On Plausibility V

"Proof is an idol before whom the pure mathematician tortures himself. In physics we are generally content to sacrifice before the lesser shrine of Plausibility." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)

"[Science fiction is] that class of prose narrative treating of a situation that could not arise in the world we know, but which is hypothesised on the basis of some innovation in science or technology, or pseudo-science or pseudo-technology, whether human or extra-terrestrial in origin. It is distinguished from pure fantasy by its need to achieve verisimilitude and win the 'willing suspension of disbelief' through scientific plausibility." (Kingsley Amis, "New Maps of Hell", 1960)

"[…] the social scientist who lacks a mathematical mind and regards a mathematical formula as a magic recipe, rather than as the formulation of a supposition, does not hold forth much promise. A mathematical formula is never more than a precise statement. It must not be made into a Procrustean bed - and that is what one is driven to by the desire to quantify at any cost. It is utterly implausible that a mathematical formula should make the future known to us, and those who think it can, would once have believed in witchcraft. The chief merit of mathematicization is that it compels us to become conscious of what we are assuming." (Bertrand de Jouvenel, "The Art of Conjecture", 1967)

"Mathematical knowledge is fixed securely by means of demonstrative reasoning, but the approaches to such knowledge are strewn with plausible modes of reasoning." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"The degree of confirmation assigned to any given hypothesis is sensitive to properties of the entire belief system [...] simplicity, plausibility, and conservatism are properties that theories have in virtue of their relation to the whole structure of scientific beliefs taken collectively. A measure of conservatism or simplicity would be a metric over global properties of belief systems." (Jerry Fodor, "Modularity of Mind", 1983)

"Therefore, mathematical ecology does not deal directly with natural objects. Instead, it deals with the mathematical objects and operations we offer as analogs of nature and natural processes. These mathematical models do not contain all information about nature that we may know, but only what we think are the most pertinent for the problem at hand. In mathematical modeling, we have abstracted nature into simpler form so that we have some chance of understanding it. Mathematical ecology helps us understand the logic of our thinking about nature to help us avoid making plausible arguments that may not be true or only true under certain restrictions. It helps us avoid wishful thinking about how we would like nature to be in favor of rigorous thinking about how nature might actually work. (John Pastor, "Mathematical Ecology of Populations and Ecosystems", 2008)

"Since we cannot completely eliminate uncertainty, we need to model it. In real life when we are faced with uncertainty, we use plausible reasoning. We adjust our belief about something, based on the occurrence or nonoccurrence of something else." (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3^rd Ed., 2017)

On Plausibility IV

"Devising the plan of the solution, we should not be too afraid of merely plausible, heuristic reasoning. Anything is right that leads to the right idea. But we have to change this standpoint when we start carrying out the plan and then we should accept only conclusive, strict arguments." (George Pólya, "How to solve it", 1945)

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to solve it", 1945)

"From the outset it was clear that the two kinds of reasoning have different tasks. From the outset. they appeared very different: demonstrative reasoning as definite, final, 'machinelike'; and plausible reasoning as vague, provisional, specifically 'human'. Now we may see the difference a little more distinctly. In opposition to demonstrative inference, plausible inference leaves indeterminate a highly relevant point: the 'strength' or the 'weight' of the conclusion. This weight may depend not only on clarified grounds such as those expressed in the premises, hut also on unclarified unexpressed grounds somewhere on the background of the person who draws the conclusion. A person has a background, a machine has not. Indeed, you can build a machine to draw demonstrative conclusions for you, but I think you can never build a machine that will draw plausible inferences." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"We secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning. A mathematical proof is demonstrative reasoning, but the inductive evidence of the physicist, the circumstantial evidence of the lawyer, the documentary evidence of the historian, and the statistical evidence of the economist belong to plausible reasoning." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"On the other hand, the 'subjective' school of thought, regards probabilities as expressions of human ignorance; the probability of an event is merely a formal expression of our expectation that the event will or did occur, based on whatever information is available. To the subjectivist, the purpose of probability theory is to help us in forming plausible conclusions in cases where there is not enough information available to lead to certain conclusions; thus detailed verification is not expected. The test of a good subjective probability distribution is does it correctly represent our state of knowledge as to the value of x?" (Edwin T Jaynes, "Information Theory and Statistical Mechanics" I, 1956)

