I can’t sleep again. If you don’t know history, you are doomed to repeat it. My diary reads mostly like any text by Berkeley even though 300 years have passed. Is understanding inherently mathematical? One of the things scientism showed me, is that the mere notion of the world as a problem can be very misleading. On a Sartrean notion, we are completely free to make up our categories. This is a great example: the world doesn’t necessarily present itself as a problem to be solved. A difficulty with this view of the world is that the solution would be part of the world, and so the problem itself as well. Is there even a state of the world when it could be called solved if we view all the possible configurations as part of some universal state space? When does a theory end? Must the end of a theory be found within the theory of ends, seeking eradication of itself?
In physics, understanding something means finding mathematical analogy to get right (like fitting into the Standard Model) calculations, in short understanding means to calculate – that fits with the common view of intelligence as optimization. The problem really gets emphasized by recalling the problem of dark matter and dark energy. We can say that science roughly works in the phase of normal science (Kuhnian terminology) at the current era, and I assume there is an epistemological gap between different phases of scientific works. (These phases are probably ideologized, and there could be overlaps between them). Understanding something starts with considering something a problem on both psychological and philosophical level. So how do we understand the world?
When we proceed to understand dark matter and dark energy, do we need to alter the theory of gravity, the theory of dark matter itself, or is it a mathematical problem only? How do we decide? Let’s say in advance that this is not about my view of physics, I will mostly think about the differences between these attitudes while the actual and practical solution is always in the domain of physics itself. With attitude I mainly mean the problem of science as an universal solution that is pretty much an ongoing debate heated up mostly by Feyerabend. Is there a single scientific method you can isolate or only scientific methods with little in common?
Whenever we apply a mathematical apparatus onto certain arguments, it’s clear that the equations themselves are indifferent to their inputs and we can only decide if they are “right” when they are consistent within a system or theory. I really think that the search for a priori principles is not even near to being finished (this is basically The Kantian project itself), because mathematical truths work within their corresponding frameworks only (that makes them a great example of the coherence theory of truth). What is understanding then? If it’s not algorithmic, then it must lie within the framework itself and we could just as well say it’s a matter of perspective. The full understanding may come with both framework and algorithmic part, but the fundamental parts are the essential ones – and it’s not in the algorithms. Why?
Maybe Scott Aaronson framed it the best about the problem of complexity in “Reasons to believe“:
“If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in “creative leaps,” no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss; everyone who could recognize a good investment strategy would be Warren Buffett.”
Even with mathematics getting more and more abstract, including more and more previously metamathematical questions into it’s framework (see Type Theory), the main questions remain the same. Why do some analogies work while others don’t? Which part of a framework is fundamental for understanding? Where does the “meaning” of everything comes from? But as I used to frame the question: how do we understand the world?
This was not just my weird dream and strange categorization. The so-called theory of everything is about this exact problem, a very fashionable problem these days that is kind of strange in a post-Gödelian scientific community. If everything is calculations, that is not more meaningful than saying that everything is in some kind of space.
As it is very imminently grasped after reading a very good summary of the related problem in a paper from 1959, the problem is mostly shadowed by our presuppositions. The mind is not an universal intelligence either, you are always bounded by a framework and as I wrote about it earlier I don’t think that the classical category of consciousness is very helpful to present these questions either. Viewing the world as a problem leads to infinite regression and other known problems that can’t be solved within the given bounds. The biological mind may be different, but it is nowhere universal either, and to solve the universe as a problem you would need an universal intelligence that is probably one of the most paradoxical things one can think of.