Comment by tsimionescu

You've translated my informal problem statement into a quasi-formal process, using your inherent natural language processing skills, and your knowledge of general human concepts like size. But you haven't explained the formal process you followed to go from my problem statement to this pseudocode.

And your pseudocode template only works for one particular kind of informal problem statement. If I instead have the problem "how much money do I need to buy this house and this chair?", or "does this byte fit in my mouth?", your general form will not work.

And what's more, you haven't actually produced a formally solvable problem definition, that we could analyze for complexity and computability, because you rely on two completely unspecified functions. Where is the formal defintion of a button? Is it a physical push button or a UI control or a clothing button? What does it mean that it is bigger or smaller? When do we know it's big enough, is that computable? And how do we scale it up? Do we increase its volume? Its surface area? One of its sides? Or maybe the radius? And how do we go about doing that? All of these, and many more, need to be explicitly defined in order to apply any kind of formal analysis to this problem. And there is no formal way to do so in a way that matches the intent of whoever posed the problem.

What is the repeatable method by which you came to that conclusion? That is what needs to be formalized for your response to make sense.

At best, this is a formal process for manipulating certain kinds of statements. But the general problem, "take a human's statement of a problem and translate it into a formal statement of a problem that, if solved, will address what the human was asking for" is far harder and more nebulous. Ultimately, it's exactly the problem that LLMs have been invented for, so it has been studied in that sense (and there is a broad literature in AI for NLP, algorithm finding, expert systems, etc). But no one would claim that they are even close to having a formal specification of this problem that they could analyze the complexity of.

I'm not saying that the statement "I want this button to be bigger" can't be formalized. I'm saying that there is no formal process you can follow to get from this problem to a formal problem that is equivalent. There isn't even a formal process you can use to check if a formal definition is equivalent to this problem.

Consider that if someone asked you solve this problem for them with just this statement, either of the following could be a sketch of a more formal statement of what they actually want:

1. In a given web page, the css class used for a particular <button> element should be changed to make the button's height larger by 10%, without changing any other <button> element on the page, or any other dimension.

2. For a particular piece of garment that you are given, the top most button must be replaced with a different button that appears to have the same color and finish to a human eye, and that has the same 3D shape up to human observational precision, but that has a radius large enough to not slip through the opposing hole under certain forces that are commonly encountered, but not so large that it doesn't fit in the hole when pushed with certain forces that are comfortable for humans.

I think you would agree that (a) someone who intended you to solve either of these problems might reasonably describe them with the statement I suggested, and (b), that it would be very hard to devise a formal mathematical process to go from that statement to exactly one of these statements.

But here's the thing, it's not that the statement isn't formalisable, it's the method that you used to formalise it isn't formalisable.

Yeah you could make it one pixel bigger but if someone asked you that, is that what they actually want?