Mathematical Clarity Quotes

We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.

A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.

I myself, a professional mathematician, on re-reading my own work find it strains my mental powers to recall to mind from the figures the meanings of the demonstrations, meanings which I myself originally put into the figures and the text from my mind. But when I attempt to remedy the obscurity of the material by putting in extra words, I see myself falling into the opposite fault of becoming chatty in something mathematical.

That sometimes clear ... and sometimes vague stuff ... which is ... mathematics.

That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.

Mathematics is so much easier than words mathematics makes things clear that words merely muddle and confuse and mess up.

The confusion, the difficulties, the contradictions which, in consequence of a want of accurate distinctions in this particular, have up to even a recent period encumbered mathematics in all those branches involving the consideration of negative and impossible quantities, will at once occur to the reader who is at all versed in this science, and would alone suffice to justify dwelling somewhat on the point, in connexion with any subject so peculiarly fitted to give forcible illustration of it as the Analytical Engine.

The effort of the economist is to "see," to picture the interplay of economic elements. The more clearly cut these elements appear in his vision, the better; the more elements he can grasp and hold in his mind at once, the better. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looms up in firm, bold outlines. The old phantasmagoria disappear. We see better. We also see further.

In math, you're either right or you're wrong.

[In the Royal Society, there] has been, a constant Resolution, to reject all the amplifications, digressions, and swellings of style: to return back to the primitive purity, and shortness, when men deliver'd so many things, almost in an equal number of words. They have exacted from all their members, a close, naked, natural way of speaking; positive expressions; clear senses; a native easiness: bringing all things as near the Mathematical plainness, as they can: and preferring the language of Artizans, Countrymen, and Merchants, before that, of Wits, or Scholars.

How thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which, at the same time, assist in understanding earlier theories and in casting aside some more complicated developments.

Words are a pretty fuzzy substitute for mathematical equations.

I would advise you, Sir, to study algebra, if you are not an adept already in it: your head would get less muddy, and you will leave off tormenting your neighbours about paper and packthread, while we all live together in a world that is bursting with sin and sorrow.

Here I beg you to observe in passing that the scruples that prevented ancient writers from using arithmetical terms in geometry, and which can only be a consequence of their inability to perceive clearly the relation between these two subjects, introduced much obscurity and confusion into their explanations.

Cam was so much clearer, easier to figure out. Like he was algebra and Daniel was calculus.

Mathematicians themselves set up standards of generality and elegance in their exposition which are a bar to understand.