Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
Interpretation
Mathematics is both a rational and creative endeavor, merging logical analysis with intuitive thought to achieve beauty and understanding.
This quote by Richard Courant emphasizes that mathematics is not only a logical discipline but also an artistic one that reflects human creativity and intellect. It highlights how the combination of different elements—such as the interplay between logic and intuition—contributes to the richness and value of mathematical science, suggesting that the synthesis of contrasting forces is essential for understanding its true beauty and utility.
In practice
This quote can be used in a lecture on the philosophy of mathematics to illustrate its dual nature.
It is generally believed that our science is empirical and that we draw our concepts and our mathematical constructs from the empirical data. If this were the whole truth, we should, when entering into a new field, introduce only such quantities as can directly be observed, and formulate natural laws only by means of these quantities.
I ... began my career as a wireless amateur. After 43 years in radio, I do not mind confessing that I am still an amateur. Despite many great achievements in the science of radio and electronics, what we know today is far less than what we have still to learn.
No matter how counter-intuitive it may seem, basic research has proven over and over to be the lifeline of practical advances in medicine.
If the tribal peoples actually represented Western origins at a much earlier time, it was exceedingly valuable that they should be studied intensely for clues about the nature and origin of human society. Consequently it was an injury to science and human knowledge to allow the military to simply exterminate them.
People who look for the first time through a microscope say, 'Now I see this, and then I see that,' and even a skilled observer can be fooled. On these observations I have spent more time than many will believe, but I have done them with joy, and I have taken no notice of those who have said, 'Why take so much trouble,' and, 'What good is it?'
A discovery is like falling in love and reaching the top of a mountain after a hard climb all in one, an ecstasy not induced by drugs but by the revelation of a face of nature that no one has seen before and that often turns out to be more subtle and wonderful than anyone had imagined.
Subscribe for the occasional hand-picked quote. No noise.