User rhett butler - MathOverflow

Answer by Rhett Butler for Is there a good reason why a^{2b} + b^{2a} <= 1 when a+b=1?

2013-05-23T10:50:51Z

Maybe there is a small trick yet.

For $a + b = 1$ we can write the sum as $$a^{2(1-a)} + b^{2(1-b)} = (\frac{a}{a^a})^2+ (\frac{b}{b^b})^2 $$.

Obviously the sum is 1 for $(a, b) = (0,1), (\frac{1}{2},\frac{1}{2}), (1, 0)$. The question is whether at $(\frac{1}{2},\frac{1}{2})$ there is a maximum or a minimum, i.e. whether the second derivative of

$$(\frac{x}{x^x})^2+ (\frac{1-x}{(1-x)^{1-x}})^2$$

is positive or negative. (By symmetry only minimum or maximum can occur there.)

Using the fact that, for $x = \frac{1}{2}$,

$$\frac{d^2}{dx^2}(\frac{1-x}{(1-x)^{1-x}}) = \frac{d^2}{(-dx)^2}(\frac{x}{x^x})^2 = \frac{d^2}{dx^2}(\frac{x}{x^x})^2$$

it is sufficient to prove

$$2\frac{d^2}{dx^2}(\frac{x}{x^x})^2 < 0$$

for $x = \frac{1}{2}$ which in fact is easily demonstrated.

Answer by Rhett Butler for Sum of odd number is a square, whos theorem is this?

2013-04-19T21:05:04Z

The original source, attributing this theorem to the early pythagoreans is Theon of Smyrna. Eduard Hiller: Expositio Rerum Mathematicarum ad legendum Platonem utilium. Rec. Theon Smyrnaeus, reprinted by Teubner, Stuttgart, 1995.

Further Aristoteles writes, with reference to the early pythagoreans too: "the gnomons are placed round the one" explaining in a somewhat dark manner the geometric aspect of the sum of odd numbers, placed around the 1 Physics, book 3, chapter 4 as sketched in the answer by Stefan .

So there is no chance to find an individual name of the first inventor other than Pythagoras himself. But it is not clear what he really did. No documents of his are left. (Nicomachus, Aristotle, and Archimedes definitively lived too late.)

Answer by Rhett Butler for Did any new mathematics arise from Ruffini's work on the quintic equation?

2013-03-11T19:41:17Z

Pierre L. Wantzel acknowledges the works of Niels H. Abel and Paolo Ruffini in the introduction to his complete proof of the insolubility of higher polynomial equations: In meditating on the researches of these two mathematicians, and with the aid of principles we established in an earlier paper, we have arrived at a form of proof which appears so strict as to remove all doubt on this important part of the theory of equations.

Heinrich Burkhard in Die Anfänge der Gruppentheorie holds the opinion that work of the Italien mathematician Pietro Abbati Marescotti could have been inspired by Paolo Ruffini's results. Vice versa it cannot be excluded that Abbati mentioned the first ideas of group theory to Ruffini, who subsequently expanded it.

Whether Ruffini himself or his friend Abbati had the first idea to apply group theory, partially based upon Lagrange's work on permutations, is not clear from their preserved correspondence but since Ruffini determined nearly all subgroups of the symmetric group $S_5$ his result belongs to the foundation of group theory. So his influence extends to class field theory (although modern terminology is connected with the name of Abel). This is certainly an accomplishment important enough to be called "significant new mathematical concepts or ideas".

Answer by Rhett Butler for Who first cared about singular points?

2013-04-22T19:52:32Z

Gauss' Disquisitiones generales circa superficies curvas, read at the Royal Society in Goettingen on 8 Oct. 1827, contain many termini like punctis singularibus or singulis punctis. Already on the first page we read cuius singula puncta repraesentare.

Dirichlet's collected works, edited by Kronecker and Fuchs, contain often the phrase "singular cases", and on pag. 365 punctum singulare, (written around 1850).

Riemann's collected works, edited by Weber and Dedekind, contain the first mentioning of "singulärer Punkt" on page 389 that I know of in German. It is taken from a paper about linear differential equations of 1857.

The Earliest Known Uses of Some of the Words of Mathematics supply the following dates:

SINGULAR POINT appears in a paper by George Green published in 1828. The paper also contains the synonymous phrase "singular value" [James A. Landau].

Singular point appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young.

In An Elementary Treatise on Curves, Functions and Forces (1846), Benjamin Peirce writes, "Those points of a curve, which present any peculiarity as to curvature or discontinuity, are called singular points."

Additional remark

Hermann, a correspondent of Leibniz, wrote about singulis locis and singulis punctis in letters to Leibniz of 11 Jan. 1711 and Jun 1712, respectively. [C. I. Gerhardt (ed.): "Leibnizens mathematische Schriften", Halle (1859) p. 364 and 368]

And we should not forget that l'Hospital's famous theorem (1696) has been designed for singular points of functions only.

Answer by Rhett Butler for What is the history of $\sqrt{}$

2013-02-08T16:22:15Z

This question has already been completely answered, but here is a bit more on history.

The symbol has its origin in the Latin letter R as an abbreviation of radix (latin: root). It has been used by Leonardo de Pisa (Fibonacci) in his seminal book Liber Abaci (1202) where he treats square roots and cubic roots in chapter 14 and 15 (as well as in his later less well known Liber Quadratorum (1225)).

