On Principles V: Identity

"The topics of ontology, or metaphysic, are cause, effect, action, passion, identity, opposition, subject, adjunct, and sign." (Isaac Watts, "Logic, or The right use of reason, in the inquiry after truth", 1725)

"Science arises from the discovery of Identity amid Diversity." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle." (Henri Poincaré, "Science and Hypothesis", 1901)

"Metaphors deny distinctions between things: problems often arise from taking structural metaphors too literally. Because unexamined metaphors lead us to assume the identity of unidentical things, conflicts can arise which can only be resolved by understanding the metaphor (which requires its recognition as such), which means reconstructing the analogy on which it is based. […] The unexplained extension of concepts can too often result in the destruction rather than the expansion of meaning." (David Pimm,"Metaphor and Analogy in Mathematics", For the Learning of Mathematics Vol. 1 (3), 1981)

"The idea that one can 'introduce' a kind of objects simply by laying down an identity criterion for them really inverts the proper order of explanation. As Locke clearly understood, one must first have a clear conception of what kind of objects one is dealing with in order to extract a criterion of identity for them from that conception. […] So, rather than 'abstract' a kind of object from a criterion of identity, one must in general 'extract' a criterion of identity from a metaphysically defensible conception of a given kind of objects." (Edward J Lowe, The metaphysics of abstract objects, Journal of Philosophy 92 (10), 1995) 

"There is no unique, global, and universal relation of identity for abstract objects. [...] Abstract objects are of different sorts and this should mean, almost by definition, that there is no global, universal identity for sorts. Each sort X is equipped with an internal relation of identity but there is no identity relation that would apply to all sorts." (Jean-Pierre Marquis," Categorical foundations of mathematics, or how to provide foundations for abstract mathematics", The Review of Symbolic Logic Vol. 6 (1), 2012) 

"Reason is indeed all about identity, or, rather, tautology. Mathematics is the eternal, necessary system of rational, analytic tautology. Tautology is not 'empty', as it is so often characterized by philosophers. It is in fact the fullest thing there, the analytic ground of existence, and the basis of everything. Mathematical tautology has infinite masks to wear, hence delivers infinite variety. Mathematical tautology provides Leibniz’s world that is 'simplest in hypothesis and the richest in phenomena'. No hypothesis cold be simpler than the one revolving around tautologies concerning 'nothing'. There is something - existence - because nothing is tautologous, and 'something' is how that tautology is expressed. If we write x = 0, where x is any expression that has zero as its net result, then we have a world of infinite possibilities where something ('x') equals nothing (0)." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

On Syllogism II

"For all who argue per impossibile infer by syllogism a false conclusion, and prove the original conclusion hypothetically when something impossible follows from a contradictory assumption, as, for example, that the diagonal [of a square] is incommensurable [with the side] because odd numbers are equal to even if it is assumed to be commensurate. It is inferred by syllogism that odd numbers are equal to even, and proved hypothetically that the diagonal is incommensurate, since a false conclusion follows from the contradictory assumption." (Aristotle, "Laws")

"It must be granted that in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii [assumption of premises]. When we say, All men are mortal Socrates is a man therefore Socrates is mortal; it is unanswerably urged by the adversaries of the syllogistic theory, that the proposition, Socrates is mortal." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

"The maxim is, that whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of everything included in the class. This axiom, supposed to be the basis of the syllogistic theory, is termed by logicians the dictum de omni et nullo [the maxim of all and none]." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1858)

"Does the mathematical method proceed from particular to the general, and, if so, how can it be called deductive? […] If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from a syllogism." (Henri Poincaré, "Science and Hypothesis", 1901)

"Syllogistic reasoning remains incapable of adding anything to the data that are given it; the data are reduced to axioms, and that is all we should find in the conclusions." (Henri Poincaré, "Science and Hypothesis", 1901)

"The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinite number of syllogisms. […] We see, then, that in reasoning by recurrence we confine ourselves to the enunciation of the minor of the first syllogism, and the general formula which contains as particular cases all the majors. This unending series of syllogisms is thus reduced to a phrase of a few lines." (Henri Poincaré, "Science and Hypothesis", 1901)

"The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle." (Henri Poincaré, "Science and Hypothesis", 1901)

