Stephen Wolfram

JUNE 28, 2023

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 ?”

Computer 117

Computer Calculus Construction Mathematics 117

Stephen Wolfram

JANUARY 9, 2024

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.

Computer 103

Computer Calculus Construction Accessibility 103

Stephen Wolfram

NOVEMBER 10, 2021

It’s yet another surprising construct that’s arisen from our Physics Project. And—it should be said at the outset—we’re still only at the very beginning of nailing down those technical details and setting up the difficult mathematics and formalism they involve.) In some ways it’s a bit like our efforts to construct the ruliad.

Physics 122

Physics Mathematics Computer Construction 122

Stephen Wolfram

SEPTEMBER 23, 2021

And at first I did so in the main scientific paradigm I knew : models based on mathematics and mathematical equations. From mathematics. Mathematical physics. By the late 1970s, though, there were other initiatives emerging, particularly coming from mathematics and mathematical physics. Synergetics.

Computer 55

Computer Mathematics Science Sciences 55

Stephen Wolfram

FEBRUARY 14, 2023

Then we might make a mathematical guess, like that perhaps we should use a straight line as a model: We could pick different straight lines. It’s just something that’s mathematically simple, and we’re used to the fact that lots of data we measure turns out to be well fit by mathematically simple things.

Computer 145

Computer Architecture Mathematics Construction 145

Stephen Wolfram

JUNE 29, 2022

we have a new symbolic construct, Threaded , that effectively allows you to easily generalize listability. Why is such a construct needed? Now “characters” could be 16-bit constructs, with nearly 65536 possible “glyphs” allocated across different languages and uses (including some mathematical symbols that we introduced).

Calculus 114

Calculus Construction Computer Argumentation 114

Stephen Wolfram

JANUARY 31, 2023

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. There were still mathematical loose ends, as well as issues such as its application to living systems and to systems involving gravity.

Energy 88

Energy Physics Argumentation Mathematics 88

Stephen Wolfram

SEPTEMBER 9, 2021

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. In the mathematical paradigm one imagines having a mathematical equation and then separately somehow solving it.

Science 65

Science Sciences Physics Mathematics 65

Stephen Wolfram

SEPTEMBER 9, 2021

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. In the mathematical paradigm one imagines having a mathematical equation and then separately somehow solving it.

Physics 64

Physics Science Sciences Mathematics 64

Stephen Wolfram

SEPTEMBER 29, 2023

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.

Physics 108

Physics Computer Argumentation Construction 108

Stephen Wolfram

FEBRUARY 3, 2023

But it is only now—with ideas that have emerged from our Physics Project —that I think I can pull all the pieces together and finally be able to construct a proper framework to explain why—and to what extent—the Second Law is true. In some types of rules it’s basically always there , by construction.

Computer 92

Computer Energy Physics Construction 92

Stephen Wolfram

JUNE 8, 2022

But with the multicomputational paradigm there are now some more general concepts that can be applied to these constructs. And once again we can construct a multiway graph that represents all possible sequences of “moves” in the game. And once again we can construct a multiway graph for it. After 10 steps the graph is. &#10005.

Physics 71

Physics Computer Construction Equality 71

Robert Talbert, Ph.D.

APRIL 15, 2022

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. But no matter the form, they are are typically one-and-done: Students turn them in, and the work is graded, and that's that. Limit the frequency of reassessments.

Academic Standards 52

Academic Standards Communications Calculus Argumentation 52

Stephen Wolfram

AUGUST 22, 2023

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”. So what happened is Marvin [Minsky] and I basically fleshed out the idea of a mathematical thing.

Computer 67

Computer Physics Science Sciences 67

Stephen Wolfram

DECEMBER 13, 2021

the same integral could still be done, but only in terms of elliptic integrals : Mathematical Functions: A Milestone Is Reached. Special functions are in a sense a way of packaging mathematical knowledge : once you know that the solution to your equation is a Lamé function , that immediately tells you lots of mathematical things about it.

Construction 66

Construction Computer Physics Accessibility 66