Good problems and mushrooms of certain kinds have something in common; they grow in clusters.
We must be careful not to discourage our twelve-year-olds by making them waste the best years of their lives preparing for examinations.
Humans are actually spirit beings who each possess a mind and a body and what we do with them is entirely up to us.
A picture held us captive. And we could not get outside it, for it lay in our language and language seemed to repeat it to us inexorably.
I give no sources, because it is indifferent to me whether what I have thought has already been thought before me by another.
Logic is the last scientific ingredient of Philosophy; its extraction leaves behind only a confusion of non-scientific, pseudo problems.
If someone does not believe in fairies, he does not need to teach his children 'There are no fairies'; he can omit to teach them the word 'fairy'.
It is obvious that an imagined world, however different it may be from the real one, must have something - a form - in common with it.
Philosophy should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred.
Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing.
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.
Creating a body of mathematics is about intellectual labor, not some kind of transcendental revelation. There are plenty of important components of European fractal geometry that are missing from the African version.
I just toured around looking for fractals, and when I found something that had a scaling geometry, I would ask the folks what was going on - why they had made it that way.
We have the duty of formulating, of summarizing, and of communicating our conclusions, in intelligible form, in recognition of the right of other free minds to utilize them in making their own decisions.
It would be very discouraging if somewhere down the line you could ask a computer if the Riemann hypothesis is correct and it said, 'Yes, it is true, but you won't be able to understand the proof.'
One of the issues facing us today is that there are countries where there is a serious lack of resources, the standards of living are very low, and this creates a fundamental unease and discomfort in entire populations.
It is precisely because neither individuals nor small groups can be fully self-sufficient that cooperation is necessary to human survival and flourishing.
Libertarians recognize the difference between adults and children, as well as differences between normal adults and adults who are insane or mentally hindered or retarded.
But there is no obvious reason for holding that some normal adults are entitled to make choices for other normal adults, as paternalists of both left and right believe.
Libertarians typically argue that particular obligations, at least under normal circumstances, must be created by consent; they cannot be unilaterally imposed by others.
To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
