^CategoryMathematics

3D illustration of Escher's inspired stairs

^{Type
post

Author
Denyse O'Leary
Date
January 9, 2023

Categorized
Arts & Culture, Mathematics, Physics

Tagged
Closer to Truth, M. C. Escher (1898-1972), Night and Day (Escher 1938), Relativity (Escher 1953), Robert Lawrence Kuhn, Roger Penrose}

How Surreal Artist MC Escher Influenced Physicist Roger Penrose

_{Escher’s mathematical art was all the more remarkable because he had no formal training in mathematics} _{Denyse O'Leary

January 9, 2023

4

Arts & Culture, Mathematics, Physics}

Last month, Robert Lawrence Kuhn interviewed eminent British mathematical physicist Sir Roger Penrose on a number of topics, including the influence of surrealist artist M. C. Escher (December 9, 2022/32:00 min). Here is a transcribed selection from the second part of the discussion in Part 1 above*, beginning around the 12-minute mark, with some notes: Robert Lawrence Kuhn: We talked about the impossible Penrose triangle which really opens up another area of your life in terms of visual representations of remarkable things. Penrose tiling really new ways of thing of seeing visual representation of fiery fundamental geometric and algebraic transformations and things. But what I wanted to ask you is, as youdeveloped that you had this interaction with the artist Read More ›

Lost and confused man walks puzzled on a penrose triangle. Perplexed person looks disoriented ahead, don't know which way to choose. Surreal and conceptual scene, mental maze, optical illusion

^{Type
post

Author
Denyse O'Leary
Date
January 8, 2023

Categorized
Mathematics, Philosophy of Mind, Physics

Tagged
Closer to Truth, Computationalism, Consciousness, Idealism (philosophy), Penrose Triangle, Physicalism, Robert Lawrence Kuhn, Roger Penrose, Supervenience}

Nobelist Roger Penrose Talks About His Impossible Triangle

_{At Closer to Truth, the mathematical physicist explains to Robert Lawrence Kuhn how he understands the relationship between mathematics, the mind, and the physical world} _{Denyse O'Leary

January 8, 2023

8

Mathematics, Philosophy of Mind, Physics}

Last month, Robert Lawrence Kuhn interviewed eminent British mathematical physicist Sir Roger Penrose on the relationship between mathematics, the mind, and the physical universe (December 9, 2022/32:00 min). Penrose likes to illustrate the relationship between the three with an “impossible” triangle (see below). Here are a couple of transcribed selections from the first part of the discussion in Part 1*, concerning the Penrose Triangle, with some notes: Robert Lawrence Kuhn: Let’s start with your grand metaphysical framework, your three worlds — three mysteries: the physical world, the mental world, the platonic or mathematical world — each connected to the other two in your famous diagram of an equilateral triangle. What’s the origin of this vision of yours of foundational reality? Read More ›

Abstract futuristic stripe line printed circuit board pattern with gear wheel and math fornula on blue color background. Math science engineered drawn project plot concept

^{Type
post

Author
Michael Egnor
Date
December 29, 2022

Categorized
Mathematics, Philosophy, Religion

Tagged
Augustine of Hippo, Augustinian proof (of God), Featured, Gottfried Wilhelm Leibniz, Jerry Coyne on God, Platonic forms, Realism (in philosophy), Scholastic realism}

Mathematics Can Prove the Existence of God

_{Atheist biologist Jerry Coyne finds that difficult to believe but it’s really a matter of logic} _{Michael Egnor

December 29, 2022

6

Mathematics, Philosophy, Religion}

This story was #3 in 2022 at Mind Matters News in terms of reader numbers. As we approach the New Year, we are rerunning the top ten stories of 2022, based on reader interest. In “Mathematics can prove the existence of God” (July 31, 2022), neurosurgeon Michael Egnor offers this thought: Because mathematics can show infinity, eternity, and omnipotence, it can only have proceeded from a mind with those characteristics. That’s God. In a recent post, atheist biologist Jerry Coyne takes issue with a commenter who asserts that God exists in the same sort of way mathematics exists. Here’s the analogy the commenter offered, as quoted by Coyne: Think of numbers for example, or mathematical equations, these are metaphysical things, Read More ›