"Trust is the core of human relationships, of gregariousness among men. Friendship, a puzzle to the syllogistic and critical mentality, is not based on experiments or tests of another person's qualities but on trust. It is not critical knowledge but a risk of the heart which initiates affection and preserves loyalty in our fellow men." (Abraham J Heschel, "Moral Grandeur and Spiritual Audacity: Essays", 1997) 

On Syllogism I

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction." (Francis Bacon, The New Organon, 1620)

"[…] mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediaeval slowness of the syllogisms expressed in our words." (Charles Nordmann, "Einstein and the Universe", 1922)

"Knowledge is ours only if, at the moment of need, it offers itself to the mind without syllogisms or demonstrations for which there is no time." (André Maurois, "Un Art de Vivre" ["The Art of Living"], 1939)

"A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules and syllogisms, without motivation or goal." (Richard Courant & Herbert Robbins, "What Is Mathematics?", 1941)

"The construction of hypotheses is a creative act of inspiration, intuition, invention; its essence is the vision of something new in familiar material. The process must be discussed in psychological, not logical, categories; studied in autobiographies and biographies, not treatises on scientific method; and promoted by maxim and example, not syllogism or theorem." (Milton Friedman, "Essays in Positive Economics", 1953)

"[…] the distinction between rigorous thinking and more vague ‘imaginings’; even in mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing. After all, most thinking is a synthesis or juxtaposition of advances along a line of syllogisms - perhaps in a continuous and persistent ‘forward'’ movement, with searching, so to speak ‘sideways’, in directions which are not necessarily present from the very beginning and which I describe as ‘sending out exploratory patrols’ and trying alternative routes." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"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)

"Whenever I have talked about mental models, audiences have readily grasped that a layout of concrete objects can be represented by an internal spatial array, that a syllogism can be represented by a model of individuals and identities between them, and that a physical process can be represented by a three-dimensional dynamic model. Many people, however, have been puzzled by the representation of abstract discourse; they cannot understand how terms denoting abstract entities, properties or relations can be similarly encoded, and therefore they argue that these terms can have only 'verbal' or propositional representations." (Philip Johnson-Laird, "Mental Models: Towards a Cognitive Science of Language, Inference and Consciousness", 1983)

"Formal logic and the logical syllogism encapsulate connectedness in reasoning." (Marshall McLuhan & Eric McLuhan, "Laws of Media: The New Science", 1988)

"Metaphorizing is a manner of thinking, not a property of thinking. It is a capacity of thought, not its quality. It represents a mental operation by which a previously existing entity is described in the characteristics of another one on the basis of some similarity or by reasoning. When we say that something is (like) something else, we have already performed a mental operation. This operation includes elements such as comparison, paralleling and shaping of the new image by ignoring its less satisfactory traits in order that this image obtains an aesthetic value. By this process, for an instant we invent a device, which serves as the pole vault for the comparison’s jump. Once the jump is made the pole vault is removed. This device could be a lightning-speed logical syllogism, or a momentary created term, which successfully merges the traits of the compared objects." (Ivan Mladenov, "Conceptualizing Metaphors: On Charles Peirce’s marginalia", 2006)

On Induction (-1849)

"The only possible way to conceive universal is by induction, since we come to know abstractions by induction. But unless we have sense experience, we cannot make inductions. Even though sense perception relates to particular things, scientific knowledge concerning such can only be constructed by the successive steps of sense perception, induction, and formulation of universals." (Aristotle, "Posterior Analytics", cca. 350 BC)

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction." (Francis Bacon, "The New Organon", 1620)

"In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions." (Isaac Newton, "The Principia: Mathematical Principles of Natural Philosophy", 1687)

"As in Mathematics, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions but such as are taken from Experiments, or other certain Truths." (Sir Isaac Newton, "Opticks", 1704)

"It is often in our Power to obtain an Analogy where we cannot have an Induction." (David Hartley, "Observations on Man, His Frame, His Duty, and His Expectations", 1749)

"Especially when we investigate the general laws of Nature, induction has very great power; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law; although, whether they can be made to agree in this way better than in any Other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law." (Roger J Boscovich, "De Lege Continuitatis" ["On the law of continuity"], 1754)

"A discovery in mathematics, or a successful induction of facts, when once completed, cannot be too soon given to the world. But […] an hypothesis is a work of fancy, useless in science, and fit only for the amusement of a vacant hour." (Henry Brougham, Edinburgh Review 1, 1803)

