Mind Matters Natural and Artificial Intelligence News and Analysis

CategoryMathematics

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Two piles of coins

Are Our Neurons Really Wired for Numbers?

Some neuroscientists say they have shown hardwiring in studies of crows and macaques but others say no, these life forms differ too much

University College London cognitive neuroscientist Brian Butterworth, author of a forthcoming book, Can fish count? (Basic Books, 2022), reckons that, one way or another, in a modern urban society, we process about 16,000 numbers in an average day. Numbers create conceptual relationships between vastly different things. From the publisher’s introduction to his book, we learn, “The philosopher Bertrand Russell once observed that realizing that a pair of apples and the passage of two days could somehow both be represented by the concept we call “two” was one of the most astonishing discoveries anyone had ever made.” At The Scientist, Catherine Offord, discussing his work, offers a critical distinction between estimations of quantity and actual counting: “Our perception of quantity, separate…

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Small kitten

In an Infinity of Universes, Countless Ones Are Run by Cats…

Daniel Díaz notes that most of the talk about the multiverse started to appear once it was realized that there was fine-tuning in nature

Walter Bradley Center director Robert J. Marks has been doing a series of podcasts with Swedish mathematician Ola Hössjer, and Colombian biostatistician Daniel Díaz in connection with a recent co-authored paper on the fine-tuning of the universe for life in the Journal of Cosmology and Astroparticle Physics. In the first portion of this episode, podcast 153, “Why is there fine-tuning everywhere?” they look at whether life was seeded in our universe by advanced life forms (directed panspermia), as advocated by some prominent scientists. In the second portion, they discuss the view — again, held by prominent science figures — that our universe is an advanced computer sim. In the third segment, they tackle the idea that there is nothing to…

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Robot eyes closeup

Sure, AI Could Run the World — Except for Its Fundamental Limits

But many of the basic errors, problems, and limitations have no easy solution

We are told that not only will AI take our jobs but it will take our bosses’ jobs and their bosses’ jobs and pretty soon., AI will be running the world… We can see those films on Netflix any night. Science writer and science fiction author Charles Q. Choi offers, in a longish piece at the Institute of Electrical and Electronic Engineers’ online magazine, Spectrum, talking about the real world where “Neural networks can be disastrously brittle, forgetful, and surprisingly bad at math.” AI frequently flubs and it is not clear how to make it flub less. Here are brief notes on three examples of the seven he offers: ➤“Brittle” 97% of AIs could not identify a school bus flipped…

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Soup With Letter Noodles On Spoon

Can There Be a General Theory for Fine-Tuning?

If you make a bowl of alphabet soup and the letters arrange themselves and say, good morning, that is specified. What are the probabilities?

In Episode 2, the first part, (September 9, 2021), Swedish mathematician Ola Hössjer discusses fine tuning in biology with Walter Bradley Center director Robert J. Marks the way “Life is so finely tuned that it is frightening,” Put another way, the billions of cells in our bodies are each like a city. Not as a group but each of them. No wonder we feel so sick when things are going wrong with our cells. It is like billions of dysfunctional cities… Anyway, Hössjer has been working on a general theory for fine-tuning: https://episodes.castos.com/mindmatters/Mind-Matters-Episode-151-Hossjer-Diaz.mp3 This portion begins at 12:07 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: Ola, you came up with a general theory. We…

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Vaccine or flu shot in injection needle. Doctor working with patient's arm. Physician or nurse giving vaccination and immunity to virus, influenza or HPV with syringe. Appointment with medical expert.

COVID-19, Bayes’ Rule, and Simpson’s paradox

Israeli data, when studied carefully, confirm the effectiveness of COVID-19 vaccines

Israel has a very high COVID-19 vaccination rate and yet, on August 15, 2021, 58% of those Israelis hospitalized for COVID-19 were fully vaccinated — suggesting that vaccinations are ineffective or even harmful. This is a great example of two common statistical traps. The first is confusion about inverse probabilities. One hundred doctors were once asked this hypothetical question: In a routine examination, you find a lump in a female patient’s breast. In your experience, only 1 out of 100 such lumps turn out to be malignant, but, to be safe, you order a mammogram X-ray. If the lump is malignant, there is a 0.80 probability that the mammogram will identify it as malignant; if the lump is benign, there is a 0.90 probability that…

