“Some observers have professed to detect, in the variety and freedom of today’s mathematics, symptoms of decadence and decline. They tell us that mathematics has fragmented into unrelated specialties, has lost its sense of unity, and has no idea where it is going. They speak of a ‘crisis’ in mathematics, as if the whole subject has collectively taken a wrong turning. There is no crisis. Today’s mathematics is healthy, vigorous, unified, and as relevant to the rest of human culture as it ever was… If there appears to be a crisis, it is because the subject has become too large for any single person to grasp… today’s mathematics is not some outlandish aberration: it is a natural continuation of the mathematical mainstream. It is abstract and general, and rigorously logical, not out of perversity, but because this appears to be the only way to get the job done properly. It contains numerous specialties, like most sciences nowadays, because it has flourished and grown. Today’s mathematics has succeeded in solving problems that baffled the greatest minds of past centuries. Its most abstract theories are currently finding new applications to fundamental questions in physics, chemistry, biology, computing, and engineering. Is this decadence and decline? I doubt it.”
Sunday, September 17, 2017
This Sunday reflection from Ian Stewart in the 2nd edition (1992) of “The Problems of Mathematics”:
Wednesday, September 13, 2017
Fantastic article from Quanta Magazine (Kevin Hartnett) about new findings/proof of the equivalency of two variant infinities — actually findings published a year ago; am amazed it’s just now reaching the wider press (at least I’d not heard about this ’til now!):
Part of what makes the proof interesting (IF I understand matters correctly) is that it didn't require any re-statement of fundamental set theory, but only a bringing together of disparate math models that had not been linked up before. Even if you (like me) don't understand the details of the finding, just recognizing that a 50+ year problem has been resolved is very exciting. The solvers, Maryanthe Malliaris and Saharon Shelah, received the Hausdorff Medal for their work earlier this year.
…In a bit of irony, the above article got tweeted out yesterday on the very anniversary of the death of David Foster Wallace whose book on infinity, “Everything and More,” I've discussed earlier here:
Tuesday, September 12, 2017
“According to a recent study, 36 percent of college students don’t significantly improve in critical thinking during their four-year tenure. 'These students had trouble distinguishing fact from opinion, and cause from correlation,' Goldin explained.”
The above words from mathematician/statistician Rebecca Goldin come near the beginning of this new piece in Quanta Magazine:
The title of the piece is “Why Math Is the Best Way To Make Sense of the World.” I fear the title may be the very sort that turns people away from it, or at least many of those who most need to read it — just mention 'math' in some sort of positive light and a lot of the ‘I-was-never-any-good-at-math’ folks will turn away out of disinterest :(
And if college-bound students aren’t gaining critical thinking skills over their 4-year sojourn, what can we expect of the non-college crowd who may have even less opportunity to be exposed to critical-thinking skills?
But critical thinking shouldn’t even begin with college; it should begin back in elementary school with language skills, which are themselves fundamentally entwined in critical thinking. Nonetheless, the above article (and interview with Goldin) is excellent and focused on the societal value of math and science at the university level -- there are several lines in it I’d love to quote, but just go read it for yourself and take to heart this central message: “…if we don’t have the ability to process quantitative information, we can often make decisions that are more based on our beliefs and our fears than based on reality.”Interestingly, this article appears at a time that topics like critical thinking, quantitative reasoning, innumeracy and the like are getting a fair amount of discussion in society, though I’m not confident that we’re even close to dispensing such skills to the population-at-large, nor to upcoming generations. In fact I fear quite the opposite; it may be too little too late in a digital world of speed, simplification, and reality-manipulation... hope I'm wrong, but the Machiavellians who plotted the path of our current Oval Office interloper knew all-too-well that appeals to base instincts could overcome appeals to critical thought. :(
Sunday, September 10, 2017
Wednesday, September 6, 2017
ICYMI, the more hardcore among you may want to see John Baez's recent commentary (and the comments that follow) on Mochizuki's "proof" of the ABC conjecture:
Mathematician Go Yamashita has written a 294-page "summary" of Mochizuki's 500-page inscrutable(?) proof... if that's any encouragement to you ;)
Here's a few lines of the summary as quoted by Baez:
"By combining a relative anabelian result (relative Grothendieck Conjecture over sub-p-adic felds (Theorem B.1)) and "hidden endomorphism" diagram (EllCusp) (resp. "hidden endomorphism" diagram (BelyiCusp)), we show absolute anabelian results: the elliptic cuspidalisation (Theorem 3.7) (resp. Belyi cuspidalisation (Theorem 3.8)). By using Belyi cuspidalisations, we obtain an absolute mono-anabelian reconstruction of the NF-portion of the base field and the function field (resp. the base field) of hyperbolic curves of strictly Belyi type over sub-p-adic fields (Theorem 3.17) (resp. over mixed characteristic local fields (Corollary 3.19))."...Have at it!
Sunday, September 3, 2017
We’ve ended another wrenching week with this current unfit, anti-science, truth-warping, money-worshipping, law-disrespecting, ignorant, imperious, corrupt, coarse, Kafka-esque, foul, faux-Christian, scam-loving, dysfunctional, deplorable, delusional, democracy-dismantling, ill-principled, nepotistic, police-state-leaning, patronizing, power-grasping, plutocratic, pompous, prissy, prevaricating, pathological, piggish, petty, Putin-obeying, phony, pathetic, prickish, press-bashing, cerebrally-challenged, pseudo-American, scared little, self-absorbed, simple-minded, non-stable, tweet-obsessed, whistling-in-the-dark, roguish, Russian-colluding, Alpha-malevolent, amateurish (and impeachable?), Aryan-embracing, routinely-ridiculed-as-clueless, tin-pot dictatorial, pussy-grabbing Regime (...but you didn't hear any of that from me), and somehow I feel compelled to again run this classic Jacob Bronowski clip: