Foundations Of Geometry Answer Key: Fill & Download for Free

Honestly, who knows?I have written about Pythagoras before: Why is Pythagoras important?There are a few key things to take away from my answer there. First, there are various concepts that the Pythagoreans came up with that we still use today, such as that of odd, even, square, and cube numbers. Secondly, the Pythagorean theorem itself is important, but there is no way of knowing whether the Pythagoreans were the first to come up with a proof of it—I would argue that they probably weren’t, and theirs is simply the earliest historical account that has survived. Finally, it is incredibly difficult to separate legends of Pythagoras (many of which are contradictory) with the actual historical facts of the man himself.For example, there is no way to know whether it was Pythagoras who came up with a proof of the eponymous theorem, or if it was one of his students. In fact, since all of the writings that we have about Pythagoras were written centuries after he died, there is no way to know if he was responsible for any of the things that are commonly attributed to him.The one thing that I can say with confidence is that he inspired a great many Greek mathematicians who laid the foundations of geometry, number theory, and axiomatic mathematics. I would say that makes him pretty great.

A2A FrankI doubt mathematical creativity can be measured on a 1-dimensional scale so I shall list a few who seem to be very accomplished, but inevitably my list is biased by my own interests (and I’m not quite a professional mathematician either).Historically, we may never know who lay behind some of the ideas written up by e.g. Euclid - Wikipedia and Proclus - Wikipedia or the origin of zero…More recently, in roughly historical sequence:Gerolamo Cardano - Wikipedia an Italian polymath, whose interests and proficiencies ranged from being a mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler.He was one of the most influential mathematicians of the Renaissance, and was one of the key figures in the foundation of probability and the earliest introducer of the binomial coefficients and the binomial theorem in the Western world…. He made the first systematic use of negative numbers in Europe, published with attribution the solutions of other mathematicians for the cubic and quartic equations, and acknowledged the existence of imaginary numbers.2. Isaac Newton - Wikipedia for laying the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.3. Hermann Grassmann - Wikipedia In 1844, Grassmann published his masterpiece, his Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik [The Theory of Linear Extension, a New Branch of Mathematics], which translates as "theory of extension" or "theory of extensive magnitudes." Since it proposed a new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. (Grassmann’s work influenced Hamilton and Clifford, in terms of whose work much modern theoretical physics can be cast)4. Henri Poincaré - Wikipedia As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.5. Élie Cartan - Wikipedia was an influential French mathematician who did fundamental work in the theories of Spinors and of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century.6. David Hilbert - Wikipedia was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and foundations of mathematics (particularly proof theory). Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. Some of these were solved within a short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some even continue to this day to remain a challenge for mathematicians.7. Srinivasa Ramanujan - Wikipedia During his short life, Ramanujan independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research. Nearly all his claims have now been proven correct.8. Emmy Noether - Wikipedia (already advanced in Comet’s Answer prior to this one)9. Kurt Gödel - Wikipedia Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor.10. Hermann Weyl - Wikipedia His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.11. Paul Erdős - Wikipedia was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his social practice of mathematics (he engaged more than 500 collaborators) and for his eccentric lifestyle (Time magazine called him The Oddball's Oddball). [NOT the Oddball character in the film, Kelly’s Heroes] He devoted his waking hours to mathematics, even into his later years—indeed, his death came only hours after he solved a geometry problem at a conference in Warsaw.However, it is often said that theoretical physicists need to be accomplished as both mathematicians and physicists, so I would also promote for mathematical creativity:Paul Dirac - WikipediaRichard Feynman - WikipediaMurray Gell-Mann - WikipediaEdward Witten - WikipediaNB I omit Albert Einstein - Wikipedia Great and superlatively creative though he undoubtedly was in physics, apparently he was not an outstanding mathematician on the scale of the great mathematicians, and relied to some extent on Hermann Minkowski - Wikipedia for mathematical support - but this is discussed more widely at Do you think Einstein was "bad" at math compared to the other people in the physics community? I saw in a documentary that it took Einstein 7 years to solve a problem a mathematician did in a couple of months. .Interested to learn others’ views of who I’ve omitted as >= to those listed here.