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Guaranteed Likely Probable Certainty Measuring Confidence Level

Fine-tuning? How Bayesian Statistics Could Help Break a Deadlock

Bayesian statistics are used, for example, in spam filter technology, identifying probable spam by examining vast masses of previous messages

In the earlier part of podcast episode 150, “Ours is a finely tuned — and No Free Lunch — universe,” Swedish mathematician Ola Hössjer and University of Miami biostatistician Daniel Andrés Díaz-Pachón discussed with Walter Bradley Center director Robert J. Marks the many ways in which the universe is finely tuned for life. Many theorists are not happy with the idea of fine-tuning because they are uncomfortable with its theistic implications. In this second portion of the episode, they discuss how a method of estimating probability called Bayesian statistics or Bayes theorem could help break a deadlock around fine-tuning: https://episodes.castos.com/mindmatters/Mind-Matters-Episode-150-Hossjer-Diaz-.mp3 This portion begins at 13:00 min. A partial transcript, Show Notes, and Additional Resources follow. Robert J. Marks: Bayes’ theorem…

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Set of numbers 1, 2, 3 made of leather. 3D render font with skin texture isolated on black background.

Further Dispatches From the War on Math

Discussions of social policy where math is relevant can be useful. But a student who does not understand how an equation works will fail at both math AND social policy

Earlier this year, I reposted an article that originally ran at Salvo on the war on the teaching of mathematics as a discipline in publicly funded schools in North America. The war continues so here are some updates: Recently, three mathematicians who immigrated to the United States weighed in: The United States has been dominant in the mathematical sciences since the mass exodus of European scientists in the 1930s. Because mathematics is the basis of science—as well as virtually all major technological advances, including scientific computing, climate modelling, artificial intelligence, cybersecurity, and robotics—US leadership in math has supplied our country with an enormous strategic advantage. But for various reasons, three of which we set out below, the United States is…

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Piano tuning process. closeup of hand and tools of tuner working on grand piano. Detailed view of Upright Piano during a tuning

Ours Is a Finely Tuned — and No Free Lunch — Universe

Mathematician Ola Hössjer and biostatistician Daniel Andrés Díaz-Pachón explain to Walter Bradley Center director Robert J. Marks why nature works so seamlessly

Our Walter Bradley Center director Robert J. Marks had a chance, recently, to talk with Swedish mathematician Ola Hössjer and University of Miami biostatistician Daniel Andrés Díaz-Pachón on the many ways in which the universe is finely tuned for life. This is Part 1. Part 2 will shortly follow. https://episodes.castos.com/mindmatters/Mind-Matters-Episode-150-Hossjer-Diaz-.mp3 This portion begins at 00:00 min. A partial transcript, Show Notes, and Additional Resources follow. Today on Mind Matters news on the podcast, we’re going to talk about fine-tuning of the universe for life. Scientists know that Earth is is finely tuned for life to come into existence. For example, pronounced atheist Sir Fred Hoyle (1915–2001). Hoyle was a great astronomer, maybe known best for his coining the term, Big…

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Joke

Why Animals Can Count But Can’t Do Math

A numerical cognition researcher outlines the differences between recognizing numbers and doing math

At The Conversation, psychology professor Silke Goebel offers some perspective on the differences between the way animals and small children process numbers and the way adults do. As a researcher in numerical cognition, she tries to focus on how brains process numbers. Her research shows that Humans and animals actually share some remarkable numerical abilities – helping them make smart decisions about where to feed and where to take shelter. But as soon as language enters the picture, humans begin outperforming animals, revealing how words and digits underpin our advanced mathematical world. Silke Goebel, “Why animals recognise numbers but only humans can do maths” at The Conversation (July 28, 2021) Essentially, she says, there are two different types of counting…

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solving algebra equation on whiteboard in classroom

How Eccentric Mathematician Kurt Gödel Opened the World

Science writer: As often happens, few people understood the significance of what had just happened. The one exception was John von Neumann.

