Chentsov Theorem Quotes

Science in its everyday practice is much closer to art than to philosophy. When I look at Godel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral. Godel took Hilbert's formalized axioms of mathematics as his building blocks and built out of them a lofty structure of ideas into which he could finally insert his undecidable arithmetical statement as the keystone of the arch. The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Godel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalism into unexplored territory. ~ Freeman Dyson

I carried this problem around in my head basically the whole time. I would wake up with it first thing in the morning, I would be thinking about it all day, and I would be thinking about it when I went to sleep. Without distraction I would have the same thing going round and round in my mind.
(Recalling the degree of focus and determination that eventually yielded the proof of Fermat's Last Theorem.) ~ Andrew John Wiles

Humanism ... is not a single hypothesis or theorem, and it dwells on no new facts. It is rather a slow shifting in the philosophic perspective, making things appear as from a new centre of interest or point of sight. ~ William James

The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains. ~ G.H. Hardy

I know this may sound like an excuse," he said. "But tensor functions in higher differential topology, as exemplified by application of the Gauss-Bonnett Theorem to Todd Polynomials, indicate that cohometric axial rotation in nonadiabatic thermal upwelling can, by random inference derived from translational equilibrium aggregates, array in obverse transitional order the thermodynamic characteristics of a transactional plasma undergoing negative entropy conversions."
"Why don't you just shut up," said Hardesty. ~ Mark Helprin

Before beginning [to try to prove Fermat's Last Theorem] I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure. ~ David Hilbert

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. ~ G.H. Hardy

Note for a Textbook

The question is never answered, never resolved,
The circle of love and anger never squared,
The stubborn instinct not translated yet
In decimals, accurate and predictable
In union dues or payroll cuts or blood...

Always a symbol lost in the lovely theorem,
A fraction that will not fit in the sum of the system,
A jutting thrust in the graph of the commissar's forecast,
A troublesome blank in gauleiter's careful accounting,
An awkward hitch in the plans of the second vice-president.

The answers worked on the slate are never the same,
And the answers proved in the back of the book are wrong. ~ Charles Bruce

Stephen Hawking has proven a general theorem stating that all solutions of Einstein's equations that allow faster-than-light travel must involve negative matter or energy. ~ Michio Kaku

And I believe that the Binomial Theorem and a Bach Fugue are, in the long run, more important than all the battles of history. ~ James Hilton

The primes are the raw material out of which we have to build arithmetic, and Euclid's theorem assures us that we have plenty of material for the task. ~ G.H. Hardy

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means ... ~ Henri Poincare

The most famous Diophantine equation in history is the one known as Fermat's last Theorem, the celebrated statement by Pierre de Fermat (1601-55) that there are no whole number solutions to the equation x^n + y^n = z^n, where n is any number greater than 2. When n = 2, there are many solutions (in fact an infinite number). For instance, 3^2 + 4^2 = 5^2 (9 + 16 = 25); or 12^2 +5^2 = 13^2 (144 + 25 = 169). Miraculously, when we go from n = 2 to n = 3, there are no whole numbers x,y,z that satisfy x^3 + y^3 = z^3, and the same is true for any other value of n that is greater than 2. Appropriately, it was in the margin of the second book of Diophantus's Arithmetica, which Fermat was eagerly reading, that he wrote his extraordinary claim-one that took no fewer than 356 years to prove. ~ Mario Livio

Colin,
I hate to fulfill the Theorem, but I don't think we should be involved romantically.
The problem is that I secretly in love with Hassan.
I can't help myself. I hold your bony shoulder blades in my hands and think of his fleshy back.
I kiss your stomach and I think of his awe-inspiring gut. I like you,
Colin, I really do. But-I'm sorry. It's just not going to work.
I hope we can still be friends.
Sincerely,
Lindsey Lee Wells
P.S. Just kidding. ~ John Green

Words are the children of reason and, therefore, can't explain it. They really can't translate feeling because they're not part of it. That's why it bugs me when people try to analyze jazz as an intellectual theorem. It's not. It's feeling. ~ Bill Evans

That is why one day I said my game will be like the Pythagorean Theorem - hard to figure out. A lot of people really don't know the Pythagorean Theory. They don't make them like me anymore. They don't want to make them like that anymore. ~ Shaquille O'Neal

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann. ~ George Polya

The future is the result of actions, and actions are the result of behavior, and behavior is the result of prediction. ~ David McRaney

Fain would I turn back the clock and devote to French or some other language the hours I spent upon algebra, geometry, and trigonometry, of which not one principle remains with me. Stay! There is one theorem painfully drummed into my head which seems to have inhabited some corner of my brain since that early time: "The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides!" There it sticks, but what of it, ye gods, what of it? ~ Jessie Belle Rittenhouse

