It may happen that small differences in the initial conditions produce very great ones in the final phenomena.
Interpretation
Small initial differences can lead to significant outcomes over time.
This quote by Henri PoincarΓ© highlights the concept of sensitivity to initial conditions, often associated with chaos theory. It suggests that even minimal variations in the starting point of a system can exponentially influence its trajectory and end result, emphasizing the importance of precision and care in scientific endeavors and in understanding complex systems.
In practice
In a discussion about climate change, one might use this quote to illustrate how small changes now can radically affect future weather patterns.
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.
Any objective look at what science has to say about climate change ought to be sufficient to persuade reasonable people that the climate is changing and that humans are responsible for a substantial part of that - and that these changes are doing harm and will continue to do more harm unless we start to reduce our emissions.
When the wrong question is being asked, it usually turns out to be because the right question is too difficult. Scientists ask questions they can answer. That is, it is often the case that the operations of a science are not a consequence of the problematic of that science, but that the problematic is induced by the available means.
In less than eight years "The Origin of Species" has produced conviction in the minds of a majority of the most eminent living men of science. New facts, new problems, new difficulties as they arise are accepted, solved, or removed by this theory; and its principles are illustrated by the progress and conclusions of every well established branch of human knowledge.
To a synthetic chemist, the complex molecules of nature are as beautiful as any of her other creations. The perception of that beauty depends on the understanding of chemical structures and their transformations, and, as with a treasured work of art, deepens as the subject is studied, perhaps even to a level approaching romance.
When it comes to the things that people really want in science fiction - like space travel - the simplest things end up causing them not to happen. Humans are 100-pound bags of water, built to live on Earth.
We need more science in the world. Train me.
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