William Rowan Hamilton Quotes

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

When I have needed solace and I have had to depend on my own resources, the mathematics has been there. I am grateful.

In their field they [mathematicians] do what we ought to be doing in ours. Therein lies the significant lesson … of their existence. They are an analogy for the intellectual of the future.

Our ignorance of this beautiful subject -- a tree of ideas with ancient roots and modern fruit -- is profound and beset with fear, superstition, and misinformation.

Nearly always it is the recondite and complicated which is elaborated first; and it is only when some relatively unsophisticated mind attacks a problem that its deep simplicity is revealed.

There is a wide distinction between the degree of mathematical acquirement necessary for making discoveries, and that which is requisite for understanding what other have done.

Mathematics ought properly to be a model of logical clarity. In actual fact there are perhaps no scientific works where you will find more wrong expressions, and consequently wrong thoughts, than in mathematical ones.

Mathematics is principally a tool to meditate, rather than to compute.

What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. ...[P]eople know that it deals with numbers and figures, with patterns, relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes. ... that it purports to form the foundations of all rational thought. ...The aesthetic side of mathematics has been of overwhelming importance throughout its growth. It is not so much whether a theorem is useful that matters, but how elegant it is. ...One can ...look conversely at ...the homely side of mathematics ...having to be punctilious ...having to make sure of every step. ...[O]ne cannot stop at drawing with a big, wide brush; all the details have to be filled in ...Mathematicians ...fool themselves ...when they think their main business is to prove theorems without at least indicating why they may be important. If left entirely to the aesthetic criteria, doesn't it compound the mystery? ...[I]n the decades to come there will be more understanding ...of the degree of beauty, though ...the criteria may have shifted ...[to] a super beauty in unanalyzable higher levels. ...It has to appeal to connections with other theories of the external world or to the history of the development of the human brain, or else it is purely aesthetic and very subjective in the sense that music is. ...[E]ven the quality of music will be analyzable ...by mathematizing the idea of analogy.

I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.

There is no mathematical substitute for philosophy.

If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.

...[A]ll the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced.

You don’t need anybody’s permission to be a great mathematician!

Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation. In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means.