Stephen Wolfram

MARCH 7, 2022

One can view a symbolic expression such as f[g[x][y, h[z]], w] as a hierarchical or tree structure , in which at every level some particular “head” (like f ) is “applied to” one or more arguments. So how about logic, or, more specifically Boolean algebra ? We’ve looked at axioms for group theory and for Boolean algebra.

Mathematics 115

Mathematics Physics Computer Algebra 115

Futurum

SEPTEMBER 12, 2023

Library and research skills cover areas such as knowing how to reference and cite authors properly, being able to discern between reliable and unreliable sources of information, accessing scientific literature and giving accurate evidence-based arguments when writing scientific essays and reports. What do students learn from studying this?

Biology 81

Biology STEM Chemistry Critical Thinking 81

Stephen Wolfram

JUNE 28, 2023

And, yes, when you try to run the function, it’ll notice it doesn’t have correct arguments and options specified. But what if we ask a question where the answer is some algebraic expression? Here’s an example: We’ve got an argument n that’s declared as being of type MachineInteger. And now in Version 13.3

Computer 119

Computer Calculus Construction Mathematics 119

Stephen Wolfram

SEPTEMBER 29, 2023

Since the standard Wolfram Language evaluator evaluates arguments first (“leftmost-innermost evaluation”), it therefore won’t terminate in this case—even though there are branches in the multiway evaluation (corresponding to “outermost evaluation”) that do terminate. As the Version 1.0

Physics 101

Physics Computer Argumentation Construction 101

Magic EdTech

MAY 13, 2022

In language arts, students can create two contrasting media messages that employ persuasive techniques to capture opposing sides of an issue, instead of just examining the impact of persuasive techniques in a formal argument. The curriculum for algebra classes, for example, will move at a faster and more efficient pace.

Curriculum 52

Curriculum Algebra Argumentation Personalized Learning 52

Stephen Wolfram

SEPTEMBER 27, 2024

Some involve alternate functional forms; others involve introducing additional functions, or allowing multiple arguments to our function f. But it turns out that the fact that this can happen depends critically on the Ackermann function having more than one argument—so that one can construct the “diagonal” f [ m , m , m ].

Computer 108

Computer Mathematics Argumentation Science 108

Stephen Wolfram

JANUARY 23, 2025

Introducing Tabular Manipulating Data in Tabular Getting Data into Tabular Cleaning Data for Tabular The Structure of Tabular Tabular Everywhere Algebra with Symbolic Arrays Language Tune-Ups Brightening Our Colors; Spiffing Up for 2025 LLM Streamlining & Streaming Streamlining Parallel Computation: Launch All the Machines!

Computer 62

Computer Algebra Data Science Construction 62

Robert Talbert, Ph.D.

MAY 19, 2023

This is especially important if you’re writing an article involving multiple sources, or asking one source to critique the arguments of another: It’s quite likely that they aren’t talking about the same thing. Why would an instructor use ungrading? I will speak for myself here.

Publication 73

Publication Communications Construction Academic Standards 73

Stephen Wolfram

JANUARY 9, 2024

The function Map takes a function f and “maps it” over a list: Comap does the “mathematically co-” version of this, taking a list of functions and “comapping” them onto a single argument: Why is this useful? But we wanted to be able to compute hundreds of different functions to arbitrary precision for any complex values of their arguments.

Computer 96

Computer Calculus Construction Accessibility 96

Robert Talbert, Ph.D.

MARCH 30, 2022

Last semester, when I learned I would be teaching Modern Algebra a third-year level course on number theory, rings, and fields in January, I knew I wanted to make some changes to how I'd taught it in the past. I wrote about that in my previous post. One of the changes I decided to make was to make a full-throated leap into ungrading.

Mathematics 52

Mathematics Algebra Argumentation Communications 52

Robert Talbert, Ph.D.

