Brouwer Quotes & Sayings

Beyond the natural numbers, addition, multiplication, and mathematical induction are intuitively clear. — L. E. J. Brouwer

Mathematics is nothing more, nothing less, than the exact part of our thinking. — L. E. J. Brouwer

The construction itself is an art, its application to the world an evil parasite. — L. E. J. Brouwer

Yet truth is a diamond; even mishandles, smeared with grease, or buried in mud, it cannot be marred and waits for one with a cloth to polish it clean. — Sigmund Brouwer

Here are the basic principles of Constructivism as practiced by Kronecker and codified by J.H. Poincare and L.E.J. Brouwer and other major figures in Intuitionism: (1) Any mathematical statement or theorem that is more complicated or abstract than plain old integer-style arithmetic must be explicitly derived (i.e. 'constructed') from integer arithmetic via a finite number of purely deductive steps. (2) The only valid proofs in math are constructive ones, with the adjective here meaning that the proof provides a method for finding (i.e., 'constructing') whatever mathematical entities it's concerned with. — David Foster Wallace

We mourn His suffering when, instead, we should mourn the reason He suffered. — Sigmund Brouwer

Oil is seldom found where it is most needed, and seldom most needed where it is found. — L. E. J. Brouwer

True men of this world do not wish. They change that which they know can be changed, they accept what cannot, and they always strive for the wisdom to know the difference. — Sigmund Brouwer

An incorrect theory, even if it cannot be inhibited by any contradiction that would refute it, is none the less incorrect, just as a criminal policy is none the less criminal even if it cannot be inhibited by any court that would curb it. — L. E. J. Brouwer

Outside, most people knew that decades ago, the religious fundamentalists lost the ability to transform society when they became a political movement. Their boycotts and protests were commonplace, any outcry against anything beyond the narrow range of what they saw as biblically acceptable was dismissed as a knee-jerk reaction. — Sigmund Brouwer

Without a sharp turnaround toward democracy and equality in the United States, Europe will be virtually alone in its commitment to social democracy. The pressures of low-wage immigrant labor, cheap imports from Eastern Europe and Asia, and free-market practices of governments are already threatening once secure areas of employment and causing right-wing populism to pop up in various Western European countries. Surprising numbers of middle-class and working-class voters have supported ultranationalist, neofascist parties throughout Europe because, like white male workers in the United States, they see their status slipping. — Steve Brouwer

Are you insane? You still not convinced that chances are better with me? — Sigmund Brouwer

Life is difficult more often that it is not. To live means to face difficulties. It's what you learn from those difficulties that matters. — Sigmund Brouwer

Astute social commentators had been anticipating this rightward shift since the early 1980s. Bertram Gross predicted, in his book Friendly Fascism, that the United States might arrive at a gentler form of the virulent ultranationalism, antilabor activity, and racism, which coalesced into fascism in Europe in the 1930s. Corporate America would tolerate such a rightward drift, so the argument went, because more government restrictions on personal freedom would enhance business efforts to discipline the labor force and increase corporate profits. — Steve Brouwer

Our greatest temptations generally arise from the areas closest to our hearts. — Sigmund Brouwer

One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics. — L. E. J. Brouwer

I hope it doesn't get worse. But even if it gets worse, I won't regret it. I would rather be dead than live in the factory anymore. Not much difference as they just want you to work to death anyway. And you can't even think there or talk. but I have to think. I have to talk. I have to talk about what I think. — Sigmund Brouwer

For each of us, time is a thief of flory. What give meaning to our lives and deaths is love and hope, if we are willing to share and accept — Sigmund Brouwer

The subject for which I am asking your attention deals with the foundations of mathematics. To understand the development of the opposing theories existing in this field one must first gain a clear understnding of the concept "science"; for it is as a part of science that mathematics originally took its place in human thought. — L. E. J. Brouwer

Think, guys. What was strange about the story Uncle Goerge told us?"
"That a woman outsmarted the men", Mike said. For which he received a scowl that did a thunderstorm justice. — Sigmund Brouwer

Alas, the spheres of truth are less transparent than those of
illusion. — L. E. J. Brouwer