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The whole of arithmetic now appeared within the grasp of mechanism.
--
Passages from the Life of a Philosopher (1864), ch. 8 "Of the Analytical Engine"

 

Charles Babbage

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If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability. The abacus, with its beads strung on parallel wires, led the Arabs to positional numeration and the concept of zero many centuries before the rest of the world; and it was a useful tool— so useful that it still exists.

 

Vannevar Bush

 

In the early days of the computer revolution computer designers and numerical analysts worked closely together and indeed were often the same people. Now there is a regrettable tendency for numerical analysts to opt out of any responsibility for the design of the arithmetic facilities and a failure to influence the more basic features of software. It is often said that the use of computers for scientific work represents a small part of the market and numerical analysts have resigned themselves to accepting facilities "designed" for other purposes and making the best of them. [...] One of the main virtues of an electronic computer from the point of view of the numerical analyst is its ability to "do arithmetic fast." Need the arithmetic be so bad!

 

James H. Wilkinson

 

How can one be free of the images that one has? First of all, I must find out how these images come into being, what is the mechanism that creates them. You can see that at the moment of actual relationship, that is, when you are talking, when there are arguments, when there are insults and brutality, if you are not completely attentive at that moment, then the mechanism of building an image starts. That is, when the mind is not completely attentive at the moment of action, then the mechanism of building images is set in motion. When you say something to me which I do not like — or which I like — if at that moment I am not completely attentive, then the mechanism starts. If I am attentive, aware, then there is no building of images.

 

Jiddu Krishnamurti

 

A convincing demonstration of correctness being impossible as long as the mechanism is regarded as a black box, our only hope lies in not regarding the mechanism as a black box.

 

Edsger W. Dijkstra

 

Suppose I think, after doing my accounts, that I have a large balance at the bank. And suppose you want to find out whether this belief of mine is “wishful thinking.” You can never come to any conclusion by examining my psychological condition. Your only chance of finding out is to sit down and work through the sum yourself. When you have checked my figures, then, and then only, will you know whether I have that balance or not. If you find my arithmetic correct, then no amount of vapouring about my psychological condition can be anything but a waste of time. If you find my arithmetic wrong, then it may be relevant to explain psychologically how I came to be so bad at my arithmetic, and the doctrine of the concealed wish will become relevant—but only after you have yourself done the sum and discovered me to be wrong on purely arithmetical grounds. It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must first find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error.

 

C. S. Lewis