"It is widely recognized that the word 'probability' has two very different main senses. In its original meaning, which is still the popular meaning, the word is roughly synonymous with plausibility. It has reference to reasonableness of belief or expectation. If 'logic' is interpreted in a broad sense, then this kind of probability belongs to logic. In its other meaning, which is that usually attributed to it by statisticians, the word has reference to a type of physical phenomena, known as random or chance phenomena. If 'physics' is interpreted in a broad sense, then this kind of probability belongs to physics. Physical probabilities can be determined empirically by noting the proportion of successes in some trials. (The determination is inexact and unsure, like all other physical determinations.)" (Francis J Anscombe & Robert J Aumann, "A Definition of Subjective Probability", The Annals of Mathematical Statistics Vol. 34 (1), 1963)

"Probability theory, for us, is not so much a part of mathematics as a part of logic, inductive logic, really. It provides a consistent framework for reasoning about statements whose correctness or incorrectness cannot be deduced from the hypothesis. The information available is sufficient only to make the inferences 'plausible' to a greater or lesser extent." (Ralph Baierlein, "Atoms and Information Theory: An Introduction to Statistical Mechanics", 1971)

"By common consensus in the mathematical world, a good proof displays three essential characteristics: a good proof is (1) convincing, (2) surveyable, and (3) formalizable. The first requirement means simply that most mathematicians believe it when they see it. […] Most mathematicians and philosophers of mathematics demand more than mere plausibility, or even belief. A proof must be able to be understood, studied, communicated, and verified by rational analysis. In short, it must be surveyable. Finally, formalizability means we can always find a suitable formal system in which an informal proof can be embedded and fleshed out into a formal proof." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Given any collection of infinite sets the Axiom of Choice tells us that there exists a set which has one element in common with each of the sets in the collection. Choice, which seems to be an intuitively sound principle, is equivalent to the much less plausible statement that every set has a well-ordering. Although many tried to prove Choice, they only seemed to be able to find equivalent statements which were just as difficult to prove." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Objections to the Axiom of Choice, either the strong or the weak version, are typically either philosophical, based on the intuitive temporal implausibility of making an infinite number of choices, or on the non-constructive nature of the axiom, or are based on a peculiar identification of continuum-based models of physics with the physical objects being modelled; properties of the model which are implied by the Axiom of Choice are deemed to be counterintuitive because the physical objects they model don’t have these properties. Motivated by these objections, or just for curiosity, several alternatives to Choice have been explored." (Barnaby Sheppard, "The Logic of Infinity", 2014)

On Plausibility III

"You have to guess the mathematical theorem before you prove it: you have to guess the idea of the proof before you carry through the details. You have to combine observations and follow analogies: you have to try and try again. The result of the mathematician’s creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing" (George Polya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"The degree of confirmation assigned to any given hypothesis is sensitive to properties of the entire belief system [...] simplicity, plausibility, and conservatism are properties that theories have in virtue of their relation to the whole structure of scientific beliefs taken collectively. A measure of conservatism or simplicity would be a metric over global properties of belief systems." (Jerry Fodor, "Modularity of Mind", 1983)

"The point we want to make here is that sensemaking is about plausibility, coherence, and reasonableness. Sensemaking is about accounts that are socially acceptable and credible... It would be nice if these accounts were also accurate. But in an equivocal, postmodern world, infused with the politics of interpretation and conflicting interests and inhabited by people with multiple shifting identities, an obsession with accuracy seems fruitless, and not of much practical help, either." (Karl E Weick, "Sensemaking in Organizations", 1995)

"Nonetheless, the basic principles regarding correlations between variables are not that difficult to understand. We must look for patterns that reveal potential relationships and for evidence that variables are actually related. But when we do spot those relationships, we should not jump to conclusions about causality. Instead, we need to weigh the strength of the relationship and the plausibility of our theory, and we must always try to discount the possibility of spuriousness." (Joel Best, "More Damned Lies and Statistics: How numbers confuse public issues", 2004)