There isn’t a single “next big thing” mathematicians are trying to solve. There are many next big things, and sometimes the next big thing that actually takes place isn’t any of the next big things anyone expected. But we can try, cautiously, to portray some of the most prominent challenges ahead, which are likely to attract attention and effort.How can we identify these challenges? There are a few reasonable sources:Hilbert’s famous list of 23 problems[1] is the closest thing we’ve ever had to an explicit portrayal of the greatest mathematical problems facing us. This was in 1900, but several of the problems are still only partially solved, or not at all.The Clay Mathematics Institute’s seven Millennium Problems[2] addressed a similar need in the year 2000. Only one of the seven has been solved so far.Books, articles and other publications addressing themes for the future of mathematics, such as the tome Mathematics Unlimited – 2001 and beyond[3] by Engquist and Schmid (eds.)Anyone attempting to write down such a list is bound to be at least somewhat biased, and likely often wrong. I will likely be more than somewhat biased and more than often wrong, but I’ll do my best, for whatever that’s worth.Complexity, Computability and LogicThe foundational questions of what can be computed and how efficiently are intimately linked to questions of mathematical logic: proofs, models and the boundaries of the mathematical endeavor. Indeed, one of the meta-questions here is, in my mind: what is the future of mathematics, now that computers have begun to augment our own abilities? This question will get answered, one way or another, in the coming 100 years.There are, certainly, more concrete problems we are facing. The most famous one is, of course, P vs NP,[4][4][4][4]which is one of the Millennium problems (and, in my humble opinion, the most profound one). Apart from this mother-of-all-problems, making any significant dent in the polynomial hierarchy and the complexity zoo is “the next big thing” for an entire industry of thinkers, working on a dazzling array of complexity classes.[5][5][5][5]Specifically, understanding the theoretical and practical significance of quantum computation is a major endeavor for the coming decades.One of the most intriguing (if speculative) directions in the area of computability, provability and foundations are Homotopy Type Theory[6][6][6][6]and Univalent Foundations[7][7][7][7] whose development was profoundly and tragically set back by the untimely death, in September 2017, of Fields medalist Vladimir Voevodsky.[8][8][8][8]Photo by Andrea KaneVoevodsky was, among many other things, a cautious intuitionist, or at least he was intrigued by intuitionism and finitism as a possible foundation for aspects of mathematics. My personal belief is that finitist methods and ideas will play an important role in the future of mathematics, not because they are somehow philosophically or morally superior, but because they are interesting.We’ve learned a lot about provability, but I think we’ve only skimmed the surface of proof complexity. What if a number-theoretic conjecture is false, but holds until some vast, uncomputable number? What if a statement is provable in ZFC, but the shortest proof is TREE(999) characters long? I believe that elucidating the fascinating frontiers of feasible thought will continue to define central pieces of mathematics for decades, or centuries.The Langlands ProgramRobert Langlands[9][9][9][9] is another person whose vision is deep and far-reaching enough to create an entire program, one that has guided the lives of many people since the late 1960s and is far from running out of steam. The Langlands program has been called “A Grand Unified Theory of Mathematics” by Edward Frenkel,[10][10][10][10] and I think this is only a slight exaggeration. It is an incredibly daring program, and it is peculiar that it’s absent from some of the sources I had mentioned.Describing the program is a daunting task. In An Elementary Introduction to the Langlands Program,[11][11][11][11]Stephen Gelbart writes:Herein lies the agony as well as the ecstasy of Langlands' program. To merely state the conjectures correctly requires much of the machinery of class field theory, the structure theory of algebraic groups, the representation theory of real and [math]p[/math]-adic groups, and (at least) the language of algebraic geometry. In other words, though the promised rewards are great, the initiation process is forbidding.Very, very roughly, the Langlands Program proposes that all [math]L[/math]-series, traditional objects of central importance in number theory, “come from” representations of certain