Category: Mathematics | Page 7

Abstract virtual binary code illustration on blurry modern office building background. Big data and coding concept. Multiexposure

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News
Date
March 28, 2021

Categorized
Mathematics

Tagged
___longform, Chaitin's number, Elegant program, Featured, Gregory Chaitin, Law of non-contradiction, Robert J. Marks}

Why the Unknowable Number Exists But Is Uncomputable

_{Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible} _{News
March 28, 2021

17

Mathematics}

In this week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin on his “unknowable number.” That’s the topic of this series, based on the fourth podcast. Last week, we tried getting to know the unknowable number. Today, let’s look at the question of how we know that the number is unknowable — instead of merely non-computable. Lots of things are non-computable but we do not expect that to be true of numbers. Let’s see what’s happening here, as Chaitin offers a walk through his proof that it really is unknowable: https://episodes.castos.com/mindmatters/Mind-Matters-127-Gregory-Chaitin.mp3 This portion begins at 09:43 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: Read More ›

matrix made up of math formulas and mathematical equations - illustration rendering

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News
Date
March 26, 2021

Categorized
Mathematics

Tagged
Elegant program, Featured, Gregory Chaitin, Numbers (types), Real numbers, Robert J. Marks, Unknowable number}

Getting To Know the Unknowable Number (More or Less)

_{Only an infinite mind could calculate each bit} _{News
March 26, 2021

10

Mathematics}

In this week’s podcast, “The Chaitin Interview IV: Knowability and Unknowability,” Walter Bradley Center director Robert J. Marks interviewed mathematician Gregory Chaitin on his discovery of the “unknowable number.” How can a number that is unknowable exist? Some numbers go on indefinitely (.999999999… ) but we can describe them accurately even if they don’t seem to come to an end anywhere. Some numbers, like pi (π), are irrational — pi goes on and on but its digits form no pattern. However, what does it mean to say that a number exists if it is unknowable? How do we even know it exists? That’s the topic of this series, based on the fourth podcast between Dr. Marks and Gregory Chaitin. Note: Read More ›

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News
Date
March 23, 2021

Categorized
Mathematics, Peer Review, Science

Tagged
Elon Musk, Featured, Gregory Chaitin, IBM (and innovation), Robert J. Marks}

Gregory Chaitin on How Bureaucracy Chokes Science Today

_{He complains, They’re managing to make it impossible for anybody to do any real research. You have to say in advance what you’re going to accomplish. You have to have milestones, reports} _{News
March 23, 2021

10

Mathematics, Peer Review, Science}

In last week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on how Stephen Wolfram’s software has taken much of the drudgery out of math. At the same time, in Chaitin’s view, a threat looms: A new, more bureaucratic, mindset threatens to take the creativity out of science, technology, and math: https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 19:45 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: I was sitting down tallying, I think, the intellectual giants that have introduced new mathematical ideas, brand new. I was thinking of people like Claude Shannon, Lotfi Zadeh, yourself… I don’t know if we Read More ›

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News
Date
March 22, 2021

Categorized
Artificial Intelligence, Mathematics, Programming

Tagged
Creativity (and computers), Elon Musk, Featured, Gottfried Wilhelm Leibniz, Gregory Chaitin, Robert J. Marks, Stephen Wolfram}

How Stephen Wolfram Revolutionized Math Computing

_{Wolfram has not made computers creative but he certainly took a lot of the drudgery out of the profession} _{News
March 22, 2021

11

Artificial Intelligence, Mathematics, Programming}

In last week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things mathematical, including why math or engineering geniuses (Elon Musk came to mind, of course) can’t just follow the rules. This week, we look at Stephen Wolfram’s new program that checks your hard math. What can — and can’t — it do for mathematicians? https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 13:22 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: Now, there is what I regard as a piece of AI, so it might be interesting to talk about it. My friend Stephen Wolfram (pictured), the system he’s created, Read More ›

SpaceX Concept Spacecraft in orbit of the Earth. SpaceX Elon Musk Mars programm 3d render

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News
Date
March 21, 2021

Categorized
Artificial Intelligence, Mathematics, Neuroscience, Programming

Tagged
Brain-Computer Interface (BCI), Computer sim universe, Creativity, Elon Musk, Featured, Gregory Chaitin, Kurt Gödel, Neuralink, Robert J. Marks}

