for Sunday reflection... just an old, classic joke:
“There are three kinds of people in the world: those who are good at mathematics and those who aren’t.”
...Links for math buffs and number-luvin' laymen
“There are three kinds of people in the world: those who are good at mathematics and those who aren’t.”
“You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.”
“Meanwhile, Nelson Goodman kept sharpening the knife of nominalism. In 1951 he published The Structure of Appearances. This book contains a logic of parts and wholes. Goodman denies that there are sets. Instead, there are fusions built up from smaller things. Unlike a set, a fusion has a position in space and time. You can touch a fusion. I’m a fusion. So are you. Goodman’s ‘calculus of individuals’ says that there are only finitely many atomic individuals and that any combination of atoms is an individual. Objects do not need to have all their parts connected, for instance, Alaska and Hawaii are parts of the United States of America. Goodman does not let human intuition dictate what counts as an object; he also thinks that there is the fusion of his ear and the moon. In a seminar Goodman taught at the University of Pennsylvania around 1965, John Robison pointed out that The Structure of Appearances implies an answer to ‘Is the number of individuals in the universe odd or even?’ Since there are only finitely many atoms and each individual is identical to a combination of atoms, there are exactly as many individuals as there are combinations of atoms. If there are n atoms, there are 2n - 1 combinations of individuals. No matter which number we choose for n, 2n - 1 is an odd number. Therefore, the number of individuals in the universe is odd! The exclamation point is not for the oddness per se. Aside from those who think the universe is infinite, people agree that the universe contains either an odd number of individuals or an even number of individuals. What they find absurd is that there could be a proof that the number of individuals is odd. ‘Is the number of individuals in the universe odd or even?’ illustrates the possibility of one good answer being too many. Our expectation is that this question is unanswerable. The lone good answer confounds beliefs about what arguments can accomplish.”Anyway, seems like an interesting thought exercise to play with.
“The minds of brilliant mathematicians are of perennial fascination. But in the onrushing era of synthetic neurobiology and genomic reconfiguration, the possibility that genius and mental illness are intertwined takes on monumental significance. If scientists are eventually able to alter living brains or edit human embryos with an eye to mitigating conditions such as autism and schizophrenia, do we risk excising brilliant outliers from the gene pool? Isaac Newton, John Nash, and Alexander Grothendieck are low-frequency, high-impact minds; they advanced civilization in the domain on which they trained their high beams. It is worth turning the high beams of scientific inquiry on those same unusual minds.”
“It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible. From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.”
“Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.” —E. T. Bell
"Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in concepts that unify our thinking... Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time. I happen to be a frog, but many of my best friends are birds."
“ErdÃ¶s was an avowed atheist, and his friends at Notre Dame enjoyed teasing him about his working at a Roman Catholic university. ‘He said in all seriousness that he liked being there very much,’ Melvin Henriksen, a colleague from those days, recalled, ‘and especially enjoyed discussions with the [priests].’ Only one thing bothered him. ‘There were too many plus signs,’ he irreverently remarked."
“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.”
“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.”
"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!
“…the world of today’s mathematician is one not only in which truth is not synonymous with logical proof but also in which merely trusting in the validity of a logical proof is itself a matter of faith. This is because GÃ¶del not only showed that any logical system is unable to prove all the mathematical statements that are actually true, but also that any system of logic is unable to prove its own logical consistency. Believing in logic, in other words, is no less subjective a frame of mind than believing in, say, a secular or mystical principle of faith, because even logic itself cannot be verified logically or objectively.”
"This is how category theory arose, as a new piece of math to study math. In a way, category theory is an ultimate abstraction. To study the world abstractly you use science; to study math abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory."
"Mathematicians are people who like solving problems, and have the persistence to work on problems that take time to solve, and have collected a mental tool-kit consisting of methods that have helped them solve problems in the past. Some mathematicians distinguish between methods and tricks. A method is a tool that solves more than one problem, while a trick is a tool that applies to only one. Under this definition, I’d say that there are no tricks in math, and part of the discipline of getting good at math is to study every trick you encounter until you see the method hiding inside it."
“The ideology of mathematical certainty and objectivity is our most potent weapon; we should not allow it to be used to undermine democracy. With regard to mathematical modeling, we should constantly remind anyone who is willing to listen that a model is not objective or scientific just because it is mathematical.”