Personally, I've always believed in a cognitive linkage between math and music, in part because of the number of individuals I've encountered who were either math majors and music minors, or vice-versa, music majors with math minors... and there are other reasons as well. Apparently, though, some folks doubt the connection, which this study now seems to back up:
https://phys.org/news/2020-11-strong-links-music-math.html
Fascinating review by Michael Harris of Marcus du Sautoy's latest, "The Creativity Code":
https://siliconreckoner.substack.com/p/book-review-the-creativity-code-by
...and somewhat related, this recent piece on "common sense" and AI:
A recent Dave Richeson tweet generated these two links to great pieces having to do with lexicography and the circularity of words (including mathematical or computational aspects):
https://pointatinfinityblog.wordpress.com/2020/05/13/circles-1-circular-definitions/
http://bit-player.org/wp-content/extras/bph-publications/AmSci-1999-03-Hayes-Wordnet.pdf
“In note of Richard Feynman’s memoirs he mentions an experience with a room-mate. One of them found he could keep counting while he was reading, and the other could keep count while he was talking, but neither could do both. They discovered that for one of them perceiving his numbers meant hearing them (so talking blocked his counting), while the other was seeing them (and so couldn’t simultaneously read).”
"[Jacques] Hadamard deconstructed the process of mathematical discovery into four successive stages: initiation, incubation, illumination, and verification. Initiation covers all the preparatory work, the deliberate conscious exploration of a problem. This frontal attack, unfortunately, often remains fruitless -- but all may not be lost, for it launches the unconscious mind on a quest. The incubation phase -- an invisible brewing period during which the mind remains vaguely preoccupied with the problem but shows no conscious sign of working hard on it -- can start. Incubation would remain undetected, were it not for its effects. Suddenly, after a good night's sleep or a relaxing walk, illumination occurs: the solution appears in all its glory and invades the mathematician's conscious mind. More often than not, it is correct. However, a slow and effortful process of conscious verification is nevertheless required to nail all the details down."
"You could say that the 'failed prediction' of Aumann’s Theorem is no surprise, since virtually all human beings are irrational cretins, or liars. Except for you, of course: you’re perfectly rational and honest. And if you ever met anyone else as rational and honest as you, maybe you and they could have an Aumannian conversation. But since such a person probably doesn’t exist, you’re totally justified to stand your ground, discount all opinions that differ from yours, etc."Anyway, give it a gander; you'll probably know before you're half-way through it if it's the sort of mind-stretching thought-exercise that strikes your fancy or not. (I suspect I may still be re-reading it a week from now, trying to better grasp parts!) The piece also contains some key links to related material.
"...under the right conditions, brain damage may paradoxically result in brain enhancement. Small-world, scale-free networking— focused, intensified, overclocked— might turbocharge a fragment of a brain into acting like the whole thing."Check it out:
"Finally, scientists score higher than non-scientists, and within the sciences, mathematics, physical scientists, computer scientists, and engineers score higher than the more human or life-centred sciences of medicine (including veterinary science) and biology. This latter finding replicates our earlier studies finding a link between autism spectrum conditions and occupations/skills in maths, physics, and engineering."Of course all this plays into the stereotypical view people often hold of nerdy, geeky mathematicians... I don't really have too great a problem with that, except to caution that all generalizations are mushy, and "mathematician," like any other category includes a wide range of individuals and personalities.
"If the human brain were so simple that we could understand it, we would be so simple that we couldn't." -- Emerson M. Pugh
1) A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?
2) If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
3) In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?People of a certain impulsive cognitive bent will tend to jump to the wrong answers on most of these questions, while those with a more patient cognitive style will take more time, arriving at correct answers. And these cognitive styles of course have further consequences.
"Many people think of mathematics as something akin to pure logic, cold reckoning, soulless computation. But as the mathematician and educator Paul Lockhart has put it, 'There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics.' The chilly analogies win out, Lockhart argues, because mathematics is misrepresented in our schools, with curricula that often favour dry, technical and repetitive tasks over any emphasis on the 'private, personal experience of being a struggling artist.'"An interview with Daniel about the book here:
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"Influential scholars have advanced the thesis that many of the building blocks of mathematics are 'hard-wired' in the human mind through millions of years of evolution. And a number of different sources of evidence do suggest that humans naturally associate numbers with space"…
"Our study shows, for the first time, that the number-line concept is not a 'universal intuition' but a particular cultural tool that requires training and education to master. Also, we document that precise number concepts can exist independently of linear or other metric-driven spatial representations."
"Mathematics all over the world -- from Europe to Asia to the Americas -- is largely taught dogmatically, as objective fact, black and white, right/wrong, but our work shows that there are meaningful human ideas in math, ingenious solutions and designs that have been mediated by writing and notational devices, like the number line… Mathematics is neither hardwired, nor 'out there.'"
"These findings suggest that how we think about abstract concepts is even more flexible than previously thought and is profoundly affected by language, culture and environment."
"Our familiar notions on 'fundamental' concepts such as time and number are so deeply ingrained that they feel natural to us, as though they couldn't be any other way."
