Mathematical Infinity Quotes

Some infinities are bigger than other infinities.

The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers.

I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.

There are infinite numbers between 0 and 1.

I entertain no doubts as to the truths of the tranfinites, which I recognized with God's help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly disimilar to, and I might even say in priciple the same as, my method described above of introducing trasfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.

What is history? Its beginning is that of the centuries of systematic work devoted to the solution of the enigma of death, so that death itself may eventually be overcome. That is why people write symphonies, and why they discover mathematical infinity and electromagnetic waves.

Regarding the fundamental investigations of mathematics, there is no final ending ... no first beginning.

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.

But let us remember that we are dealing with infinities and indivisibles both of which transcend our finite understanding, the former on account of their magnitude, the latter because of their smallness.

From a mathematical standpoint it is possible to have infinite space. In a mathematical sense space is manifoldness, or combinations of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, the 10-dimension system.

Now what is history? It is the centuries of systematic explorations of the riddle of death, with a view to overcoming death. That's why people discover mathematical infinity and electromagnetic waves, that's why they write symphonies.

The infinite in mathematics is alway unruly unless it is properly treated.

Something is infinite if, taking it quantity by quantity, we can always take something outside.

What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending lader of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.

A googolplex is precisely as far from infinity as is the number 1... no matter what number you have in mind, infinity is larger still.

Two infinities: the one that stretches to the beginning but never touches-when you halve and halve and halve, infinitely-and then the one that spreads out into the endless, endless future, the endless, endless, distance.The set of infinities that is itself infinite.

The solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast.

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

So many little lives, amounting to nothing. I ask you: What is infinity multiplied by zero? It is hardly worth our discussion.

Infinities are not for casual exploration. You could fall and keep falling. You could get lost.

There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.