...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
Paul HalmosRead
[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing - one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.
Interpretation
Mathematics is a unified field representing truth, beauty, and interconnection in knowledge.
In this quote, Paul Halmos emphasizes that mathematics is not just a collection of separate disciplines, but rather a cohesive and interconnected body of knowledge. He views mathematics as embodying essential qualities such as security, certainty, and beauty, highlighting the intrinsic relationships among its various branches, suggesting that the essence of mathematics lies in its ability to reveal profound truths about the world.
In practice
During a seminar on the beauty of mathematics, this quote can be shared to inspire appreciation for mathematical concepts.
...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.
Scientific truth is too beautiful to be sacrificed for the sake of light entertainment or money. Astrology is an aesthetic affront. It cheapens astronomy, like using Beethoven for commercial jingles.
Science is feasible when the variables are few and can be enumerated; when their combinations are distinct and clear. We are tending toward the condition of science and aspiring to do it. The artist works out his own formulas; the interest of science lies in the art of making science.
Radio has no future." "X-rays are clearly a hoax". "The aeroplane is scientifically impossible.
The overwhelming majority of theories are rejected because they contain bad explanations, not because they fail experimental tests.
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.
The upshot of all this is that we live in a universe whose age we can't quite compute, surrounded by stars whose distances we don't altogether know, filled with matter we can't identify, operating in conformance with physical laws whose properties we don’t truly understand.
Subscribe for the occasional hand-picked quote. No noise.