Tuesday, April 25, 2017
Blogging may continue to be a bit slow while I'm catching up on a number of things (and waiting very patiently for impeachment hearings ;), so may just re-run some old posts in the meantime. Anyway, will start by referencing this favorite old "Gauss Facts" site that's always good for a chuckle:
Sunday, April 23, 2017
Physicist David Deutsch reflecting on mathematics (from his "The Fabric of Reality"):
[There is] "...an ancient and widespread confusion between the methods of mathematics and its subject-matter. Let me explain. Unlike the relationships between physical entities, relationships between abstract entities are independent of any contingent facts and any laws of physics. They are determined absolutely and objectively by the autonomous properties of the abstract entities themselves. Mathematics, the study of these relationships and properties, is therefore the study of absolutely necessary truths. In other words, the truths that mathematics studies are absolutely certain. But that does not mean that our knowledge of those necessary truths is itself certain, nor does it mean that the methods of mathematics confer necessary truth on their conclusions. After all, mathematics also studies falsehoods and paradoxes. And that does not mean that the conclusions of such study are necessarily false or paradoxical.
"Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics. The objective of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be mathematical explanation."
Sunday, April 16, 2017
Sunday reflection... from Rebecca Goldstein in "Incompleteness: the proof and paradox of Kurt Gödel":
"So the question is: Whence certainty? What is our source for mathematical certainty? The bedrock of empirical knowledge consists of sense perceptions: what I am directly given to know -- or at least to think -- of the external world through my senses of sight and hearing and touch and smell. Sense perception allows us to make contact with what's out there in physical reality. What is the bedrock of mathematical knowledge? Is there something like sense perception in mathematics? Do mathematical intuitions constitute this bedrock? Is our faculty for intuition the means for making contact with what's out there in mathematical reality? Or is there just no 'there'?"
Tuesday, April 11, 2017
At a recent large used book sale I headed to the math section and picked up a few older volumes, including John Allen Paulos’ “Once Upon a Number” from 1998. Of course mathematics is timeless, but I was pleasantly surprised to see how much of the less-mathematical content of this volume is still relevant today (heck, maybe even more so since Nov. 8, 2016); much of it concerning logic, reasoning, meaning, information, clear/critical thinking and the like.
Anyway, I’ll put all that aside to only pass along this non-math joke Paulos tosses in at one point (the book is sprinkled with his typical humor):A young man is on vacation and calls home to speak to his brother.
“How’s Oscar the cat?”
“The cat’s dead, died this morning.”
“That’s terrible. You know how attached I was to him. Couldn’t you have broken the news more gently?”
“You could’ve said that he’s on the roof. Then the next time I called you could have said that you haven’t been able to get him down, and gradually like this you could’ve broken the news."
’‘Okay, I see. Sorry.”
“Anyway, how’s Mom?”
“She’s on the roof.”
Sunday, April 9, 2017
“It feels like there are two opposite things that the public thinks about science: that it’s a magic wand that turns everything it touches to truth, or that it’s all bullshit because what we used to think has changed… The truth is in between. Science is a process of uncertainty reduction. If you don’t show that uncertainty is part of the process, you allow doubt-makers to take genuine uncertainty and use it to undermine things…
“And it’s absolutely crucial that we continue to call out bad science. If this environment forces scientists to be more rigorous, that’s not a bad thing.”
— Christie Aschwanden (of FiveThirtyEight )
Wednesday, April 5, 2017
Lots of interesting mathy stuff out there this week, but hey, you can’t go wrong with the crown jewel of number theory, so I’ll direct you to two pieces on the Riemann Hypothesis, if you’ve not seen them:
First, a brief interview with Barry Mazur and William Stein, authors of “Prime Numbers and the Riemann Hypothesis” (one of my favorite 2016 books):
…and then the incomparable Natalie Wolchover summarizing the latest intriguing approach from physicists to Riemann’s 150+ year-old, million-dollar conundrum:
The actual (physics) work was published last year but is just now being widely disseminated on popular media:
There are a great many other introductions to RH on the internet, including some video ones such as these:
…and from 3 Blue1Brown:
Sunday, April 2, 2017
For a beautiful Sunday reflection, the ending words from Eugenia Cheng in "Beyond Infinity":
"The most beautiful things to me are the things just beyond that boundary of logic. It's the things we can get quite a long way towrd explaining, but then in the end they just elude us. I can get quite a long way toward explaining why a certain piece of music makes me cry, but after a certain point there's something my analysis can't explain. The same goes for why looking at the sea makes me so ecstatic. Or why love is so glorious. Or why infinity is so fascinating. There are things we can't even get close to explaining, in the realm far from the logical center of our universe of ideas. But for me all the beauty is right there on that boundary. As we move more and more things into the realm of logic, the sphere of logic grows, and so its surface grows. That interface between the inside and the outside grows, and so we actually have access to more and more beauty. That, for me, is what this is all about.
"In life and in mathematics there is often a trade-off between beauty and practicality, along with a contrast between dreams and reality, between the explicable and the inexplicable. Infinity is a beautiful dream, inside the beautiful dream that is mathematics."