Stephen Wolfram

SEPTEMBER 27, 2024

Some involve alternate functional forms; others involve introducing additional functions, or allowing multiple arguments to our function f. But functions that for example test whether a Turing machine will ever halt (or give the state that it achieves if and when it does halt) are not in general primitive recursive.

Computer 112

Computer Mathematics Argumentation Science 112

Stephen Wolfram

JANUARY 9, 2024

So many discoveries, so many inventions, so much achieved, so much learned. And key to everything we do is leveraging what we have already done—often taking what in earlier years was a pinnacle of technical achievement, and now using it as a routine building block to reach a level that could barely even be imagined before.

Computer 100

Computer Calculus Construction Accessibility 100

Robert Talbert, Ph.D.

APRIL 1, 2022

Both grades and marks are shorthand indicators of some level of achievement; the mark is given at the level of an individual piece of student work. This is the concept behind ungrading , which both David and I have tried and have written about ( here’s David’s article , here’s mine ).

Primary 52

Primary Mathematics Algebra Argumentation 52

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 121

Physics Mathematics Computer Construction 121

Stephen Wolfram

SEPTEMBER 9, 2021

Part of what this achieves is to generalize beyond traditional mathematics the kind of constructs that can appear in models. Events are like functions, whose “arguments” are incoming tokens, and whose output is one or more outgoing tokens. But there is something else too—and it’s from this that the full computational paradigm emerges.

Physics 67

Physics Science Sciences Mathematics 67

Stephen Wolfram

SEPTEMBER 9, 2021

Part of what this achieves is to generalize beyond traditional mathematics the kind of constructs that can appear in models. Events are like functions, whose “arguments” are incoming tokens, and whose output is one or more outgoing tokens. But there is something else too—and it’s from this that the full computational paradigm emerges.

Science 61

Science Sciences Physics Mathematics 61

Stephen Wolfram

MAY 17, 2021

Traditional blockchains achieve consensus through what amounts to a centralized mechanism (even though there are multiple “decentralized” copies of the blockchain that is produced). In both these cases, the rule successfully achieves “global consensus”. So what other cellular automaton rules achieve consensus like this?

Computer 52

Computer Achievement Mathematics Physics 52

Stephen Wolfram

MARCH 7, 2022

And in what follows we’ll see the great power that arises from using this to combine the achievements and intuitions of physics and mathematics—and how this lets us think about new “general laws of mathematics”, and view the ultimate foundations of mathematics in a different light. So how about logic, or, more specifically Boolean algebra ?

Mathematics 119

Mathematics Physics Computer Algebra 119

Stephen Wolfram

MAY 20, 2021

1”) releases that deliver our latest R&D achievements—both fully fleshed out, and partly as “coming attractions”—much more frequently. Particularly notable is significantly faster rendering on Windows platforms, achieved by using DirectWrite and making use of GPU capabilities. We released Version 12.2 on December 16, 2020.

Computer 52

Computer Construction Physics Technology 52

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 Argumentation 81

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”. An idea that was someone’s great achievement had been buried and lost to the world. A test example coming soon is whether I can easily explain math ideas like algebra and calculus this way.)

Physics 114

Physics Computer Biology Mathematics 114

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

Science 120

Science Sciences Computer Mathematics 120

Stephen Wolfram

AUGUST 22, 2023

Ed was never officially a “test pilot”, but he told me stories about figuring out how to take his plane higher than anyone else—and achieving weightlessness by flying his plane in a perfect free-fall trajectory by maintaining an eraser floating in midair in front of him. Then McCarthy started to explain ways a computer could do algebra.

Computer 70

Computer Physics Science Sciences 70