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 118

Computer Calculus Construction Mathematics 118

Stephen Wolfram

MARCH 7, 2022

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.

Mathematics 120

Mathematics Physics Computer Algebra 120

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. But what’s important for our purposes here is that in the setup Carnot constructed he basically ended up introducing the Second Law.

Energy 88

Energy Physics Argumentation Mathematics 88

Stephen Wolfram

NOVEMBER 10, 2021

It’s yet another surprising construct that’s arisen from our Physics Project. For integers, the obvious notion of equivalence is numerical equality. In some ways it’s a bit like our efforts to construct the ruliad. The whole continuum of all real numbers is “from the outside” in many ways a simple construct.

Physics 121

Physics Mathematics Computer Construction 121

Stephen Wolfram

JUNE 8, 2022

But with the multicomputational paradigm there are now some more general concepts that can be applied to these constructs. Imagine that rather than playing a specific game, we instead at each step just make every possible move with equal probability. The Icosian Game & Some Relatives. After 10 steps the graph is. &#10005.

Physics 71

Physics Computer Construction Equality 71

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

DECEMBER 13, 2021

When you do operations on Around numbers the “errors” are combined using a certain calculus of errors that’s effectively based on Gaussian distributions—and the results you get are always in some sense statistical. Also in the area of calculus we’ve added various conveniences to the handling of differential equations.

Construction 67

Construction Computer Physics Accessibility 67