Why Elon Musk and Other Geniuses Can’t Afford To Follow Rules

_{Mathematician Gregory Chaitin explains why Elon Musk is, perhaps unexpectedly, his hero} _{News
March 21, 2021

12

Artificial Intelligence, Mathematics, Neuroscience, Programming}

In last week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things mathematical, including why great books on math, advancing new theorems, aren’t written much any more. This week, we look at why geniuses like Musk (whose proposed Mars Orbiter is our featured image above) simply can’t just follow the rules, for better or worse: https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 7:57 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: Look at Elon Musk (pictured). He’s my great hero. He’s a wonderful engineer and he’s a wonderful entrepreneur and he doesn’t follow the rules. Robert J. Marks: He doesn’t, Read More ›

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News
Date
March 20, 2021

Categorized
Mathematics, Peer Review

Tagged
Andrew Wiles, Elon Musk, Featured, G. H. Hardy (1877-1947), Gregory Chaitin, Karl Popper, publish or perish, Robert J. Marks, Srinivasa Ramanujan, William Sealy Gosset}

Why Don’t We See Many Great Books on Math Any More?

_{Decades ago, Gregory Chaitin reminds us, mathematicians were not forced by the rules of the academic establishment to keep producing papers, so they could write key books.} _{News
March 20, 2021

9

Mathematics, Peer Review}

In our most recent podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on many things mathematical, including whether math is invented or discovered. This time out, Chaitin talks about why he thinks great books on math, advancing new theorems, aren’t written much any more: https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 02:49 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: You don’t hear the word “scholarship” very much anymore in academia. Gregory Chaitin: And people don’t write books. In the past, some wonderful mathematicians like G. H. Hardy (1877–1947, pictured in 1927) would write wonderful books like A Mathematician’s Apology (1940) Read More ›

Red and Blue Spiral Fractal Background Image, Illustration - Vortex repeating spiral pattern, Symmetrical repeating geometric patterns. Abstract background

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News
Date
March 19, 2021

Categorized
Mathematics, Peer Review, Philosophy

Tagged
Featured, Gregory Chaitin, publish or perish, Robert J. Marks}

Mathematics: Did We Invent It Or Did We Merely Discover It?

_{What does it say about our universe if the deeper mathematics has always been there for us to find, if we can?} _{News
March 19, 2021

7

Mathematics, Peer Review, Philosophy}

In this week’s podcast, “The Chaitin Interview III: The Changing Landscape for Mathematics,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin (pictured) on how math presents us with a challenging philosophical question: Does math image deep truth about our universe? Or do we just make up these math rules in our own minds to help us understand nature? https://episodes.castos.com/mindmatters/Mind-Matters-126-Gregory-Chaitin.mp3 This portion begins at 00:39 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: Deep philosophical questions have many answers, sometimes contradictory answers even, that different people believe in. Some mathematics, I think, is definitely invented, not discovered. That tends to be trivial mathematics — papers that fill in much-needed gaps because Read More ›

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News
Date
March 16, 2021

Categorized
Arts & Culture, Mathematics

Tagged
Andrew Wiles, Featured, Fermat's Last Tango (musical), Fermat's Last Theorem, Gregory Chaitin, Pierre de Fermat, Robert J. Marks}

Hard Math Can Be Entertaining — With the Right Musical Score!

_{Gregory Chaitin discusses with Robert J. Marks the fun side of solving hard math problems, some of which come with million-dollar prizes} _{News
March 16, 2021

8

Arts & Culture, Mathematics}

In last week’s podcast,, “The Chaitin Interview II: Defining Randomness,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on his method of describing true randomness:. If no theory is simpler than the data you are trying to explain, then the data is random. They also discussed the work of true randomness but also on how Ray Solomonoff (1926–2009), another algorithmic information theory founder, who pursued the “shortest effective string of information that describes an object.” But now, for a lighter touch, we learn that a musical comedy was made of Fermat’s Last Theorem. https://episodes.castos.com/mindmatters/Mind-Matters-125-Gregory-Chaitin.mp3 This portion begins at 19:24 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: If you Read More ›

Infinite random numbers, original 3d rendering background, technology and science concepts

