Some Classic Thoughts

"The enormous usefulness of mathematics in natural sciences is something bordering on the mysterious, and there is no rational explanation for it. It is not at all natural that 'laws of nature' exist, much less that man is able to discover them. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve."  -- Eugene P. Wigner


"I believe that scientific knowledge has fractal properties, that no matter how much we learn, whatever is left, however small it may seem, is just as infinitely complex as the whole was to start with. That, I think, is the secret of the Universe."   -- Isaac Asimov


"I think it's important to regard science not as an enterprise for the purpose of making predictions but as an enterprise for the purpose of discovering what the world is really like, what is really there, how it behaves and why. Which is tested by observation. But it's absolutely amazing that the tiny little parochial and weak and error-prone access that we have to observations is capable of testing theories and knowledge of the whole of reality, which has tremendous reach far beyond our experience. And yet we know about it. That's the amazing thing about science. That's the aspect of science that I want to pursue."   
-- David Deutsch

No Largest Prime Gap

I've reported on this in the distant past, but since Mike Lawler recently asked bloggers to post some entries that might be of interest to both mathematicians and students, I’ll re-run this simple, old demonstration that you can have ANY size gap between two prime numbers that you want. I’ve always liked it, for its simplicity, since first seeing it in a popular 1984 volume from Laurie Buxton called “Mathematics For Everyone.” It runs like this (using Buxton’s example):

Hopefully you know what 600! means, i.e. the product of 600 x 599 x 598 x ….. x 2 x 1.

A pretty large number, but we need not actually multiply it out. Now consider the following string of consecutive numbers:

600! + 2

600! + 3

600! + 4

600! + 5

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600! + 600

The above produces a list of 599 consecutive integers, NONE of which can be prime. Every number here will be divisible by at least the number on the right (because 600! is divisible, without remainder, by every number UP TO 600, and adding anything between 1 and 600 simply includes one of those divisors). Thus, in this example we have a gap of at least 599 integers without a prime appearing. BUT clearly one need not start with 600. One can start with a number as large as one likes in order to generate a prime gap as large as one wants. There will never be a largest gap. Simple and convincing!

Multiplying It Out

I’m currently in rerun mode, just replaying some posts from the past. Here’s a previous puzzle from an old “Scam School” episode. You may get it right away (it’s simple), or if you don’t, you’ll facepalm yourself when you see the answer (below).

Here goes:

You are to multiply together a long sequence as follows:

(a-x) X (b-x) X (c-x) X (d-x)...... (y-x) X (z-x)  i.e., utilizing ALL the letters of the alphabet once.

What will be the end product of this sequence when multiplied out???

.Answer below

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Answer = 0 ...just before the final two sequence entries listed, would be (x-x)

"Gauss can recite all of pi -- backwards"

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:

http://www.gaussfacts.com/top

Truth, Certainty, Explanation... and Mathematics

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."

"Whence Certainty?"

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'?"

For Your Funnybone…

 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?”

“How?”

“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.”

Science As Uncertainty Reduction

Sunday reflection:

“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 )

Riemann In the News...

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):

http://www.cambridgeblog.org/2017/04/a-brief-interview-with-barry-mazur-and-william-stein/

…and then the incomparable Natalie Wolchover summarizing the latest intriguing approach from physicists to Riemann’s 150+ year-old, million-dollar conundrum:

https://www.quantamagazine.org/20170404-quantum-physicists-attack-the-riemann-hypothesis/

The actual (physics) work was published last year but is just now being widely disseminated on popular media:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130201

There are a great many other introductions to RH on the internet, including some video ones such as these:

From Numberphile:

https://www.youtube.com/watch?v=d6c6uIyieoo

…and from 3 Blue1Brown:

https://www.youtube.com/watch?v=sD0NjbwqlYw

Beyond the Boundary of Logic

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 toward 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."