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That’s the argument of Peter Liljedahl, a professor of mathematics education at Simon Fraser University in Vancouver, who has spent years researching what works in teaching. These are the students who end up hitting a wall when math courses move from easier algebra to more advanced concepts in, say, calculus, he argues. “At
Line, Surface and Contour Integration “Find the integral of the function ” is a typical core thing one wants to do in calculus. But particularly in applications of calculus, it’s common to want to ask slightly more elaborate questions, like “What’s the integral of over the region ?”, or “What’s the integral of along the line ?”
It began partly as an empirical law, and partly as something abstractly constructed on the basis of the idea of molecules, that nobody at the time knew for sure existed. But what’s important for our purposes here is that in the setup Carnot constructed he basically ended up introducing the Second Law.
Then for each function (or other construct in the language) there are pages that explain the function, with extensive examples. So did that mean we were “finished” with calculus? Somewhere along the way we built out discrete calculus , asymptotic expansions and integral transforms. But even now there are still frontiers.
And if we treat these as equivalent and merge them we now get: (The question of “state equivalence” is a subtle one, that ultimately depends on the operation of the observer, and how the observer constructs their perception of what’s going on. It’s a new kind of fundamentally multiway construct.
Just like in our original f [0] = 1 case, we can construct “blue graph trees” rooted at each of the initial conditions. Some involve alternate functional forms; others involve introducing additional functions, or allowing multiple arguments to our function f. So what about the behavior of f [ n ] for large n ?
It’s yet another surprising construct that’s arisen from our Physics Project. In some ways it’s a bit like our efforts to construct the ruliad. In constructing it, one can imagine using Turing machines or hypergraph rewriting systems or indeed any other kind of computational system. As an analogy, consider the real numbers.
When most working mathematicians do mathematics it seems to be typical for them to reason as if the constructs they’re dealing with (whether they be numbers or sets or whatever) are “real things”. And we can think of that ultimate machine code as operating on things that are in effect just abstract constructs—very much like in mathematics.
Its key idea is to think of things in the world as being constructed from some kind of simple-to-describe elements—say geometrical objects—and then to use something like logical reasoning to work out what will happen with them. It’s not difficult to construct multiway system models. There are multiway Turing machines.
Sometimes textbooks will gloss over everything; sometimes they’ll give some kind of “common-sense-but-outside-of-physics argument”. In some types of rules it’s basically always there , by construction. But one never quite gets there ; it always seems to need something extra. But the mystery of the Second Law has never gone away.
Those assessments can take on various forms, and in well-constructed courses they do have varying forms, corresponding to different levels of Bloom's Taxonomy. An argument for traditional grading goes like this: Sure, a single assessment might have a grade on it that doesn't accurately reflect student understanding.
Its key idea is to think of things in the world as being constructed from some kind of simple-to-describe elements—say geometrical objects—and then to use something like logical reasoning to work out what will happen with them. It’s not difficult to construct multiway system models. There are multiway Turing machines.
Even if I hadn’t been able to imagine quite what could be built on them, I’d been able to construct solid foundations, that successfully encapsulated things in the cleanest and simplest ways. In the end—after all sorts of philosophical arguments, and an analysis of actual historical data —the answer was: “It’s Complicated”.
It didn’t help that his knowledge of physics was at best spotty (and, for example, I don’t think he ever really learned calculus). He was going for what he saw as the big prize: using them to “construct the universe”. Richard Feynman and I would get into very fierce arguments. But Ed wasn’t interested. It’s just my nature.
we have a new symbolic construct, Threaded , that effectively allows you to easily generalize listability. You can give Threaded as an argument to any listable function, not just Plus and Times : ✕. we’re adding SymmetricDifference : find elements that (in the 2-argument case) are in one list or the other, but not both.
there’s a new construct NetExternalObject that allows you to run trained neural nets “from the wild” in the same integrated framework used for actual Wolfram-Language-specified neural nets. Calculus & Its Generalizations. Is there still more to do in calculus? In Version 13.2 Another new tree feature in Version 13.2
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