From Davis and Hersh (in "The Mathematical Experience"):

"To the philosopher, there is all the difference in the world between a proof that depends on the reliability of a machine and a proof that depends only on human reason. To the mathematician, the fallibility of reason is such a familiar fact of life that he welcomes the computer as a more reliable calculator than he himself can hope to be."

“Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can’t be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere – no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what’s also amazing is that we own all of them. No one can patent a mathematical formula, it’s ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It’s too precious to be given away to the 'initiated few.' It belongs to all of us.”

The continuum hypothesis involves a well-known conundrum (considered undecidable without new set theory) among mathematicians over whether any infinite sets exist between aleph-zero and aleph-one. But I don't recall hearing any arguments over whether there could be infinite sets between any other alephs (say, aleph-three and aleph-four) — of course these higher sets are all power sets, but I can’t recall ever reading any “proof” that there can be no set in-between… I assume this is a long-settled simple question, but am not sure what the simple answer is! or is it somehow axiomatic with no real proof... or does it not even much matter since there's already an infinity of infinite sets???

ADDENDUM: I've now posed this question to a local retired math professor/number-theorist and he didn't know the answer, so at least I feel better at this point that it may not be a simple or dumb question to ask! (but have not heard from anyone else)

2) Next, no math, just a question that I asked on Twitter but got little response to, so will try here:

Google search has given me sporadically crappy results for several weeks (sometimes NO results, and sometimes results having no or little bearing on what I’m searching for), so I’ve switched over to Bing for now, but just wondering what search engines other math-types are happy with (it’s not a privacy issue or any other concern, strictly quality/relevance of results)? And are others experiencing issues with Google search — seems like some real glitch involved?

3) Finally, just a note… Two of the the greatest television shows ever when I was younger were Carl Sagan’s “Cosmos” and Jacob Bronowski’s “The Ascent of Man.” I’ve sometimes used Bronowski quotes and videos here on the blog, and just as “Cosmos” was re-done a few years back I feel like “Ascent of Man” should be re-run or re-done for each new generation.

Anyway, I stumbled across Bronowski on the Web a few days back and suddenly realized that he was trained as a mathematician (somehow I had him pegged in my mind as a physical scientist). Also didn’t realize he had died at the age of 66 (much younger than I thought) just one year after “Ascent…” was completed in 1973. Additionally discovered that “Ascent…” was originally commissioned by Sir David Attenborough. Just all interesting tidbits to me… and here is Dr. Bronowski voicing some of his thoughts about mathematics:

“Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion—not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”

Today, am just re-citing 2 of my very favorite long, rich blogposts that I’ve previously linked to, one from Lior Pachter a couple years back and an even older one from Tim Gowers. New people/teachers are constantly entering the math blogosphere, so for any who may have missed these earlier on I think they bear re-posting/re-reading:

Over 30 years ago I was delighted to see bubblemeister Tom Noddy perform his bubble tricks/creations on TV, and so was re-delighted to see he’s still around, appearing yesterday on CBS’s “Sunday Morning” show:

In turn, looking him up on YouTube I discovered that a new generation of bubble (or "bubbleology") artists are now out there performing, carrying on the fun for children and adults alike:

Though these show-meisters don’t get into the math involved, I can’t help but think that if Richard Feynman was in attendance watching he’d be hurriedly getting out a pen and pad to scribble down equations. ;)

If you do want a little more mathematical discussion, check out this take:

…seems like “bubbles” might be a good subject for Numberphile, Tadashi Tokieda perhaps, or possibly Mike Lawler and the boys could do something with them (apologies if any of you have already done so and I missed it). Bubbles represent a simple (or, maybe not-so-simple) everyday intersection of math and physics. Here's ZomeTool put to use exploring bubbles:

*************************

ADDENDUM:Mike L. sends me a link to several videos he has indeed done previously with soap bubbles :)

“What one learns about mathematics in primary school corresponds to the alphabet. What one learns in high school corresponds to the sentences of a primer. What one learns in elementary college courses corresponds to simple little stories. Scholars alone are aware of the mathematics that corresponds to literature.”