Blue glowing 4 dimensional object in space abstract fractal background

^{Type
post

Author
News
Date
December 28, 2022

Categorized
Mathematics, Philosophy of Mind

Tagged
Flatland (short novel), Modes of Sentience (book), Peter Sjöstedt-Hughes, Tesseract}

Hard Problem of Consciousness Solved?: A 4th Spatial Dimension?

_{Philosopher Peter Sjöstedt-Hughes argues that higher spatial dimensions might hold the key} _{News

December 28, 2022

2

Mathematics, Philosophy of Mind}

This story was #4 in 2022 at Mind Matters News in terms of reader numbers. As we approach the New Year, we are rerunning the top ten Mind Matters News stories of 2022, based on reader interest. In “Hard problem of consciousness solved?: A 4th spatial dimension?” (April 20, 2022), our News division looks at philosopher Peter Sjöstedt-Hughes’ view that higher spatial dimensions might hold the key to the uniqueness of human consciousness. In an abridged chapter of his recent book Modes of Sentience (2021), University of Exeter philosopher of mind, Peter Sjöstedt-Hughes, argues that higher spatial dimensions might hold the key to the hard problem of consciousness:” He is a fan of the More–Broad–Smithies theory of consciousness: The word tesseract was Read More ›

Spiral aloe vera with water drops, closeup

^{Type
post

Author
News
Date
December 23, 2022

Categorized
Mathematics, Science

Tagged
Eugene Wigner, Golden Ratio, Ketaki Bepat, Mathematics as invented or discovered, Sunny Labh, Unreasonable Effectiveness of Mathematics (1960)}

Our Physical Universe Is Based on Patterns in Mathematics

_{In that sense, the underlying basis of our universe is immaterial, not material} _{News

December 23, 2022

3

Mathematics, Science}

Sunny Labh, science writer and “budding astrophysicist,” reminds us that, in our universe, everything from snowflakes to black holes can be described by mathematics: To have a basic understanding of how accurate and loyal this universe is to math, we could visualize our solar system. Back in 1905, scientists discovered that the orbit of Uranus and Neptune was not following Newton’s Gravitational laws, and using mathematical expressions, predicted that there must be some other heavenly body that was influencing their orbit. In 1930 Clyde Tombaugh used the method of comparative plating of astronomical photographs also known as the blink comparator technique. The calculations were made precise to a point where we could even tell where to point our telescope in Read More ›

^{Type
post

Author
Robert J. Marks
Date
December 19, 2022

Categorized
Economics, Mathematics

Tagged
Everett Dirksen, Featured, U.S. deficit in personal terms, U.S. spending and deficits}

Celebrating My 2 Billionth Birth-Second: What Big Numbers Mean

_{Let’s see if we can give a clearer, sharper personality to these big numbers} _{Robert J. Marks

December 19, 2022

4

Economics, Mathematics}

I have lived for over two billion seconds. In 2013, I celebrated my 2 billionth birth-second. The party did not last long. Today US spending and deficits are going through the roof. References to billions and trillions of dollars of spending and deficit are everywhere. The late US Senator Everett Dirksen of Illinois is purported to have said “A billion here, a billion there; pretty soon you’re talking about real money.” He said this in the middle of the last century. Today we can replace “billion” in Dirksen’s quote with “trillion.” Let’s see if we can give a clearer, sharper personality to these big numbers. A trillion is a thousand times bigger than a billion. If we scale a trillion Read More ›

A sad young female student sitting at the table, studying.

^{Type
post

Author
News
Date
November 1, 2022

Categorized
Education, Mathematics, Neuroscience

Tagged
Barbara Oakley, Fred Bech, Kumon math, Mathematics education (difficulty learning math), Mathematics education and practice, Neuroplasticity and learning new skills}

Can We Rewire Our Brains To Be More Fluent in Math?

