[lisper]

Let's go back to the original formulation so we don't lose the plot here:

Me: As an analogy, consider a professional tennis or baseball player.

YeGoblynQueenne: humans e.g. playing baseball do not find solutions to kinematic equations, but instead use simple heuristics that exploit our senses and body configuration, like placing their hands in front of their eyes so that they line up with the ball etc.

At the risk of stating the obvious, being a professional tennis or baseball player involves a lot more than "simple heuristics ... like placing their hands in front of their eyes so that they line up with the ball." That simple heuristic might work for one specific skill -- catching a ball that happens to be heading in your direction. But it won't help much for moving a bat or a raquet in such a way that it will hit a ball moving past you at close to 100mph in such a way that the ball ends up traveling on some desired trajectory.

But even just moving your hand in front of your eyes is nowhere near as trivial as YeGoblynQueenne implies. To do that you have to control seven degrees of freedom: two at your shoulder, two at your elbow, and two at your wrist. Solving those kinematic equations even to find a static solution is elementary but non-trivial, a skill that is solidly at the undergraduate level.

Now consider running to catch a ball. That involves controlling about 20 or 30 degrees of freedom (two arms, two legs, neck, waist, two eyes...) in real time in a situation that involves not just kinematics but also dynamics. Solving that analytically was an unsolved research problem for a long time (maybe still is, I haven't been keeping up with recent developments). A child can learn to do it. But they do have to learn to do it. It's not a skill humans are born with.

It seems pretty obvious to me that the process of learning how to catch a ball while running is very different than the process of learning how to do math. And yet, there must be a mapping between them because the movements required for catching a ball are the solutions to kinematic equations.

[avmich]

I suspect there's a terminological difference.

> being a professional tennis or baseball player involves a lot more than "simple heuristics ...

Mmm, a combination of simple heuristics, all of which are of course learned, but still simple heuristics, could in itself be a simple heuristic. Yet it could allow performing pretty complex-looking actions, including those you described. Simple heuristic here could be a linear or low degree polynomial approximation of a good solution to kinematic equation - not precise, but enough to get to the goal, while learnable and explainable. But still without actual full-blown abstract, mathematically correct complete solution.

[YeGoblynQueenne]

What's meant by heuristics is some times unclear. I wonder if by heuristics you mean a shortcut. In CS and AI our model of a shortcut is the heuristic cost functions in heuristic search algorithms like A* and its variants.

It's interesting because I've thought along the following lines. A* is a pathfinding algorithm so it's a natural choice for path planning -the process of planning a path through some environment for an autonomous agent to follow. The funny thing is, as it turns out, pathfinding can be abstracted as finding a "path" through a graph: a set of nodes connected by edges; and that's a great abstraction for general task planning - the task of achieving any arbitrary objective - so A* is also widely used for task planning.

Well, isn't path planning an almost universal ability of intelligent animals? Most animals are motile for some part of their lives and they seem to use their intelligence at the very least to navigate their environment. So is it that far-fetched to think that an ancestral ability for path planning, essentially identical to a heuristic search algorithm like A*, evolved into general intelligence? And wouldn't that mean that general intelligence can be, ultimately, modeled as some kind of heuristic search?

The answer I think is: no, and that's a dangerous way to think. A model is a model, it's not the process it models. And I think that's my fundamental disagreement with lisper, disregarding my confusion about the meaning of "kinematics".

[YeGoblynQueenne]

>> But even just moving your hand in front of your eyes is nowhere near as trivial as YeGoblynQueenne implies.

Yeah, I was actually thinking of kinematics as in classical mechanics. I think you were speaking about kinematic equations as in robotics. My bad, I misunderstood.

I agree that moving your hand in the right place is not a simple problem, and I don't actually have an insight into that, but I think it's easier than calculating the trajectory of an object, let alone many at once (think juggling). Maybe that's another source of our disagreement- but see my comment about having multiple models for a process.

[lisper]

> Yeah, I was actually thinking of kinematics as in classical mechanics. I think you were speaking about kinematic equations as in robotics.

What do you see as the relevant difference?

> I think it's easier than calculating the trajectory of an object, let alone many at once

Well, yeah, but there isn't anything fundamentally more difficult about juggling. It all boils down to Newton's laws.

My point is that there are two different ways that human brains can apply Newton's laws. We can do it intuitively, without even being consciously aware of Newton's laws, which is why humans were able to throw and catch objects before 1687. Or we can do it consciously by manipulating symbolic representations of the equations of motion. Those two activities are in some sense equivalent because they both involve producing a model of a physical system in our brains and using that model to make accurate predictions about that system. But they are also obviously radically different in other ways, and being skilled at one in now way implies being skilled at the other.

[YeGoblynQueenne]

>> What do you see as the relevant difference?

I'm not an expert in either, so I'm possibly overemphasizing the difference.

[Edit: As far as I understand it, one is useful in predicting the movement of objects outside the body, the other the position of the limbs etc.]

>> My point is that there are two different ways that human brains can apply Newton's laws. We can do it intuitively, without even being consciously aware of Newton's laws, which is why humans were able to throw and catch objects before 1687. Or we can do it consciously by manipulating symbolic representations of the equations of motion. Those two activities are in some sense equivalent because they both involve producing a model of a physical system in our brains and using that model to make accurate predictions about that system. But they are also obviously radically different in other ways, and being skilled at one in now way implies being skilled at the other.

