There Are Paradoxes

Sunday reflection:

“Perhaps the greatest paradox of all is that there are paradoxes in mathematics. We are not surprised to discover inconsistencies in the experimental sciences, which periodically undergo such revolutionary changes that although only a short time ago we believed ourselves descended from the gods, we now visit the zoo with the same friendly interest with which we call on distant relatives. Similarly, the fundamental and age-old distinction between matter and energy is vanishing, while relativity physics is shattering our basic concepts of time and space. Indeed, the testament of science is so continuously in a flux that the heresy of yesterday is the gospel of today and the fundamentalism of tomorrow. Paraphrasing Hamlet -- what was once a paradox is one no longer, but may again become one. Yet, because mathematics builds on the old but does not discard it, because it is the most conservative of the sciences, because its theorems are deduced from postulates by the methods of logic, in spite of its having undergone revolutionary changes we do not suspect it of being a discipline capable of engendering paradoxes.” 

— Edward Kasner and James R. Newman ("Mathematics and the Imagination")

[ ==> This will be the LAST of the "Sunday Reflections" here, as I continue streamlining  blogging activity, however be sure to return here tomorrow morning for a NEW timely interview.]

Getting Into Trouble

Just an old classic this week, for Sunday reflection:

"It ain't what you don't know that gets you into trouble. It's what you know for sure that just ain't so." -- Mark Twain

Indeed!...

In Their Own World

Sunday reflection via Jim Simons, mathematician and billionaire hedge-fund manager:

“When you’re really thinking hard about mathematics, you’re in your own world, and you’re cushioned from other things.” 

n-body problems

Sunday reflection:

"...a lot of effort was devoted to the three-body problem: the motion of a system consisting of three point masses (such as Sun, Earth, Moon) moving under Newtonian gravitation. It's easy enough to write down the appropriate equations of motion; but immensely harder to solve them... As an aside: it has been said that one can gauge the progress of science by the value of n for which the n-body problem cannot be solved. In Newtonian mechanics the 3-body problem appears to be insoluble. In Relativity, it is the 2-body problem that causes trouble. Quantum Theory gets hung up on the 1-body problem (a particle); and Relativistic Quantum Field Theory runs into trouble with the 0-body problem (the vacuum)!"
                                                                                -- Ian Stewart

Of Math and Cults

Sunday reflection:

“One of the most painful aspects of teaching mathematics is seeing my students damaged by the cult of the genius. That cult tells students that it’s not worth doing math unless you’re the best at math — because those special few are the only ones whose contributions really count.”

                                                                                             -- Jordan Ellenberg 

Teach Programming

Sunday reflection:

“My basic idea is that programming is the most powerful medium of developing the sophisticated and rigorous thinking needed for mathematics, for grammar, for physics, for statistics, for all the 'hard' subjects.... In short, I believe more than ever that programming should be a key part of the intellectual development of people growing up.”

— Seymour Papert

new appreciation....

Mathematician Richard Schwartz on his changed view of things:

"...one thing I appreciate more now is that the state of human knowledge is full of holes. When you’re young you have the impression that almost everything is known, but now I have this feeling that almost everything is unknown about mathematics. There are these very thin channels that people have gone along, like ants following each other along a trail. You find these long thin trails of things, and most things are undeveloped. I have more of a sense of the openness of it."

Model-T Algorithms...

Sunday reflection:

“Algorithms don’t make things fair, if you just blithely, blindly apply them. They don’t make things fair. They automate the status quo…

“Science is our only hope and I feel like we’ve created a field we call ‘data science,’ but there’s no science in it. We have not demanded evidence. The sort of hallmark of science is that it needs to be tested and testable, and we need to see the evidence, and we need to test every assumption. And we just haven’t done any of that. We’ve just been driving blind in our Model-T algorithms.”

