I know that this sounds like a very rediculous question, but I have very good reason to believe that dogs are alot smarter than we actually think. Here, read THIS and tell me what you can make of it.
Your answer is in your own cite:
Remember that you’re talking about creatures who can’t figure out how to unwind themselves from a wound up tether.
I know, but some dogs are smarter than others.
My Siberian Husky knows how not to get the tether tangled – when we’re walking, if he goes around a post or tree the wrong way, he stops, backs up and goes around the right way.
OTOH, when I was teaching calculus, I knwo that a number of the students in the class were dogs.
Is this the same kind of thing where, when you catch a ball that has been thrown to you, you have effectively solved a higher order equation, regardless of the fact you couldn’t write out a proof or anything?
I’m not entirely sure I follow that; however, when I taught, there were a few foxes in my classes.
Regardless, I’m sure Mathochist will come tell us about the Turing-Something Thesis about how all computation scheme thingies are identical and how he doesn’t subscribe to it.
Until then, this is not a new question. Economists have, for some time now, been experimenting w/ animals and find them to choose rationally and, therefore, optimally. But to find an optimum often requires calculus to do it explicitly—especially with the sorts of problems economists deal with, since they often deal w/ curves. How would a rat know how to alter his root beer consumption optimally in response to a change in price?!
Personally, it seems reasonable that nature, i.e. evolution, wouldn’t favor wasteful animals. In response to a GQ question I posted once, I was informed that as little as 3% difference in survivability can cause extinction in the less “fit” species. It makes sense that animals would have evolved to be non-wasteful, and that would imply they behave optimally. I would suggest, without argument or proof, that there is some sort of built-in math coprocessor, or whatever its called, to help us make these optimizing decisions. Obviously, dogs don’t “know” calculus—ever met a raven that could count past six?—but surely nature could hard-wire the brain so that optimal solutions are implicitly solvable, no?
The paper linked to above deals with something dogs are designed to do: Chasing prey. Chasing prey must be done efficiently because wild dogs don’t have much energy to waste, or that’s what I’d assume to be the case. It would be interesting to see a dog making a decision that isn’t its bread and butter, and see whether it can solve the problem optimally. I have no clue what that’d be, but it would be interesting. Especially if the dog was cute.
I think the solution to this apparent dilemma is that the dog (and us for that matter) solve the problem in a very different method when we catch a ball than when we solve it via a calculus equation. I do not know what that method is, but it would be quite a surprise to me if our subconscious brains used the same mathematical methods. Hell, maybe the method it uses can’t even be called a form of math.
Granted that they’re not the same models, but why would they not be mathematic? Okay, this may be a completely different question: If the dog knows that it is slower in water, isn’t adjusting for that inhherently mathematical? Or am being over-broad in my use of “math”?
Related: Don’t dogs, when chasing a hare who’s path is initially perpendicular to the dog’s path, take a cycloid path, and is that optimal?
Well, I only bring the possibility that it’s not mathematical because I have no idea what’s going on there.
Saying “dogs know calculus” because they are able to catch a ball or catch prey using fairly complicated maneuvers is putting the cart before the horse. Calculus doesn’t govern anything; it simply describes behaviors in the material world. It’s a written set of mathematical notations that are a shorthand - a complex shorthand, to be sure - for describing physical phenomena, whether they be the cycloid path of a dog chasing a rabbit or the sequence of points on a piece of graph paper.
Dogs, or humans, or shrews, or any other living being, don’t have to understand the concepts of calculus - to “know” it - to get anything done. Witness our amazing ability to survive as a species for countless millennia before the births of Newton and Leibnitz. Calculus isn’t an a priori concept - humans invented it in order to further their understanding of the world.
js_africanus, I think there’d be a stronger argument for some sort of “compressor” if a dog were able to catch an airborne ball unerringly every single time it was thrown, or if predators brought down their prey every time they went hunting. That doesn’t happen - more often that not the dog misses and ends up chasing the ball after it lands, and prey escapes fairly frequently. As it is, the only thing at work in either case is instinct, and nothing more.
Could be a lookup table with approximations. That’s not mathematical.
No, I’m not entirely joking, either. Basically, you have to put it down to the fact that ‘thinking’ has a few tens of thousands of years of evolution, while bodies have a few million. Whatever they’re doing, be it a lookup table, a calculus function, an analog computer loop describing the function… it has no relationship to concious thought. Instinct, you know?
On the other hand, my standard poodle can open swinging, latched, and knobbed doors, and figured out how to turn on the television with the remote control. We taught him none of this.
That’s impressive.
Turing-Church thesis? What’s the objection to it?
Coprocessor, not compressor.
The prey question doesn’t fly since they’re involved in a game and subject to different constraints. You would no more expect a predator to catch every prey than you’d expect every prey to escape every predator.
The ball catching question is interesting since that’s a problem to solve that dogs really haven’t evolved to do, isn’t it? Unfortunately, there is the question of skill, which may have nothing to do with the question of finding an optimum.
I do like the look-up table idea. Maybe beavers do that—they are nature’s engineers after all.
+MDI, you’ll have to ask Mathochist that, I don’t know what they are.
Neural nets are probably capable of solving differential equations. That could explain it all.
I think the phrase you’re looking for is “Turing Complete”. Basically, anything you can program in one Turing Complete language can be programmed in any other Turing Complete language. What freaks people out is that there are only a handful of required functions required to be Turing Complete… add one, subtract one, move forward one register, move back one register, and a few control functions.