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News
Date
March 12, 2021

Categorized
Mathematics, Programming

Tagged
Algorithmic Information Theory, David Hilbert, Featured, Gottfried Wilhelm Leibniz, Gregory Chaitin, Hermann Weyl, Karl Popper, Randomness, Robert J. Marks}

Chaitin’s Discovery of a Way of Describing True Randomness

_{He found that concepts from computer programming worked well because, if the data is not random, the program should be smaller than the data} _{News
March 12, 2021

12

Mathematics, Programming}

In this week’s podcast, “The Chaitin Interview II: Defining Randomness,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on randomness. It’s a subject on which Chaitin has thought deeply since his teenage years (!), when he published a journal paper on the subject. How do we measure randomness? Chaitin begins by reflecting on his 1969 paper: https://episodes.castos.com/mindmatters/Mind-Matters-125-Gregory-Chaitin.mp3 This portion begins at 1:12 min. A partial transcript, Show Notes, and Additional Resources follow. Gregory Chaitin: In particular, my paper looks at the size of computer programs in bits. More technically you ask, what is the size in bits of the smallest computer program you need to calculate a given digital object? That’s called the program Read More ›

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Jonathan Bartlett
Date
March 8, 2021

Categorized
Mathematics

Tagged
Algebra, calculus, George Berkeley (1685–1753), Mathematics}

The Needless Complexity of Modern Calculus

_{How 18th century mathematicians complicated calculus to avoid the criticisms of a bishop} _{Jonathan Bartlett
March 8, 2021

7

Mathematics}

Would you be surprised if I told you that the essentials of calculus are actually very straightforward and simple? Read More ›

White, grey and pink mandelbrot fractal.

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News
Date
March 7, 2021

Categorized
Mathematics, Religion

Tagged
Anselm of Canterbury, Evolution, Featured, Gregory Chaitin, Incompleteness theorems (Gödel), Kurt Gödel (belief in God), Robert J. Marks, Saint Anselm}

How Kurt Gödel Destroyed a Popular Form of Atheism

_{We don’t hear much about logical positivism now but it was very fashionable in the early twentieth century} _{News
March 7, 2021

6

Mathematics, Religion}

In this week’s podcast, “The Chaitin interview I: Chaitin chats with Kurt Gödel,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin. Earlier, we noted his comments on the almost supernatural awareness that the great mathematicians had of the foundations of reality in the mathematics of our universe. Yesterday, we heard Chaitin’s recollection of how he (almost) met the eccentric genius Kurt Gödel (1906–1978). One way that Gödel stood out from many of his contemporaries was that he believed in God. He even wrote a mathematical proof of the existence of God. https://episodes.castos.com/mindmatters/Mind-Matters-124-Gregory-Chaitin.mp3 This portion begins at 17:16 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: One of the things Read More ›

mandelbrot set abstract mathematics computer generated

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News
Date
March 6, 2021

Categorized
Mathematics

Tagged
Featured, Gregory Chaitin, Kurt Gödel, Robert J. Marks}

Gregory Chaitin’s “Almost” Meeting With Kurt Gödel

_{This hard-to-find anecdote gives some sense of the encouraging but eccentric math genius} _{News
March 6, 2021

7

Mathematics}

In this week’s podcast, “The Chaitin interview I: Chaitin chats with Kurt Gödel,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin. Yesterday, we noted his comments on the almost supernatural awareness that the great mathematicians had of the foundations of reality in the mathematics of our universe. This time out, Chaitin recounts how he (almost) met the eccentric genius Kurt Gödel (1906–1978): https://episodes.castos.com/mindmatters/Mind-Matters-124-Gregory-Chaitin.mp3 This portion begins at 12:42 min. A partial transcript, Show Notes and Additional Resources follow. Robert J. Marks: You mentioned that you read the article about Gödel’s Incompleteness Theorem in Scientific American I also know that you had a near brush with Gödel and I’ve heard the story from you. But Read More ›

Silhouette of human with universe and physical, mathematical formulas

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News
Date
March 5, 2021

Categorized
Artificial Intelligence, Mathematics, Religion

Tagged
Carl Friedrich Gauss, Featured, Georg Cantor, God's existence, Gregory Chaitin, infinity, Kurt Gödel, Leonhard Euler, Robert J. Marks, Srinivasa Ramanujan}