groups. It connects together automorphic forms, the theory of adeles, representation theory and many other threads of geometry, algebra, analysis and number theory.The Langlands Program is not a single question or conjecture. It is an entire web of ideas. Ngô Bảo Châu’s[12][12][12][12] proof of the “Fundamental Lemma”[13][13][13][13]and Laurent Lafforgue’s[14][14][14][14] proof of the Langlands Correspondence over function fields were tremendous achievements that landed them both a Fields medal, but they are only steps in the grand program. Much is yet to be discovered and done, and I’m pretty sure that progress in the program will continue to happen over the coming decades.Children’s DrawingsI know this seems a little tongue-in-cheek, but it is only slightly so. The idea of “dessins d’enfants”[15][15][15][15]is one of Alexander Grothendieck’s[16][16][16][16] legacies, and I submit that elucidating, clarifying and building upon Grothendieck’s ideas is a significant outstanding challenge for mathematics.Specifically, dessins d’enfants offer a perspective on an object which (in my personal view) is one of the most mysterious and magnificent things in the deep, true universe (the one that is indifferent to the shackles of our actual, random physical world). It is perhaps the single most mysterious and magnificent of all: the absolute Galois group [math]\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})[/math]. Whatever progress is being made on understanding this profound group – and progress will, I am sure, be made – will shed light on the deepest questions in Number Theory and beyond. Grothendieck’s idea is not the only direction, but it is a promising and exciting one, like most everything this singular man came up with.Another thread of Grothendieck’s ideas, that of higher categories and topos theory, is pursued by many incredible and dedicated people, perhaps most vigorously by Jacob Lurie.[17][17][17][17]I don’t think we can expect him to finish it all on his own, wildly ingenious as he may be. It is hard to predict, but it is possible that higher category theory will grow in significance to play an absolutely central role in future math. It is also, by the way, closely connected to some of the ideas I mentioned in the first section (univalent foundations and so on).The Mathematics of PhysicsOne of the most ambitious, and most vague, of Hilbert’s 23 problems is the sixth.[18][18][18][18]It is phrased:6. Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.Remember: this is 1900. Before Einstein’s 1905,[19][19][19][19]before General Relativity, before Quantum Mechanics, before the Standard Model and Quantum Field theory and Superstring Theory. Hilbert couldn’t have known how far his physics colleagues were from understanding physics, let alone axiomatizing it. And the challenge remains: to form a coherent, unified mathematical framework that describes our physical world.From this massive endeavor, the Clay Mathematics Institute picked a much more concrete, yet still profound, challenge for the Millennium Problems: the problem[20][20][20][20] of establishing a Yang-Mills theory corresponding to any gauge group, and showing that it has a minimal mass. A different way to phrase this challenge in a mathematical context is, perhaps: organize, axiomatize and understand topological quantum field theories.In a wonderful survey describing the Yang-Mills problem, Jaffe and Witten wrote:[21][21][21][21]…one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precise definition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so!To be sure, physics poses many other additional challenges for mathematics. Another Millennium Problem[22][22][22][22] seeks a proof (or refutation) that the Navier-Stokes[23][23][23][23] equations admit a smooth solution given any smooth boundary conditions.These are equations which describe fluid motion; they are not quantum or relativistic or anything, but they are fundamental for our understanding of the macroscopic world. Also, they are a key example of nonlinear partial differential equations (PDEs), and understanding nonlinear PDEs is a huge and multifaceted mathematical challenge. We really, really want to understand if, despite their chaotic nature, such equations always have smooth solutions or if they can “break down” given the right initial conditions.Specifically for Navier-Stokes, I think many people are just waiting for Terry Tao to dispose of the damn thing already, despite the fact that he’s so far made progress on showing why the problem is harder than it may seem. Incidentally, the way he does that hints at establishing “sufficiently strong” computational models within