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There is proof enough furnished by every science, but by none more than geometry, that the world to which we have been allotted is peculiarly adapted to our minds, and admirably fitted to promote our intellectual progress. There can be no reasonable doubt that it was part of the Creator's plan. How easily might the whole order have been transposed! How readily might we have been assigned to some complicated system which our feeble and finite powers could not have unravelled!

 

Benjamin Peirce

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The movement of doubt consisted precisely in this: that at one moment he was supposed to be in the right, the next moment in the wrong, to a degree in the right, to a degree in the wrong, and this was supposed to mark his relationship with God; but such a relationship with God is not relationship, and this was the sustenance of doubt. In his relationship with another person, it certainly was possible that he could be partly in the wrong, partly in the right, to a degree in the wrong, to a degree in the right, because he himself and every human being is finite, and their relationship is a finite relationship that consists in a more or less. Therefore as long as doubt would make the infinite relationship finite, and as long as wisdom would full up the infinite relationship with the finite-just so long he would remain in doubt. Thus every time doubt wants to trouble him about the particular, tell him that he is suffering too much or is being tested beyond his powers, he forget the finite in the infinite, that he is always in the wrong. Every time the cares of doubt want to make him sad, he lifts himself above the finite into the infinite, because this thought, that he is always in the wrong, is the wings upon which he soars over the finite. This is the longing with which he seeks God; this is the love which he finds God.

 

Soren Aabye Kierkegaard

 

Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i.e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.

 

Hans Reichenbach

 

Rousseau's writings are so admirably adapted to touch both these classes that the effect they produced, especially in France, is easily intelligible.

 

Thomas Henry Huxley

 

I believe there are two opposing theories of history, and you have to make your choice. Either you believe that this kind of individual greatness does exist and can be nurtured and developed, that such great individuals can be part of a cooperative community while they continue to be their happy, flourishing, contributing selves — or else you believe that there is some mystical, cyclical, overriding, predetermined, cultural law — a historic determinism.
The great contribution of science is to say that this second theory is nonsense. The great contribution of science is to demonstrate that a person can regard the world as chaos, but can find in himself a method of perceiving, within that chaos, small arrangements of order, that out of himself, and out of the order that previous scientists have generated, he can make things that are exciting and thrilling to make, that are deeply spiritual contributions to himself and to his friends. The scientist comes to the world and says, "I do not understand the divine source, but I know, in a way that I don't understand, that out of chaos I can make order, out of loneliness I can make friendship, out of ugliness I can make beauty."
I believe that men are born this way — that all men are born this way. I know that each of the undergraduates with whom I talked shares this belief. Each of these men felt secretly — it was his very special secret and his deepest secret — that he could be great.
But not many undergraduates come through our present educational system retaining this hope. Our young people, for the most part — unless they are geniuses — after a very short time in college give up any hope of being individually great. They plan, instead, to be good. They plan to be effective, They plan to do their job. They plan to take their healthy place in the community. We might say that today it takes a genius to come out great, and a great man, a merely great man, cannot survive. It has become our habit, therefore, to think that the age of greatness has passed, that the age of the great man is gone, that this is the day of group research, that this is the day of community progress. Yet the very essence of democracy is the absolute faith that while people must cooperate, the first function of democracy, its peculiar gift, is to develop each individual into everything that he might be. But I submit to you that when in each man the dream of personal greatness dies, democracy loses the real source of its future strength.

 

Edwin H. Land

 

I think the first thing that led me toward philosophy (though at that time the word 'philosophy' was still unknown to me) occurred at the age of eleven. My childhood was mainly solitary as my only brother was seven years older than I was. No doubt as a result of much solitude I became rather solemn, with a great deal of time for thinking but not much knowledge for my thoughtfulness to exercise itself upon. I had, though I was not yet aware of it, the pleasure in demonstrations which is typical of the mathematical mind. After I grew up I found others who felt as I did on this matter. My friend G. H. Hardy, who was professor of pure mathematics, enjoyed this pleasure in a very high degree. He told me once that if he could find a proof that I was going to die in five minutes he would of course be sorry to lose me, but this sorrow would be quite outweighed by pleasure in the proof. I entirely sympathized with him and was not at all offended. Before I began the study of geometry somebody had told me that it proved things and this caused me to feel delight when my brother said he would teach it to me. Geometry in those days was still 'Euclid.' My brother began at the beginning with the definitions. These I accepted readily enough. But he came next to the axioms. 'These,' he said, 'can't be proved, but they have to be assumed before the rest can be proved.' At these words my hopes crumbled. I had thought it would be wonderful to find something that one could prove, and then it turned out that this could only be done by means of assumptions of which there was no proof. I looked at my brother with a sort of indignation and said: 'But why should I admit these things if they can't be proved?' He replied, 'Well, if you won't, we can't go on.'

 

Bertrand Russell