Gregory Chaitin on the Great Mathematicians, East and West

_{Himself a “game-changer” in mathematics, Chaitin muses on what made the great thinkers stand out} _{News
March 5, 2021

12

Artificial Intelligence, Mathematics, Religion}

In this week’s podcast, “The Chaitin interview I: Chaitin chats with Kurt Gödel,” Walter Bradley Center director Robert J. Marks interviewed mathematician and computer scientist Gregory Chaitin on the almost supernatural awareness that the great mathematicians had of the foundations of reality in the mathematics of our universe: https://episodes.castos.com/mindmatters/Mind-Matters-124-Gregory-Chaitin.mp3 This discussion begins at 8:26 min. A partial transcript, Show Notes and Additional Resources follow. Robert J. Marks: There are few people who can be credited without any controversy with the founding of a game changing field of mathematics. We are really fortunate today to talk to Gregory Chaitin (pictured) who has that distinction. Professor Chaitin is a co-founder of the Field of Algorithmic Information Theory that explores the properties of Read More ›

Schoolchildren writing on chalkboard in classroom during math lesson

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Jonathan Bartlett
Date
February 26, 2021

Categorized
Education, Mathematics

Tagged
education, Featured, math, Racism}

Antiracism In Math Promotes Racism and Bad Math

_{If you are scratching your head over how math might be racist, you are not alone} _{Jonathan Bartlett
February 26, 2021

6

Education, Mathematics}

Recently, a conglomeration of California education associations got together to work on a series of resources for mathematics teachers. The goal? Eliminate racism in mathematics classes by promoting Equitable Math. If you are struggling to imagine how mathematics could be racist, you are not alone. I am certain there exist racist teachers, and probably teachers who exhibit racist expectations of their students. I would support any reasonable action to get rid of or reform such teachers. But that is not the primary goal of these resources. The website, equitablemath.org, instead believes that the very way that mathematics is commonly taught is not just racist, but is specifically white supremacist. While I consider myself to be somewhat of a mathematics reformer Read More ›

Educational kids math toy wooden board stick game counting set in kids math class kindergarten. Math toy kids concept.

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Denyse O'Leary
Date
February 16, 2021

Categorized
Education, Mathematics

Tagged
1984 (book), Algebra, Douglas Murray, Edward Feser, G. K. Chesterton, George Orwell, James Lindsay, Nancy Pearcey, Rochelle Gutierrez, War on math}

Yes, There Really Is a War on Math in Our Schools

_{Pundits differ as to the causes but here are some facts parents should know} _{Denyse O'Leary
February 16, 2021

10

Education, Mathematics}

The Oregon Department of Education (ODE) recently encouraged teachers to register for training that encourages “ethnomathematics,” an education trend that argues, “among other things, that White supremacy manifests itself in the focus on finding the right answer”: “The concept of mathematics being purely objective is unequivocally false, and teaching it is even much less so,” the document for the “Equitable Math” toolkit reads. “Upholding the idea that there are always right and wrong answers perpetuate objectivity as well as fear of open conflict.” … An associated “Dismantling Racism” workbook, linked within the toolkit, similarly identifies “objectivity” — described as “the belief that there is such a thing as being objective or ‘neutral’” — as a characteristic of White supremacy. Instead Read More ›

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Richard W. Stevens
Date
February 15, 2021

Categorized
Mathematics, Science

Tagged
Birds, Dinosaurs, Featured, Muscle memory, Neo-Darwinism}

Random Evolution Doesn’t Produce Algorithmic Functions in Animals

_{A bird does not fly just because it has wings; it needs a “flight” program in its brain. Explanations of the evolution of flight do not account for that.} _{Richard W. Stevens
February 15, 2021

7

Mathematics, Science}

In a recent article “Evolution and artificial intelligence face the same basic problem,” Eric Holloway addressed the conundrum faced by artificial intelligence theorists: How can “a random process with no insight into the environment… increase information about that environment within evolving DNA sequences and/or artificial intelligence programs. By what mechanism can randomness ‘know’ anything?” Dr. Holloway’s challenge goes to the heart of the problem with the materialist worldview regarding origins, evolution, and ultimately intelligence. Software vs. hardware in your body Imagine you knew absolutely nothing about roller skates. Then you awoke this morning to find your ankles and feet permanently installed into roller skates. Instantly, everything you understood about walking and running is worthless. Getting onto your feet at all Read More ›

