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.
Interpretation
The quote reflects on the ability of abstract concepts to feel as vibrant and real as living beings.
Henri Poincaré's quote emphasizes the unique perspective of M. Hermite, who engages in discussions that may not conjure specific images but instead illustrate the vitality and existence of abstract ideas. This suggests that while these concepts may lack tangibility, they can evoke deep understanding and appreciation, akin to interacting with living entities.
In practice
In a lecture about the nature of mathematics, one could quote Poincaré to explain the significance of abstract concepts.
When the logician has resolved each demonstration into a host of elementary operations, all of them correct, he will not yet be in possession of the whole reality, that indefinable something that constitutes the unity ... Now pure logic cannot give us this view of the whole; it is to intuition that we must look for it.
It is a misfortune for a science to be born too late when the means of observation have become too perfect. That is what is happening at this moment with respect to physical chemistry; the founders are hampered in their general grasp by third and fourth decimal places.
A scientist worthy of his name, about all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.
. . . by natural selection our mind has adapted itself to the conditions of the external world. It has adopted the geometry most advantageous to the species or, in other words, the most convenient. Geometry is not true, it is advantageous.
The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another.
What is a good definition? For the philosopher or the scientist, it is a definition which applies to all the objects to be defined, and applies only to them; it is that which satisfies the rules of logic. But in education it is not that; it is one that can be understood by the pupils.
Drugs. If they did not exist our governors would have invented them in order to prohibit them and so make much of the population vulnerable to arrest, imprisonment, seizure of property, and so on.
It is spiritual poverty, not material lack, that lies at the core of all human suffering.
The world is governed by opinion.
We can no longer take our own way of life for granted - we know that it may be challenged. And we know this, too - and know it ever more deeply - we know that freedom and democracy are not just big words mouthed by orators but the rain and the wind and the sun, the air and the light by which we breathe and live.
.. that which renders morality an active principle and constitutes virtue our happiness, and vice our misery: it is probable, I say, that this final sentence depends on some internal sense or feeling, which nature has made universal in the whole species.
The invention of ethical and political doctrines, which blossomed into our own social sciences, is a product of times when things appeared manageable. The same goes for the criticism of those doctrines, though as a voice from the past, this criticism proved prophetic.
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