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Is the bounty system effective? - Meta Stack Exchange
could do some kinda econometric analysis using the data explorer to determine this once and for all:
maybe some kinda RDD in time, or difference-in-differences?
I don't think answer quality/quantity by time meets the common trend assumption for DD, tho... Questions that eventually receive bounty are prob higher quality in the first place, and higher quality answers accumulate more and better answers regardless. Hmm.
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november 2019 by nhaliday
exponential function - Feynman's Trick for Approximating $e^x$ - Mathematics Stack Exchange
1. e^2.3 ~ 10
2. e^.7 ~ 2
3. e^x ~ 1+x
e = 2.71828...

errors (absolute, relative):
1. +0.0258, 0.26%
2. -0.0138, -0.68%
3. 1 + x approximates e^x on [-.3, .3] with absolute error < .05, and relative error < 5.6% (3.7% for [0, .3]).
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october 2019 by nhaliday
The Future of Mathematics? [video] | Hacker News
Kevin Buzzard (the Lean guy)

- general reflection on proof asssistants/theorem provers
- Kevin Hale's formal abstracts project, etc
- thinks of available theorem provers, Lean is "[the only one currently available that may be capable of formalizing all of mathematics eventually]" (goes into more detail right at the end, eg, quotient types)
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october 2019 by nhaliday
online resources - How to write special set notation by hand? - Mathematics Stack Exchange
Your ℕN is “incorrect” in that a capital N in any serif font has the diagonal thickened, not the verticals. In fact, the rule (in Latin alphabet) is that negative slopes are thick, positive ones are thin. Verticals are sometimes thin, sometimes thick. Unique exception: Z. Just look in a newspaper at A, V, X, M, and N.
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october 2019 by nhaliday
Carryover vs “Far Transfer” | West Hunter
It used to be thought that studying certain subjects ( like Latin) made you better at learning others, or smarter generally – “They supple the mind, sir; they render it pliant and receptive.” This doesn’t appear to be the case, certainly not for Latin – although it seems to me that math can help you understand other subjects?

A different question: to what extent does being (some flavor of) crazy, or crazy about one subject, or being really painfully wrong about some subject, predict how likely you are to be wrong on other things? We know that someone can be strange, downright crazy, or utterly unsound on some topic and still do good mathematics… but that is not the same as saying that there is no statistical tendency for people on crazy-train A to be more likely to be wrong about subject B. What do the data suggest?
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october 2019 by nhaliday
Shuffling - Wikipedia
The Gilbert–Shannon–Reeds model provides a mathematical model of the random outcomes of riffling, that has been shown experimentally to be a good fit to human shuffling[2] and that forms the basis for a recommendation that card decks be riffled seven times in order to randomize them thoroughly.[3] Later, mathematicians Lloyd M. Trefethen and Lloyd N. Trefethen authored a paper using a tweaked version of the Gilbert-Shannon-Reeds model showing that the minimum number of riffles for total randomization could also be 5, if the method of defining randomness is changed.[4][5]
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august 2019 by nhaliday
Sage: Open Source Mathematics Software: You don't really think that Sage has failed, do you?
> P.S. You don't _really_ think that Sage has failed, do you?

After almost exactly 10 years of working on the Sage project, I absolutely do think it has failed to accomplish the stated goal of the mission statement: "Create a viable free open source alternative to Magma, Maple, Mathematica and Matlab.".     When it was only a few years into the project, it was really hard to evaluate progress against such a lofty mission statement.  However, after 10 years, it's clear to me that not only have we not got there, we are not going to ever get there before I retire.   And that's definitely a failure.   
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july 2019 by nhaliday
Alon Amit's answer to Why is there no formal definition for a set in math? How can we make any statement about sets (and therefore all of math) if we don’t even know what it is? - Quora
In the realm of mathematics, an object is what it does (I keep quoting Tim Gowers with this phrase, and I will likely do so many more times). The only thing that matters about points, lines, real numbers, sets, functions, groups and tempered distributions is the properties and features and rules they obey. What they “are” is of no concern.

