Alternate Realities: Mathematical Models of Nature and Man

New mathematical model can help save endangered species
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The question is: Where do the rules come from? As we grow and learn through everyday experience, each of us develops a system of common sense CS concepts about how the world works. To evaluate introductory physics instruction, the Force Concept Inventory FCI was developed to detect differences in student thinking between CS concepts and Newtonian concepts about motion and its causes [6].

Results from applying the FCI were stunning from the get-go! First, the differences were huge before instruction.

Second, the change was small after instruction. These results have been replicated thousands of times from high school to Harvard and in 25 different languages. The FCI is by far the most cited reference in the physics education literature, and it is widely used today to evaluate the effectiveness of teaching reforms. Here we are interested in what the FCI tells us about human cognition. These concepts are overlooked or summarily dismissed as misconceptions by most physicists.

However, they are common outcomes from everyday experience, and they are quite serviceable for dealing with physical objects. Moreover, central CS concepts in the 5 categories have been clearly articulated and discussed by major intellects of the pre-Newtonian age, including Newton himself before the Principia [7].

So CS concepts should be regarded as alternative hypotheses about the physical world that, when clearly formulated, can be tested empirically. One consequence of all this is that in a conventional physics course students systematically misinterpret what they hear and see in class, which goes a long way to account for the typical disastrous student performance on exams.

Ability to distinguish between CS concepts and scientific concepts in the FCI or elsewhere is not a matter of intelligence but of experience. It is acquired only by engagement with science itself, usually through academics. Remarkably, physicists seldom recall any event in their own transitions to Newtonian thinking.

Typically, they presume that the world of classical physics is given directly by experience, in contrast to the subtlety and weirdness of quantum mechanics. They are blind to the subtle revolution in their own thinking that came from learning physics; for the FCI tells us that classical physics differs from common sense in almost every detail. These facts suggest that the transition from common sense to scientific thinking is not a replacement of CS concepts with scientific concepts, but rather a realignment of intuition with experience. Science does not replace common sense.

Rather, as Modeling Theory aims to show, science is a refinement of common sense differing in respect to:. Kant is unsurpassed in using introspection to analyze his own thinking. But introspection was dismissed as subjective and unreliable by behaviorists in the twentieth century, who claimed that scientific objectivity requires psychology to take its data from observable behavior under controlled conditions. However, the behaviorist straight jacket has been cast off in recent decades by the emergence of cognitive science , which draws its data and insights from many independent academic disciplines.

Disciplinary barriers are crossed with increasing frequency, largely due to the speed and ease of electronic communication. Human perception, memory and cognition are being studied in many different ways. The problem, as ever in science, is to identify reproducible patterns in the results. Here follows a sampling of approaches and results with high relevance to Modeling Theory. The Learning Sciences : Research on teaching and learning is emerging as a coherent science with independent branches like physics education research PER devoted to a single discipline.

The most robust finding in the field is that effective teaching requires matching the method to the subject matter, and that requires research embedded within each discipline. As he explains,. Evidently this intuitive structure is abstracted from experience in pushing objects. More generally, diSessa argues that this structure is fundamental to qualitative reasoning. This reasoning structure is often evoked for explanatory purposes in everyday experience. However he does not recognize it as a basic aspectual schema for verb structure, which has been studied at length in cognitive grammar [9].

Aspectual concepts are generally about event structure, where events are changes of state and causes or causal agents induce events. One can argue instead that the fundamental Laws and Principles of science are discovered as general patterns in experience and simply adopted as postulates in our theories.

All this has bearing on other domains of cognitive science, for example, the psychology of perception. In general, we learn about the world around us from our interactions perception-actions with it. Note that the intuitive causal syntax discussed above can be construed by metaphorical projection at least as. When the symbols are words, this provides an intuitive base for verb structure expressing the action of mental agents on mental objects.

The same idea has emerged independently in cognitive linguistics see below. Also, the operator syntax provides an intuitive base for the mathematical concept of function though probably not the only one. Narrative Comprehension : Readers of stories construct mental models of the situation and characters described [10]. Recently visited as well as imagined locations are also activated for several seconds.

