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.
It is a good principle in science not to believe any 'fact'---however well attested---until it fits into some accepted frame of reference. Occasionally, of course, an observation can shatter the frame and force the construction of a new one, but that is extremely rare. Galileos and Einsteins seldom appear more than once per century, which is just as well for the equanimity of mankind.
I profess to learn and to teach anatomy not from books but from dissections, not from the tenets of Philosophers but from the fabric of Nature.
As we leave the Moon at Taurus-Littrow, we leave as we came, and God willing, as we shall return, with peace and hope for all mankind.
When it comes to how neuroscience could help the wider public, the worst thing is when we make advances in, say, mindfulness, and then decide that everybody can potentially think their way to curing themselves or develop their own psycho-neuro-immune mechanisms for boosting cancer defenses.
In 1903, I finished my doctor's thesis and obtained the degree. At the end of the same year, the Nobel prize was awarded jointly to Becquerel, my husband and me for the discovery of radioactivity and new radioactive elements.
The real reason why general relativity is widely accepted is because it made predictions that were borne out by experimental observations.
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