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Robert J. Marks
Date
January 21, 2021

Categorized
Mathematics

Tagged
David Hilbert, Featured, Real numbers}

Most Real Numbers Are Not Real, or Not in the Way You Think

_{Typical real numbers contain an encoding of all of the books in the US Library of Congress} _{Robert J. Marks
January 21, 2021

5

Mathematics}

Pick a random real number between zero and one. The number you choose, with probability one, will contain an encoding of all of the books in the US Library of Congress. This sounds absurd, but real numbers require infinite precision and every time you deal with the infinite, things get absurd. Infinities, including the infinite number of digits to express almost every real number, don’t exist. Curiously then, real numbers are not real. How do we choose a random number between zero and one? The easiest way to explain is using binary decimals. The binary number 0.1000… with zeros forever denotes the number ½ or, in base 10 notation, 0.5. The binary decimal 0.01000… with zeros forever is the number Read More ›

Young serious female latin math school teacher wearing glasses holding writing equation on whiteboard in classroom. Hispanic university college tutor, graduate student learning, teaching during class.

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Jonathan Bartlett
Date
January 18, 2021

Categorized
Mathematics

Tagged
Featured, Hyperreal numbers, infinity, Irrational numbers, Real numbers}

Can We Add New Numbers to Mathematics?

_{We can work with hyperreal numbers using conventional methods. It could start in high school} _{Jonathan Bartlett
January 18, 2021

6

Mathematics}

Sometimes mathematics is moved forward by the discovery of new formulas and solutions to problems. However, sometimes mathematics grows by adding new kinds of numbers to the number system. In the early days of mathematics, it was thought that whole numbers were the only kind that existed. Sure, there were fractions, but fractions are merely ratios of whole numbers. It was thought that every possible number could be written in terms of whole numbers. These numbers were called rational numbers because they could be written as a ratio. There is a story about a Greek philosopher, Hippasus who discovered, roughly 2500 years ago, that certain numbers (specifically the square root of two) could not be written in terms of ratios Read More ›

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Jonathan Bartlett
Date
December 1, 2020

Categorized
Mathematics

Tagged
Bayes' Theorem, David Hume, Featured, Statistics, Thomas Bayes}

How Bayes’ Math Rule Can Counter Unreasonable Skepticism

_{Mathematics is much more interesting if we know a bit about the players and their positions} _{Jonathan Bartlett
December 1, 2020

4

Mathematics}

Yesterday, we discussed the importance of Bayes’ rule in statistical reasoning. We used the example of a person who goes for a battery of screening tests and comes up positive for HIV. Let’s say she is surprised (and alarmed) because she is not at any known risk for HIV. But, it turns out, the risk of false positives for the test is several times greater than the incidence of HIV in the population. In that case, it is reasonable for her to suspect—on a statistics science basis, not just wishful thinking—that the test is a false positive. The formula we used is part of Bayesian reasoning, originally developed by an eighteen-century British clergyman and mathematician Thomas Bayes (1702–1761), but now Read More ›

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Jonathan Bartlett
Date
November 30, 2020

Categorized
Artificial Intelligence, Mathematics

Tagged
False positives, Featured, HIV, Probability, Thomas Bayes}

Can an 18th Century Statistician Help Us Think More Clearly?

_{Distinguishing between types of probability can help us worry less and do more} _{Jonathan Bartlett
November 30, 2020

6

Artificial Intelligence, Mathematics}

Thomas Bayes (1702–1761) (pictured), a statistician and clergyman, developed a theory of decision-making which was only discussed after his death and only became important in the 20th century. It is now a significant topic in philosophy, in the form of Bayesian epistemology. Understanding Bayes’ Rule may be essential to making good decisions. Let’s say that you are a generally healthy person and have no symptoms of any illness and no specific risk factors for any illness. Acting on a friend’s suggestion, you get screened for a variety of diseases, just to be sure. Of the diseases you test for, the HIV test comes back positive. You read on the package that the test is 99.6% accurate. Are you more likely Read More ›