Albert Einstein, Jogalekar tells us, considered it a privilege to walk home with Gödel every day. Why?: In an exceptionally elegant essay, science writer Ashutosh Jogalekar (no stranger to controversy) talks about the huge difference Kurt Gödel (1906–1978) made by eliminating the idea that some single, simple explanation would put an end to all questioning about the nature of the universe in favor of some simple materialism. In a review of Stephen Budiansky’s biography of Gödel, Journey to the Edge of Reason (Harvard 2021), Jogalekar explains how Gödel dashed such hopes: In September 1930, a big conference was going to be organized in Königsberg. German mathematics had been harmed because of Germany’s instigation of the Great War, and Hilbert’s decency…

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Close up of math formulas on a blackboard

Is Our “Number Sense” Biology, Culture — or Something Else?

It’s a surprisingly controversial question with a — perhaps unsettling — answer

British science writer Philip Ball, author of How to Grow a Human, offers an even-handed account of a controversy on the origin of our ability to understand numbers (numeracy). Numeracy is the beginning of mathematics, the most abstract of all human pursuits. It isn’t possible to get very far in mathematics without some ability to abstract. Ball cites as an example the difference between 152 and 153. Many life forms, competing for a pile of food items, can distinguish between 2 and 3. But distinguishing between 152 and 153 clearly requires abstraction. It’s the same principle as the chiliagon, a geometric figure like a triangle except that it has 1000 sides. A triangle can be envisioned concretely. A chiliagon can…

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Father And Son Competing In Video Games At Home

Why Did Video Gamers Uncover Fraud More Easily Than Scientists?

Video gamers are subject, a psychologist tells us, to much more rigorous constraints than scientists

In a recent article at The Atlantic, King’s College psychologist Stuart Ritchie, author of Science Fictions: How Fraud, Bias, Negligence and Hype Undermine the Search for Truth (2020), has noted a curious fact: Video gamers are much quicker to spot fraud than scientists. The video game fraud he focuses on involved a gamer’s claim that he had finished a round of Minecraft in a little over 19 minutes, a feat he attributed, Ritchie tells us, to “an incredible stretch of good luck.” “Incredible” is the right choice of word here. “Dream,” as the player was known, later admitted — in the face of skepticism — that he had “inadvertently” left some software running that improved his game — thus disqualifying…

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Space and Galaxy light speed travel. Elements of this image furnished by NASA.

No Free Lunches: Bernoulli is Right, Keynes is Wrong

What the Big Bang teaches us about nothing

Jacob Bernoulli made a now obvious observation about probability over three-and-a-half centuries ago: If nothing is known about the outcome of a random event, all outcomes can be assumed to be equally probable. Bernoulli’s Principle of Insufficient Reason (PrOIR) is commonly used. Throw a fair die. There are six outcomes, one for each face of the cube. The chance of getting five pips showing on the roll of a die is therefore one sixth. If a million lottery tickets are sold and you buy one ticket, the chances of winning are one in a million. This reasoning is intuitively obvious.  The assumption about the die is wrong if the die is loaded. But you don’t know that. You know nothing. So Bernoulli’s PrIOR…

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Statue of Saint Anselm and the towers of the Cathedral of Aosta, the Cattedrale di Aosta de Corso Pere-Laurent in Aosta. Aosta Valley. Italy. Europe

Gödel Says God Exists and Proves It

Here is a line-by-line explanation of his proof

Kurt Gödel, an intellectual giant of the 20th century, offered a mathematical proof that God exists. Those who suffer from math anxiety admire what the theorem (shown below) claims to do, but have absolutely no idea what it means. Our goal is to explain, in English, what Gödel’s existence of God proof says. Gödel’s proof shows the existence of God is a necessary truth. The idea behind the truth is not new and dates back to Saint Anselm of Canterbury (1033-1109). Great scientists and philosophers, including Descartes and Leibniz, have reconsidered and refined Anselm’s argument. Gödel appears to be the first, however, to present the argument using mathematical logic. Lexicography In any development of a mathematical theory, there are foundational axioms…

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A cute girl playing with wooden abacus. Educational concept for kids. Learing to count. Math for children. Math for children.

The Mystery of Numeracy: How DID We Learn To Count?