"The most important questions of life are, for the most part, really only problems of probability. Strictly speaking one may even say that nearly all our knowledge is problematical; and in the small number of things which we are able to know with certainty, even in the mathematical sciences themselves, induction and analogy, the principal means for discovering truth, are based on probabilities, so that the entire system of human knowledge is connected with this theory." (Pierre-Simon Laplace, "Theorie Analytique des Probabilités", 1812)

"Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorem of the binomial and the principle of universal gravity." (Pierre-Simon Laplace, "Philosophical Essay on Probabilities”, 1814)

"Induction, analogy, hypotheses founded upon facts and rectified continually by new observations, a happy tact given by nature and strengthened by numerous comparisons of its indications with experience, such are the principal means for arriving at truth." (Pierre-Simon Laplace, "A Philosophical Essay on Probabilities", 1814)

"One may even say, strictly speaking, that almost all our knowledge is only probable; and in the small number of things that we are able to know with certainty, in the mathematical sciences themselves, the principal means of arriving at the truth - induction and analogy - are based on probabilities, so that the whole system of human knowledge is tied up with the theory set out in this essay." (Pierre-Simon Laplace, "Philosophical Essay on Probabilities", 1814)

"It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations." (Carl Friedrich Gauss, 1817)

"Such is the tendency of the human mind to speculation, that on the least idea of an analogy between a few phenomena, it leaps forward, as it were, to a cause or law, to the temporary neglect of all the rest; so that, in fact, almost all our principal inductions must be regarded as a series of ascents and descents, and of conclusions from a few cases, verified by trial on many." (Sir John Herschel, "A Preliminary Discourse on the Study of Natural Philosophy" , 1830)

"We have here spoken of the prediction of facts of the same kind as those from which our rule was collected. But the evidence in favour of our induction is of a much higher and more forcible character when it enables us to explain and determine cases of a kind different from those which were contemplated in the formation of our hypothesis. The instances in which this has occurred, indeed, impress us with a conviction that the truth of our hypothesis is certain. No accident could give rise to such an extraordinary coincidence. No false supposition could, after being adjusted to one class of phenomena, so exactly represent a different class, when the agreement was unforeseen and contemplated. That rules springing from remote and unconnected quarters should thus leap to the same point, can only arise from that being where truth resides." (William Whewell, "The Philosophy of the Inductive Sciences" Vol. 2, 1840)

"There is in every step of an arithmetical or algebraical calculation a real induction, a real inference from facts to facts, and what disguises the induction is simply its comprehensive nature, and the consequent extreme generality of its language." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)

On Facts (-1799)

"While those whom devotion to abstract discussions has rendered unobservant of the facts are too ready to dogmatize on the basis of a few observations." (Aristotle, "De Caelo" ["On the Heavens"], cca. 350 BC)

"Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth." (Marcus Aurelius, "Meditations", cca. 2nd century)

"The Syllogism consists of propositions, propositions consist of words, words are symbols of notions. Therefore if the notions themselves (which is the root of the matter) are confused and over-hastily abstracted from the facts, there can be no firmness in the superstructure. Our only hope therefore lies in a true induction." (Francis Bacon, "The New Organon", 1620)

"There are two kinds of truths: those of reasoning and those of fact. The truths of reasoning are necessary and their opposite is impossible; the truths of fact are contingent and their opposites are possible." (Gottfried W Leibniz, "Monadology", 1714)

"We have three principal means: observation of nature, reflection, and experiment. Observation gathers the facts reflection combines them, experiment verifies the result of the combination. It is essential that the observation of nature be assiduous, that reflection be profound, and that experimentation be exact. Rarely does one see these abilities in combination. And so, creative geniuses are not common." (Denis Diderot, "On the Interpretation of Nature", 1753)

"Facts, observations, experiments - these are the materials of a great edifice, but in assembling them we must combine them into classes, distinguish which belongs to which order and to which part of the whole each pertains." (Antoine L Lavoisier, "Mémoires de l’Académie Royale des Sciences", 1777)

"The impossibility of separating the nomenclature of a science from the science itself is owing to this, that every branch of physical science must consist of three things: the series of facts which are the objects of the science, the ideas which represent these facts, and the words by which these ideas are expressed. Like three impressions of the same seal, the word ought to produce the idea, and the idea to be a picture of the fact." (Antoine L Lavoisier, "Elements of Chemistry in a New Systematic Order", 1790)

