Associate Professor of Mathematics
Preserving the look-and-feel of the World Wide Web as it was, in 1998.
Some Favorite Quotes
We are passing through a period of great commercial prosperity,
and such a period is as sure as adversity itself
to bring mutterings of discontent.
At a time when most men prosper somewhat,
some men always prosper greatly;
and it is as true now as when the tower of Siloam fell upon all alike,
that good fortune does not come solely to the just,
nor bad fortune solely to the unjust.
When the weather is good for crops,
it is good for weeds.
Moreover, not only do the wicked flourish
when the times are such that most men flourish,
but, what is worse,
the spirit of envy and jealousy springs up in the breasts of those who,
though they may be doing fairly well themselves,
see others no more deserving who do better.
Wise laws and fearless and upright administration of the laws
can give the opportunity for such prosperity as we see about us.
But that is all that they can do.
When the conditions have been created which make prosperity possible,
then each individual man must achieve it for himself
by his own energy and thrift and business intelligence.
There are some things that we cannot remember.
There are some things that we cannot forget.
Most things are in between.
Knowing what math is used for helps us decide
What bits of math to retain and which to forget.
Don't forget that most men with nothing
would rather protect the possibility of becoming rich
than face the reality of being poor.
If you believe that just sitting in a math classroom is going to make you good at math, then you must believe that sitting in a garage would turn you into a car.
The ignoraunte multitude doeth, but as it was euer wonte, enuie that knoweledge, whiche thei can not attaine, and wishe all men ignoraunt, like unto themself. . .
Yea, the pointe in Geometrie, and the unitie in Arithmetike, though bothe be undiuisible, doe make greater woorkes, & increase greater multitudes, then the brutishe bande of ignoraunce is hable to withstande. . .
But yet one commoditie moare. . . I can not omitte. That is the
filying, sharpenyng, and quickenyng of the witte, that by practice
of Arithmetike doeth insue. It teacheth menne and accustometh
them, so certainly to remember thynges paste: So circumspectly
to consider thynges presente: And so prouidently to forsee thynges
that followe: that it maie truelie bee called the File of witte.
Math is built with facts as a house is built with bricks, but a collection of facts cannot be called mathematics anymore than a pile of bricks can be called a house.
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
Let no one ignorant of geometry enter here.
A mathematician is a machine for turning coffee into theorems; but American coffee is only good enough for a lemma.
"Suppose now, Glaucon, someone were to ask them, 'My good friends, what numbers are these you are talking about, in which the one is such as you postulate, each unity equal to every other without the slightest difference and admitting no division into parts?' What do you think would be their answer?"
"This, I think--that they are speaking of units which can only be conceived by thought, and which it is not possible to deal with in any other way."
"You see, then, my friend," said I, "that this branch of study really seems to be indispensable for us, since it plainly compels the soul to employ pure thought with a view to truth itself."
"It most emphatically does."
"Again, have you ever noticed this, that natural reckoners are by nature quick in virtually all their studies? And the slow, if they are trained and drilled in this, even if no other benefit results, all improve and become quicker than they were1?"
"It is so," he said.
"And, further, as I believe, studies that demand more toil in the learning and practice than this we shall not discover easily nor find many of them."
"You will not, in fact."
"Then, for all these reasons, we must not neglect this study, but must use it in the education of the best endowed natures..."
"I agree", he said.
"Assuming this one point to be established," I said, "let us in the second place consider whether the study that comes next is suited to our purpose."
"What is that? Do you mean geometry," he said.
"Precisely that," said I.
"So much of it," he said, "as applies to the conduct of war is obviously suitable. For in dealing with encampments and the occupation of strong places and the bringing of troops into column and line and all the other formations of an army in actual battle and on the march, an officer who had studied geometry would be a very different person from what he would be if he had not."
"But still," I said, "for such purposes a slight modicum of geometry and calculation would suffice. What we have to consider is whether the greater and more advanced part of it tends to facilitate the apprehension of the idea of good. That tendency, we affirm, is to be found in all studies that force the soul to turn its vision round to the region where dwells the most blessed part of reality, which it is imperative that it should behold."
"You are right," he said.
"Then if it compels the soul to contemplate essence, it is suitable; if genesis, it is not."
"So we affirm."