I've seen this idea in a lot of different places
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july 2019 by nhaliday
Rational Sines of Rational Multiples of p
For which rational multiples of p is the sine rational? We have the three trivial cases
[0, pi/2, pi/6]
and we wish to show that these are essentially the only distinct rational sines of rational multiples of p.

The assertion about rational sines of rational multiples of p follows from two fundamental lemmas. The first is

Lemma 1: For any rational number q the value of sin(qp) is a root of a monic polynomial with integer coefficients.

[Pf uses some ideas unfamiliar to me: similarity parameter of Moebius (linear fraction) transformations, and finding a polynomial for a desired root by constructing a Moebius transformation with a finite period.]


Lemma 2: Any root of a monic polynomial f(x) with integer coefficients must either be an integer or irrational.

[Gauss's Lemma, cf Dummit-Foote.]

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july 2019 by nhaliday
The Existential Risk of Math Errors -
How big is this upper bound? Mathematicians have often made errors in proofs. But it’s rarer for ideas to be accepted for a long time and then rejected. But we can divide errors into 2 basic cases corresponding to type I and type II errors:

1. Mistakes where the theorem is still true, but the proof was incorrect (type I)
2. Mistakes where the theorem was false, and the proof was also necessarily incorrect (type II)

Before someone comes up with a final answer, a mathematician may have many levels of intuition in formulating & working on the problem, but we’ll consider the final end-product where the mathematician feels satisfied that he has solved it. Case 1 is perhaps the most common case, with innumerable examples; this is sometimes due to mistakes in the proof that anyone would accept is a mistake, but many of these cases are due to changing standards of proof. For example, when David Hilbert discovered errors in Euclid’s proofs which no one noticed before, the theorems were still true, and the gaps more due to Hilbert being a modern mathematician thinking in terms of formal systems (which of course Euclid did not think in). (David Hilbert himself turns out to be a useful example of the other kind of error: his famous list of 23 problems was accompanied by definite opinions on the outcome of each problem and sometimes timings, several of which were wrong or questionable5.) Similarly, early calculus used ‘infinitesimals’ which were sometimes treated as being 0 and sometimes treated as an indefinitely small non-zero number; this was incoherent and strictly speaking, practically all of the calculus results were wrong because they relied on an incoherent concept - but of course the results were some of the greatest mathematical work ever conducted6 and when later mathematicians put calculus on a more rigorous footing, they immediately re-derived those results (sometimes with important qualifications), and doubtless as modern math evolves other fields have sometimes needed to go back and clean up the foundations and will in the future.7


Isaac Newton, incidentally, gave two proofs of the same solution to a problem in probability, one via enumeration and the other more abstract; the enumeration was correct, but the other proof totally wrong and this was not noticed for a long time, leading Stigler to remark:


“Lefschetz was a purely intuitive mathematician. It was said of him that he had never given a completely correct proof, but had never made a wrong guess either.”
- Gian-Carlo Rota13

Case 2 is disturbing, since it is a case in which we wind up with false beliefs and also false beliefs about our beliefs (we no longer know that we don’t know). Case 2 could lead to extinction.


Except, errors do not seem to be evenly & randomly distributed between case 1 and case 2. There seem to be far more case 1s than case 2s, as already mentioned in the early calculus example: far more than 50% of the early calculus results were correct when checked more rigorously. Richard Hamming attributes to Ralph Boas a comment that while editing Mathematical Reviews that “of the new results in the papers reviewed most are true but the corresponding proofs are perhaps half the time plain wrong”.


Gian-Carlo Rota gives us an example with Hilbert:


Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems. There was one exception, a paper Hilbert wrote in his old age, which could not be fixed; it was a purported proof of the continuum hypothesis, you will find it in a volume of the Mathematische Annalen of the early thirties.


