Styles of reasoning
At the end of the nineteenth century, Charles Sanders Pierce, a founder of the American school of pragmatist philosophy, distinguished three broad styles of reasoning.
Deductive reasoning reaches logical conclusions from stated premises. For example, ‘Evangelical Christians are Republican. Republicans voted for Donald Trump. Evangelical Christians voted for Donald Trump.’ This syllogism is descriptive of a small world. As soon as one adds the word ‘most’ before either evangelical Christians or Republicans, the introduction of the inevitable vagueness of the larger world modifies the conclusion.
Inductive reasoning is of the form ‘analysis of election results shows that they normally favour incumbent parties in favourable economic circumstances and opposition parties in adverse economic circumstances’. Since economic conditions in the United States in 2016 were neither particularly favourable nor unfavourable, we might reasonably have anticipated a close result. Inductive reasoning seeks to generalise from observations, and may be supported or refuted by subsequent experience.
Abductive reasoning seeks to provide the best explanation of a unique event. For example, an abductive approach might assert that Donald Trump won the 2016 presidential election because of concerns in particular swing states over economic conditions and identity, and because his opponent was widely disliked.
Deductive, inductive and abductive reasoning each have a role to play in understanding the world, and as we move to larger worlds the role of the inductive and abductive increases relative to the deductive. And when events are essentially one-of-a-kind, which is often the case in the world of radical uncertainty, abductive reasoning is indispensable. Although the term ‘abductive reasoning’ may be unfamiliar, we constantly reason in this way, searching for the best explanation of what we see: ‘I think the bus is late because of congestion in Oxford Street’. But the methods of decision analysis we have described in earlier chapters are derived almost entirely from the deductive reasoning which is relevant only in small worlds. (Kay, John. Radical Uncertainty: Decision-Making Beyond the Numbers (pp. 137-138). W. W. Norton & Company. Kindle Edition.)
(….) Most problems we confront in life are typically not well defined and do not have single analytic solutions.
(….) But logic derived from reasonably maintained premises can only ever take us so far. Under radical uncertainty, the premises from which we reason will never represent a complete description of the world. There will be different actions which might properly be described as ‘rational’ given any particular set of beliefs about the world. As soon as any element of subjectivity is attached either to the probabilities or to the valuation of the outcomes, problems cease to have any objectively correct solution.(Kay, John. Radical Uncertainty: Decision-Making Beyond the Numbers (pp. 137-139). W. W. Norton & Company. Kindle Edition.)
DIFFERENT WAYS OF USING THE MIND
Mathematics has something to teach us, all of us, whether or not we like mathematics or use it very much. This lesson has to do with thinking, the way we use our minds to draw conclusions about the world around us. When most people think about mathematics they think about the logic of mathematics. They think that mathematics is characterized by a certain mode of using the mind, a mode I shall henceforth refer to as “algorithmic.” By this I mean a step-by-step, rule-based procedure for going from old truths to new ones through a process of logical reasoning. But is this really the only way that we think in mathematics? Is this the way that new mathematical truths are brought into being? Most people are not aware that there are, in fact, other ways of using the mind that are at play in mathematics. After all, where do the new ideas come from? Do they come from logic or from algorithmic processes? In mathematical research, logic is used in a most complex way, as a constraint on what is possible, as a goad to creativity, or as a kind of verification device, a way of checking whether some conjecture is valid. Nevertheless, the creativity of mathematics—the turning on of the light switch—cannot be reduced to its logical structure. (Byers, William. How Mathematicians Think (p. 5). Princeton University Press. Kindle Edition.)
Where does mathematical creativity come from? This book will point toward a certain kind of situation that produces creative insights. This situation, which I call “ambiguity,” also provides a mechanism for acts of creativity. The “ambiguous” could be contrasted to the “deductive,” yet the two are not mutually exclusive. Strictly speaking, the “logical” should be contrasted to the “intuitive.” The ambiguous situation may contain elements of the logical and the intuitive, but it is not restricted to such elements. An ambiguous situation may even involve the contradictory, but it would be wrong to say that the ambiguous is necessarily illogical.(Byers, William. How Mathematicians Think (pp. 5-6). Princeton University Press. Kindle Edition.)
Science has always had (…) a metaphoric function — that is, it generates an important part of culture’s symbolic vocabulary and provides some of the metaphysical bases and philosophical orientations of our ideology. As a consequence the methods of argument of science, its conceptions and its models, have permeated first the intellectual life of the time, then the tenets and usages of everyday life. All philosophies share with science the need to work with concepts such as space, time, quantity, matter, order, law, causality, verification, reality. (Holton 2000, 43, in Einstein, History and Other Passions)
Our discussion of the nature of physical concepts has shown that a main reason for formulating concepts is to use them in connection with mathematically stated laws. It is tempting to go one step further and to demand that practicing scientists deal only with ideas corresponding to strict measurables, that they formulate only concepts reducible to the least ambiguous of all data: numbers and measurements. The history of science would indeed furnish examples to show the great advances that followed from the formation of strictly quantitative concepts. (Holton and Brush 2001, 170)
(….) The nineteenth-century physicist Lord Kelvin commended this attitude in the famous statement:
I often say that when you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of Science, whatever the matter may be. (“Electrical Units of Measurement”)
Useful though this trend is within its limits [emphasis added], there is an entirely different aspect to scientific concepts: indeed it is probable that science would stop if every scientist were to avoid anything other than strictly quantitative concepts. We shall find that a position like Lord Kelvin’s (which is similar to that held at present by some thinkers in the social sciences) does justice neither to the complexity and fertility of the human mind nor to the needs of contemporary physical science itselfnot to scientists nor to science. Quite apart from the practical impossibility of demanding of one’s mind that at all times it identify such concepts as electron only with the measurable aspects of that construct, there are specifically two main objections: First, this position misunderstands how scientists as individuals do their work, and second, it misunderstands how science as a system grows out of the contribution of individuals. (Holton and Brush 2001, 170-171)
(….) While a scientist struggles with a problem, there can be little conscious limitation on his free and at times audacious constructions. Depending on his field, his problem, his training, and his temperament, he may allow himself to be guided by a logical sequence based on more or less provisional hypotheses, or equally likely by “feelings for things,” by likely analogy, by some promising guess, or he may follow a judicious trial-and-error procedure.
The well-planned experiment is, of course, by far the most frequent one in modern science and generally has the best chance of success; but some men and women in science have often not even mapped out a tentative plan of attack on the problems, but have instead let their enthusiasms, their hunches, and their sheer joy of discovery suggest the line of work. Sometimes, therefore, the discovery of a new effect or tool or technique is followed by a period of trying out one or the other applications in a manner that superficially almost seems playful. Even the philosophical orientation of scientists is far less rigidly prescribed than might be supposed. (Holton and Brush 2001, 170-171)