And over five yrs ago I ran this just-for-fun post (entitled: "There is no title for this post."):

*****************************************

This is the first sentence of the post titled, 'There is no title for this post.' This appears to be the sentence that follows sentence #1 of that post. This is the sentence following the previous sentence, but preceding the next sentence. This is the next sentence... or is it? Apparently this is sentence #5. This is the sentence you just finished reading. The last sentence of this post will come at the end. Thus, this is NOT the last sentence of this post. It is untrue that the prior sentence was false. This sentence begins with the word "this," followed by the word "sentence," followed by the word "begins," followed by the word "with," followed by the word "the," followed by the word "word," ...AND also ends with the word "word." And this is the sentence that informs you that the very next sentence is the final sentence of this post. This is the last sentence of the post, but why oh why does it end with a question-mark?

*****************************************

We'll end with more humor, starting with a well-known, geeky aphorism:

"In order to understand recursion, one must first understand recursion."

...which reminds me in turn of one of mathematicians' favorite jokes:

Q: What does the "B" in "Benoit B. Mandelbrot" stand for?

A: Benoit B. Mandelbrot

Then, there is this thoughtful quote that I've used before:

"If my mental processes are determined wholly by the motion of atoms in my brain, I have no reason to believe that my beliefs are true... and hence I have no reason for supposing my brain to be composed of atoms."

--- J.B.S. Haldane, "Possible Worlds" (1927)

In a slightly similar vein, this famous refrain out of AI:

"If the brain were so simple that we could understand it, then we would be so simple that we couldn't."

There is always xkcd's classic treatment of self-reference:

“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.”

An entranced Arthur Koestler, in "The Invisible Writing":

“I went on to recall Euclid’s proof that the number of primes is infinite… the scribbled symbols on the wall represented one of the rare cases where a meaningful and comprehensive statement about the infinite is arrived at by precise and finite means. I must have stood there for some minutes, entranced, with a wordless awareness that ‘this is perfect, perfect’, until I noticed some slight mental discomfort nagging at the back of my mind, some trivial circumstance that marred the perfection of the moment. Then I remembered the nature of that irrelevant annoyance: I was, of course, in prison and might be shot. But this was immediately answered by a feeling whose verbal translation would be: 'So what? Is that all? Have you got nothing more serious to worry about?' -- an answer so spontaneous, fresh and amused as if the intruding annoyance had been the loss of a collar-stud.”

By coincidence, shortly before our Il Duce was referencing “shithole” countries I was re-reading an old Presh Talwalkar post from a few years back that I always enjoyed on ‘Nigerian’ email scammers. It explains, as many know by now, why ’Nigerian’ email scams got stupider and stupider over the years, full of misspellings, bad grammar, poor English, outrageous narratives, etc. — the scammers wanted to make their messages SO obviously fraudulent that only the most gullible, naive, unthinking people would even respond (why waste time on thinking-folks wary enough to not follow through with the scam):

Anyway, I have a suggestion for how the scammers can be even more efficient: Just buy a copy of the mailing lists used by the Republican National Committee -- boy, talk about a sucker-list…

Yo, logic enthusiasts, when I saw a post entitled, “Smullyan and the President’s Sanity” listed on the mathbogging.org feed this morning it caught my attention. Have fun:

I'm a number-luvin' primate; hope you are too! ... "Shecky Riemann" is the fanciful pseudonym of a former psychology major and lab-tech (clinical genetics), now cheerleading for mathematics! A product of the 60's he remains proud of his first Presidential vote for George McGovern ;-) ...Cats, cockatoos, & shetland sheepdogs revere him. ...now addicted to pickleball.
Li'l more bio here.

............................... --In partial remembrance of Martin Gardner (1914-2010) who, in the words of mathematician Ronald Graham, “...turned 1000s of children into mathematicians, and 1000s of mathematicians into children.” :-) ............................... Rob Gluck