On Assertions

"Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning. Experience sees that the assertions are true, but she sees not how profound and absolute is their truth. She unhesitatingly assents to the laws which geometry delivers, but she does not pretend to see the origin of their obligation. She is always ready to acknowledge the sway of pure scientific principles as a matter of fact, but she does not dream of offering her opinion on their authority as a matter of right; still less can she justly claim to herself the source of that authority." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"It would be a mistake to suppose that a science consists entirely of strictly proved theses, and it would be unjust to require this. […] Science has only a few apodeictic propositions in its catechism: the rest are assertions promoted by it to some particular degree of probability. It is actually a sign of a scientific mode of thought to find satisfaction in these approximations to certainty and to be able to pursue constructive work further in spite of the absence of final confirmation." (Sigmund Freud, "Introductory Lectures on Psycho-Analysis", 1916)

"Mathematicians do not know what they are talking about because pure mathematics is not concerned with physical meaning. Mathematicians never know whether what they are saying is true because, as pure mathematicians, they make no effort to ascertain whether their theorems are true assertions about the physical world." (Morris Kline, “Mathematics in Western Culture”, 1953)

"For foundations it is important to know what we are talking about; we make the subject as specific as possible. In this way we have a chance to make strong assertions. For practice, to make a proof intelligible, we want to eliminate all properties which are not relevant to the result proved, in other words, we make the subject matter less specific." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"Most people imagine mathematics to be a deductive science in which all theorems, results and facts are obtained via logical reasoning by proceeding from certain starting axioms, primal assertions, assumed to be self-evident or not requiring any proof." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure. […]  a person might use his cognitive map to derive explanations of the past, make predictions for the future, and choose policies in the present." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"The concepts a person uses are represented as points, and the causal links between these concepts are represented as arrows between these points. This gives a pictorial representation of the causal assertions of a person as a graph of points and arrows. This kind of representation of assertions as a graph will be called a cognitive map. The policy alternatives, all of the various causes and effects, the goals, and the ultimate utility of the decision maker can all be thought of as concept variables, and represented as points in the cognitive map. The real power of this approach ap pears when a cognitive map is pictured in graph form; it is then relatively easy to see how each of the concepts and causal relation ships relate to each other, and to see the overall structure of the whole set of portrayed assertions." (Robert Axelrod, "The Cognitive Mapping Approach to Decision Making" [in "Structure of Decision: The Cognitive Maps of Political Elites"], 1976)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements."  (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

"Because mathematical proofs are long, they are also difficult to invent. One has to construct, without making any mistakes, long chains of assertions, and see what one is doing, see where one is going. To see means to be able to guess what is true and what is false, what is useful and what is not. To see means to have a feeling for which definitions one should introduce, and what the key assertions are that will allow one to develop a theory in a natural manner." (David Ruelle, "Chance and Chaos", 1991)

"Virtually all mathematical theorems are assertions about the existence or nonexistence of certain entities. For example, theorems assert the existence of a solution to a differential equation, a root of a polynomial, or the nonexistence of an algorithm for the Halting Problem. A platonist is one who believes that these objects enjoy a real existence in some mystical realm beyond space and time. To such a person, a mathematician is like an explorer who discovers already existing things. On the other hand, a formalist is one who feels we construct these objects by our rules of logical inference, and that until we actually produce a chain of reasoning leading to one of these objects they have no meaningful existence, at all." (John L Casti, "Reality Rules: Picturing the world in mathematics" Vol. II, 1992)

"Sometimes in math the best way to make progress is to introduce simplifications [...]. The simplifications are not assertions about the outside world: they are ways to restrict the domain of discourse, to keep it manageable." (Ian Stewart, "Letters to a Young Mathematician", 2006)

"A theory is an organizational form of scientific knowledge about a certain set of objects, representing a system of interconnected assertions and proofs and containing methods of explanation and prediction of phenomena and processes in a given problem domain, i.e., of all phenomena and processes described by this theory." (Dmitry A Novikov, "Cybernetics: From Past to Future", 2016)