My favourite fellow of the Royal Society is the Reverend Thomas Bayes, an obscure 18th-century Kent clergyman and a brilliant mathematician who devised a complex equation known as the Bayes theorem, which can be used to work out probability distributions. It had no practical application in his lifetime, but today, thanks to computers, is routinely used in the modelling of climate change, astrophysics and stock-market analysis. ~ Bill Bryson

Any effect, constant, theorem or equation named after Professor X was first discovered by Professor Y , for some value of Y not equal to X. ~ John C. Baez

In mathematics, in place of characters, you have variables or unknowns. If I'm trying to plot a theorem, I try to imagine these variables interacting with each other. The boundary of their interaction is the theorem. ~ Manil Suri

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules ... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms. ~ Kurt Godel

Bell's theorem ... proves that quantum theory requires connections that appear to resemble telepathic communication. ~ Gary Zukav

Noether's theorem fused together symmetries and conservation laws-these two giant pillars of physics are actually nothing but different facets of the same fundamental property. ~ Mario Livio

There's no answer for my offense, just like the polythagorean theorem. ~ Shaquille O'Neal

Explanation is always incomplete: we can always raise another Why-questions. And the new why-questions may lead to a new theory which not only "explains" the old theory but corrects it.
This is why the evolution of Physics is likely to be an endless process of correction and better approximation. And even if one day we should reach a stage where our theories were no longer open to correction, because they are simply true, they would still not be complete - and we should know it. For Godel's famous incompleteness theorem would come into play: in view of the Mathematical background of Physics, at best an infinite sequence of such true theories would be needed in order to answer the problems which any given (formalized) theory would be undecidable.
Such considerations do not prove that the objective physical world is incomplete, or undetermined: they only show the essential incompleteness of our efforts. But they also show that it's barely possible (if possible at all) for science to reach a stage in which it can provide genuine support for the view that the physical world is deterministic. Why, the, should we not accept the verdict of common sense- at least until these arguments have been refuted? ~ Karl Popper

Number is the ruler of forms and ideas, and the cause of gods and demons. ~ Pythagoras

Here are the basic principles of Constructivism as practiced by Kronecker and codified by J.H. Poincare and L.E.J. Brouwer and other major figures in Intuitionism: (1) Any mathematical statement or theorem that is more complicated or abstract than plain old integer-style arithmetic must be explicitly derived (i.e. 'constructed') from integer arithmetic via a finite number of purely deductive steps. (2) The only valid proofs in math are constructive ones, with the adjective here meaning that the proof provides a method for finding (i.e., 'constructing') whatever mathematical entities it's concerned with. ~ David Foster Wallace

MacPherson told me that my theorem can be viewed as blah blah blah Grothendieck blah blah blah, which makes it much more respectable. ~ Jim Propp

Max Boot theorem: Don't invade without planning for a long occupation. ~ Max Boot

Identifying a problem is the real theorem and fixing it is only a corollary. ~ Balan Gothandaraman

You wouldn't think there was a need for a Coase Theorem, really. ~ Ronald Coase

In 1975, ... [speaking with Shiing Shen Chern], I told him I had finally learned ... the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added, "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." He immediately protested: "No, no. These concepts were not dreamed up. They were natural and real." ~ Chen-Ning Yang

A Fairy must make her own way in the world, for the world will never make way for her. That, incidentally, is the First Theorem of Questing Physicks, which you'll learn all about when you're older and don't care anymore. ~ Catherynne M Valente

In a companion essay to "Continuity and I.N." that's usually translated as "The Nature and Meaning of Numbers," Dedekind evinces a remarkable proof for his "Theorem 66. There exist infinite systems," which runs thus: "My own realm of thoughts, i.e., the totality S of all things which can be objects of my thought, is infinite. For if s signifies an element of S, then the thought s', that s can be an object of my thought, is itself an element of S,..." and so on, meaning that the infinite series ([s] + [s is an object of thought]+['s is an object of thought' is an object of thought] + ...) exists in the Gedankenwelt, which entails that the Gedankenwelt is itself infinite. With respect to this proof, notice (a) how closely it resembles the various Zeno-like VIR back in paragraph 2a, and (b) how easily we could object that the proof establishes only that Dedekind's Gedankenwelt is 'potentially infinite' (and in precisely Aristotle's sense of the term), since nobody can ever actually sit down and think a whole infinite series of (s+s'+s")-type thoughts-i.e., the series is a total abstraction. ~ David Foster Wallace

