomorphic to parts of reality.
If it could be shown that this relation exists, then this might be of some interest to the philosophy of science. In particular with regard to the question: why can we even understand reality (to the degree that we actually do understand it)?
If a theory where, say, merely a way to summarize the observed course of events, then the question is difficult even to approach. If, on the other hand, a successful theory is in principle related to reality by preservation of operations, we might find that this is connected to whatever allows understanding.
Thank you for your helpful comments, you have given me a lot to think about.
Best,
Sven Beecken
'Are successful theories homomorphic to the actual structure of reality?'
he theory that describes reality contains some concepts depending on the scale of the description. The transition from one scale to another involves the modification of certain concepts and the introduction of new ones. See, for example, classical and quantum mechanics. I think that just in the behavior of certain concepts is a positive answer to the above question:-)
Dear Jerzy,
Thank you for your answer. Unfortunately, I have to admit that I don't entirely understand your argument. Let me try to clarify:
(1.) and (2.) are certainly true, as your examples show. What I don't understand is how the behavior of concepts shows the existence of a homomorphism. I think that there is a mapping between concepts and entities in world within successful theories, but I don't see how this relates to preservation of operations. Would you be so kind to elaborate on this a bit?
Best,
Sven Beecken
The] “preservation of certain concepts” shows that theories are homomorphic to the structure of reality. I'm sorry for the mistake, Sven, instead of 'behavior', it should be 'preservation'
Not necessarily...First of all, there is no way to check. Second, a functional system - one that permits you to do things in a repeatable way - can be functional only within certain limits (as in the iconic example of Newtonian physics) or might contain unnecessary complexities that nevertheless "work".
Dear Jerzy, Michael and readers,
The clarification helps a lot, thank you Jerzy. I would like to address the issues raised by Michael. The most serious issue is the possibility that the claim could not be falsified. If that is the case, we have left the realm of empirical inquiry. I think this can be answered. If we assume that we understand reality insofar as the best theories allow us to, then we can state the claim on a par with the Gallistel King- model. That is to say, we assume a mapping between “symbols”, i. e. physically realized entities within the mind/brain of an individual and entities in the world. Under this assumption, one can argue that theories can be included in the model. Hence, we would get an empirical claim that could – in principle – be falsified. I say in principle, because to the best of my knowledge the contemporary understanding of the mind/brain is not advanced enough.
The second issue seems less problematic - if I understand it correctly, that is. That particular theories, i. e. functional systems, have limits would only entail that there are limits to what we can understand. Within these limits, we can have a homomorphism. That's what I had in mind when I sad the we might have to think of parts of reality, not “the reality” as such. The unnecessary complexities might be understood within the Gallistel King- model. After all, simplification seem to be correlated to understanding.
Best,
Sven Beecken
Doesn't a "homomorphism between the structure of the brain and the actual (?) structure of the world" imply that both structures can, at least in principle, be computed? My 'question to your question', Sven, is: are you sure that memory is not an analogous system?
Doesn't a "homomorphism between the structure of the brain and the actual (?) structure of the world" imply that both structures can, at least in principle, be amenable to a digital approach ? Thus, my "question to your question", Sven, is: how can we be sure that memory and thought aren't analogous processes (i.e., not amenable to a digital approach: remember Hofstadter and Sander's subtitle to their Surfaces and Essenses, 1913: "Analogy as the fuel and fire or thinking")?
Dear Auguste,
I'm not a mathematician, I'm afraid. So, it might be that a homomorphism between two formal systems, one representing the brain (or a particular system within the brain/mind) and parts of the structure of reality (as far as science can discover the “actual” structure) entails that both systems are computable. Let us assume that this is correct. If the brain is in some sense analogous, then we would have a contradiction to the thesis that there is a homomorphism between theories and the world.
I'm not familiar with the publication that you cite, but the thesis that the brain/mind is a computational system is a long standing issue within the philosophy of mind. Interestingly, current approaches, such as connectionist models seem to stand in conflict with the idea that the processes in the brain/mind are discrete. The reason is that connectionist models take synapses into account which seems to be best thought of as analogous (see the entry on The Computational Theory of Mind in The Stanford Encyclopedia of Philosophy).
These are hard empirical questions. As I have said, I'm rather skeptical about our understanding of the brain/mind at this point. It might be of some interest that Gallistel and King state that they “are profoundly skeptical about the hypothesis that the physical basis of memory is some form of synaptic plasticity, the only hypothesis that has ever been seriously considered by the neuroscience community. The obvious question is: Well, if it’s not synaptic plasticity, what is it? Here, we refuse to be drawn. We do not think we know what the mechanism of an addressable read/write memory is, and we have no faith in our ability to conjecture a correct answer.” (Gallistel and King, p. xvi)
Best,
Sven Beecken
https://plato.stanford.edu/entries/computational-mind/
A system functions in certain ways prior to human analytical capacities, and are then correctly deduced by human analytical capacities: while accepting the production of evidence of the early structure of the universe, I'm not sure we can truthfully know. We know the system we experience, but one we didn't? That must surely remain in the realms of speculation?
Dear Stanley,
I did not mean to suggest that we can go beyond our best scientific theories. Hence, we can be as certain as these theories allow us to be. One way to put the problem is this: Given that we are able to correctly deduce how a particular system functions, the question becomes why are we able to do this?
We might find an answer by looking at us in the same way we look at other systems in nature and then ask: What system is responsible for our analytic capacities and how is this system related to the systems that our best theories describe?
Best,
Sven Beecken
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