the framework of fluid dynamics, which brings us back to the domain of computability and proofs. If solutions to nonlinear PDEs are rich enough to accommodate non-trivial logic, it may be that certain problems in PDE are as non-solvable as the halting problem.Whether it’s Tao or someone else, making progress here would be huge.The (Generalized) Riemann HypothesisAnother major candidate for “the next big thing” is the Riemann Hypothesis, the only Millennium Problem which has survived intact since Hilbert’s list in 1900. It may not be “the next big thing” in terms of being solved – heaven knows how long this might take – but it will surely continue to consume people’s minds, hopes and dreams.The Riemann Hypothesis seems kind of exotic: it deals with a specific function, the Riemann zeta function [math]\zeta(s)[/math], and asks about the location of its roots (other than a series of “trivial” ones). They are all expected to lie on the line [math]\Re(s)=\frac{1}{2}[/math].Specific functions and their roots aren’t usually a matter of universal interest, but this one is a huge exception. The Riemann zeta function has central importance in several fields of mathematics (most obviously, Number Theory). Furthermore, there’s a natural generalization of the Riemann Hypothesis which deals with other zeta functions, and has applications even more far-reaching than RH alone.This problem has now stood for over 150 years, and I don’t think it’s going away any time soon. Whenever anyone proves it, or even significantly enhances our understanding of the difficulty, that’s a Next Big Thing indeed.The ABC ConjectureSince its introduction in the mid-1980s, the ABC conjecture[24][24][24][24] was found to offer a unified way of understanding many old and new problems in Number Theory. In fact its proof would imply[25][25][25][25] many of the Fields-winning problems of the past 50 years, and papers regularly show up showing that it implies this or that open problem.The problem is unsolved. I’m pointing this out since there’s currently a pretty confusing state of affairs in which a proposed proof by Shinichi Mochizuki[26][26][26][26] has stood unconfirmed since 2012, and so far only a very small number of mathematicians believe that it does indeed prove the ABC Conjecture. Many questions about this have[27][27][27][27] been[28][28][28][28] asked[29][29][29][29] on[30][30][30][30] Quora.[31][31][31][31](This situation saddens me. The ABC Conjecture is a thing of beauty, and the deep work done around it is truly exciting and intriguing. But the general public is attracted to drama, and the drama obscures the real essence of the theory. There’s no reason to be excited by Mochizuki’s proof. Not yet, at any rate.)Developing Mochizuki’s “Inter-Universal Teichmüller Theory”, if at all possible, will be a monumental progress. Alternatively, finding other ways to attack the ABC Conjecure and the closely related Szpiro’s Conjecture[32][32][32][32] would certainly qualify as a Big Thing.There is no shortage of huge open problems in many other areas of mathematics, any one of which could be the Next Big Thing. We don’t know where things will lead us, or which dramatic new innovations would transform some or all of math. The last century was transformative; it’s entirely reasonable that this one will be, too.Footnotes[1] Hilbert's problems - Wikipedia[2] Clay Mathematics Institute[3] Mathematics Unlimited - 2001 and Beyond | Björn Engquist | Springer[4] Clay Mathematics Institute[4] Clay Mathematics Institute[4] Clay Mathematics Institute[4] Clay Mathematics Institute[5] Complexity Zoo[5] Complexity Zoo[5] Complexity Zoo[5] Complexity Zoo[6] Homotopy type theory - Wikipedia[6] Homotopy type theory - Wikipedia[6] Homotopy type theory - Wikipedia[6] Homotopy type theory - Wikipedia[7] Univalent foundations - Wikipedia[7] Univalent foundations - Wikipedia[7] Univalent foundations - Wikipedia[7] Univalent foundations - Wikipedia[8] Vladimir Voevodsky 1966–2017[8] Vladimir Voevodsky 1966–2017[8] Vladimir Voevodsky 1966–2017[8] Vladimir Voevodsky 1966–2017[9] Robert Langlands - Wikipedia[9] Robert Langlands - Wikipedia[9] Robert Langlands - Wikipedia[9] Robert Langlands - Wikipedia[10] Edward Frenkel - Wikipedia[10] Edward Frenkel - Wikipedia[10] Edward Frenkel - Wikipedia[10] Edward Frenkel - Wikipedia[11] Bulletin of the American Mathematical Society[11] Bulletin of the American Mathematical Society[11] Bulletin of the American Mathematical Society[11] Bulletin of the American Mathematical Society[12] Ngô Bảo Châu - Wikipedia[12] Ngô Bảo Châu - Wikipedia[12] Ngô Bảo Châu - Wikipedia[12] Ngô Bảo Châu - Wikipedia[13] Fundamental lemma (Langlands