Alexander Grothendieck Quotes

I do think from time to time that conceptual questions arise: What do we mean by equilibrium? What do we mean by this concept and that concept?

One can imagine that the ultimate mathematician is one who can see analogies between analogies.

Lately I've been working to convince myself that everything is a computation.

What I knew was I liked math and science, and I never wanted to memorize everything. I wanted to understand where it came from.

Math research is more like a marathon.

The math helps you have better understanding and helps you have more creative ideas, but you can't replace the creative ideas

I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.

But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.

There are things which seem incredible to most men who have not studied Mathematics.

Consider the concepts referred to in the words 'where', 'when', 'why', 'being', to the elucidation of which innumerable volumes of philosophy have been devoted. We fare no better in our speculations than a fish which should strive to become clear as to what is water.

At first it seems obvious, but the more you think about it the stranger the deductions from this axiom seem to become; in the end you cease to understand what is meant by it.

Only Numbers. Pure math. You have to accustom yourself to thinking that way.

It is always the case, with mathematics, that a little direct experience of thinking over things on your own can provide a much deeper understanding than merely reading about them.

There is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being.

If you can’t program it, you haven’t understood it.

The ability of nonintelligent people to understand the most complicated mechanisms and to use them has always been to me a cause of astonishment: their inability to understand simple questions is even more astonishing. The general acceptance of simple ideas is difficult and rare, and yet it is only when simple, fundamental, ideas have been accepted that further progress becomes possible on a higher level.

I have often noticed that when people come to understand a mathematical proposition in some other way than that of the ordinary demonstration, they promptly say, "Oh, I see. That's how it must be." This is a sign that they explain it to themselves from within their own system.

But in the new (math) approach, the important thing is to understand what you're doing, rather than to get the right answer.

Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures.

The physicist may be satisfied when he has the mathematical scheme and knows how to use for the interpretation of the experiments. But he has to speak about his results also to non-physicists who will not be satisfied unless some explanation is given in plain language. Even for the physicist the description in plain language will be the criterion of the degree of understanding that has been reached.

Strange! I don't understand how it is that we can write mathematical expressions and calculate what the thing is going to do without being able to picture it.

This idea that there is generality in the specific is of far-reaching importance.

Making mental connections is our most crucial learning tool, the essence of human intelligence; to forge links; to go beyond the given; to see patterns, relationships, context.

For the past several generations we've forgotten what the psychologists call our archaic understanding, a willingness to know things in their deepest, most mythic sense. We're all born with archaic understanding, and I'd guess that the loss of it goes directly along with the loss of ourselves as creators.

One cannot understand... the universality of laws of nature, the relationship of things, without an understanding of mathematics. There is no other way to do it.