"We must trust to nothing but facts: These are presented to us by Nature, and cannot deceive. We ought, in every instance, to submit our reasoning to the test of experiment, and never to search for truth but by the natural road of experiment and observation." (Antoin-Laurent de Lavoisiere, "Elements of Chemistry", 1790)

"[…] the speculative propositions of mathematics do not relate to facts; […] all that we are convinced of by any demonstration in the science, is of a necessary connection subsisting between certain suppositions and certain conclusions. When we find these suppositions actually take place in a particular instance, the demonstration forces us to apply the conclusion. Thus, if I could form a triangle, the three sides of which were accurately mathematical lines, I might affirm of this individual figure, that its three angles are equal to two right angles; but as the imperfection of my senses puts it out of my power to be, in any case, certain of the exact correspondence of the diagram which I delineate, with the definitions given in the elements of geometry, I never can apply with confidence to a particular figure, a mathematical theorem." (Dugald Stewart, "Elements of the Philosophy of the Human Mind", 1792)

"It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind." (Nicolas de Condorcet, "Outlines Of An Historical View Of The Progress Of The Human Mind", 1795)

Mental Models XLVII (Limitations VI)

"Every presentation of philosophy, whether oral or written, is to be taken and can only be taken in the sense of a means. Every system is only an expression or image of reason, and hence only an object of reason, an object which reason - a living power that procreates itself in new thinking beings - distinguishes from itself and posits as an object of criticism. Every system that is not recognized and appropriated as just a means, limits and warps the mind for it sets up the indirect and formal thought in the place of the direct, original and material thought." (Ludwig A Feuerbach, "Towards a Critique of Hegel's Philosophy", 1839) 

"[…] we can only study Nature through our senses - that is […] we can only study the model of Nature that our senses enable our minds to construct; we cannot decide whether that model, consistent though it be, represents truly the real structure of Nature; whether, indeed, there be any Nature as an ultimate reality behind its phenomena." (William C Dampier, "The Recent Development of Physical Science", 1904)

"Most mistakes in philosophy and logic occur because the human mind is apt to take the symbol for the reality." (Albert Einstein, "Cosmic Religion: With Other Opinions and Aphorisms", 1931) 

"The model of the natural world we build in our minds by such a process will forever be inadequate, just a little cathedral in the mountains. Still it is better than no model at all." (Timothy Ferris, "The Red Limit: The Search for the Edge of the Universe", 1977)

"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) 

"Whenever I have talked about mental models, audiences have readily grasped that a layout of concrete objects can be represented by an internal spatial array, that a syllogism can be represented by a model of individuals and identities between them, and that a physical process can be represented by a three-dimensional dynamic model. Many people, however, have been puzzled by the representation of abstract discourse; they cannot understand how terms denoting abstract entities, properties or relations can be similarly encoded, and therefore they argue that these terms can have only 'verbal' or propositional representations." (Philip Johnson-Laird,"Mental Models: Towards a Cognitive Science of Language, Inference and Consciousness", 1983)

"Perhaps we all lose our sense of reality to the precise degree to which we are engrossed in our own work, and perhaps that is why we see in the increasing complexity of our mental constructs a means for greater understanding, even while intuitively we know that we shall never be able to fathom the imponderables that govern our course through life." (Winfried G Sebald, "The Rings of Saturn", 1995) 

"Faced with the overwhelming complexity of the real world, time pressure, and limited cognitive capabilities, we are forced to fall back on rote procedures, habits, rules of thumb, and simple mental models to make decisions. Though we sometimes strive to make the best decisions we can, bounded rationality means we often systematically fall short, limiting our ability to learn from experience." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"The robustness of the misperceptions of feedback and the poor performance they cause are due to two basic and related deficiencies in our mental model. First, our cognitive maps of the causal structure of systems are vastly simplified compared to the complexity of the systems themselves. Second, we are unable to infer correctly the dynamics of all but the simplest causal maps. Both are direct consequences of bounded rationality, that is, the many limitations of attention, memory, recall, information processing capability, and time that constrain human decision making." (John D Sterman, "Business Dynamics: Systems thinking and modeling for a complex world", 2000)

"A general limitation of the human mind is its imperfect ability to reconstruct past states of knowledge, or beliefs that have changed. Once you adopt a new view of the world (or any part of it), you immediately lose much of your ability to recall what you used to believe before your mind changed." (Daniel Kahneman, "Thinking, Fast and Slow", 2011)