There is a theorem that colloquially translates, You cannot comb the hair on a bowling ball ... Clearly, none of these mathematicians had Afros, because to comb an Afro is to pick it straight away from the scalp. If bowling balls had Afros, then yes, they could be combed without violation of mathematical theorems. ~ Neil DeGrasse Tyson

I swear the older I get, the more I value bad examples over good ones. It's a good thing too, because most people are egotistical, neurotic, self-absorbed peons, insistent on wearing near-sighted glasses in a far-sighted world. And it's this exact sort of myopic ignorance that has led to my groundbreaking new theory. I call it Mim's Theorem of Monkey See Monkey Don't, and what it boils down to is this: it is my belief that there are some people whose sole purpose of existence is to show the rest of how not to act. ~ David Arnold

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. ~ Johannes Kepler

You're right: if there's sentient life behind the border, it probably won't share my goals. Unlike the people in this room, who all want exactly the same things in life as I do, and have precisely the same tastes in food, art, music, and sex. Unlike the people of Schur, and Cartan, and Zapata - who I came here in the hope of protecting, after losing my own home - who doubtless celebrate all the same festivals, delight in the same songs and stories, and gather every fortieth night to watch actors perform the same plays, in the same language, from the same undisputed canon, as the people I left behind.

"If there's sentient life behind the border, of course we couldn't empathize with it. These creatures are unlikely to possess cute mammalian neonate faces, or anything else we might mistake for human features. None of us could have the imagination to get over such insurmountable barriers, or the wit to apply such difficult abstractions as the General Intelligence theorem - though since every twelve-year-old on my home world was required to master that result, it must be universally known on this side of the border.

"You're right: we should give up responsibility for making any difficult moral judgments, and surrender to the dictates of natural selection. Evolution cares so much about our happiness that no one who's obeyed an inherited urge has ever suffered a moment's regret for it. History is full of joyful case studies of people who followed their na ~ Greg Egan

There are lots of things I don't understand - say, the latest debates over whether neutrinos have mass or the way that Fermat's last theorem was (apparently) proven recently. But from 50 years in this game, I have learned two things: (1) I can ask friends who work in these areas to explain it to me at a level that I can understand, and they can do so, without particular difficulty; (2) if I'm interested, I can proceed to learn more so that I will come to understand it. Now Derrida, Lacan, Lyotard, Kristeva, etc. -- even Foucault, whom I knew and liked, and who was somewhat different from the rest -- write things that I also don't understand, but (1) and (2) don't hold: no one who says they do understand can explain it to me and I haven't a clue as to how to proceed to overcome my failures. That leaves one of two possibilities: (a) some new advance in intellectual life has been made, perhaps some sudden genetic mutation, which has created a form of "theory" that is beyond quantum theory, topology, etc., in depth and profundity; or (b) ... I won't spell it out. ~ Noam Chomsky

In any event, Socrates' proof of prenatal immortality is that one of Meno's uneducated slave boys actually comes up with the Pythagorean theorem without ever having studied geometry! Therefore, he must be remembering it. You recall that theorem: in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Huh? We can barely remember that from tenth grade, let alone from before we were born. ~ Thomas Cathcart

The missing piece in his stomach hurt so much-and eventually he stopped thinking about the Theorem and wondered only how something that isn't there can hurt you. ~ John Green

Combinatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum and ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late 18th century. ~ Richard Askey

It is a common misconception, in the spirit of the sentiments expressed in Q16, that Godel's theorem shows that there are many different kinds of arithmetic, each of which is equally valid. The particular arithmetic that we may happen to choose to work with would, accordingly, be defined merely by some arbitrarily chosen formal system. Godel's theorem shows that none of these formal systems, if consistent, can be complete; so-it is argued-we can keep adjoining new axioms, according to our whim, and obtain all kinds of alternative consistent systems within which we may choose to work. The comparison is sometimes made with the situation that occurred with Euclidean geometry. For some 21 centuries it was believed that Euclidean geometry was the only geometry possible. But when, in the eighteenth century, mathematicians such as Gauss, Lobachevsky, and Bolyai showed that indeed there are alternatives that are equally possible, the matter of geometry was seemingly removed from the absolute to the arbitrary. Likewise, it is often argued, Godel showed that arithmetic, also, is a matter of arbitrary choice, any one set of consistent axioms being as good as any other. ~ Roger Penrose

Poetry is a theorem of a yellow-silk handkerchief knotted with riddles, sealed in a balloon tied to the tail of a kite flying in a white wind against a blue sky in spring. ~ Carl Sandburg