Fibonacci, like Euclid, did not invent all the mathematics he reports, but took much of it from the Arabian world, mainly from al-Khwarizmi and Omar Khayyam. The Arabian word for "root" had been used by al-Khwarizmi already; his word is rendered radix in translations from the Arabic to Latin by Robert of Chester, John of Seville, and Gerard of Cremona. It appears also in Alexandre de Villedieu's Carmen de Algorismo (1240) and in Sacrobosco's Algorismus (1250). By the way Fibonacci calculates approximations but considers roots as exact numbers, even if they cannot be expressed as integers or fractions.

Following the tradition of medieval writers, Nicolas Chuquet used the uppercase Latin letter R with a small stroke, looking very similar to Px when written close together. Both, R and R$^2$ indicate the square root, R$^3$ indicates the cubic root, R$^4$ the forth root and so on. Regiomontanus (1464), Luca Pacioli (1494), and Estienne de la Roche (1520) adopted this sign.

The hook-like symbol √ that resembles a small r was introduced by Christoff Rudolff in his book "Die Coß" (1525), the first German book on algebra. He used c√ for cubic root and √√ for fourths root. (By the way he introduced also the convention $x^0$ = 1). English, French and Italien writers were slow in adopting that German sign. Even in Germany the symbol "l" for latus (side of the square) was long in use. After Michael Stifel had published the second edition of Rudolff's Coß (1553) the symbol became more and more accepted.

René Descartes (1596 bis 1650) invented (or extended) the bar above the radicand (this word being first used in 1889) in order to indicate what symbols belong to the radicand.

Moritz Cantor: "Vorlesungen über Geschichte der Mathematik", Teubner, Leipzig (1894) http://archive.org/stream/vorlesungenber02cantuoft#page/n5/mode/2up

Florian Cajori: "A History of Mathematics" MacMillan, London (1909) http://www.gutenberg.org/files/31061/31061-pdf.pdf

David Eugene Smith: "History of Mathematics, vol. 2", Dover Publications (1958) http://books.google.de/books/about/History_of_Mathematics.html?id=uTytJGnTf1kC&redir_esc=y

http://jeff560.tripod.com/r.html

http://www-history.mcs.st-and.ac.uk/Biographies/Fibonacci.html

Edit As Paul Taylor said Florian Cajori favours another root of the root symbol, namely the generation of an upstroke from a row of points. In fact there may have been many sources playing together. Moritz Cantor points out that Alkasadi (or Alkasawi) an Arab living in Spain (died 1477 or 1486) wrote a book which is known under different titles like Lifting the veils of the science of the Gubar (Gubar means "dust" or "calculating with digits") where he not only abbreviated the Arab word for root dschidr by writing its first letter, but wrote it not right of the radicand (in Arabic, meaning in front of the radicand) but above. The jim can be seen in the column Initial in the table Arabic letters usage in Literary Arabic. This could also be a source for our root symbol.

Answer by Rhett Butler for Can different bicycles leave the same tracks?

2013-05-05T16:58:08Z

I would answer no, based on a very simple argument. (But in contrast to the Sean Howe's answer I restrict my argument to finite curves.) Consider a bicycle ridden on a circle. Its radius $r_1$ is defined by the angle of the frontwheel with respect to the bicycle. The rearwheel will draw a smaller circle with radius $r_2$. The difference of radii will depend on the distance $d$ of the axes such that $d^2 = r_1^2 - r_2^2$.

Since every curve can be considered as an approximation of a circle, this implies that different distances of the axes will supply different curves for every given $r_1$, except when the curve is a straight line, both radii being infinite.

Answer by Rhett Butler for The eliminant of a system of differential equations

2013-05-04T14:39:53Z

In "Leopold Kroneckers Werke, 3. Band", Teubner (1899), published by K. Hensel, we find on p. 179 in a section about applications of modulsystems the phrase: "Die Resultante der Elimination von ...".

This suggests that Thorny's above conjecture is correct and the eliminant is also called resultant. The due reference given by José Figueroa-O'Farrill is confirmed by the McGraw-Hill Dictionary of Scientific & Technical Terms.

The eliminant has originally been defined for algebraic equations. See Otto Biermann's comprehensive and detailed paper: "Über die Bildung der Eliminanten eines Systems algebraischer Gleichungen", Monatshefte für Mathematik und Physik (1894) pp. 17-32, referring (without giving sources) to Salmon-Fiedler, Günther, Sylvester and Cayley. The mode of application to linear differential operators has been suggested here by Charles Siegel, and application to linear differential equations becomes obvious when the characteristic polynoms are formed in the common way by means of the exponential function.

Application of eliminant to differential equations can be found in Paul Funk's paper "Beiträge zur zweidimensionalen Finsler'schen Geometrie", Monatshefte für Mathematik 52 (1948) pp. 194-216, and also in modern literature, namely in A.P. Alexandrov's arXiv-paper: "Dynamic systems with quantum behaviour" on p. 99.

Apropos The word "eliminante" has acquired a general use in mathematics. This can be seen by the completely independent application of the word by Hermann Weyl in his paper "Reine Infinitesimalgeometrie" Mathematische Zeitschrift 2 (1918) pp 384-411.

