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.
Interpretation
Mathematics can help us understand physical laws by revealing connections between seemingly unrelated facts.
Henri Poincaré emphasizes the significance of mathematical facts that have the potential to illuminate physical laws through their relationships with other truths. He argues that by recognizing these analogies, we can uncover deep connections between facts that were previously thought to be unrelated, enriching our understanding of both mathematics and the physical universe.
In practice
During a lecture on the relationship between mathematics and natural sciences.
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.
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.
Most striking at first is the appearance of sudden illumination, a manifest sign of long unconscious prior work.
If ever there was a slamming of the door in the face of constructive investigation, it is the word miracle. To a medieval peasant, a radio would have seemed like a miracle.
From the point of view of basic physics, the most interesting phenomena are, of course, in the new places, the places where the rules do not work - not the places where they do work! That is the way in which we discover new rules.
All you are is a bag of particles acting out the laws of physics. That to me is pretty clear.
Of course, if one ignores contradictory observations, one can claim to have an "elegant" or "robust" theory. But it isn't science.
We're at a point in history were we have to become a part of the neighborhood of inhabited planets, like a neighborhood of a community, which we have not even acknowledged that that community exists up until this point.
The chance that higher life forms might have emerged through evolutionary processes is comparable with the chance that a tornado sweeping through a junk yard might assemble a Boeing 747 from the material therein.
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