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Michael Atiyah

65 quotes

“Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“In the broad light of day mathematicians check their equations and proofs, leaving no stone unturned in their search for rigor. But at night, under the full moon, they dream. They float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life.”
Michael Atiyah (1929–2019) first seen 2008 wikiquote, CC BY-SA 4.0 source
Recorded in: Dreams" from "Sir Michael Atiyah - From Algebraic Geometry to Physics - a Personal Perspective [2010]" 1:00:41, a YouTube video (Jun 15, 2018) from the Gaduate Mathematics channel. See also: 1) Jean-François Dars, Annic…
“My own supervisor, William Hodge, the creator of the fertile theory of harmonic forms, was not a genius like Ramanujan but resembled Lefschetz.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“This 'Hodge conjecture' has by now achieved a considerable status, almost on a par with the Riemann hypothesis or the Poincaré conjecture.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“As individuals, our personality resides entirely in our memory. ...The same is true of civilization. ...Without a knowledge of our past, we... are nothing, but the present... [T]hat applies... to mathematics. You must... understand the historical background of any part of mathematics, any problem, where it comes from, how it progresses and where it might lead to in the future.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[M]athematics is not unconnected with the real world... [I]t has a very strong, intimate and long relationship with physics, which is pertinent in the context of geometry and of the analysis that often goes with it, and that... is true of the Poincaré conjecture.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[J]ust as the past is important, so the future is important, although the future is a little harder to predict.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[B]y the 19th century, end of the 18th century people were studying curved surfaces, and the since then, has to a great extent, been the study of the notion of curvature. What is curvature? How do you define it precisely? What does it mean, and how does it change as you increase the number of dimensions..?”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“In dimension one, along a straight line or on the edge of a circle there is no real geometry. All you can do is to measure how far along the path you are.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Geometry starts in dimension 2, and it was Gauss... who defined... Gaussian... or scaler curvature of any surface... which varies from point to point. ...[T]his is a notion of ...intrinsic curvature to distinguish it from extrinsic curvature.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[A]lthough the sphere... is curved intrinsically because you can't draw a precise map of the atlas of the world on a piece of paper without distortion, if you have something on a scroll.. you... unroll that and there's no distortion. A cylinder looks curved, but it's... flat from the point of view of Gaussian curvature. It curves in one direction, but not in the other, and it's important that... both change.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[T]he 19th century... started... with Gauss and went on with the development of the notion of curvature in 2 dimensions, by one function.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“In the 20th century we moved... from 2 dimensions to 3...”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“The 19th century finished with a very very deep theory of 2 dimensional surfaces, in all aspects: geometrical, analytical, complex, real.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“The 20th century... was devoted to ideas of 3 dimensional geometry, and this is the culmination... today...”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“In 3 dimensions the curvature is more complicated... It's determined not by by a single number r, but by a number R that depends on 2 directions i and j, called the ...”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“4 dimensions is... very important, even if you don't believe in it. There are lots of ways in which 4 dimensions affects our world, especially if you include time. That is now... the center of attraction for mathematicians in the 21st century, and it may be not until the end of the 21st century that it is... better understood... [H]ere the curvature first introduced by Riemann... depends on 4 indices, or 4 directions. ...[A]fter that.., it doesn't get more complicated with 5 or more dimensions. It becomes simpler. These are the dimensions of interest.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[M]athematics is created by individual people... with real names.., faces and... personalities.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“The genus is the number of holes. The sphere [surface] has no hole... The torus is a sphere with a hole bored through...”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Mathematics is a very ancient subject... mainly created by the young, and... it's cumulative. ...How doews this happen?”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Mathematics was well known to the Babylonians. ...[A]bout 2,000 BC ...this tablet shows how sophisticated the mathematics was ...An approximation to \sqrt{2} ...accurate to 4 or 5 decimal places ...written in decimal notation ...[P]eople think the decimal system was created by the Arabs, Indians, Chinese, but ...the Babylonians had a system ...based on 60. So when they wanted to describe a number, they did it with 1/60, 1/602 ...[etc.]”