Leslie Lamport advocates for machine-checked proofs and a more rigorous style of proofs similar to natural deduction, noting a mathematician acquaintance guesses at a broad error rate of 1/329 and that he routinely found mistakes in his own proofs and, worse, believed false conjectures30.

[more on these "structured proofs":

We can probably add software to that list: early software engineering work found that, dismayingly, bug rates seem to be simply a function of lines of code, and one would expect diseconomies of scale. So one would expect that in going from the ~4,000 lines of code of the Microsoft DOS operating system kernel to the ~50,000,000 lines of code in Windows Server 2003 (with full systems of applications and libraries being even larger: the comprehensive Debian repository in 2007 contained ~323,551,126 lines of code) that the number of active bugs at any time would be… fairly large. Mathematical software is hopefully better, but practitioners still run into issues (eg Durán et al 2014, Fonseca et al 2017) and I don’t know of any research pinning down how buggy key mathematical systems like Mathematica are or how much published mathematics may be erroneous due to bugs. This general problem led to predictions of doom and spurred much research into automated proof-checking, static analysis, and functional languages31.

I don't know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn't.

Define sinc𝑥=(sin𝑥)/𝑥.

Someone found the following result in an algebra package: ∫∞0𝑑𝑥sinc𝑥=𝜋/2
They then found the following results:


So of course when they got:


Which means that nobody knows Fourier analysis nowdays. Very sad and discouraging story... – fedja Jan 29 '10 at 18:47


Because the most popular systems are all commercial, they tend to guard their bug database rather closely -- making them public would seriously cut their sales. For example, for the open source project Sage (which is quite young), you can get a list of all the known bugs from this page. 1582 known issues on Feb.16th 2010 (which includes feature requests, problems with documentation, etc).

That is an order of magnitude less than the commercial systems. And it's not because it is better, it is because it is younger and smaller. It might be better, but until SAGE does a lot of analysis (about 40% of CAS bugs are there) and a fancy user interface (another 40%), it is too hard to compare.

I once ran a graduate course whose core topic was studying the fundamental disconnect between the algebraic nature of CAS and the analytic nature of the what it is mostly used for. There are issues of logic -- CASes work more or less in an intensional logic, while most of analysis is stated in a purely extensional fashion. There is no well-defined 'denotational semantics' for expressions-as-functions, which strongly contributes to the deeper bugs in CASes.]


Should such widely-believed conjectures as P≠NP or the Riemann hypothesis turn out be false, then because they are assumed by so many existing proofs, a far larger math holocaust would ensue38 - and our previous estimates of error rates will turn out to have been substantial underestimates. But it may be a cloud with a silver lining, if it doesn’t come at a time of danger.

more on formal methods in programming:
Update: measured effort
In the October 2018 issue of Communications of the ACM there is an interesting article about Formally verified software in the real world with some estimates of the effort.

Interestingly (based on OS development for military equipment), it seems that producing formally proved software requires 3.3 times more effort than with traditional engineering techniques. So it's really costly.

On the other hand, it requires 2.3 times less effort to get high security software this way than with traditionally engineered software if you add the effort to make such software certified at a high security level (EAL 7). So if you have high reliability or security requirements there is definitively a business case for going formal.

You can see examples of how all of these look at Let’s Prove Leftpad. HOL4 and Isabelle are good examples of “independent theorem” specs, SPARK and Dafny have “embedded assertion” specs, and Coq and Agda have “dependent type” specs.6

If you squint a bit it looks like these three forms of code spec map to the three main domains of automated correctness checking: tests, contracts, and types. This is not a coincidence. Correctness is a spectrum, and formal verification is one extreme of that spectrum. As we reduce the rigour (and effort) of our verification we get simpler and narrower checks, whether that means limiting the explored state space, using weaker types, or pushing verification to the runtime. Any means of total specification then becomes a means of partial specification, and vice versa: many consider Cleanroom a formal verification technique, which primarily works by pushing code review far beyond what’s humanly possible.