Jene fünf Identitäten stehen in engstem Zusammenhang mit den sog. Erhaltungssätzen, nämlich dem (einkomponentigen) Satz yon der Erhalung der Elektrizität und dem (vierkomponentigen) Energie-Impulsprinzip. Sie lehren nämlich: die Erhaltungssätze (auf deren Gültigkeit die Mechanik beruht) folgen auf doppelte Weise aus den elektromagnetischen sowie den Gravitationsgleichungen; man möchte sie daher als die gemeinsame Eliminante dieser beiden Gesetzesgruppen bezeichnen.

Briefly: The conservation principles follow from the electromagnetic and the gravitational equations; one is tempted to denote them as common eliminant of these two groups of laws.

What is the oldest known evidence of application of mathematics?

2013-05-05T09:26:45Z

According to Wikipedia the Lebombo bone (age 35 KY) and the Ishango bone (age at least 20 KY) presently are believed to show the first evidence for application of mathematics by humans. (Possibly African women, constructing a moon calendar, were the first mathematicians.) Is this the present state of the art? Or are there newer discoveries of older mathematical artifacts?

Remark: This question is not asking for some examples but for the present state of historical research in paleo-mathematics.

Answer by Rhett Butler for What does ! above = mean

2013-05-04T16:42:42Z

In my class I use $f'(x) \stackrel{!}{=} 0 $ to show that we look for a zero of the derivative of $f$ in order to find a local extremum or a stationary point. Other requirements like the definition of a normalization factor are possible candidates.

Large numbers in small systems

2013-02-02T12:42:23Z

Can we ever know the sum of the first $10^{10^{100}}$ digits of $\pi$?

Can we calgulate the $n$th digit of $\pi$ when the Kolmogorov-complexity of $n$ is larger than the complexity of the calculating system?

Example: We cannot calculate digit number 314159265358 of $\pi$ on a typical pocket calculator.

EDIT: As Goldstern remarks, we cannot write down all the digits simultaneously. But can we know every desired shorter string of digits, for instance the first billion following the digit number $10^{{10}^{90}}$ and then note their sum, and so on?

What is the best general triangle?

2013-04-25T13:41:44Z

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more equal angles. Further all angles should be less than $\pi/2$. Has anybody optimized this old problem of geometry-teachers?

Answer by Rhett Butler for Math History Question about the exponential function

2013-04-28T11:35:42Z

Rafaele Bombelli in l'Algebra (1572) considered imaginary roots of cubic equations as numbers since they might cancel out each other in the course of a following calculation. So he was the first who dared to use imaginary numbers in calculations. How much this calculations were "formalized" is a matter of opinon. Certainly this was not a formalization in modern sense. However, from Pythagoras over Archimedes to Gauß formalization has been sufficient to obtain correct results. Therefore Bombelli is at least one of the inventors of imaginary numbers. Sometimes also Cardano is nominated.

This knowledge spread soon. Albert Girard in Invention nouvelle en l'algèbre (1629) treated roots of negative numbers already without any ado. He distinguished the roots 3 and -3 of 9 and said of the root of -9 that it cannot be decided as positive or negative.

René Descartes baptized these new numbers in 1637. In his Géométrie he talks about imaginary roots (radices imaginariae) of an equation in contrast to true and false roots (radices vera, radices falsae) the latter denoting negative numbers, which was a common expression at that time.

As the inventors of logarithms we can name Jost Bürgi (his Arithmetische und geometrische Progresstabulen were published in 1620, but discovered already about 1590), and John Napier whose Descriptio appeared in 1614.

The word function, not yet used in modern sense (even Euler required a single formula to define a function), has been invented by Leibniz in his Methodus tangentium inversa, seu de fuctionibus (1673).

So the original question can be answered: First there were imaginary numbers, later came functions and logarithms. And logarithms of negative numbers came last.

Leibniz himself denied the existence of such logarithms and had a controversy with Johann Bernoulli about that topic in an exchange of letters in 1712 to 1713. Bernoulli defended his logarithme imaginaire. Finally Euler solved the problem in full generality. But that is a well known story.

Answer by Rhett Butler for Periods and commas in mathematical writing

2013-04-29T18:33:13Z

Plain text needs some punctuation. Otherwise it can easily appear confusing. But a displayed formula should be burnt into the brain of the reader as it is. Every additional punctuation is disturbing this aim. I think that the new paragraph starting below the formula is sufficient to mark an interruption if there is any.

Answer by Rhett Butler for History of the high-dimensional volume paradox

2013-04-26T12:50:32Z

Consideration of higher-dimensional spheres at least goes back to the 19th century.

In his paper: "Über verschiedene Theoreme aus der Theorie der Punktmengen in einem $n$-fach ausgedehnten stetigen Raume $G_n$. Zweite Mitteilung." Acta Mathematica 7 (1885) 105-124, Cantor uses "$n$-dimensionale Vollkugeln" ($n$-dimensional solid spheres) frequently. His calculation of the volume has first been mentioned in a letter to Felix Klein. See J. W. Dauben: "Georg Cantor His Mathematics and Philosophy of the Infinite", Princeton University Press (1990) p.326:

In a letter to Felix Klein of June 6, 1882, Cantor explained the details of his more accurate determination of the volume of the unit sphere of dimension $n$ in a space of dimension $n + 1$. It was true that the volume was always less than or equal to $2^n\pi$. But equality was true only for $n$ = 1, $n$ = 2.

