I've always liked the dog analogy ever since I came across it in Stephen Weinberg's book Dreams of a Final Theory: "...it may be that humans are simply not int

I've always liked the dog analogy ever since I came across it in Stephen Weinberg's book Dreams of a Final Theory: "...it may be that humans are simply not intelligent enough to discover or to understand the final theory. It is possible to train dogs to do all sorts of clever things, but I doubt that anyone will ever train a dog to use quantum mechanics to calculate atomic energy levels." The impossibility of dogs doing Calculus is an equally nice example, but I think we could push the point further. Calculus is not only hard and unnatural for dogs; it is hard and unnatural for humans as well. There were anatomically modern humans equipped with culture and language for something like 30,000 years before even one of them understood calculus. It's truly extraordinary that this thing which was at the absolute limits of the human intellect in the 17th century is now taught routinely in high schools and colleges across the world, and used for such mundane purposes as building bridges or stock trading. Those inclined to optimism could read in this a story about how humans are not like dogs, because culture. (i.e.: "culture allows us to consolidate and simply the understanding of previous generations so future generations can build on it towards almost limitless horizons", etc.). But I'm a pessimist: calculus is still damn hard for nearly all human beings (just ask your average college student). Heck, I found it pretty tough to learn and I ended up getting a masters in math. And - to get back to the article - I'm not convinced that current topics at the edge of human understanding today (string theory, or whatever) are ever going to get 'mundane'. It takes many years of special training and extraordinarily single-minded dedication to even get to the current frontier of knowledge in these areas. At some point, as the article says, perhaps the math just gets too hard. Fortunately, just because the universe might not be comprehensible by humans (we never get to a successful theory of quantum gravity, or whatever) doesn't mean science itself will have a end. There's plenty to do in other fields.