The question, then: “is 90/95/99% correct significantly cheaper than 100% correct?” The answer is very yes. We all are comfortable saying that a codebase we’ve well-tested and well-typed is mostly correct modulo a few fixes in prod, and we’re even writing more than four lines of code a day. In fact, the vast… [more]
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july 2019 by nhaliday
Factorization of polynomials over finite fields - Wikipedia
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.

All factorization algorithms, including the case of multivariate polynomials over the rational numbers, reduce the problem to this case; see polynomial factorization. It is also used for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.

As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.


In the algorithms that follow, the complexities are expressed in terms of number of arithmetic operations in Fq, using classical algorithms for the arithmetic of polynomials.

[ed.: Interesting choice...]


Factoring algorithms
Many algorithms for factoring polynomials over finite fields include the following three stages:

Square-free factorization
Distinct-degree factorization
Equal-degree factorization
An important exception is Berlekamp's algorithm, which combines stages 2 and 3.

Berlekamp's algorithm
Main article: Berlekamp's algorithm
The Berlekamp's algorithm is historically important as being the first factorization algorithm, which works well in practice. However, it contains a loop on the elements of the ground field, which implies that it is practicable only over small finite fields. For a fixed ground field, its time complexity is polynomial, but, for general ground fields, the complexity is exponential in the size of the ground field.

[ed.: This actually looks fairly implementable.]
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july 2019 by nhaliday
About - Project Euler
I've written my program but should it take days to get to the answer?
Absolutely not! Each problem has been designed according to a "one-minute rule", which means that although it may take several hours to design a successful algorithm with more difficult problems, an efficient implementation will allow a solution to be obtained on a modestly powered computer in less than one minute.
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june 2019 by nhaliday
soft question - What are good non-English languages for mathematicians to know? - MathOverflow
I'm with Deane here: I think learning foreign languages is not a very mathematically productive thing to do; of course, there are lots of good reasons to learn foreign languages, but doing mathematics is not one of them. Not only are there few modern mathematics papers written in languages other than English, but the primary other language they are written (French) in is pretty easy to read without actually knowing it.

Even though I've been to France several times, my spoken French mostly consists of "merci," "si vous plait," "d'accord" and some food words; I've still skimmed 100 page long papers in French without a lot of trouble.

If nothing else, think of reading a paper in French as a good opportunity to teach Google Translate some mathematical French.
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february 2019 by nhaliday
Theory of Self-Reproducing Automata - John von Neumann

Comparisons between computing machines and the nervous systems. Estimates of size for computing machines, present and near future.

Estimates for size for the human central nervous system. Excursus about the “mixed” character of living organisms. Analog and digital elements. Observations about the “mixed” character of all componentry, artificial as well as natural. Interpretation of the position to be taken with respect to these.

Evaluation of the discrepancy in size between artificial and natural automata. Interpretation of this discrepancy in terms of physical factors. Nature of the materials used.

The probability of the presence of other intellectual factors. The role of complication and the theoretical penetration that it requires.

Questions of reliability and errors reconsidered. Probability of individual errors and length of procedure. Typical lengths of procedure for computing machines and for living organisms--that is, for artificial and for natural automata. Upper limits on acceptable probability of error in individual operations. Compensation by checking and self-correcting features.

Differences of principle in the way in which errors are dealt with in artificial and in natural automata. The “single error” principle in artificial automata. Crudeness of our approach in this case, due to the lack of adequate theory. More sophisticated treatment of this problem in natural automata: The role of the autonomy of parts. Connections between this autonomy and evolution.