Answer by Rhett Butler for sequence, such that sum of any combinations in the sequence does not equal another

2013-04-25T12:39:39Z

Other sequences that immediately come to mind are vector-like expressions $$(1, 0, 0, 0, 0, ...), (0, 1, 0, 0, 0, ...) , (0, 0, 1, 0, 0, ...), ...$$ (or matrices or tensors) or the sequence $(a_n)$ of approximations of transcendentals like Liouville's number $$a_n = \sum_{k=0}^{n} \frac{1}{k!}$$ or just terminating rationals with different points of termination behind the decimal point $$0.1, 0.11, 0.111, ...$$ or before$$1, 10, 100, ... $$

But that's already very close to the first answer.

Answer by Rhett Butler for Did Hermite really prove "Hermite's Theorem" on number field discriminants?

2013-04-23T18:47:23Z

Hermite and Minkowski used completely different ideas.

Hermite, by induction on $n$, proved that a positive definite quadratic form $q = X^TAX$ of determinant $ |A| = 1$ and $n$-dimensional column-vector $X$ takes a nonzero value $q \leq (4/3)^{(n-1)/2}$ for some integer vector.

Minkowsky, by a purely geometrical consideration, discovering that spheres of radius of half the distance of two nearest lattice-points could be replaced by any symmetric convex body, achieved his estimation which is better than Hermite's for large $n$. Hilbert, in his obituary, praised: Minkowski's proof of a deep number theoretic theorem without computational help, mainly on geometrical vizualization, is a pearl of Minkowski's inventiveness.

A masterly and/but very readable presentation of both ideas has been given by C.L. Siegel: Lectures on Quadratic Forms

Answer by Rhett Butler for Numerical Methods for ODEs - History

2013-04-21T21:04:07Z

Here are the sources:

Leonhard Euler: Institutiones calculi integralis (1768)

Augustin Louis Cauchy: Cours d'Analyse: Equations différentielles ordinaires et aux dérivées partielles (1824)

The Adams-Bashforth multistep-method as well as the Adams–Moulton methods are described here

Carl Runge: Über die numerische Auflösung von Differentialgleichungen, Math. Ann. 46 (1895) 167-178

W. Kutta: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys. 46 (1901) 435-453. (Remark: The name is Martin Wilhelm Kutta, the correct year is 1901.)

Lewis Fry Richardson: The approximate arithmetical solution by finite differences of physical problems involving differential equations,with an application to the stresses in a masonry dam Phil. Trans. R. Soc. London Ser. A 210 (1910) 307–57

Of interest in this connection is also E.J. Nyström: Über die numerische Integration von Differentialgleichungen, Acta Soc. Sci. Fennicae 50, 13 (1925) 55

Charles F. Curtiss and Joseph Oakland Hirschfelder: Integration of stiff equations, Proc Natl Acad Sci U S A, 38,3 (1952) 235–243

Answer by Rhett Butler for History of Irrationality results

2013-04-18T18:12:53Z

One of the Pythogoreans, probably Hippasos of Metapont, showed the irrationality of $\sqrt 2$ about 500 BC. According to Platon, Theodorus of Cyrene knew that the square roots of all integers up to 17 are integers or irrationals, and Theaetetus proved this for all integers. Much of what Euclid writes about incommensurable magnitudes, distinguishing different kinds of irrationalities, stems from these scholars and Eudoxus of Knidos who also invented the theory of exhaustion. (Note that Euclid's understanding of irrational numbers is not the same as today, but this would deviate too much from answering this question.). Appolonius of Perga extended this theory, as can be obtained from an Arabic translation of a commentary of Pappus on Euclid's book X. No details, however, are known.

The Indian mathematician Bhāskarāchārya, a contemporary of Fibonacci, calculated square roots of sums of rational and irrational numbers and, like Fibonacci, treated polynomial equations of higher than second degree. Also in the 13th century Johannes de Sacrobosco (Johannes Campanus) proved the irrationality of the golden ratio (this result had been known to the old Greek but lost in the pass of times) by the method of descente infinie.

Simon Stevin, the inventor of the decimal system, knows that there are two cases which do not allow an exact decimal representation, fractions like 5/6 and irrationals. Marin Mersenne in a critique of a paper of Alfons Anton de Sarasa, mentions the difficulty to obtain by geometrical means the logarithm of a certain quantity, if two other logarithms are given, may they be rational or irrational. The correspondence between Leibniz and Newton is abundant with irrationalities and the praise of the own method to handle them better. But proofs of irrationalities are not contained. May it be that enough irrational numbers were already available, or that the proof of irrationality in case of logarithms is so easy (Euler mentiones it en passant).

The French algebraist de Lagny showed that a certain kind of polynomial equations have irrational roots (Histoire de l'Académie de Sciences, 1705, p. 294).