— Cathy O’Neil on TED Radio Hour

Too Good Not to Pass Along (...from Jim Holt)

For Sunday reflection, from Jim Holt's fabulous new volume, "When Einstein Walked with Gödel":

"As for space and time, according to current speculative theories they, too, could well have a discontinuous sand-like structure on the tiniest scale, with the minimum length being the Planck length of 10^-33 centimeters and the minimum time being the Planck time of 10^-43 seconds (exactly the time, it has been observed, that it takes a New York cabbie to honk after the light turns green)."

(p.s.... over at MathTango this morning there is a new interview)

Of God and the Devil

For Sunday reflection, this classic old Andre Weil:

 "God exists since mathematics is consistent, and the Devil exists since we cannot prove it."

Delight and Exaltation

Sunday reflection...

“The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.”

— Bertrand Russell

Infallibility... Nope

Today's Sunday reflection:

“Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.”

— Johann Wolfgang von Goethe

'devoid of factual, empirical content'

Sunday reflection from Carl Hempel:

“The nature of the peculiar certainty of mathematics is now clear: A mathematical theorem is certain relatively to the set of postulates from which it is derived; i.e., it is necessarily true if those postulates are true; and this is so because the theorem, if rigorously proved, simply re-asserts part of what has been stipulated in the postulates… A mathematical truth is irrefutably certain just because it is devoid of factual, or empirical content.”

Liberation...

For today’s Sunday reflection, a little Rudy Rucker:

“…the ultimate success will never be ours [scientists]. Nowhere in the castle of science is there a final exit to absolute truth.
“This seems terribly depressing. But, paradoxically, to understand Gödel’s proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel’s name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.”

The Pleasure of Proof

For Sunday reflection, Bertrand Russell quoted in David Wells’ “The Penguin Book of Curious and Interesting Mathematics”:

“My friend G.H. Hardy, who was a professor of pure mathematics… told me once that if he could find proof that I was going to die in five minutes he would of course be sorry to lose me, but this sorrow would be quite outweighed by pleasure in the proof. I entirely sympathised with him and was not at all offended.”

Experiencing Pleasure... Music and Math

Sunday reflection via Gottfried Leibniz:

"Music is the pleasure the human mind experiences from counting without being aware that it is counting."

Physics and Common Sense

Succinct from Bertrand Russell:

"Common sense implies physics and physics refutes common sense."

[...meanwhile, a new book blurb over at MathTango today]

"When things get tough..."

Sunday reflection from physicist Victor Weisskopf:

“When things get tough, there are two things that make life worth living: Mozart, and quantum mechanics.”

Improvising... Music and Math

For Sunday reflection, from Stephon Alexander’s “The Jazz of Physics”:

“Humans are the only creatures that can discover advanced mathematics, and the only creatures that can create and formalize music. If the beauty and physics of the universe, and the beauty and physics of music are linked, the links exist uniquely in human brains… What makes us uniquely able to do what nonhuman brains cannot: appreciate music and understand mathematics? And to create new things under the sun: compose, improvise, discover new mathematical facts about the universe?   

 “A few musicians, like Coltrane, have an uncanny ability to improvise, to find the hidden patterns and regularities underlying harmonic forms and to use those insights to generate brand-new kinds of melodic sequences. And a few scientists, like Einstein, can find regularities that have eluded even other great scientists — such as taking the Maxwell equations and reducing them into a single unifying formulation.”

Bernie's Self-control...

Sunday reflection:

"[Walter] Mischel has priceless videos from some of the early experiments that demonstrate the difficulty kids had in exerting self-control. There is one kid I am particularly curious about. He was in the toughest setup, in which the bigger prize, three delicious Oreo cookies, was sitting right in front of him. After a brief wait, he could not stand it anymore. But rather than ring the bell, he carefully opened each cookie, licked out the yummy white filling, and then put the cookie back together, arranging the three cookies as best he could to avoid detection. In my imagination, this kid grows up to be Bernie Madoff."

                                                             -- Richard Thaler in "Misbehaving"