_{An artsy who flunked math — but later became an electrical engineering prof — says yes} _{News

November 1, 2022

5

Education, Mathematics, Neuroscience}

Barbara Oakley, a self-confessed math phobe, nonetheless became a professor of electrical engineering at Oakland University in Michigan, as well as an author. In 2014, she offered some secrets: at Nautilus. Be warned: Her secrets are not “Forget homework!” or “Math is a tool of oppression!” No, this is quite a different message. It’s about neuroplasticity, the ways our brains adapt to our circumstances, to give us the tools we need. But to adapt, the brain needs practice: Japan has become seen as a much-admired and emulated exemplar of these active, “understanding-centered” teaching methods. But what’s often missing from the discussion is the rest of the story: Japan is also home of the Kumon method of teaching mathematics, which emphasizes Read More ›

Stock Trader Sleeping At Multiple Computer's Desk

^{Type
post

Author
Gary Smith
Date
October 17, 2022

Categorized
Business and Finance, Economics, Mathematics

Tagged
Alan Sokal, Black box algorithms (in decision-making), Data pruning, Featured, Harris Friedman, Internal rate of return (IRR), Lemonade insurance company, Losada line, Marcial Losada, Mathematics and comprehensibility, Net present value (NPV), Nick Brown, NPV vs. IRR, Principal components regression}

More Hard Math Does Not Necessarily Mean More Useful Solutions

_{It is sometimes tempting to overemphasize the math and underemphasize the relevance} _{Gary Smith

October 17, 2022

7

Business and Finance, Economics, Mathematics}

Math is said to be the language of science in that most (but definitely not all) scientific models of the world involve mathematical equations. The Pythagorean theorem, the normal distribution, Einstein’s energy-mass equivalence, Newton’s second law of motion, Newton’s universal law of gravitation, Planck’s equation. How could any of these remarkable models be expressed without math? Unfortunately, it is sometimes tempting to overemphasize the math and underemphasize the relevance. The brilliance of the models listed above lies not in mathematical pyrotechnics but, if anything, in their breath-taking simplicity. Useful models help us understand the world and make reliable predictions. Math for the sake of math does neither. Examples of mindless math are legion. I will give three very different examples. Read More ›

Funny boy and his dog looking at piles of coins

^{Type
post

Author
News
Date
October 14, 2022

Categorized
Mathematics, Natural Intelligence, Neuroscience

Tagged
Brian Butterworth, Dyscalculia, Túngara frog counting, Zebrafish number sense}

Researchers Are Zeroing In on Animal Number Sense

_{We’re beginning to find out more about how animals that don’t really “think” much can keep track of numbers, when needed} _{News

October 14, 2022

4

Mathematics, Natural Intelligence, Neuroscience}

University College cognitive psychology prof Brian Butterworth, author of Can fish count? (Basic Books, 2022), talks about animal number sense in a recent article in Psyche: He offers many examples of animals counting single digit numbers but then helpfully addresses the question of how they do it. We are talking here about a variety of very different types of neurological equipment — insects vs. amphibians, for example. Neuroscientists are beginning to pinpoint specific brain functions associated with counting for specific tasks: Female túngara frogs benefit by mating with the male that can produce six croaks in one breath, over the male that can manage only five, because this is an indicator of respiratory fitness. Naturally, the male will try to Read More ›

solving algebra equation on whiteboard in classroom

^{Type
post

Author
News
Date
September 19, 2022

Categorized
Education, Mathematics

Tagged
Mathematics (war on math), Mathematics and racism claims, mathematics education, Mathematics testing, War on math}

The War on Math Is Becoming an Entrenched Ground War

_{If math skills are rooted in white supremacy, as alleged, one current solution is tests that don’t require math skills} _{News