"Reality dishes out experiences using probability, not plausibility." (Eliezer S Yudkowsky, "A Technical Explanation of Technical Explanation", 2005)

"Facts and values are entangled in science. It's not because scientists are biased, not because they are partial or influenced by other kinds of interests, but because of a commitment to reason, consistency, coherence, plausibility and replicability. These are value commitments." (Alva Noë)

On Plausibility II

"From the outset it was clear that the two kinds of reasoning have different tasks. From the outset. they appeared very different: demonstrative reasoning as definite, final, 'machinelike'; and plausible reasoning as vague, provisional, specifically 'human'. Now we may see the difference a little more distinctly. In opposition to demonstrative inference, plausible inference leaves indeterminate a highly relevant point: the 'strength' or the 'weight' of the conclusion. This weight may depend not only on clarified grounds such as those expressed in the premises, hut also on unclarified unexpressed grounds somewhere on the background of the person who draws the conclusion. A person has a background, a machine has not. Indeed, you can build a machine to draw demonstrative conclusions for you, but I think you can never build a machine that will draw plausible inferences." (George Pólya, "Mathematics and Plausible Reasoning", 1954)

"One feature [...] which requires much more justification than is usually given, is the setting up of unplausible null hypotheses. For example, a statistician may set out a test to see whether two drugs have exactly the same effect, or whether a regression line is exactly straight. These hypotheses can scarcely be taken literally." (Cedric A B Smith, "Book review of Norman T. J. Bailey: Statistical Methods in Biology", Applied Statistics 9, 1960)

"[…] the social scientist who lacks a mathematical mind and regards a mathematical formula as a magic recipe, rather than as the formulation of a supposition, does not hold forth much promise. A mathematical formula is never more than a precise statement. It must not be made into a Procrustean bed - and that is what one is driven to by the desire to quantify at any cost. It is utterly implausible that a mathematical formula should make the future known to us, and those who think it can, would once have believed in witchcraft. The chief merit of mathematicization is that it compels us to become conscious of what we are assuming." (Bertrand de Jouvenel, "The Art of Conjecture", 1967)

"Demonstrative reasoning differs from plausible reasoning just as the fact differs from the supposition, just as actual existence differs from the possibility of existence. Demonstrative reasoning is reliable, incontrovertible and final. Plausible reasoning is conditional, arguable and oft-times risky." (Yakov Khurgin, "Did You Say Mathematics?", 1974)

"Philosophical objections may be raised by the logical implications of building a mathematical structure on the premise of fuzziness, since it seems (at least superficially) necessary to require that an object be or not be an element of a given set. From an aesthetic viewpoint, this may be the most satisfactory state of affairs, but to the extent that mathematical structures are used to model physical actualities, it is often an unrealistic requirement. [...] Fuzzy sets have an intuitively plausible philosophical basis. Once this is accepted, analytical and practical considerations concerning fuzzy sets are in most respects quite orthodox." (James Bezdek, 1981)

"In all scientific fields, theory is frequently more important than experimental data. Scientists are generally reluctant to accept the existence of a phenomenon when they do not know how to explain it. On the other hand, they will often accept a theory that is especially plausible before there exists any data to support it." (Richard Morris, 1983)

"The systems' basic components are treated as sets of rules. The systems rely on three key mechanisms: parallelism, competition, and recombination. Parallelism permits the system to use individual rules as building blocks, activating sets of rules to describe and act upon the changing situations. Competition allows the system to marshal its rules as the situation demands, providing flexibility and transfer of experience. This is vital in realistic environments, where the agent receives a torrent of information, most of it irrelevant to current decisions. The procedures for adaptation - credit assignment and rule discovery - extract useful, repeatable events from this torrent, incorporating them as new building blocks. Recombination plays a key role in the discovery process, generating plausible new rules from parts of tested rules. It implements the heuristic that building blocks useful in the past will prove useful in new, similar contexts." (John H Holland, "Complex Adaptive Systems", Daedalus Vol. 121 (1), 1992)

