A Quote by Edward Kasner
Perhaps the greatest paradox of all is that there are paradoxes in mathematics.
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You are not supposed to get it. It's a paradox. All of mathematics is built on paradoxes. That's the biggest paradox of all-all this orderliness, and at the heart, impossibility. Contradiction. Heaven built on the foundations of hell.
A society, in the process of moving forward, often appears to be tearing itself apart. Certainly, an age of rapid change, such as ours, produces many paradoxes. But perhaps the most tragic paradox of our time is to be found in the failure of nation-states to recognize the imperatives of internationalism.
Even in pure mathematics they can't remove all paradox, and the rest of us should also recognize we are going to have to endure a lot of paradox, like it or not.
When Pico [Iyer] talks about home being a place of isolation, I think he's right. But it's the paradox. I think that's why I so love Great Salt Lake. Every day when I look out at that lake, I think, "Ah, paradox" - a body of water than no one can drink. It's the liquid lie of the desert. But I think we have those paradoxes within us and certainly the whole idea of home is windswept with paradox.
On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.
When the mathematician says that such and such a proposition is true of one thing, it may be interesting, and it is surely safe. But when he tries to extend his proposition to everything, though it is much more interesting, it is also much more dangerous. In the transition from one to all, from the specific to the general, mathematics has made its greatest progress, and suffered its most serious setbacks, of which the logical paradoxes constitute the most important part. For, if mathematics is to advance securely and confidently, it must first set its affairs in order at home.
There are two kinds of paradoxes. They are not so much the good and the bad, nor even the true and the false. Rather they are the fruitful and the barren; the paradoxes which produce life and the paradoxes that merely announce death. Nearly all modern paradoxes merely announce death.
A genius may perhaps be a century ahead of his age and hence stands there as a paradox, but in the end, the race will assimilate what was once a paradox, so it is no longer paradoxical.
One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.
People who express themselves in paradoxes are in a strong position; and the more outrageous the paradox, in general the stronger the position.
I think I am intrigued by paradoxes. If something seems to be a paradox, it has something deeper, something worth exploring.
The 20th century gave rise to one of the greatest and most distressing paradoxes of human history: that the greatest intolerance and violence of that century were practiced by those who believed that religion caused intolerance and violence.
This is the greatest paradox: the emotions cannot be trusted; yet it is the emotions that tell us the greatest truths.
The paradox is really the pathos of intellectual life and just as only great souls are exposed to passions it is only the great thinker who is exposed to what I call paradoxes, which are nothing else than grandiose thoughts in embryo.
Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
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