These patterns of temporary activation facilitate comprehension. Evolutionary Psychology [11] tells us that human brains evolved adaptively to enable navigation to find food and respond to threats. There can be little doubt that narrative emerged in human prehistory. The practice of storytelling is ancient, pre-dating not only the advent of writing, but of agriculture and permanent settlement as well. Language, an obvious prerequisite for storytelling, is likely to have emerged between 50, and , years ago.

Cognitive linguistics see below aims to ascertain what language can tell us about evolved cognitive abilities. Cognitive Psychology : Psychologist Philip Johnson-Laird [12] is a pioneer in studying human inference by manipulating mental models. His research supports the claim that most human reasoning is inference from mental models. We can distinguish several types of model-based reasoning :.

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Justification of model-based reasoning requires translation from mental models to inference from conceptual models that can be publicly shared, like the scientific models discussed below. In contrast, formal reasoning is computational, using axioms, production rules and other procedures.

It is the foundation for rigorous proof in mathematics and formal logic. However, Modeling theory see below holds that mathematicians and even logicians reason mostly from mental models. Model-based reasoning is more general and powerful than propositional logic, as it integrates multiple representations of information propositions, maps, diagrams, equations into a coherently structured mental model. Rules and procedures are central to the formal concept of inference, but they can be understood as prescriptions for operations on mental models as well as on symbols.

Psychology of Spatial Perception : Everyone has imagination, the ability to conjure up an image of a situation from a description or memory. What can that tell us about mental models? Some people report images that are picture-like, similar to actual visual images. However, others deny such experience, and blind people are perfectly capable of imagination.

Classical research in this domain found support for the view that mental imagery is internalized perception, but not without critics. Barbara Tversky and collaborators [13] have tested the classical view by comparison to mental model alternatives. Among other things, they compared individual accounts of a visual scene generated from narrative with accounts generated from direct observation and found that they are functionally equivalent. A crucial difference is that perceptions have a fixed point of view, while mental models allow change in point of view. Furthermore, spatial mental models are more schematic and categorical than images, capturing some features of the object but not all and incorporating information about the world that is not purely perceptual.

The general conclusion is that mental models represent states of the world as conceived, not perceived. To know a thing is to form a mental model of it. Major characteristics of spatial mental models are summarized in the following list. The best fit to data is a spatial framework model , where each object has an egocentric frame consisting of mental extensions with three body axes.

The details in this list are abundantly supported by other lines of research, especially in cognitive linguistics, to which we now turn. Cognitive Linguistics : The most extensive and coherent body of evidence comes from cognitive linguistics [14], supporting the revolutionary thesis : Language does not refer directly to the world, but rather to mental models and components thereof!

Words serve to activate, elaborate or modify mental models, as in comprehension of a narrative. This thesis rejects all previous versions of semantics, which located the referents of language outside the mind, in favor of cognitive semantics , which locates referents inside the mind. I see the evidence supporting cognitive semantics as overwhelming, but it must be admitted that some linguists are not convinced, and many research questions remain. Two pillars of cognitive linguistics deserve mention here. This should be contrasted with the classical concept of a formal category for which membership is determined by a set of defining properties, a noteworthy generalization of the container metaphor.

This distinction between category types is supported by a mountain of empirical evidence on natural language use. The second pillar is the notion of image schema introduced by Mark Johnson and George Lakoff. Image schemas are basic structural units gestalts that provide structure to natural language and presumably cognition. There are too many to discuss here. Many are discussed in [15] as structural elements in mathematical thinking, including four grounding metaphors for arithmetic. Cognitive Neuroscience : Human brain structures have evolved to support perception, memory and movement, that is, all components needed to execute the perception-action cycle.

But no distinct component for cognition has been identified. It seems reasonable, therefore, to conclude that cognition is executed by coopting drivers of the perception-action for internal planning and simulation. The boundary between hardwired and learned mathematical abilities continues to be a rich area for further research. To deepen this insight and coordinate empirical results, we need a scientific theory, to which we now turn.