- 10^10 neurons in brain, 10^4 vacuum tubes in largest computer at time
- machines faster: 5 ms from neuron potential to neuron potential, 10^-3 ms for vacuum tubes
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april 2018 by nhaliday
John Dee - Wikipedia
John Dee (13 July 1527 – 1608 or 1609) was an English mathematician, astronomer, astrologer, occult philosopher,[5] and advisor to Queen Elizabeth I. He devoted much of his life to the study of alchemy, divination, and Hermetic philosophy. He was also an advocate of England's imperial expansion into a "British Empire", a term he is generally credited with coining.[6]

Dee straddled the worlds of modern science and magic just as the former was emerging. One of the most learned men of his age, he had been invited to lecture on the geometry of Euclid at the University of Paris while still in his early twenties. Dee was an ardent promoter of mathematics and a respected astronomer, as well as a leading expert in navigation, having trained many of those who would conduct England's voyages of discovery.

Simultaneously with these efforts, Dee immersed himself in the worlds of magic, astrology and Hermetic philosophy. He devoted much time and effort in the last thirty years or so of his life to attempting to commune with angels in order to learn the universal language of creation and bring about the pre-apocalyptic unity of mankind. However, Robert Hooke suggested in the chapter Of Dr. Dee's Book of Spirits, that John Dee made use of Trithemian steganography, to conceal his communication with Elizabeth I.[7] A student of the Renaissance Neo-Platonism of Marsilio Ficino, Dee did not draw distinctions between his mathematical research and his investigations into Hermetic magic, angel summoning and divination. Instead he considered all of his activities to constitute different facets of the same quest: the search for a transcendent understanding of the divine forms which underlie the visible world, which Dee called "pure verities".

In his lifetime, Dee amassed one of the largest libraries in England. His high status as a scholar also allowed him to play a role in Elizabethan politics. He served as an occasional advisor and tutor to Elizabeth I and nurtured relationships with her ministers Francis Walsingham and William Cecil. Dee also tutored and enjoyed patronage relationships with Sir Philip Sidney, his uncle Robert Dudley, 1st Earl of Leicester, and Edward Dyer. He also enjoyed patronage from Sir Christopher Hatton.
mind meld

Leave Me Alone! Misanthropic Writings from the Anti-Social Edge
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april 2018 by nhaliday
Hyperbolic angle - Wikipedia
A unit circle {\displaystyle x^{2}+y^{2}=1} x^2 + y^2 = 1 has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola {\displaystyle x^{2}-y^{2}=1} {\displaystyle x^{2}-y^{2}=1} has a hyperbolic sector with an area half of the hyperbolic angle.
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november 2017 by nhaliday
Stability of the Solar System - Wikipedia
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others) the Solar System is chaotic,[1] and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years.[2]

The Solar System is stable in human terms, and far beyond, given that it is unlikely any of the planets will collide with each other or be ejected from the system in the next few billion years,[3] and the Earth's orbit will be relatively stable.[4]

Since Newton's law of gravitation (1687), mathematicians and astronomers (such as Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Vladimir Arnold and Jürgen Moser) have searched for evidence for the stability of the planetary motions, and this quest led to many mathematical developments, and several successive 'proofs' of stability of the Solar System.[5]


The planets' orbits are chaotic over longer timescales, such that the whole Solar System possesses a Lyapunov time in the range of 2–230 million years.[3] In all cases this means that the position of a planet along its orbit ultimately becomes impossible to predict with any certainty (so, for example, the timing of winter and summer become uncertain), but in some cases the orbits themselves may change dramatically. Such chaos manifests most strongly as changes in eccentricity, with some planets' orbits becoming significantly more—or less—elliptical.[7]

Is the Solar System Stable?:

Is the Solar System Stable?:
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november 2017 by nhaliday
gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof? - MathOverflow
In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says "... every pair of curves in the square joining different pairs of opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go about proving this rigorously. I have thought of a proof using Brouwer's Fixed Point Theorem which I describe below. I would greatly appreciate the group's comments on whether this proof is right and if a simpler proof is possible.


Since the full Jordan curve theorem is quite subtle, it might be worth pointing out that theorem in question reduces to the Jordan curve theorem for polygons, which is easier.

