"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1 (1), 1980)
"Perhaps the best way to approach the question of what mathematics is, is to start at the beginning. In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics." (Richard Hamming, "The Unreasonable Effectiveness of Mathematics", The American Mathematical Monthly Vol. 87" (2), 1980)
"The advantage of semantic networks over standard logic is that some selected set of the possible inferences can be made in a specialized and efficient way. If these correspond to the inferences that people make naturally, then the system will be able to do a more natural sort of reasoning than can be easily achieved using formal logical deduction." (Avron Barr, Natural Language Understanding, AI Magazine Vol. 1" (1), 1980)
"The invalid assumption that correlation implies cause is probably among the two or three most serious and common errors of human reasoning." (Stephen J Gould, "The Mismeasure of Man", 1980)
"A person who thinks by images becomes less and less capable of thinking by reasoning, and vice versa. The intellectual process based on images is contradictory to the intellectual process of reasoning that is related to the word. There are two different ways of dealing with an object. They involve not only different approaches, but even more important, opposing mental attitudes. This is not a matter of complementary processes, such as analysis and synthesis or logic and dialectic. These processes lack any qualitative common denominator." (Jacques Ellul, "The Humiliation of the Word", 1981)
"All advances of scientific understanding, at every level, begin with a speculative adventure, an imaginative preconception of what might be true - a preconception that always, and necessarily, goes a little way" (sometimes a long way) beyond anything which we have logical or factual authority to believe in. It is the invention of a possible world, or of a tiny fraction of that world. The conjecture is then exposed to criticism to find out whether or not that imagined world is anything like the real one. Scientific reasoning is therefore at all levels an interaction between two episodes of thought - a dialogue between two voices, the one imaginative and the other critical; a dialogue, as I have put it, between the possible and the actual, between proposal and disposal, conjecture and criticism, between what might be true and what is in fact the case." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)
"Scientific theories" (I have said) begin as imaginative constructions. The begin, if you like, as stories, and the purpose of the critical or rectifying episode in scientific reasoning is precisely to find out whether or not these stories are stories about real life. Literal or empiric truthfulness is not therefore the starting-point of scientific enquiry, but rather the direction in which scientific reasoning moves. If this is a fair statement, it follows that scientific and poetic or imaginative accounts of the world are not distinguishable in their origins. They start in parallel, but diverge from one another at some later stge. We all tell stories, but the stories differ in the purposes we expect them to fulfil and in the kinds of evaluations to which they are exposed." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)
"A mental model is a collection of 'connected' autonomous objects. Running a mental model corresponds to modifying the parameters of the model by propagating information using the internal rules and specified topology. Running a mental model can also occur when autonomous objects change state. For us the definition of state is distinct from the current parameter values of an object. A state change consists of the replacement of one set of behavior rules with another." (Michael D Williams et al, "Human Reasoning About a Simple Physical System", [in "Mental Models", Ed(s). Dedre Gentner & Albert L Stevens], 1983)
"Since mental models can take many forms and serve many purposes, their contents are very varied. They can contain nothing but tokens that represent individuals and identities between them, as in the sorts of models that are required for syllogistic reasoning. They can represent spatial relations between entities, and the temporal or causal relations between events. A rich imaginary model of the world can be used to compute the projective relations required for an image. Models have a content and form that fits them to their purpose, whether it be to explain, to predict, or to control." (Philip Johnson-Laird, "Mental models: Toward a cognitive science of language, inference, and consciousness", 1983)
“[…] mathematics is not just a symbolism, a set of conventions for the use of special, formal vocabularies, but is intimately connected with the structure of rational thought, with reasoning practices. [...] mathematics is not just a language, and of refusing the foundationalist move of trying to reduce mathematics to logic, instead seeing mathematics as providing rational frameworks for science, is to set science against a background of rational structures and rational methods which itself has a built-in dynamics. The rational framework of science is itself historically conditioned, for it changes with developments in mathematics." (Mary Tiles,"Bachelard: Science and Objectivity", 1984)