It is controversial whether Euler has implicitly proved the irrationality of $e$ and $\pi$ by means of continued fractions. Anyhow, his introductio in analysin infinitorum is full of irrationalities. Vol. 1, contains, in chapter 6, the assertion that with exception of the powers of the base, the logarithm of a number $h$ is not rational (and cannot be irrational, hence must be transcendental). In § 508 of vol. 2, he explains: The algebraic equations are either rational and do not contain other than integer exponents or irrational with broken exponents. But in the latter case, they can be made rational. If an equation of a graph neither is rational nor can be made rational, it is transcendental. If an equation contains powers, the exponents of which are neither integers nor fractions, it cannot be made rational. The graphs of these equations are the first and, so to say, simplest kind of transcendental graphs, namely such resulting from equations with irrational exponents. § 509 starts with the example $y = x^{\sqrt 2}$.

And then came Lambert.

Etymology

In Earliest Known Uses of Some of the Words of Mathematics we read: Cajori (1919, page 68) writes, "It is worthy of note that Cassiodorius was the first writer to use the terms 'rational' and 'irrational' in the sense now current in arithmetic and algebra." The first citation of rational in the OED2 is by John Wallis in 1685. Irrational is used in English by Robert Recorde in 1551 in The Pathwaie to Knowledge: "Numbres and quantitees surde or irrationall."

M. Cantor in his Vorlesungen über die Geschichte der Mathematik, vol 2, credits the Italien (living is Spain) Gerard of Cremona (c. 1114 - 1187), the English mathematician and bishop Thomas Bradwardine (c. 1290 - 1349), the French mathematician and bishop Nicole d'Oresme (c. 1320 – 1382), and the German mathematician and bishop Albert of Saxony (c. 1320 - 1390) with early use of the word "irrational" in mathematical context, arguing how fast this notion spread in the world of mathematics in the 14th century. Nevertheless Vieta, Fermat, Newton and others used the word "asymetriae" or "quantitates surdae".

Answer by Rhett Butler for Lapses of "the early proponents of the doctrine of limits"

2013-04-04T12:06:11Z

It seems that Robinson addressed some gross mistakes that happend in the early times of calculus and that were passed with silence by the posterity, the inventors of more rigor in analysis, in particular Weierstraß and his school. To give few examples:

Leibniz, Jakob Bernoulli, and Euler accepted

$\frac{1}{1-(-1)}= 1 - 1 + 1 - 1 +-... = \frac {1}{2} $

Leibniz subtracted two harmonic series, with the correct result though,

$\sum_{k \geq 2}^{\infty} \frac{1}{k^2-1} = \frac{1}{2} \sum_{k \geq 2}^{\infty} \frac{1}{k-1} - \frac{1}{2} \sum_{k \geq 2}^{\infty} \frac{1}{k+1} = \frac{3}{4} $

but in a way that today certainly would not be tolerated.

Wallis and Euler accepted

$\frac{1}{1-2}= 1 + 2 + 4 + ... = \frac {1}{-1} > \frac {1}{0} > \frac {1}{1} $

Euler calculated

$1 + 2 + 3 + ... = \frac{-1}{12}$

which, irrespective of the analytic continuation of the $\zeta$-function and Ramanujan's rediscovering it, is as wrong as the sum of the geometric series above.

With respect to such mistakes, those of d'Alembert, touched in a comment by the author of this question, seem to be negligible. In my opinion they are not the target of Robinson's remark. Some gaps in d'Alembert's proof of the fundamental theorem of algebra have been remedied by Gauss. However, even Gauss left gaps in his first proof. And nobody knows what future generations will have to criticize in proofs that presently are accepted as complete. Further d'Alembert held the opinion that irrationalities are not numbers. But that has to be understood out of his time where "number" denoted a string of digits that can be read from a sheet of paper.

Answer by Rhett Butler for Alexander John Thompson - Logarithmetica Britannica

2013-04-14T12:01:09Z

Neither the British Library nor his publisher Cambridge University Press know the date of death of the late Alexander John Thompson. From Cambridge University Press I got the following notice:

As strange as it may sound one of the reasons that his date of death does not appear on Wikipedia is that no one is entirely certain of the precise date that he died, and even our records do not contain this fact. The best any one can guess is that he died sometime between 1968 and 1975. Unfortunately I can not be more precise than that.

So the narrowest interval is a guess by Denis Roegel (2012): "A reconstruction of the tables of Thompson’s Logarithmetica Britannica (1952)" "He possibly died in 1968 or 1973."

Was Desargues more an Euclid or an Eudoxos?

2013-04-12T08:37:25Z

In the course of preparing lessons on projective geometry I want to give an account on the historical development. It is easy to obtain an overview of the history starting with G. Desargues. And with respect to older sources http://en.wikipedia.org/wiki/Projective_geometry is of great help. But what I am not sure about is the question, whether Desargues was more an Euclid or an Eudoxos. Was he more collecting and reproducing the knowledge of scholars and artists like Filippo Brunelleschi, Ambrogio Lorenzetti, Pietro Perugino? Or did he invent himself most of what he wrote about? And what were his relations to his contemporary Johannes Kepler who worked in the same field?

Answer by Rhett Butler for Is there an observer dependent mathematics?

2013-04-11T12:32:21Z

To answer your question on the observer-dependence of mathematics, we have at first to define what mathematics is. Since many brave and intelligent men have failed to accomplish this goal, I will consider only three different domains, basic subjects, basic tools, and advanced mathematics which appear to belong to mathematics by general consensus.