Suppose on the contrary that the curves A,BA,B joining opposite corners do not meet. Since A,BA,B are closed sets, their minimum distance apart is some ε>0ε>0. By compactness, each of A,BA,B can be partitioned into finitely many arcs, each of which lies in a disk of diameter <ε/3<ε/3. Then, by a homotopy inside each disk we can replace A,BA,B by polygonal paths A′,B′A′,B′ that join the opposite corners of the square and are still disjoint.

Also, we can replace A′,B′A′,B′ by simple polygonal paths A″,B″A″,B″ by omitting loops. Now we can close A″A″ to a polygon, and B″B″ goes from its "inside" to "outside" without meeting it, contrary to the Jordan curve theorem for polygons.

- John Stillwell
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october 2017 by nhaliday
multivariate analysis - Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? - Cross Validated
The bivariate normal distribution is the exception, not the rule!

It is important to recognize that "almost all" joint distributions with normal marginals are not the bivariate normal distribution. That is, the common viewpoint that joint distributions with normal marginals that are not the bivariate normal are somehow "pathological", is a bit misguided.

Certainly, the multivariate normal is extremely important due to its stability under linear transformations, and so receives the bulk of attention in applications.

note: there is a multivariate central limit theorem, so those such applications have no problem
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october 2017 by nhaliday
Best Topology Olympiad ***EVER*** - Affine Mess - Quora
Most people take courses in topology, algebraic topology, knot theory, differential topology and what have you without once doing anything with a finite topological space. There may have been some quirky questions about such spaces early on in a point-set topology course, but most of us come out of these courses thinking that finite topological spaces are either discrete or only useful as an exotic counterexample to some standard separation property. The mere idea of calculating the fundamental group for a 4-point space seems ludicrous.

Only it’s not. This is a genuine question, not a joke, and I find it both hilarious and super educational. DO IT!!
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october 2017 by nhaliday
Variance of product of multiple random variables - Cross Validated
prod_i (var[X_i] + (E[X_i])^2) - prod_i (E[X_i])^2

two variable case: var[X] var[Y] + var[X] (E[Y])^2 + (E[X])^2 var[Y]
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october 2017 by nhaliday
Power of a point - Wikipedia
The power of point P (see in Figure 1) can be defined equivalently as the product of distances from the point P to the two intersection points of any ray emanating from P.
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september 2017 by nhaliday
Isaac Newton: the first physicist.
[...] More fundamentally, Newton's mathematical approach has become so basic to all of physics that he is generally regarded as _the father of the clockwork universe_: the first, and perhaps the greatest, physicist.

The Alchemist

In fact, Newton was deeply opposed to the mechanistic conception of the world. A secretive alchemist [...]. His written work on the subject ran to more than a million words, far more than he ever produced on calculus or mechanics [21]. Obsessively religious, he spent years correlating biblical prophecy with historical events [319ff]. He became deeply convinced that Christian doctrine had been deliberately corrupted by _the false notion of the trinity_, and developed a vicious contempt for conventional (trinitarian) Christianity and for Roman Catholicism in particular [324]. [...] He believed that God mediated the gravitational force [511](353), and opposed any attempt to give a mechanistic explanation of chemistry or gravity, since that would diminish the role of God [646]. He consequently conceived such _a hatred of Descartes_, on whose foundations so many of his achievements were built, that at times _he refused even to write his name_ [399,401].

The Man

Newton was rigorously puritanical: when one of his few friends told him "a loose story about a nun", he ended their friendship (267). [...] He thought of himself as the sole inventor of the calculus, and hence the greatest mathematician since the ancients, and left behind a huge corpus of unpublished work, mostly alchemy and biblical exegesis, that he believed future generations would appreciate more than his own (199,511).