Some animals can do rough figuring but only humans count

At a time when many people see numbers, arithmetic, and mathematics as mere oppression, others are intrigued by the basic question, how did the human race learn to do math? There are some clues, as a new project called Quanta is trying to establish. First, while no animals use abstract numbers in the wild, some animals can do rough figuring: Although researchers once thought that humans were the only species with a sense of quantity, studies since the mid-twentieth century have revealed that many animals share the ability. For instance, fish, bees and newborn chicks can instantly recognize quantities up to four, a skill known as subitizing. Some animals are also capable of ‘large-quantity discrimination’: they can appreciate the difference…

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Hausschwein

Did an AI Disprove 5 Math Conjectures With No Human Help?

Yes and no. Read the fine print…

Mathematicians have thought that five long-standing conjectures in graph theory might be true but they have not been able to prove them: Wagner programmed a neural network to create random examples and use these measures to assess their suitability as a counterexample. The AI discarded the worst scoring ones and then replaced them with more random examples before starting again. In dozens of cases the AI was unable to find an example that disproved the theory, but in five cases it landed on a solution which showed that the conjecture must be false. Matthew Sparkes, “An AI has disproved five mathematical conjectures with no human help” at New Scientist (May 20, 2021 A subscription is required to read the whole…

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Education and maths concept

Will AI Change — or Eliminate — the Mind-Body Problem?

Can an artificial intelligence program that calculates really understand mathematics? Or is that a “hard ceiling” for AI?

In last week’s podcast, Walter Bradley Center director Robert J. Marks interviewed Concordia University philosopher Angus Menuge on the difficult mind–body problem: Dr. Menuge sees mind–body interaction as a transmission of information between two realms; our minds and bodies are one integrated system with a translation function… like developing and then writing down an idea. But what about artificial intelligence? We are told that artificial general intelligence (AGI) is now pushing towards a machine that can totally duplicate the functions of the human mind. But what if the mind is not simply a mechanical function of the brain? What if it is non-algorithmic and non-computable? This portion begins at 29:04 min. A partial transcript, Show Notes, and Additional Resources follow.…

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Circle of people. Teamwork. Business meeting. Negotiations, reaching consensus in disagreements. Joint problem solving. Conflict resolution through dialogue. Compromise. Cooperation and collaboration

Consensus Gives Us Information Only If We Are Free to Doubt

There are so many credentialed people on the internet with sufficiently differing views that it sometimes seems as if we could find an expert somewhere to support almost any harebrained idea. So how does a non-expert figure out the truth? Most of us lack the time, training, and inclination to investigate most subjects sufficiently so we are often urged to adopt the consensus opinion. While an individual expert may have wild and crazy ideas, the consensus will most likely be an average informed view. But it’s not that simple. Most of the time it is impossible for the public to determine the consensus opinion. What is usually labeled as consensus opinion is what media believe it to be. And the…

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Big brother

How Orwell’s 1984 Can Be Seen As an Argument for God’s Existence

Atheism is not only fundamental to the power of the Party in 1984 but is also its central weakness

University of Nebraska political science prof Carson Holloway (pictured) asks, “Does discrediting the existence of God promote enlightened thinking or a lack of objective reality?” Unpacking the social structure in George Orwell’s classic totalitarian dystopia, 1984 (1949), he observes that not only does the Party have the power of life and death but the atheistic Party faithful fear death as utter annihilation: Atheism is the moral basis of the Party’s unlimited hold on its own members because it makes them terrified of death as absolute nonexistence. Like any government, the Party in 1984 has the power to kill disobedient subjects. Party members, however, view death not just as the end of bodily life, but as a complete erasure of their…

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Purple transparent molecule model over purple

Could the World Above the Level of the Particle Be Superposed?

It’s been done with molecules. After that, there are barriers

In last week’s podcast,,” our guest host, neurosurgeon Michael Egnor, interviewed idealist philosopher of science and physicistBruce Gordon on how the quantum physics that underlies our universe makes much more sense if we have a non-materialist view of reality. Even then, it challenges our conventional view of how nature “must” work: We were introduced to the quantum eraser experiment, which showed that what happens at the level of individual particles depends on whether you choose to measure it or not and to non-locality, the Cheshire Cat’s science of being in no one particular place at any time. Particles can do that even if we can’t. But wait: Is it possible that things larger than particles can in fact do that…