September 19, 2022

5

Education, Mathematics}

When we first started talking about the war on math, many readers may have thought we were joking. No. The war on 2 + 2 = 4 is getting some pushback but it continues. The basic idea is that the rules of math are rooted in white supremacy. Last December, the question was mooted at USA Today, “Is math racist?” The context was proposed changes to math education: After Ebri switched to emphasizing real-world problems and collaboration, her students, most of whom are Black, improved their scores on Florida’s math exam in 2020-21 – even with 1 in 3 learning from home. But other, bolder recommendations to make math more inclusive are blowing up the world of mathematics education. Schools Read More ›

Groups of people are connected by lines. Interdependence correlation in workflow. Interacting and joining forces with other teams. Interact to complete tasks. Formation of a more complex community.

^{Type
post

Author
Gary Smith
Date
September 19, 2022

Categorized
Mathematics, Probability

Tagged
Featured, HARKing, spurious correlations (in Big Data), Spurious Correlations (website), Stepwise regression, Stepwise regression tested}

Step Away From Stepwise Regression (and Other Data Mining)

_{Stepwise regression, which is making a comeback, is just another form of HARKing — Hypothesizing After the Results are Known} _{Gary Smith

September 19, 2022

6

Mathematics, Probability}

There is a strong correlation between the number of lawyers in Nevada and the number of people who died after tripping over their own two feet. There are similarly impressive correlations between U.S. crude oil imports and the per capita consumption of chicken — and the number of letters in the winning word in the Scripps National Spelling Bee and the number if people killed by venomous spiders. If you find these amusing (as I do), there are many more at the website Spurious Correlations. These silly statistical relationships are intended to demonstrate that correlation is not causation. But no matter how often or how loudly statisticians shout that warning, many people do not hear it. When there is a Read More ›

^{Type
post

Author
Jonathan Bartlett
Date
September 5, 2022

Categorized
Education, Mathematics

Tagged
Algebra, Clifford algebra, Complex numbers and vector algebra, Complex spinors, Featured, Gibbs vectors, Maxwell's Equations and Clifford algebra, Quaternions, Vector algebra, Vector Algebra Wars, William Kingdon Clifford, William Rowan Hamilton}

The Vector Algebra Wars: A Word in Defense of Clifford Algebra

_{A well-recognized, deep problem with using complex numbers as vectors is that they only really work with two dimensions} _{Jonathan Bartlett

September 5, 2022

5

Education, Mathematics}

Vector algebra is the manipulation of directional quantities. Vector algebra is extremely important in physics because so many of the quantities involved are directional. If two cars hit each other at an angle, the resulting direction of the cars is based not only on the speed they were traveling, but also on the specific angle they were moving at. Even if you’ve never formally taken a course in vector algebra, you probably have some experience with the easiest form of vector algebra — complex numbers (i.e., numbers that include the imaginary number i). In a complex number, you no longer have a number line, but, instead, you have a number plane. The image below shows the relationship between the real Read More ›

Set of red glass dice isolated on black background. 3d rendering - illustration.

^{Type
post

Author
Gary Smith
Date
September 5, 2022

Categorized
Business and Finance, Economics, Mathematics

Tagged
Economics and human motivation, Economics and math predictions, Featured, Great Depression economic policies, James Tobin, John Maynard Keynes, Paul Samuelson, Samuelson simulations, Stock market bubbles vs. dice rolls, Zvi Bodie}

Don’t Worship Math: Numbers Don’t Equal Insight

_{The unwarranted assumption that investing in stocks is like rolling dice has led to some erroneous conclusions and extraordinarily conservative advice} _{Gary Smith

September 5, 2022

7

Business and Finance, Economics, Mathematics}

My mentor, James Tobin, considered studying mathematics or law as a Harvard undergraduate but later explained that I studied economics and made it my career for two reasons. The subject was and is intellectually fascinating and challenging, particularly to someone with taste and talent for theoretical reasoning and quantitative analysis. At the same time it offered the hope, as it still does, that improved understanding could better the lot of mankind. I was an undergraduate math major (at Harvey Mudd, not Harvard) and chose economics for the much the same reasons. Mathematical theories and empirical data can be used to help us understand and improve the world. For example, during the Great Depression in the 1930s, governments everywhere had so Read More ›