The basics of mathematics have been obtained from physics. 2 apples and 3 pears are 5 fruits. The pythagorean theorem holds, because it is always true in reality where not too big masses are around. Two coupled oscillators do what they do because each of them computes two harmonic functions which are added or subtracted to tell the trajectory. The three-body problem is always easily solved by three bodies.

With this understanding, I can answer your question as follows: The more we can observe and the more we can think and talk about it, the more mathematics we can do. With a simple abacus, only small numbers can be calculated. The last prime number calculated by hand is $2^{127} - 1$. With increasing computing capacity, more and more prime numbers will come into reach and more and more proofs will be done by automata, proofs that we probably cannot even understand, at least that we cannot completely read (think of the four-colour theorem or the sequence of digits of $\pi$). Fibonacci proudly published the prime numbers between 1 and 100, Newton did not yet know the first 100 digits of $\pi$, Brouwer asked whether there is a sequence of nine consecutive digits 9 in the decimal expansion of $\pi$. Today billions of digits of $\pi$ are available and Brouwer's question has been decided in the affirmative. So we can state that basic mathematics strongly depends on the facilities of the observer.

An observer observes and communicates his observation to other observers. Mathematics has been defined as discourse (about the universe of discourse) but of course it occurs inside of our universe with limited ressources. We should not forget that. It is undisputed that without signals, at least inside one brain, no mathematics is possible. Our thinking and talking is limited by the media available. That is also limiting the mathematics we can do. The maximum number of steps of a proof is limited by the memory space of the proving system like the maximum number of digits of a number.

In addition to these material restrictions, we have the relativity in logic of set theory that has been discovered and advocated by Skolem, who asserted that there is no possibility of introducing something absolutely uncountable, but by a pure dogma. In 1929 he wrote already (in German): It seems that Hilbert wants to maintain Cantor's ideas in their old absolutistic sense. I find that remarkable. It is strange that he never has considered the relativism that I have proved for every finite formulation of set axiomatics.

It is rarely mentioned, but we have a similar relativity in proof theory discovered by Gödel (who did not prove that some propositions are undecidable but that their decidability is relative to the system applied). Here it may be even most obvious that mathematics depends on the observer, in that possible proofs depend on the level of theory attained by the observer.

Answer by Rhett Butler for The Origin of the Musical Isomorphisms

2013-04-10T16:51:51Z

Marcel Berger, on p. 696 of his Springer-book: A panoramic view of Riemannian Geometry, writes "Next we define the canonical musical duality", referring to a footnote: "These dualities are called musical because they are often written with symbols like ♭: V $\rightarrow$ V* and ♯: V* $\rightarrow$ V."

Even if this is not the first metioning of "musical", we can assume that Berger invented that expression, because in a panoramic viev of over 800 pages, a reference to another originator could have been expected.

Further Rob Kusner's hint here # or "sharp" is what Marcel Berger calls a "musical isomorphism" and Olivier Bégassat's comment here seem to provide sufficient evidence to determine the inventor.

To answer this question completely, one would have to read through Berger's complete works. But if we wait for a while, this task will probably be reduced to touching a button.

Answer by Rhett Butler for Uppercase Point Labels in High-School Diagrams: from Euclid?

2013-04-05T17:39:52Z

The convention of labeling points in geometric diagrams with uppercase symbols derives (at least also) from Greek mathematics. I cannot judge about the question whether this is the "ultimate" reason. But if we look into Euclid's elements, we find many diagrams where all points and also all lengths (magnitudes) are labeled with Greek uppercase symbols. It is said that Euler introduced the convention to label points of triangles with uppecase Latin letters and sides with lower case Latin letters and angles with lower case Greek letters. Probably the middle- and high-school textbooks, as many other elements of mathematics, adhere to the custom invented by Euler. But when we have a closer look into his books, first of all his INTRODUCTIO IN ANALYSIN INFINITORUM, we find that he used also lower case symbols to label points. So we should drop the word "ultimately" from the original question.

Has the controversy about fiducial distribution been settled?

2013-04-04T16:17:34Z

Has the controversy about the correct meaning of Fisher's notion fiducial distribution meanwhile been settled? And are there newer applications than quoted in the following literature?

G.P. Klimov: "On the fiducial approach in statistics", Sov. Math. Dokl. 11,2 (1970) 442–444

J.G. Pedersen: "Fiducial inference", Int. Stat. Rev. 46 (1978) 147–170

Answer by Rhett Butler for Location of Archimedes' grave in Syracuse (math/archaelogy trivia)

2013-04-04T08:41:00Z

In 1802 Johann Gottfried Seume visited Syracuse (from Germany per pedes). In his diary he mentions the place where Cicero had found Archimedes' grave: A chapel near the Greek theatre and the water pipe.

Etwas rechts weiter hinauf hat Landolina das römische Amphitheater besser aufgeräumt und hier und da Korridore zu Tage gefördert, die jetzt zu Mauleseleien dienen. Die Römer trugen ihre blutigen Schauspiele überall hin. Die Area gibt jetzt einen schönen Garten mit der üppigsten Vegetation. Weiter rechts hinauf ist das alte große griechische Theater, fast rundherum in Felsen gehauen. Rechts, wo der natürliche Felsen nicht weit genug hinausreichte, war etwas angebaut ... Die Wasserleitung geht nahe am Theater weg; vermutlich brachte sie ehemals auch das Wasser hinein. ... Gegenüber steht eine Kapelle an dem Orte, wo Cicero das Grab des Archimedes gefunden haben will. Wir fanden freilich nichts mehr; aber es ist doch schon ein eigenes Gefühl, daß wir es finden würden, wenn es noch da wäre, und daß vermutlich in dieser kleinen Peripherie der große Mann begraben liegt.

