The application of algebra to geometry ... has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences.
Interpretation
John Stuart Mill praises the impact of algebra on geometry and its significance in the development of exact sciences.
In this quote, John Stuart Mill highlights how the integration of algebraic principles into geometry fundamentally transformed the study of mathematics and paved the way for advancements in the sciences. He attributes immense importance to this intellectual leap made by RenΓ© Descartes, suggesting that it has fundamentally altered our understanding of the exact sciences, representing a monumental step in scientific progress.
In practice
In a lecture about the history of mathematics, one might use this quote to illustrate the importance of Descartes' contributions.
The peculiar evil of silencing the expression of an opinion is, that it is robbing the human race; posterity as well as the existing generation; those who dissent from the opinion, still more than those who hold it. If the opinion is right, they are deprived of the opportunity of exchanging error for truth: if wrong, they lose, what is almost as great a benefit, the clearer perception and livelier impression of truth, produced by its collision with error.
As for charity, it is a matter in which the immediate effect on the persons directly concerned, and the ultimate consequence to the general good, are apt to be at complete war with one another.
To think that because those who wield power in society wield in the end that of government, therefore it is of no use to attempt to influence the constitution of the government by acting on opinion, is to forget that opinion is itself one of the greatest active social forces. One person with a belief is a social power equal to ninety-nine who have only interests.
There should be perfect freedom, legal and social, to do the action and stand the consequences. It would be a great misunderstanding of this doctrine to suppose that it is one of selfish indifference, which pretends that human beings have no business with each other's conduct in life, and that they should not concern themselves about the well-doing or well-being of one another, unless their own interest is involved.
Political Economy, in truth, has never pretended to give advice to mankind with no lights but its own; though people who knew nothing but political economy (and therefore knew it ill) have taken upon themselves to advise, and could only do so by such lights as they had.
Marriage is the only actual bondage known to our law. There remain no legal slaves, except the mistress of every house.
The black holes of nature are the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time.
No one knows who wrote the laws of physics or where they come from. Science is based on testable, reproducible evidence, and so far we cannot test the universe before the Big Bang.
Part of knowing how to think is knowing how the laws of nature shape the world around us. Without that knowledge, without that capacity to think, you can easily become a victim of people who seek to take advantage of you.
There are many instances in science, where those closest to the intricacies of the subject have a more highly developed sense of its intractability than those at some remove. On the other hand, those at too great a distance may, I am well aware, mistake ignorance for perspective.
I don't know much about climate change. But I'm pretty sure we better figure out what to do to lessen its impact - at least its health impact - and that's not going to happen unless you have a lot of young talent interested in these topics.
Our senses enable us to perceive only a minute portion of the outside world.
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