^{Type
post

Author
Michael Egnor
Date
July 31, 2022

Categorized
Mathematics, Philosophy, Religion

Tagged
Augustine of Hippo, Augustinian proof (of God), Featured, Gottfried Wilhelm Leibniz, Jerry Coyne on God, Platonic forms, Realism (in philosophy), Scholastic realism}

Mathematics Can Prove the Existence of God

_{Atheist biologist Jerry Coyne finds that difficult to believe but it’s really a matter of logic} _{Michael Egnor

July 31, 2022

7

Mathematics, Philosophy, Religion}

In a recent post, atheist biologist Jerry Coyne takes issue with a commenter who asserts that God exists in the same sort of way mathematics exists. Here’s the analogy the commenter offered, as quoted by Coyne: Think of numbers for example, or mathematical equations, these are metaphysical things, that have not been created, however were discovered. The number 7 was the number 7 before anything at all came into existence. This is also true concerning the nature of God. He is not some material being that has come into existence, he is like a number that has always existed, (and by the way nobody will deny this logic with the number, however when someone mentions God a problem occurs). Jerry Read More ›

Abstract Computer generated Fractal design. 3D Illustration of a Beautiful infinite mathematical mandelbrot set fractal.

^{Type
post

Author
News
Date
July 17, 2022

Categorized
Mathematics, Philosophy

Tagged
Imaginary numbers, Imaginary numbers (as essential), Imaginary numbers in quantum mechanics, Infinite numbers, Karmela Padavic-Callaghan, Robert J. Marks, Roger Penrose on imaginary numbers}

Recent Research: Imaginary Numbers Are Part of the Real World

_{If we try to leave them out of quantum mechanics, our description of nature becomes faulty} _{News

July 17, 2022

5

Mathematics, Philosophy}

Imaginary numbers, beginning with the square roots of minus numbers, are part of the world in which we live, even though we can’t quite picture them. Try it. The square root of 1 is 1 (1 × 1 = 1). But what’s the square root of -1? It can’t be -1 because if we multiply -1 × -1, we still get 1. The two minus numbers cancel each other out. That’s why the square root of -1 is written as i. Now, here’s the odd part: Imaginary numbers are not just a conundrum; they are part of a science description of the world in which we live in: Though imaginary numbers have been integral to quantum theory since its very Read More ›

^{Type
post

Author
Robert J. Marks
Date
July 8, 2022

Categorized
Mathematics

Tagged
Aleph in mathematics, Featured, Georg Cantor and infinity, Georg Cantor and Pope Leo XIII, Infinity (infinities differ by size), Infinity (no biggest infinity), Infinity (not real world concept)}

Some Infinities Are Bigger Than Others But There’s No Biggest One

_{Georg Cantor came up with an ingenious proof that infinities can differ in size even though both remain infinite} _{Robert J. Marks

July 8, 2022

7

Mathematics}

When a child is asked “what is bigger than infinity,” the response is often “Infinity plus one.” No. Infinity plus one is still infinity. But we can show that the number of points on the interval zero to one is a bigger infinity than the counting numbers are. The first clue is the fact that we can’t count the number of points on a line interval. Try labeling the points on a line as points 1, 2, 3, etc. No matter what scheme you come up with, there will always be some points on the line segment that are not included in your count. Georg Cantor (1845–1918) came up with an ingenious argument to show that the infinite number of Read More ›

^{Type
post

Author
Robert J. Marks
Date
July 5, 2022

Categorized
Mathematics

Tagged
Base 2 numeral system, Featured, Infinity and irrational numbers, Infinity as not real, Irrational numbers, Irrational numbers and infinity, Library of Congress (encoded in binary), Pi as irrational number, Rational numbers and infinity, Real numbers and infinity}