Johann Gottfried Seume: Spaziergang nach Syrakus im Jahre 1802 - Kapitel 35

EDIT

1) According to Seume's report this ruin is not the grave of Archimedes.

2) This painting by Benjamin West, showing Cicero and the magistrates discovering the tomb of Archimedes, is very popular but not very reliable.

Answer by Rhett Butler for Logarithms and Ratios

2013-03-24T18:10:51Z

The most famous historical reference to differential ratios that I know of is Euler's Institutiones calculi differentialis vol. 1, caput 3. Starting on p. 64 you can find a lot of differential ratios like

$\frac{dx + dx^2}{dx} = 1$

$a\sqrt{dx} + bdx = a\sqrt{dx}$.

Answer by Rhett Butler for Earliest diagonal proof of the uncountability of the reals.

2013-03-17T13:22:16Z

The evidence that you look for can already be found in the second paragraph of Cantor's original paper. There he states "Es läßt sich aber von jenem Satze ein viel einfacherer Beweis liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist.", meaning that his diagonal argument supplies a much simpler proof of the theorem proved in his first paper on the uncountability of the real numbers. This does not only mention but declare that his method can be applied to real numbers.

Probably in order to emphasize its independence of numbers, Cantor did not use numerals 0 and 1, but m and w which (and this answers your last remark) in German are abbreviations of male and female. So he had a good substitution for 0 and 1 or up and down or yes and no - and he deliberately or unconsciously circumvented the problem that this proof without reservations, as stated in his original text, would fail on binary sequences.

Answer by Rhett Butler for Are Cantor type numbers algebraic?

2013-03-14T15:10:36Z

To answer the third question in the affirmative, consider a list containing all algebraic numbers enumerated according to Dedekind for instance. Then the diagonal is transcendental. (Therefore we can use binaries without excluding certain periodic representations.) Each of the permutations of the list will also result in a transcendental diagonal. However, these diagonals need not all be different. But in order to obtain infinitely many transcendentals, we can bolster the list by inserting some algebraic numbers repeatedly. This will result in infinitely many transcendental diagonals.

Answer by Rhett Butler for Who first proved that the value of C/d is independent of the choice of circle?

2013-03-09T13:08:28Z

My first impression when reading this question was that it rather should read "who was the first to prove that $C/d$ need not be a constant?" Before Gauss, Bolyai, Lobachevsky linearity and transformation-invariance had always been assumed as basic to geometry like $n + 1 = 1 + n$ in arithmetic. Small wonder that nobody in ancient Egypt, Babylonia, India, or China would bother to prove that $C/d$ is a constant for every circle or that $n + 1 = 1 + n$ for every natural number.

However, as early as about 430 BC Hippocrates of Chios for the first time explicitly mentioned that similar segments of circles are in the ratio of the squares on their bases and proved this by proving that the squares on the diameters have the same ratio as the (whole) circles. Compare the lunes of Hippocrates which until now belong to the curriculum of schools.

We know this from a comment to Aristotle's Physics, written by Simplicius who quotes Eudemus of Rhodes (the pupil of Aristotle who compiled the first catalogue of mathematicians, not Eudemus of Cyprus after whom Aristotle named his famous text) as reporting this in his lost History of Geometry.

Comment by Rhett Butler

2013-05-27T06:10:55Z

@Jon Peterson: A winning strategy also is one which guarantees many different people to win money in short runs - in paricular if these people cannot be distinguished.

Comment by Rhett Butler

2013-05-24T17:16:39Z

That can be excluded by the continuity of the function and the fact that the function between 0 and 1 has only three maxima.

Comment by Rhett Butler

2013-05-24T16:54:05Z

@Steven: This is my very last contribution to this topic. I think there are more than five statistics professors among the 105 upvoters. Their error lies not in understanding and accepting the correct calculation of Douglas but the incorrect application of this calculation onto a question that does not ask for that calculation.

Comment by Rhett Butler

2013-05-24T16:43:42Z

And a second thought: Take the same sequence bggbbggb and stop always when g appears. Then you get four sequences with 1/2, 0, 2/3, 0 percentages of boys (the last boy is dropped). The average percentage of boys would be less than 30 %. But the result of the sequence is 4/8 or, in the last case, 3/8 for boys. I hope you can recognize the nonsensical character of the use of the average of percentages.

Comment by Rhett Butler

2013-05-24T16:32:38Z

not answer the google-question.

Comment by Rhett Butler

2013-05-24T16:30:26Z

@Steven: What I repeatedly have said is this: It is wrong to answer the famous google question by taking the average over percentages (as you wish to do). The google question asks whether the equipartition of boys and girls can be violated by family planning. The answer is a clear no. The result is exatly 50/50. This is not a more or less precise approximation. I have this example for a random sequence: bggbbggb accidentally yielding exactly as many boys as girls. Four sequences stopping with b each have the percentages of girls 0, 2/3, 0, 2/3. The average percentage is 1/3. But this does

Comment by Rhett Butler

2013-05-24T15:32:43Z

@Steven, to be sure of the conditions of the contract: You accept that we throw dice until we observe the 4th odd number then we count the odds (4) and the evens. We repeat this 3000 times without any interruption, and then we will sum up all odds and all evens and we will have less than 47% even numbers and at least 53% odd numbers (in approximately 12000 throws). I am afraid, if you do not accept that, then you will have to excuse for some ramblings towards me. And if you accept your contracting competence will be doubted by your relatives (and a jury). And I would be charged of abuse.

