A Quote by Bertrand Russell
Formality Thus the absence of all mention of particular things or properties in logic or pure mathematics is a necessary result of the fact that this study is, as we say, "purely formal".
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Formal logic is mathematics, and there are philosophers like Wittgenstein that are very mathematical, but what they're really doing is mathematics - it's not talking about things that have affected computer science; it's mathematical logic.
In other words, the propositions of philosophy are not factual, but linguistic in character - that is, they do not describe the behaviour of physical, or even mental, objects; they express definitions, or the formal consequences of definitions. Accordingly we may say that philosophy is a department of logic. For we will see that the characteristic mark of a purely logical enquiry, is that it is concerned with the formal consequences of our definitions and not with questions of empirical fact.
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists of the analysis of Symbolic Logic itself.
[Mathematics is] purely intellectual, a pure theory of forms, which has for its objects not the combination of quantities or their images, the numbers, but things of thought to which there could correspond effective objects or relations, even though such a correspondence is not necessary.
It is difficult to remove by logic an idea not placed there by logic in the first place. By nature, we are emotional creatures. Often we live and react based on feelings, not logic. Feelings are wonderful, but when we become tied to a particular thought or belief we tend to ignore the fact that change might be necessary.
By a combination of formal training and self study, the latter continuing systematically well into the 1940s, I was able to gain a broad base of knowledge in economics and political science, together with reasonable skills in advanced mathematics, symbolic logic, and mathematical statistics.
...mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics. Every other science, even logic, especially in its early stages, is in danger of evaporating into airy nothingness, degenerating, as the Germans say, into an arachnoid film, spun from the stuff that dreams are made of. There is no such danger for pure mathematics; for that is precisely what mathematics ought to be.
One may say that mathematics talks about the things which are of no concern to men. Mathematics has the inhuman quality of starlight - brilliant, sharp but cold ... thus we are clearest where knowledge matters least: in mathematics, especially number theory.
Pure mathematics offers no mercenary inducements to its followers, who is attracted to it by the importance and beauty of the truths in contains; and the complete absence of any material advantage to be gained by means of it, adds perhaps another charm to its study.
The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. . . . Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
In [Aristotle's] formal logic, thought is organized in a manner very different from that of the Platonic dialogue. In this formal logic, thought is indifferent toward its objects. Whether they are mental or physical, whether they pertain to society or to nature, they become subject to the same general laws of organization, calculation, and conclusion - but they do so as fungible signs or symbols, in abstraction from their particular "substance." This general quality (quantitative quality) is the precondition of law and order - in logic as well as in society - the price of universal control.
If you want to be a physicist, you must do three things-first, study mathematics, second, study more mathematics, and third, do the same.
To treat programming scientifically, it must be possible to specify the required properties of programs precisely. Formality is certainly not an end in itself. The importance of formal specifications must ultimately rest in their utility -in whether or not they are used to improve the quality of software or to reduce the cost of producing and maintaining software.
Mathematics and logic have been proved to be one; a fact from which it seems to follow that mathematics may successfully deal with non-quantitative problems in a much broader sense than was suspected to be possible.
At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate.
Poincaré was a vigorous opponent of the theory that all mathematics can be rewritten in terms of the most elementary notions of classical logic; something more than logic, he believed, makes mathematics what it is.
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