program) - Wikipedia[13] Fundamental lemma (Langlands program) - Wikipedia[13] Fundamental lemma (Langlands program) - Wikipedia[13] Fundamental lemma (Langlands program) - Wikipedia[14] Laurent Lafforgue - Wikipedia[14] Laurent Lafforgue - Wikipedia[14] Laurent Lafforgue - Wikipedia[14] Laurent Lafforgue - Wikipedia[15] Dessin d'enfant - Wikipedia[15] Dessin d'enfant - Wikipedia[15] Dessin d'enfant - Wikipedia[15] Dessin d'enfant - Wikipedia[16] Alexander Grothendieck - Wikipedia[16] Alexander Grothendieck - Wikipedia[16] Alexander Grothendieck - Wikipedia[16] Alexander Grothendieck - Wikipedia[17] Jacob Lurie - Wikipedia[17] Jacob Lurie - Wikipedia[17] Jacob Lurie - Wikipedia[17] Jacob Lurie - Wikipedia[18] Hilbert's sixth problem - Wikipedia[18] Hilbert's sixth problem - Wikipedia[18] Hilbert's sixth problem - Wikipedia[18] Hilbert's sixth problem - Wikipedia[19] Annus Mirabilis papers - Wikipedia[19] Annus Mirabilis papers - Wikipedia[19] Annus Mirabilis papers - Wikipedia[19] Annus Mirabilis papers - Wikipedia[20] Yang–Mills existence and mass gap - Wikipedia[20] Yang–Mills existence and mass gap - Wikipedia[20] Yang–Mills existence and mass gap - Wikipedia[20] Yang–Mills existence and mass gap - Wikipedia[21] http://www.claymath.org/sites/default/files/yangmills.pdf[21] http://www.claymath.org/sites/default/files/yangmills.pdf[21] http://www.claymath.org/sites/default/files/yangmills.pdf[21] http://www.claymath.org/sites/default/files/yangmills.pdf[22] http://www.claymath.org/sites/default/files/navierstokes.pdf[22] http://www.claymath.org/sites/default/files/navierstokes.pdf[22] http://www.claymath.org/sites/default/files/navierstokes.pdf[22] http://www.claymath.org/sites/default/files/navierstokes.pdf[23] Navier–Stokes equations - Wikipedia[23] Navier–Stokes equations - Wikipedia[23] Navier–Stokes equations - Wikipedia[23] Navier–Stokes equations - Wikipedia[24] abc conjecture - Wikipedia[24] abc conjecture - Wikipedia[24] abc conjecture - Wikipedia[24] abc conjecture - Wikipedia[25] http://www.ams.org/notices/200210/fea-granville.pdf[25] http://www.ams.org/notices/200210/fea-granville.pdf[25] http://www.ams.org/notices/200210/fea-granville.pdf[25] http://www.ams.org/notices/200210/fea-granville.pdf[26] Shinichi Mochizuki - Wikipedia[26] Shinichi Mochizuki - Wikipedia[26] Shinichi Mochizuki - Wikipedia[26] Shinichi Mochizuki - Wikipedia[27] What is the reason why the proof for ABC conjecture of Shinichi Mochizuki has still not been generally considered as a proof passing the peer review stage as usual though it had already been examined by more than 10 mathematicians?[27] What is the reason why the proof for ABC conjecture of Shinichi Mochizuki has still not been generally considered as a proof passing the peer review stage as usual though it had already been examined by more than 10 mathematicians?[27] What is the reason why the proof for ABC conjecture of Shinichi Mochizuki has still not been generally considered as a proof passing the peer review stage as usual though it had already been examined by more than 10 mathematicians?[27] What is the reason why the proof for ABC conjecture of Shinichi Mochizuki has still not been generally considered as a proof passing the peer review stage as usual though it had already been examined by more than 10 mathematicians?[28] Did Shinichi Mochizuki solve the ABC Conjecture?[28] Did Shinichi Mochizuki solve the ABC Conjecture?[28] Did Shinichi Mochizuki solve the ABC Conjecture?[28] Did Shinichi Mochizuki solve the ABC Conjecture?[29] How difficult is it to penetrate the Mochizuki’s unverified proof of abc conjecture?[29] How difficult is it to penetrate the Mochizuki’s unverified proof of abc conjecture?[29] How difficult is it to penetrate the Mochizuki’s unverified proof of abc conjecture?[29] How difficult is it to penetrate the Mochizuki’s unverified proof of abc conjecture?[30] What does Terence Tao think of Mochizuki´s proof of the abc conjecture?[30] What does Terence Tao think of Mochizuki´s proof of the abc conjecture?[30] What does Terence Tao think of Mochizuki´s proof of the abc conjecture?[30] What does Terence Tao think of Mochizuki´s proof of the abc conjecture?[31] Has proof of the ABC conjecture, written by Shinichi Mochizuki, already been checked?[31] Has proof of the ABC conjecture, written by Shinichi Mochizuki, already been checked?[31] Has proof of the ABC conjecture, written by Shinichi Mochizuki, already been checked?[31] Has proof of the ABC conjecture, written by Shinichi Mochizuki, already been checked?[32] Szpiro's conjecture - Wikipedia[32] Szpiro's conjecture - Wikipedia[32] Szpiro's conjecture - Wikipedia[32] Szpiro's conjecture - Wikipedia