Though Modeling Theory is proposed as a general theory of Mind embracing all aspects of cognitive science, we limit our attention here to cognition in physics and mathematics. We have seen above that the study of natural languages gives us rich information about the structure of mental models in common sense cognition. Given its greater precision and coherence, we can expect complimentary and reinforcing results from studying the language of science, especially mathematics.

Indeed, after spelling out the structure of scientific models in explicit detail below, we discuss its implications for cognition in physics and mathematics. A model is a representation of structure in a given system. A system is a set of related objects , which may be real or imaginary, physical or mental, simple or composite. The structure of a system is a set of relations among its objects. The system itself is called the referent of the model. We often identify the model with its representation in a concrete inscription of words, symbols or figures such as graphs, diagrams or sketches.

But it must not be forgotten that the inscription is supplemented by a system of mostly tacit rules and conventions for encoding model structure.

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From my experience as a scientist, I have concluded that five types of structure suffice to characterize any scientific model. Although my initial analysis was based on physics, I have concluded the classification is sufficient for all other sciences as well. As this seems to be an important empirical fact, a brief description of each type is in order here.

Optimal precision in definition and analysis of structure is supplied by mathematics, the science of structure. Both the model and its referent are structured objects, but they need not be distinct. Indeed, the usual notion of a mathematical model as a representation in terms of mathematical symbols does not specify any referent, so we say it is an abstract model. Of course , it is a perfect representation of itself. To address this issue, Modeling Theory [3] posits three distinct worlds in which structured objects exist:.

This helps us make a crucial distinction between mental models and conceptual models.

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Praise for Alternate Realities Mathematical Models of Nature and Man "â?¦covers the major topics completely and accurately within the context of current. Alternate realities—mathematical models of nature and man, by John L. Casti, Institute for Econometrics, Operations Research, and System Theory, Technical.

Mental models are private constructions in the mind of an individual World 2. Thus, communication between individuals involves construction and use of shared conceptual models. Note that a conceptual model establishes an analogy between a mental model and its symbolic representation. Mathematical models are symbolic structures, and to understand one is to create a mental model with analogous structure. Actually, the structure is supplied by the mind not the symbols , which are reduced to meaningless marks without a mind to interpret them.

An analogy is defined as a mapping of structure from one domain source to another target [18]. The mapping is always partial, which means that some structure is not mapped. Science sets up many kinds of analogy between and within the three worlds [3].

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Thus, experimental testing or simply interpreting a scientific model World 3 requires a mapping to a physical system World 1 that I call a referential analogy. Material analogies relate structures of different physical objects in World 1 and this reduces to an inductive analogy when the objects are regarded as identical. And there are many more analogies with computer models World 1. There are other kinds of structure-preserving mappings such as metaphors , which Lakoff [15] defines as a projection of structure from one domain into another. In mathematics a morphism is a structure-preserving mapping : Thus the terms homomorphism preserves algebraic structure and homeomorphism preserves topological structure.

Physical intuition is accorded the same high regard by physicists that mathematicians accord to mathematical intuition. To quote unquestionable leaders in each field [2]:. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be voluntarily reproduced and combined. Modeling theory asserts that physical and mathematical intuitions are merely two different ways to relate products of imagination to the external world. Physical intuition matches structure in mental models with structure in physical systems.

Mathematical intuition matches mental structure with symbolic structure. Thus, structure in imagination is common ground for both physical and mathematical intuition. Kant reasoned in much the same way. He also took the physics and mathematics of his day as given and asked what makes them so special. His analysis is cogent even today, so key points are worth reconsidering. He began by identifying construction in intuition as a means for acquiring certain geometrical knowledge:. This representation of a universal procedure of imagination in providing an image for a concept, I entitle the schema of this concept.

Kant did not stop there. Like any good scientist he anticipated objections to his hypothesis. Many typos. Bogdanov, A. Definitions of non-thermodynamic, non-informational entropies at multiple levels of analysis. Severely criticized. Brown, G. Basis for a whole school of graphical approaches to classical logic. Reydel Campbell, Jeremy: Grammatical Man , Simon and Schuster, New York Popular treatment of many aspects of cognitive science, information theory, and linguistics.

Wiley, New York But also really about logic and mathematical description, excluded middles as "enemies"; relation of epistemics to action.


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