I totally agree with that. Can we agree that we can model whatever our brains do with kinematic equations, but we have no idea what is the true process that is being modeled?

[lisper]

OK, I'm an expert in both, so I can say the following with some authority:

No, we cannot model what our brains do with kinematic equations. Our brains operate according to the laws of neurobiology, which we do not yet fully understand, but which we know enough about to know that they bear absolutely no resemblance to the laws of kinematics. Your brain is not made of mechanical linkages.

Nonetheless, despite the fact that the laws of neurobiology and the laws of kinematics bear no resemblance to each other, our brains somehow manage to produce solutions to problems that require solving kinematic equations. Not only that, but our brains can do this in two completely different ways, one of which is conscious and deliberate (what we call "doing math") and the other of which is instinctive and subconscious (developing sensory-motor skills).

We get leverage out of doing math despite the fact that our brains can solve some of the same problems innately. Likewise, I believe that LLMs could get a lot of leverage if they were augmented with special-purpose modules for doing math and other specific tasks.

[YeGoblynQueenne]

>> No, we cannot model what our brains do with kinematic equations.

I've confused you. My apologies. What I meant with this sentence:

"we can model whatever our brains do with kinematic equations"

Was that we can model whatever our brains do _while catching a ball etc_ by means of kinematic equations. I did not mean that we can model everything our brains do, i.e. the function of the brain, in general. If we could model an entire brain just by kinematic equations we wouldn't need any AI research, and I wouldn't be arguing that we don't know what our brains do when they solve problems that we solve using kinematic equations. Our disagreement is about the solutions our brain finds to that kind of problem.

>> Not only that, but our brains can do this in two completely different ways, one of which is conscious and deliberate (what we call "doing math") and the other of which is instinctive and subconscious (developing sensory-motor skills).

That's my problem with all this - the "subconscious" part. I don't really understand what it means. When I catch a ball, I do it entirely consciously, and I know exactly what I'm doing: I'm extending my hand to catch the ball. I may not be able to articulate every little muscle movement, or describe precisely the position of my arms, my hand, my fingers, the ball, etc, but I do know with great accuracy where those objects are in space, and where they are in relation with each other. I cannot introspect into the intellectual mechanisms by which I know those things, but I do know them, so they're not "subconscious".

The difference you point out, between doing maths with pen-and-paper (or computers) and performing a task without having to do maths-with-pen-and-paper, is, I think, the difference between having a formal language that is powerful enough to describe all the objects and functions I describe above (hand position, muscle movement etc), on the one hand, and not having such a language on the other hand. Somehow humans are able to come up with formal languages with the power to describe some of the things we do, like catching balls etc, and many other things besides. As a side note, we do not have a formal language -we do not have the mathematics- to describe our ability to come up with formal languages, yet. That was be one of the original goals of AI research, although it has now fallen by the wayside, in the process of chasing benchmark performance.

I digress. When I speak of "formal languages", I mean more broadly formal systems, like mathematics (of which logic is one branch, btw). When I speak of a "model" in my earlier comment, I mean a formalism that describes various kinds of human capability, like our catching-balls example. Kinematic equations, that's one such model. But a model is not the thing it, well, models. Is my claim.

I hope this is clear and apologies if it's not. Most of our discussion is not on things of my expertise so I'm trying to find the best way to say them. Also, this is a much less technical discussion and so much less precise, than I'm used to. I hope I'm not wasting your time with needless philosophising.

On the other hand, I think this kind of conversation would be made much easier if we didn't assume human brains. Our ability to navigate, and interact with, our environment, is shared to a greater or lesser extent with many animals that aren't humans and don't have human brains, so whatever we can do with our brains thanks to that shared ability, must also share an underlying system- because we all evolved from the same, very distant, animal ancestors, ultimately, and we must have inherited the same basic firmware as it were.

[lisper]

> Was that we can model whatever our brains do _while catching a ball etc_ by means of kinematic equations.

No, we can't even do that. All we can do is observe that the results of what our brains do happen to be the solutions to kinematic equations. It does not follow that we can model the process of producing those solutions by kinematic equations. It does not even follow that the process of producing those solutions bears any resemblance to what we do when we do math to find them.

Here is an analogy: we can observe that the motions of objects obeys the principle of least action [1] and that to compute the action we have to integrate the Lagrangian. It does not follow that there is anything happening in the physical mechanism that causes particles to move that is even remotely analogous to integrating a Lagrangian.

> When I catch a ball ... I know exactly what I'm doing

No, I don't think you do. If you did, you would be able to describe what you are doing to someone else, and they would be able to reproduce your actions based on that description alone. Alternatively, you would be able to render your knowledge into computer code and build a robot that could do it. But I doubt you can actually do either of those things if your only skill is catching a ball and you are not trained in math.

By way of very stark contrast, I am absolutely terrible at hand-eye coordination tasks, but I can build a machine that is much better at it than I am [2]. Just to be clear, I didn't actually build that particular machine, but I do know how. And so I can tell you that the process of learning how to build a machine that can catch a ball is radically different than the process of learning how to catch a ball yourself.

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[1] https://en.wikipedia.org/wiki/Stationary-action_principle

[2] https://www.youtube.com/watch?v=FycDx69px8U