"Concepts are inventions of the human mind used to construct a model of the world. They package reality into discrete units for further processing, they support powerful mechanisms for doing logic, and they are indispensable for precise, extended chains of reasoning. […] A mental model is a cognitive construct that describes a person's understanding of a particular content domain in the world." (John Sown,"Conceptual Structures: Information Processing in Mind and Machine", 1984)
"Mathematical rigor is the clarification of the reasoning used in mathematics. Usually, mathematics first arises in some particular situation, and as the demand for rigor becomes apparent more careful definitions of what is being reasoned about are required, and a closer examination of the numerous 'hidden assumptions' is made." (Richard W Hamming, "Methods of Mathematics Applied to Calculus, Probability, and Statistics", 1985)
"For generations, scientists and philosophers have tried to explain ordinary reasoning in terms of logical principles - with virtually no success. I suspect this enterprise failed because it was looking in the wrong direction: common sense works so well not because it is an approximation of logic; logic is only a small part of our great accumulation of different, useful ways to chain things together." (Marvin Minsky, "The Society of Mind", 1987)
"One of the features that distinguishes applied mathematics is its interest in framing important questions about the observed world in a mathematical way. This process of translation into a mathematical form can give a better handle for certain problems than would be otherwise possible. We call this the modeling process. It combines formal reasoning with intuitive insights. Understanding the models devised by others is a first step in learning some of the skills involved, and that is how we proceed in this text, which is an informal introduction to the mathematics of dynamical systems." (Edward Beltrami,"Mathematics for Dynamic Modeling", 1987)"
" A mental model is a data structure, in a computational system, that represents a part of the real world or of a fictitious world. It is assumed that there can be mental models of abstract realms, such as that of mathematics, but little more will be said about them. A model-theoretic semanticist is free to think of the entities in his model as actual items in the world.[...] Mental model is an appropriate term for the mental representations that underlie everyday reasoning about the world. To understand the everyday world is to have a theory of how it works." (Alan Granham, "Mental Models as Representations of Discourse and Text", 1987)
"One of the features that distinguishes applied mathematics is its interest in framing important questions about the observed world in a mathematical way. This process of translation into a mathematical form can give a better handle for certain problems than would be otherwise possible. We call this the modeling process. It combines formal reasoning with intuitive insights. Understanding the models devised by others is a first step in learning some of the skills involved, and that is how we proceed in this text, which is an informal introduction to the mathematics of dynamical systems." (Edward Beltrami, "Mathematics for Dynamic Modeling", 1987)
"Despite the prevailing use of graphs as metaphors for communicating and reasoning about dependencies, the task of capturing informational dependencies by graphs is not at all trivial." (Judea Pearl, "Probabilistic Reasoning in Intelligent Systems: Network of Plausible, Inference", 1988)
“Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences." (Sir Erik C Zeeman,"Private Games", 1988)
"Formal logic and the logical syllogism encapsulate connectedness in reasoning." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)
"However, mathematics is not and cannot be anything more than a tool, and all my work rests on the conviction that, in its use, the only two really fruitful stages in the scientific approach are, firstly, a thorough examination of the initial hypotheses; and secondly, a discussion of the meaning and empirical relevance of the results obtained. What remains is but tautological calculation, which is of interest only to the mathematician, and the mathematical rigour of the reasoning can never justify a theory based on postulates if these postulates do not correspond to the true nature of the observed phenomena." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)
"Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences." (Sir Erik C Zeeman, "Private Games", 1988)
"The use of even the most sophisticated forms of mathematics can never be considered as a guarantee of quality. Mathematics is, and can only be, a means of expression and reasoning. The real substance on which the economist works remains economic and social. Indeed, one must avoid the development of a complex mathematical apparatus whenever it is not strictly indispensable. Genuine progress never consists in a purely formal exposition, but always in the discovery of the guiding ideas which underlie any proof. It is these basic ideas which must be explicitly stated and discussed." (Maurice Allais, "An Outline of My Main Contributions to Economic Science", [Noble lecture] 1988)
No comments:
Post a Comment