#pascals triangle was known for centuries

[chai-en-kaadhale]

thinking about the walls of benin and how the british destroyed them, and thinking about all of the literature of south america that the spanish destroyed just like that and then thinking about the library of alexandria

because how many more libraries were ravaged by colonialization yk? how much knowledge and architecture was destroyed for conquest?

feeling feelings

[sadquickchristmassnowman]

masterlist of episodes featuring prof duncan? I only JUST made the connection that he's played by john oliver

fantastic idea anon. I’m honestly surprised I haven't made this list yet as I am an avid duncan enjoyer lmfao and yes john oliver is great. so I’ll make the list right now. here we go:

the feat. professor duncan episode masterlist

1x01: pilot

he is the reason jeff chose to enroll at greendale

he has multiple interactions with jeff throughout the episode

1x04: social psychology

annie, troy, and abed participate in his psychological experiment in which he proves the existence of "the duncan principle"

1x05: advanced criminal law

duncan, dean pelton, and chang create "greendale court" in order to put britta's alleged academic dishonesty on trial

1x25: pascal's triangle revisited

first shown doing a therapy session with britta

constantly fighting with chang, escalating into chang punching him at the end

does his iconic rap after he gets drunk at the transfer dance

2x03: the psychology of letting go

takes over as the anthropology professor despite knowing nothing about the subject

uses the restraining order he got against chang to torment him

has a few conversations with jeff about the inevitability of death

is civil with chang by the end of the episode

2x05: messianic myths and ancient peoples

exclusively shown "teaching" the anthropology 101 class

2x11: abed's uncontrollable christmas

leads abed's "therapy" session but is generally unhelpful

2x12: asian population studies

again, just shown "teaching" anthropology 101

2x15: early 21st century romanticism

watches the liverpool vs. manchester united football game with jeff at jeff's apartment

has conversations with jeff about friendship and greendale etc.

2x21: paradigms of human memory

appears very briefly in a flashback to the events of "abed's uncontrollable christmas"

2x22: applied anthropology and culinary arts

is shown again at the head of the anthropology class, this time "administering their final" (getting drunk) but disappears very quickly as soon as the dean walks in

5x03: basic intergluteal numismatics

"helps" with the psychological aspect of the ass crack bandit investigation

(you might notice he is not in seasons 3 or 4. irl it's because he got too busy with his other projects, but in the show his character explains that he was "taking care of his sick mother")

5x05: geothermal escapism

shown briefly playing the floor is lava game, but loses very quickly after jeff knocks over the chair that he was going to jump on while chasing britta

5x06: analysis of cork-based networking

shown as a member of the save greendale committee

helps jeff, shirley, and chang choose a theme and decorate for the dance (bear down for midterms)

5x07: bondage and beta male sexuality

asks jeff to help him scheme a way into getting a date with britta, which ends up backfiring

ends up hanging out a lot with jeff. this episode re-establishes that the two of them have known each other longer than any of the other characters

5x11: g.i. jeff

appears briefly only in jeff's animated dream, as "xim xam"

(so, I guess it's not technically duncan, but it's jeff's g.i. joe dream version of duncan, so I’m counting it)

5x12: basic story

appears again as a member of the save greendale committee

5x13: basic sandwich

tries to help the save greendale committee uncover the hidden computer school but gets electrocuted lmfao

alright yep. that should be all. I personally went through the credits of every single episode to double check everything because I do not trust the lists that are already online tbh. full of discrepancies smh. also, yeah, he is indeed not in seasons 3, 4, or 6. anyway! hope this is helpful. duncan is hilarious and every single line that comes out of his mouth is comedy gold. bye

[globalincident]

Well, not exactly. In some cases the mathematicians named do not have a solid claim to the theorem. Let me break them down.

Al-Karaji was the first person to write down the binomial theorem, but the solution to the combinatorial problem associated with it was common knowledge by that point. The binomial theorem is a fairly trivial corrolary to that problem, so it's likely someone thought of it before.

The law of cosines was definitely known about before, since Nasir al-Din al-Tusi wrote down a sketch of the method several centuries earlier than Al-Kashi.

The Fibonacci sequence was known in India, but there's no obvious person to attribute it to. I guess you could call it the India sequence instead?