APRIL 1, 2022

In fact because of Butler and Nisan, there is a good argument to be made for decoupling marks from feedback and then throwing out the marks, basing the course grade just on feedback, student portfolios, and self-evaluations. So interpret the third pillar as If you give marks at all, they should indicate progress.

Primary 52

Primary Mathematics Algebra Argumentation 52

Robert Talbert, Ph.D.

APRIL 15, 2022

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. Nobody likes traditional grading because it is so soul-sucking and time-consuming, so why do it more often than necessary? This has a connection with the next point.

Academic Standards 52

Academic Standards Communications Calculus Algebra 52

Stephen Wolfram

AUGUST 29, 2024

In the end—after all sorts of philosophical arguments, and an analysis of actual historical data —the answer was: “It’s Complicated”. A test example coming soon is whether I can easily explain math ideas like algebra and calculus this way.) Infrageometry—as its name suggests—starts from something lower level than traditional geometry.

Physics 112

Physics Computer Biology Mathematics 112

Stephen Wolfram

SEPTEMBER 9, 2021

Events are like functions, whose “arguments” are incoming tokens, and whose output is one or more outgoing tokens. The systems can be based on Boolean algebra, database updating or other kinds of ultimately computational rules. At the level of individual events, ideas from the theory and practice of computation are useful.

Physics 65

Physics Science Sciences Mathematics 65

Stephen Wolfram

SEPTEMBER 9, 2021

Events are like functions, whose “arguments” are incoming tokens, and whose output is one or more outgoing tokens. The systems can be based on Boolean algebra, database updating or other kinds of ultimately computational rules. At the level of individual events, ideas from the theory and practice of computation are useful.

Sciences 60

Sciences Science Physics Mathematics 60

Stephen Wolfram

MARCH 5, 2024

In 2000 I was interested in what the simplest possible axiom system for logic (Boolean algebra) might be. of what’s now Wolfram Language —we were trying to develop algorithms to compute hundreds of mathematical special functions over very broad ranges of arguments. Back in 1987—as part of building Version 1.0

Sciences 116

Sciences Science Computer Mathematics 116

Stephen Wolfram

AUGUST 22, 2023

Then McCarthy started to explain ways a computer could do algebra. It was all algebra. And the only conclusion we can arrive at is that a person can’t do this much algebra with the hope of getting it right.” Richard Feynman and I would get into very fierce arguments. And he says “There’s a problem. It’s just my nature.

Computer 69

Computer Physics Science Sciences 69

Stephen Wolfram

NOVEMBER 10, 2021

For example, we know (as I discovered in 2000) that (( b · c ) · a ) · ( b · (( b · a ) · b )) = a is the minimal axiom system for Boolean algebra , because FindEquationalProof finds a path that proves it. And we can trace the argument for this to the Principle of Computational Equivalence.

Physics 117

Physics Mathematics Computer Construction 117

Stephen Wolfram

MAY 20, 2021

we’re connecting to “Descartes-style” analytic geometry, converting geometric descriptions to algebraic formulas. Given three symbolically specified points, GeometricTest can give the algebraic condition for them to be collinear: &#10005. Tree takes two arguments: a “payload” (which can be any expression), and a list of subtrees.

Computer 52

Computer Construction Physics Technology 52

Stephen Wolfram

MAY 17, 2021

In the basic definition of a standard cellular automaton, the rule “takes its arguments” in a definite order. But what kind of integro-differential-algebraic equation can reproduce the time evolution isn’t clear. RandomGraph[{20, 40}, EdgeStyle -> Gray, VertexStyle -> Table[i -> (RandomInteger[] /. {0

Computer 52

Computer Achievement Mathematics Physics 52

Stephen Wolfram

DECEMBER 14, 2022

Almost any algebraic computation ends up somehow involving polynomials. can be manipulated as an algebraic number, but with minimal polynomial: &#10005. And all of this makes possible a transformative update to polynomial linear algebra, i.e. operations on matrices whose elements are (univariate) polynomials. &#10005.

Computer 75

Computer Astronomy Construction Algebra 75