Formal Systems Quotes

Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.

Relying on words to lead you to the truth is like relying on an incomplete formal system to lead you to the truth. A formal system will give you some truths, but as we shall soon see, a formal system, no matter how powerful cannot lead to all truths.

Well it was not exactly a dissertation in logic, at least not the kind of logic you would find in Whitehead and Russell's Principia Mathematica for instance. It looked more like mathematics; no formalized language was used.

When I dealt with set theory, I could never make it be the music that I wanted.

What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else.

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.

One can't proceed from the informal to the formal by formal means.

Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long the relations don't change. Matter is not important, only form interests them.

All those formal systems, in mathematics and physics and the philosophy of science, which claim to give foundations for certain truth are surely mistaken. I am tempted to say that we do not look for truth, but for knowledge. But I dislike this form of words, for two reasons. First of all, we do look for truth, however we define it, it is what we find that is knowledge. And second, what we fail to find is not truth, but certainty; the nature of truth is exactly the knowledge that we do find.

Logic and mathematics are nothing but specialised linguistic structures.

Kurt Godel's achievement in modern logic is singular and monumental - indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Godel's achievement.

Mathematics deals exclusively with the relations of concepts to each other without consideration of their relation to experience.

Certain characteristics of the subject are clear. To begin with, we do not in this subject deal with particular things or particular properties: we deal formally with what can be said about any thing or any property. We are prepared to say that one and one are two, but not that Socrates and Plato are two.

Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then suchand such another proposition is true of that thing.... Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

An axiomatic system comprises axioms and theorems and requires a certain amount of hand-eye coordination before it works. A formal system comprises an explicit list of symbols, an explicit set of rules governing their cohabitation, an explicit list of axioms, and, above all, an explicit list of rules explicitly governing the steps that the mathematician may take in going from assumptions to conclusions. No appeal to meaning nor to intuition. Symbols lose their referential powers; inferences become mechanical.

Only mathematics and mathematical logic can say as little as the physicist means to say.

Axiomatization is what one does last, it's rubbish. It's the hygiene of mathematics, axiomatization.

But it is certainly not possible to insist on one hand that the formalism is complete and to insist on the other hand that its application to 'the actual' actually demands a step which cannot be derived from it.

Every mathematical discipline goes through three periods of development: the naive, the formal, and the critical.

There is an inherent dissonancebetween the quasi-formal world of computer programs - defining the programmed machine in each system - and the non-formal problem world of the system requirements.