Pascal's triangle was possibly discovered by Al-Karaji (yes, that guy again), but may have been thought of before that. I'm not sure what is meant by "the Persians and before them the Chinese", since Al-Karaji definitely got there before the Chinese.

The Pythagorean theorem was known earlier (and Pythagoras himself may never have heard of it), but again there's no obvious person to attribute it to.

It's also worth noting that we do not normally name things after the person who discovered them. We normally name them after a notable person associated with the theorem, not necessarily the person who actually discovered it or even knew about it. Gauss's law is so called because it is related to Gauss's theorem, which Gauss rediscovered (but did not fully prove) after Lagrange had already discovered it. Euler's identity is so called because it is related to Euler's formula, which was stated earlier than Euler by Roger Cotes. Bernoulli's principle was developed by Bernoulli, but first expressed in its modern form as an equation by Euler. Having bad names for things is not limited to things discovered by Muslims.

Here's the link to the video

👏 SAY 👏 IT 👏 LOUDER 👏

[liquorwinecave]

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[historicalmeanderings]

Omar Khayyam was a Persian polymath, a mathematician, philosopher, astronomer and poet. Omar Khayyam was born in 1048 A.D in the metropolis Nishapur, Iran. Omar Khayyam studied in Nishapur and Samarkand. Omar Khayyam produced a new calender, Jalali calendar which was adopted by the Seljuq Sultan Malik Shah. Omar Khayyam wrote Risalah fil-barahin ala masail al-jabr wa mubqabalah (Treatise on Demonstration of Problems of Algebra) which gave the solution of cubic equations among other topics. He wrote on geometric algebra, theory of parallels, binomial theorem where he wrote about triangular array known as Omar Khayyam's triangle which were later called Pascal's triangle. Omar Khayyam's work traveled to Europe where they influenced the early renaissance professor of Oxford university, John Wallis, six centuries later. Omar Khayyam along with being an intellectual was also a poet. He wrote Persian quatrains which centuries later translated into English by Edward Fitzgerald, Rubaiyat of Omar Khayyam. Omar Khayyam has written of mathematics, philosophy, astronomy and poetry. A lunar crater Omar Khayyam was named in his honor in 1970. Soviet astronomer Lyudmila Zhuravlyova's discovered minor planet was named 3095 Omarkhayyam. Omar Khayyam died in 1131 A.D. and is buried in tomb of Khayyam, Nishapur Iran. #omarkhayyam #omar #khayyam #samarkand #nishapur #science #jalalicalender #mathematics #astronomy #algebra #poetry #rubaiyat #philospher #persianscientist #intellectual #knowledge #scientist #johnwallis #oxford #edwardfitzgerald #muslimcivilization #calendar #binomialtheorem #khayyamstriangle #3095omarkhayyam #theoryofparallels https://www.instagram.com/p/CEJfh9-jgLL/?igshid=11cmsq4djazl9

[maier-files]

New Post has been published on The Maier Files

New Post has been published on http://the.maier-files.com/mathematics-signs-and-truth/

Mathematics, Signs and Truth

Otto Maier and his theory about waves, reality and time curves are rooted in the works of the men he looked up to, Leibniz and Descartes.

But who was Leibniz? 

Gottfried Wilhelm Leibniz (1646–1716) lived an extraordinarily rich and varied intellectual life in troubled times. Although remembered as a great thinker, he was a man who more than anything else wanted to do certain things, namely to improve the life of his fellow human beings through the advancement of science, and to establish a stable and just political order in which the divisions amongst the Christian churches could be reconciled. A surprising number of his apparently miscellaneous endeavours were aspects of a single master project tenaciously pursued throughout his life: the systematic reform and development of all the sciences, to be undertaken as a collaborative enterprise supported by an enlightened ruler. These theoretical pursuits were in turn ultimately grounded in a practical goal: the improvement of the human condition in a well-ordered world, and the celebration thereby of the glory of God in his creation. The effectiveness of natural language notwithstanding, as a tool for the advancement of science, nothing could be more powerful, Leibniz thought, than a science of signs broadly conceived. Algebra, arithmetic, and their applications to the physical world were prime examples of systems of symbols which, through rigorous rules for their manipulation, empowered discovery. Leibniz conceived his mathematical studies and his invention of the infinitesimal calculus (that is, an algorithm or sequence of rules for handling infinitesimal magnitudes) as part of this broader project.