"Therefore, mathematical ecology does not deal directly with natural objects. Instead, it deals with the mathematical objects and operations we offer as analogs of nature and natural processes. These mathematical models do not contain all information about nature that we may know, but only what we think are the most pertinent for the problem at hand. In mathematical modeling, we have abstracted nature into simpler form so that we have some chance of understanding it. Mathematical ecology helps us understand the logic of our thinking about nature to help us avoid making plausible arguments that may not be true or only true under certain restrictions. It helps us avoid wishful thinking about how we would like nature to be in favor of rigorous thinking about how nature might actually work." (John Pastor, "Mathematical Ecology of Populations and Ecosystems", 2008)

"Don’t just do the calculations. Use common sense to see whether you are answering the correct question, the assumptions are reasonable, and the results are plausible. If a statistical argument doesn’t make sense, think about it carefully - you may discover that the argument is nonsense." (Gary Smith, "Standard Deviations", 2014)

"The fundamental problem with MRA, as with all correlational methods, is self-selection. The investigator doesn’t choose the value for the independent variable for each subject (or case). This means that any number of variables correlated with the independent variable of interest have been dragged along with it. In most cases, we will fail to identify all these variables. In the case of behavioral research, it’s normally certain that we can’t be confident that we’ve identified all the plausibly relevant variables." (Richard E Nisbett, "Mindware: Tools for Smart Thinking", 2015)

On Plausibility I

"Time is that wherein there is opportunity, and opportunity is that wherein there is no great time. Healing is a matter of time, but it is sometimes also a matter of opportunity. However, knowing this, one must attend to medical practice not primarily to plausible theories, but to experience combined with reason. For a theory is a composite memory of things apprehended with sense perception." (Hippocrates of Kos, "Precepts", cca. 4th century)

"Reasoning from analogy is often most plausible and most deceptive." (Charles Simmons, "A Laconic Manual and Brief Remarker", 1852)

"I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses." (Bertrand Russell, "Principles of Mathematics", 1903)

"However successful a theory or law may have been in the past, directly it fails to interpret new discoveries its work is finished, and it must be discarded or modified. However plausible the hypothesis, it must be ever ready for sacrifice on the altar of observation." (Joseph W Mellor, "A Comprehensive Treatise on Inorganic and Theoretical Chemistry", 1922) 

"Heuristic reasoning is reasoning not regarded as final and strict but as provisional and plausible only, whose purpose is to discover the solution of the present problem. We are often obliged to use heuristic reasoning. We shall attain complete certainty when we shall have obtained the complete solution, but before obtaining certainty we must often be satisfied with a more or less plausible guess. We may need the provisional before we attain the final. We need heuristic reasoning when we construct a strict proof as we need scaffolding when we erect a building." (George Pólya, "How to Solve It", 1945)

"It has been said, often enough and certainly with good reason, that teaching mathematics affords a unique opportunity to teach demonstrative reasoning. I wish to add that teaching mathematics also affords an excellent opportunity to teach plausible reasoning. A student of mathematics should learn, of course, demonstrative reasoning; it is his profession and the distinctive mark of his science. Yet he should also learn plausible reasoning; this is the kind of reasoning on which his creative work will mainly depend, The general student should get a taste of demonstrative reasoning; he may have little opportunity to use it directly, but he should acquire a standard with which he can compare alleged evidence of all sorts aimed at him in modern life. He needs, however, in all his endeavors plausible reasoning. At any rate, an ambitious teacher of mathematics should teach both kinds of reasoning to both kinds of students." (George Pólya, "On Plausible Reasoning", Proceedings of the International Congress of Mathematics, 1950)

"The scientist who discovers a theory is usually guided to his discovery by guesses; he cannot name a method by means of which he found the theory and can only say that it appeared plausible to him, that he had the right hunch or that he saw intuitively which assumption would fit the facts." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Demonstrative reasoning penetrates the sciences just as far as mathematics does, but it is in itself (as mathematics is in itself) incapable of yielding essentially new knowledge about the world around us. Anything new that we learn about the world involves plausible reasoning, which is the only kind of reasoning for which we care in everyday affairs." (George Pólya, "Induction and Analogy in Mathematics", 1954)

"In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess." (George Pólya, "Mathematics and plausible reasoning" Vol. 1, 1954)

"The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing. If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference." (George Pólya, "Induction and Analogy in Mathematics", 1954)