4. How Almost Any Number Can Encode the Library of Congress

_{That’s a weird, counterintuitive — but quite real — consequence of the concept of infinity in math} _{Robert J. Marks

July 5, 2022

7

Mathematics}

We are used to dealing with simple numbers, like ½ and 2. Most numbers are not that simple. Most numbers, like 0.847859028378490… go on forever and ever without repeating or showing any pattern. Note that such numbers, called irrational numbers, have an infinite number of digits. And there are a lot of them. The number 0.847859028378490… for example differs from the number 0.847859023378490… (See if you can spot the difference.) If two numbers differ only at the billionth decimal and are otherwise the same, they are different numbers. Because an irrational number is infinitely long — and we have seen in the first three posts that weird things happen with infinity — we’d expect something weird to happen with irrational Read More ›

Abstract background architecture lines. modern architecture detail

^{Type
post

Author
Robert J. Marks
Date
July 3, 2022

Categorized
Mathematics

Tagged
Cantor's theory of infinity, Cantor's theory of infinity (lines and squares), Featured, Irrational numbers and infinity}

3. In Infinity, Lines and Squares Have an Equal Number of Points

_{We can demonstrate this fact with a simple diagram} _{Robert J. Marks

July 3, 2022

5

Mathematics}

In previous posts, we have established that two sets are of the same size if there is a one-to-one correspondence between the elements of both sets. Applying this principle to Cantor’s theory of infinity leads us to the weird but valid conclusion that the number of points on a line segment is the same as the number of points in a square. To show that this is true, here is a picture of a unit length line segment and a unit square. Let’s choose a point on the line segment. Let’s say 0.6917381276543… . It’s shown with a big blue dot on the line segment on the left. If this point corresponds to an irrational number, it goes on forever Read More ›

Big Bang in Space, The Birth of the Universe 3d illustration

^{Type
post

Author
Robert J. Marks
Date
July 2, 2022

Categorized
Mathematics

Tagged
Featured, Infinite past leads to absurd results, Infinity and physical universe, Slow Sam illustration, Universe (as having a beginning)}

2. Infinity Illustrates That the Universe Has a Beginning

_{The logical consequences of a literally infinite past are absurd, as a simple illustration will show} _{Robert J. Marks

July 2, 2022

5

Mathematics}

The size of a set is how many elements it contains. The set of letters {A,B,C} and the set of girls {Shirley, Goodness, Mercy} both have a cardinality of three. In a previous post, we showed that the infinities of counting numbers and even numbers are the same. Many subsets of the counting numbers have the same infinite size as the counting numbers. For example, consider the counting numbers and the set of numbers divisible by 10. and The size of the two sets is the same if there is a one-to-one mapping from one set to another. Here, 1 maps to 10, 2 maps to 20, 3 to 30, etc. This continues forever. The two sets are the same Read More ›

Full body gold dragon in infinity shape pose with 3d rendering include alpha path.

^{Type
post

Author
Robert J. Marks
Date
July 1, 2022

Categorized
Mathematics

Tagged
David Hilbert (on infinities), Featured, Georg Cantor on infinities, infinity, Infinity (as understood in math), Infinity (not found in nature), Natural numbers as infinite}

1. Why Infinity Does Not Exist in Reality

_{A few examples will show the absurd results that come from assuming that infinity exists in the world around us as it does in math} _{Robert J. Marks

July 1, 2022

7

Mathematics}

Does infinity exist in reality? There are, surprisingly, scientists who think infinity is a possibility even though they are unable to point to any example of infinity in reality. The great mathematician David Hilbert claimed that “the infinite is nowhere to be found in reality.” Nevertheless, the mathematical theory of infinity developed by Georg Cantor is beautiful. Hilbert was in awe of Cantor’s beautiful theory and said “No one shall drive us from the paradise which Cantor has created for us.” An assumption of the infinite leads to weird counterintuitive results. In this and the following four articles, various ludicrous properties of the infinite are explored. We’ll see, for example, that the entire Library of Congress is encoded somewhere in almost every Read More ›