Anyway, I’d say we just do something like a feedback loop and a weighting algorithm (a genetic algorithm, if you prefer). Successful levels of feedback (I.E. those that let us catch the ball) get weighted more heavily than those that don’t… practice practice practice and you’ve got a pretty good approximation. It’s like the game where you have to tanks shooting at each other. No one sits there and works out the angle of the gun, weight of the shell, factors in wind speed, etc. You just shoot and tweek the numbers. That’s all we do… all the “math” such as it is, is taken care of biologically, we’re just plugging different numbers into the function and keeping the ones that work to give us the answer we want.
Dogs may know calculus but I haven’t met one yet that seemed to fully comprehend the intricacies of quantum field theory! 
I basically agree with 1010011010: it boils down to pattern recognition, initial estimation, and repeated trial and error. An algorithm like that probably would’ve evolved pretty quickly; it’s (relatively) simple to implement, and the individuals that implement it well eat better than those that don’t. The ability to learn new patterns increases the overall robustness of the algorithm.
Speaking from personal experience, one of our dogs is terrible at catching things thrown to her. She starts the wrong direction, she over- or under-shoots the target, and on the rare occasions she’s in the vicinity of the target, she misses with her jaws, closing either too soon or too far away. And then she can’t even find the ball/treat on the ground afterwards. Despite that, she’s much, much better at it than she was when we first got her (she was an adult when we took her in). I suppose her brain could’ve gotten better at doing subconscious calculus (while still being pretty bad at it) but I think it’s much more likely that she’s just gotten better at guessing where the ball will go, how fast she needs to move to get there, and making the necessary corrections en route. In short, she’s learned some patterns that she didn’t know before.
D’ohh! :smack:
Not at all. The ball may be moving in a different path than a rabbit, but it’s the exact same principle - be where the target is expected to be in a very short while.
I don’t. Dogs don’t catch every airborne ball, either.
Of course dogs have evolved to catch balls, because they evolved to catch prey. They’ve evolved to catch, period.
There’s the meat of the question. Catching a ball or prey is a skill, based on practice and experience, rather than abstract calculations. Dogs only get good at it by doing it over and over again, rather than being unconsciously aware of formulas and equations that are purely a human invention.
If that were the case, beaver dams shouldn’t look as chaotic as they do - they work quite well, but there’s no common pattern to their constructions. Individual beavers may come up with constructions that are superior to others, but there’s no way of passing that information on either to the next generation or their contemporaries.
Here’s the problem as I see it. Calculus was not something spontaneously invented by a couple of scientists out of thin air - their work is based on, and an extension of, the work of the mathematicians and logicians who preceded them. That work depended on the ability, which only humans possess, of not only embodying abstract concepts in linguistic form but capturing that linguistic form in a physical form - i.e. writing ideas down. Calculus was invented because the mathematicians of the Renaissance read the ideas of the classical mathematicians, formulated their own ideas and theories, and wrote them down - and were in turn read by Newton and Leibniz, who formulated their own ideas and wrote them down.
If the invention of calculus took the collaborative effort of scores of human minds over the better part of centuries, how can it be instinctual in animals?
Rabbits turn, and successful rabbits turn randomly when under close pursuit. Even a predator that chooses the optimal chase strategy with certainty will fail to catch many prey because of this.
Also of note is that catching a ball is not like catching a rabbit. A rabbit isn’t plucked out of the air, and the same level of precision really isn’t required. Not to mention that the problem being solved is different. As noted above, a ball in the air allows for adjustment and updating as it travels in its path; however, the article in to OP talks about choosing an optimal path where no updating is allowed. The dog runs down the beach and hits the water at the optimum point with suprising consistency—that’s different from making a series of guesstimates as to the future location of a ball travelling through the air.
It seem at least arguable that the difficulty that dogs have in learning to catch objects in the air compared to the relative ease in choosing an optimum path through two different media (is that the right word?) suggests that nature has hard-wired a problem solving algorithm for path choosing.
Obviously, experimentation has to be done: Did Elvis learn to choose the optimal path previously to the experimenter’s measurements? Is this just pattern recognition from a lifetime of fetching on the beach? Or would a dog, that has only played fetch on the same flat lawn, but who also swims, make optimal choices when the game of fetch is played on a beach?
And FTR, the beaver remark was an allusion to a joke: A mathematician, a phycisist, and an engineer are asked to find the volume of a red rubber ball. The mathematician measures the diameter and does the calculus. The physicist submerges it in water and measures the water displacement. Then engineer finds the serial number and looks it up on his red rubber ball table.
Actually, I have to admit I didn’t read the article right off; I was arguing more from the nature of caculus and human thought than from the merits and weaknesses of the article itself.
Having read it, though, this particular paragraph raises a huge red flag for me:
In other words, I threw out the data that didn’t conform to the model at all, rather than trying to incorporate it into the model I was putting together. Hardly an ideal approach to science!
If a dog is hard-wired to choose optimally, which as I understand it is the base assumption underlying “knowing” calculus, then jumping into the water straight away (a “couple” of times? How many was it, really?) shouldn’t have been something Elvis did. He should have chosen optimally every time.
As it is, the best we can assume is that Elvis, like any other dog, is merely a happy, drooling poop factory that like to chase balls and really doesn’t care how he gets it as long as he gets it.
No, he just wanted the ball.
I doubt Elvis even thought about whether they were “good” choices or not. More than likely it was running after the ball while it was airborne and then swimming as soon as he saw the ball hit the water and knew for certain where it was.
That would be an interesting experiment.