Comment by Rhett Butler

2013-05-24T14:31:50Z

@Steven: I will accept the bet if you agree that throwing dice in an uninterrupted sequence until 10 odss have been gathered is as valid as the programs written by five graduate students. If you do not agree, say what ther difference is? And if you agree, be aware that you have lost because we are not trying to find the average of the percentages, but the percentage of the average. The fraction of females in the country. So donate your money to MO.

Comment by Rhett Butler

2013-05-24T14:12:41Z

@Douglas: I talked to a lot of colleagues about your false prediction of 47.5 to 52.5 in case of 10 sequences stopped by boys. Some of them had never heard of MO (I started only in February this year, and I found many interesting topics). And I guess that one of them is the questioner. Hello!

Comment by Rhett Butler

2013-05-24T14:06:42Z

And why employ graduate students for a generating a random sequence? Throw dice or visite a roulette evenening! And if a sequence has been stopped by a b (for boy or for black), then immediately start the next one. What would be the difference to making children???? It is really incredible that such blatantly false answer can gain positive votes!

Comment by Rhett Butler

2013-05-24T14:04:56Z

a country starting with, say, four couples, each having one child per year and stopping when they have a boy. We’ll let this run for a simulated 30 years and then compute the fraction of girls in the population. To guard against statistical flukes, we’ll run the experiment 3000 times and take the average of all the results. I claim the answer will be just a hair under 44%. Lubos claims 50%. Let’s say I win if the actual result is less than 46.5% and he wins if it’s greater than 46.5%. That is wrong. Cont'd

Comment by Rhett Butler

2013-05-24T14:03:35Z

@Steven, from you website: What fraction of the population is female? I say the answer depends on the number of families in the country, but in no case is it 50%. Lubos insists that the correct answer is 50%. Now the best way to settle such a dispute is to go to the mathematics. But since Lubos seems unable to follow the mathematics, the next best way is to run a simulation. So I propose the following terms: We’ll randomly choose five graduate students in computer science from among the top ten American university departments of computer science and have them write simulations for cont'd

Comment by Rhett Butler

2013-05-24T12:26:42Z

@quid: Why do you think that this particular subject cannot be decided by means of mathematics? I have talked to many colleagues about that scandal in MO - and nobody could understand how the correct answer 1/2 for the fraction of females could be doubdet. No, the incorrect answer has been given in order to answer the google-question. And now where it becomes obvious that this answer is wrong, some defenders try to find another interpretation.

Comment by Rhett Butler

2013-05-24T12:20:44Z

@Jules: Consider this statement: thebigquestions.com/2010/12/27/… What fraction of the population is female? Answer by Steven Landsburg "I say the answer depends on the number of families in the country, but in no case is it 50%. Lubos insists that the correct answer is 50%". Of course Lubos is right and Steven i wrong. PS: I also think that such a question is better suited for a beginners course in statistics. But it is incredible that this answer could survive for years in this forum and get more than hundred votes.

Comment by Rhett Butler

2013-05-24T11:49:28Z

-1. And here is the reason: The answer is exceptionally wrong because question does not concern the average of the percentages. The original question concerns the percentage of girls in the country. This is 50 %. Since the correct value is 1/2, 1/2 is not only "approximately correct, although the explanation is misleading", but it is precisely correct and the calulation "for 10 couples, the expected percentage of girls is 10 log 2 − 1627/252 = 47.51% contrary to what the official answer suggests" is simply wrong.

User rhett butler - MathOverflow

Answer by Rhett Butler for Is there a good reason why a^{2b} + b^{2a} <= 1 when a+b=1?

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Answer by Rhett Butler for sequence, such that sum of any combinations in the sequence does not equal another

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Answer by Rhett Butler for Numerical Methods for ODEs - History

Answer by Rhett Butler for History of Irrationality results

Answer by Rhett Butler for Lapses of "the early proponents of the doctrine of limits"

Answer by Rhett Butler for Alexander John Thompson - Logarithmetica Britannica

Was Desargues more an Euclid or an Eudoxos?

Answer by Rhett Butler for Is there an observer dependent mathematics?

Answer by Rhett Butler for The Origin of the Musical Isomorphisms

Answer by Rhett Butler for Uppercase Point Labels in High-School Diagrams: from Euclid?

Has the controversy about *fiducial distribution* been settled?

Answer by Rhett Butler for Location of Archimedes' grave in Syracuse (math/archaelogy trivia)

Answer by Rhett Butler for Logarithms and Ratios

Answer by Rhett Butler for Earliest diagonal proof of the uncountability of the reals.

Answer by Rhett Butler for Are Cantor type numbers algebraic?

Answer by Rhett Butler for Who first proved that the value of C/d is independent of the choice of circle?

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What is the best general triangle?

Has the controversy about fiducial distribution been settled?