A science of signs

Thus, while for Newton the calculus was basically a brilliant way to solve certain difficult mathematical problems, Leibniz regarded it as a sample of what he was hoping to achieve in logic for thought in general. In a letter of 28 December 1675 to the secretary of the Royal Society, Henry Oldenburg, penned shortly after his discovery of the infinitesimal calculus, Leibniz linked his new ‘algebra’ to the characteristica universalis:

This algebra is only part of that general theory. Yet it ensures that we cannot err even if we wish to … To be sure, I recognise that whatever algebra supplies of this sort is the fruit of a superior science which I am accustomed to call either Combinatory or Characteristic, a science very different from what might at once spring to one’s mind on hearing these words.

Leibniz’s discovery of the calculus matured during his period in Paris (1672–6). In the 17th century, the mathematical study of motion and, in particular, the issue of how to measure curves, had become central to the revolutionary programme of providing mathematical accounts of natural phenomena ushered in by the new quantitative physics championed by Galileo Galilei (1564–1642). The calculation of the area of the surface intercepted by a plane curve (a problem known as the ‘quadrature’ of a curve), and the determination of a tangent to a given plane curve, leading to the calculation of the velocity of a body at a given instant, took centre stage in mathematical studies aimed at providing a general method of calculation applicable to all kinds of known curves and variable quantities.

One of the milestones on the road to the discovery of the calculus was Leibniz’s generalization of the method of calculation introduced by Blaise Pascal (1623–62) in his work on the quadrature of the circle. As Leibniz triumphantly noted in a text of 1673, ‘the whole thing depends on a right-angled triangle with infinitely small sides, which I am accustomed to call “characteristic”, in similitude to which other triangles are constructed with assignable sides according to the properties of the figure. Then these similar triangles, when compared with the characteristic triangle, furnish many propositions for the study of the figure, through which curves of different kinds can be compared with one another’.

The characteristic triangle (ds, dy, dx). Building on these results, a few months later Leibniz formulated a general ‘transmutation theorem’.

Truth

Leibniz’s commitment to the characteristica universalis and the scientia generalis ultimately rested on his conviction that logic is a mirror of the structure of REALITY. In his view, the principles which govern thought were also the principles that govern reality. In turn, reality meant for Leibniz first and foremost God, the eternal and infinite Being encompassing all perfections. It is from him and his eternal thoughts that the story of the world in which we find ourselves begins. Logic therefore led via metaphysics to philosophical theology.

According to Leibniz, the first, fundamental principle which governs both the ideal order (the sphere of thought, studied by logic) and the real order (reality, studied by metaphysics) is the principle of non-contradiction: for any proposition ‘p’, ‘p’ and ‘not-p’ cannot both be true at the same time in the same respect. In other words, it cannot both be true at time t that I am such and such (whatever that is) and that I am not such and such. It cannot both be true that I am, right now, wearing my glasses on my nose and not wearing my glasses on my nose, even if I may not know which one of these alternatives is true.

As Leibniz puts it: Primary truths are those which assert the same [thing] of itself, or deny the opposite of its opposite. As A is A, or A is not not-A. If it is true that A is B, it is false that A is not B, or that A is not-B. Also: every thing is what it is. Each thing is like itself, or equal to itself. Nothing is greater or less than itself—and others of this sort which, though they may have their own grades of priority, can all be included under the name of ‘identities’. (A VI, 4, 1644)

It must be stressed, however, that for Leibniz the mental reality of non-existing possible beings does not ultimately depend on their being thought by individual human minds. One could take the view (as Leibniz did) that there are some truths (such as mathematical truths) which exist in some sense, or have some ‘reality’ of their own, even if no human being ever thinks of them.

Leibniz wrote in another note of 1677:

It is true, and even necessary, that the circle is the largest of isoperimetric figures. Even if no circle really existed. Likewise even if neither I nor you nor anyone else ever existed. … Since, therefore, this truth does not depend on our thought, there must be something real in it.

https://en.wikipedia.org/wiki/Isoperimetric_inequality

  Otto Maier and Leibniz