[...] Even though these unattractive qualities caused him to waste huge amounts of time and energy in ruthless vendettas against colleagues who in many cases had helped him (see below), they also drove him to the extraordinary achievements for which he is still remembered. And for all his arrogance, Newton's own summary of his life (574) was beautifully humble:

"I do not know how I may appear to the world, but to myself I seem to have been only like a boy, playing on the sea-shore, and diverting myself, in now and then finding a smoother pebble or prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

Before Newton


1. Calculus. Descartes, in 1637, pioneered the use of coordinates to turn geometric problems into algebraic ones, a method that Newton was never to accept [399]. Descartes, Fermat, and others investigated methods of calculating the tangents to arbitrary curves [28-30]. Kepler, Cavalieri, and others used infinitesimal slices to calculate volumes and areas enclosed by curves [30], but no unified treatment of these problems had yet been found.
2. Mechanics & Planetary motion. The elliptical orbits of the planets having been established by Kepler, Descartes proposed the idea of a purely mechanical heliocentric universe, following deterministic laws, and with no need of any divine agency [15], another anathema to Newton. _No one imagined, however, that a single law might explain both falling bodies and planetary motion_. Galileo invented the concept of inertia, anticipating Newton's first and second laws of motion (293), and Huygens used it to analyze collisions and circular motion [11]. Again, these pieces of progress had not been synthesized into a general method for analyzing forces and motion.
3. Light. Descartes claimed that light was a pressure wave, Gassendi that it was a stream of particles (corpuscles) [13]. As might be guessed, Newton vigorously supported the corpuscular theory. _White light was universally believed to be the pure form_, and colors were some added property bequeathed to it upon reflection from matter (150). Descartes had discovered the sine law of refraction (94), but it was not known that some colors were refracted more than others. The pattern was the familiar one: many pieces of the puzzle were in place, but the overall picture was still unclear.

The Natural Philosopher

Between 1671 and 1690, Newton was to supply definitive treatments of most of these problems. By assiduous experimentation with prisms he established that colored light was actually fundamental, and that it could be recombined to create white light. He did not publish the result for 6 years, by which time it seemed so obvious to him that he found great difficulty in responding patiently to the many misunderstandings and objections with which it met [239ff].

He invented differential and integral calculus in 1665-6, but failed to publish it. Leibniz invented it independently 10 years later, and published it first [718]. This resulted in a priority dispute which degenerated into a feud characterized by extraordinary dishonesty and venom on both sides (542).

In discovering gravitation, Newton was also _barely ahead of the rest of the pack_. Hooke was the first to realize that orbital motion was produced by a centripetal force (268), and in 1679 _he suggested an inverse square law to Newton_ [387]. Halley and Wren came to the same conclusion, and turned to Newton for a proof, which he duly supplied [402]. Newton did not stop there, however. From 1684 to 1687 he worked continuously on a grand synthesis of the whole of mechanics, the "Philosophiae Naturalis Principia Mathematica," in which he developed his three laws of motion and showed in detail that the universal force of gravitation could explain the fall of an apple as well as the precise motions of planets and comets.

The "Principia" crystallized the new conceptions of force and inertia that had gradually been emerging, and marks the beginning of theoretical physics as the mathematical field that we know today. It is not an easy read: Newton had developed the idea that geometry and equations should never be combined [399], and therefore _refused to use simple analytical techniques in his proofs_, requiring classical geometric constructions instead [428]. He even made his Principia _deliberately abstruse in order to discourage amateurs from feeling qualified to criticize it_ [459].

[...] most of the rest of his life was spent in administrative work as Master of the Mint and as President of the Royal Society, _a position he ruthlessly exploited in the pursuit of vendettas_ against Hooke (300ff,500), Leibniz (510ff), and Flamsteed (490,500), among others. He kept secret his disbelief in Christ's divinity right up until his dying moment, at which point he refused the last rites, at last openly defying the church (576). [...]
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august 2017 by nhaliday
Inscribed angle - Wikipedia
- for triangle w/ one side = a diameter, draw isosceles triangle and use supplementary angle identities
- otherwise draw second triangle w/ side = a diameter, and use above result twice
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Diophantine approximation - Wikipedia
- rationals perfectly approximated by themselves, badly approximated (eps>1/bq) by other rationals
- irrationals well-approximated (eps~1/q^2) by rationals:
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