Michael Atiyah (1929–2019) first seen 1600 BC wikiquote, CC BY-SA 4.0 source
Recorded in: Overhead: Babylonian clay tablet . Note: YBC 7289 (ca 1800–1600 BCE) is from the
“Given the antiquity.., the cumulative nature of mathematical knowledge, and the fact that the young people... produce the original ideas, how does it keep going? ...How do young mathematicians cope with with this vast amount of knowledge from the past and still turn out... new ideas? ...[T]here's an external answer and an internal answer.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Externally there's... diversification of mathematics. New ideas keep coming in... from the outside world, and they generate new problems, new theories. So you advance by expanding your frontiers as you interact with the outside world, and the outside world keeps changing.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Internally.., the mathematics... at any given time, it's a vast amount of information.., you have to organize it... in such a way that it can be packaged and passed on... [A] lot of facts in a lot of books aren't much help. ...You organize it by unification. You bring together many different parts ...into smaller, simpler packages which can ...be learned.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“You expand the content and... unify the subject.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[A]stronomy was the key driver of mathematics in the early days. Physics came along... closely with it... Then you go into chemistry... biology, economics, finance... [etc.] A... range of... fields... have contributed to the development of mathematics by posing problems.., giving ideas and concepts.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“All mathematics builds upon abstraction. When you count... people, I don't look at individual people... I just count... all people... abstracted as one...”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[C]oncepts are very important and abstraction develops new concepts, new principles, how you organize them, structure, all... enable you to climb higher... up the heirarchy. Arithmetic... is an abstraction... then... algebra... gives abstract symbols for numbers... [etc.]”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Abstraction enables you to put lots of things in the past [together] in one package, which you label "algebra" or whatever it may be.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Part of that process is... intellectual technology. Not only do we have hardware.., we... have the software... [e.g.,] the development of calculus, you can think of as an intellectual technology.., a machine of ideas and techniques in the brain... which you can... apply to... problems. The development of intellectual technology is... one of the secrets of advancing. Each generation inherits the intellectual technology... [with] more powerful tools it can push the ideas further.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[A]t the same time you have to do pruning.., you encourage growth and you remove unnecessary clutter so you can see through the jungle, and it simplifies... and... removes unnecessary dead wood.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[P]runing is a way of slimming down the past, to make it more productive... and... abstraction helps to use new principles, which enables you to climb higher in the process of moving forward.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“My favorite story... is about the high school physics teacher who... noticed somebody... was not paying attention. So he turned to him and... said, "...[W]hat is an electron?" So the boy scratched his head and said, "Oh well, I used to know sir, but I've forgotten." So the teacher say, "Ah, what a pity... Only two people in the universe knew what an electron was, you and God, and you've forgotten!" So replace "electron" by "" and you get the tone of this lecture. ...I've spent most of my life working on spinors ...and I don't know. Only God knows! Maybe Dirac, but he's no longer with us.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“I want to focus on the comparison between s and the s. Complex numbers... took mathematicians several hundred years to really feel they understood what i was. i is the of , what is called an . It did not exist, but they found it very useful. They used it in formulas, and gradually, step by step, it became accepted... [F]inally the formal definitions were given so i... became and accepted part of mathematics, but it took a long, long time.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[[w:Spinor|[S]pinor]]s are analogously... a of geometry.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“In geometry the fundamental elements... [have] to do with measurement: lengths, areas, volumes, and those are described mathematically by the , and it was used by in the differential geometry, as a fundamental tool to describe... s.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“So what is a ? Why do I say it's a square root? Because... the of the s... is the . So the spinors are, in some sense, the square root, not of one form of a given degree, but [of] the sum of them all.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“So it's a very deep notion, and it will take us... at least as long as it took to understand \sqrt{-1}, to understand what s are, and they've only been around for perhaps 100 years, so we are still in the very early days, the equivalent perhaps, of the 15th century...”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“So I don't know the answer to my question, and neither does Jean-Pierre here, and... maybe Dirac knows, but he's up there with God, so we are left on our own.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“If you have a , if the you are studying is complex, then... [by] Hodge... [etc.,] the differential forms on the manifold can be broken up into the forms of type (p,q) where p involves the dz's and q involves the d\bar{z}'s, the mixture of the two [is] the . So when you have a complex structure, you see the square root, and the square root is complex geometry. So when you have complex geometry, you've found the square root inside the real geometry. Complex structure has given you the square root.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“But s exist without the need of complex structures. So what is a spinor when there's no complex structure? That's really the question.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[[w:Complex number|[C]omplex number]]s... are... not important only in algebra. They're important in geometry, and it was known... early.., that you could do the theory of s.., fundamentally... formalized by Hermann Weyl... [T]he important thing in complex numbers is the analysis. The algebra, the \sqrt{-1} is interesting. We know that the complex numbers are the of the s... You don't need to keep going... higher up, just equations of degree 2. But the... deep part about complex numbers is their role in analysis: complex analysis, Cauchy theorem... [etc.] That's a very unexpected bonus.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“When people started using i in formulas they were thinking only in terms of solving polynomial equations... [T]hen in the hands of the analysts, Cauchy and others, it became an indispensible tool. So that's where the deep part of complex analysis is.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[S]imilarly, you expect the deep part of s, not to lie in the formal properties or algebra, but in their geometrical meaning.., and geometry here I identify, like Jean-Pierre, with analysis... You do geometry on s. You have to use analysis, differential equations.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“So spinor analysis has to be found as a substitute for complex analysis. That's the first stage in going from Cauchy's theorem in this idea of the square root of geometry.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“You start with the [in 3 dimensions], which is the simplest thing after the ... with the wave of a hand, you can construct two beautiful objects: the and its dual, the . ...[This] was first discovered by the ...virus.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“Most viruses are of the form of an ... because it's the simplest thing you can do by doubling up. ...We wouldn't be here without viruses because the virus is also related to DNA. ...[V]iruses come in 2 forms, one more like the icosahedron. ...There are lots of nice [sharp] vertices..: what the virus uses to deter attackers. ...The round one is more like... the .., the one that is used to get away.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“The most important things in life.., whether you're a football player or a virus, is to deter attackers and run away when necessary.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“There is no... rule for choosing 1 out of 5. If I give you 5 things and ask you to choose one, you can't do it... by the usual rules of either using ruler and compass, or equivalently, using only... roots... That comes under one of the most famous theorems in the history of mathematics..”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source
“[T]he quintic equation.., equations of degree 5 or more, cannot be solved using only... roots... [T]hat took mathematicians 100s of years.., painfully. ...[E]verybody knows how to solve quadratic equations.., and with a lot more work... over a century or two, people worked out how to solve s... [W]ith even... more hard work, finally they worked out how to solve equations of degree 4.., a famous landmark in mathematics, and then they got stuck. They couldn't do 5.”
Michael Atiyah (1929–2019) said by 2019 at the latest wikiquote, CC BY-SA 4.0 source