Archive for the ‘Geometry’ Category

Revisiting The Federalist Paper No. 31 by Alexander Hamilton: An Analogy from Geometry

July 10, 2018

[John Platt’s chapters on social chain reactions in his 1966 book, The Steps to Man, provoked my initial interest in looking into his work. That work appears to be an independent development of themes that appear in more well-known works by Tarde, Hayek, McLuhan, Latour, and others, which of course are of primary concern in thinking through metrological and ecosystem issues in psychological and social measurement. My interest also comes in the context of Platt’s supervision of Ben Wright in Robert Mulliken’s physics lab at the U of Chicago in 1948. However, other chapters in this book concern deeper issues of complexity and governance that cross yet more disciplinary boundaries. One of the chapters in the book, for instance, examines the Federalist Papers and remarks on a geometric analogy drawn by Alexander Hamilton concerning moral and political forms of knowledge. The parallel with my own thinking is such that I have restated Hamilton’s theme in my own words within the contemporary context. The following is my effort in this regard. No source citations are given, but a list of supporting references is included at bottom. Hamilton’s original text is available at: https://www.congress.gov/resources/display/content/The+Federalist+Papers#TheFederalistPapers-31.  ]

 

Communication requires that we rely on the shared understandings of a common language. Language puts in play combinations of words, concepts, and things that enable us to relate to one another at varying levels of complexity. Often, we need only to convey the facts of a situation in a simple denotative statement about something learned (“the cat is on the mat”). We also need to be able to think at a higher level of conceptual complexity referred to as metalinguistic, where we refer to words themselves and how we learn about what we’ve learned (“the word ‘cat’ has no fur”). At a third, metacommunicative, level of complexity, we make statements about statements, deriving theories of learning and judgments from repeated experiences of metalinguistic learning about learning (“I was joking when I said the cat was on the mat”).

Human reason moves freely between expressions of and representations of denotative facts, metalinguistic instruments like words, and metacommunicative theories. The combination of assurances obtained from the mutual supports each of these provides the others establishes the ground in which the seeds of social, political, and economic life take root and grow. Thought itself emerges from within the way the correspondence of things, words, and concepts precedes and informs the possibility of understanding and communication.

When understanding and communication fail, that failure may come about because of mistaken perceptions concerning the facts, a lack of vocabulary, or misconceptions colored by interests, passions, or prejudices, or some combination of these three.

The maxims of geometry exhibit exactly this same pattern combining concrete data on things in the world, instruments for abstract measurement, and formal theoretical concepts. Geometry is the primary and ancient example of how the beauty of aesthetic proportions teaches us to understand meaning. Contrary to common sense, which finds these kinds of discontinuities incomprehensible, philosophy since the time of Plato’s Symposium teaches how to make meaning in the face of seemingly irreconcilable differences between the local facts of a situation and the principles to which we may feel obliged to adhere. Geometry meaningfully and usefully, for instance, represents the undrawable infinite divisibility of line segments, as with the irrational length of the hypotenuse of a right isosceles triangle that has the other two sides with lengths of 1.

This apparently absurd and counter-intuitive skipping over of the facts in the construction of the triangular figure and the summary reference to the unstateable infinity of the square root of two is so widely accepted as to provide a basis for real estate property rights that are defensible in courts of law and financially fungible. And in this everyday commonplace we have a model for separating and balancing denotative facts, instrumental words, and judicial theories in moral and political domains.

Humanity has proven far less tractable than geometry over the course of its history regarding possible sciences of morals and politics. This is understandable given humanity’s involvement in its own ongoing development. As Freud put it, humanity’s Narcissistic feeling of being the center of the universe, the crown of creation, and the master of its own mind has suffered a serious of blows as it has had to come to terms with the works of Copernicus, Darwin, and Freud himself. The struggle to establish a common human identity while also celebrating individual uniqueness is an epic adventure involving billions of tragic and comedic stories of hubris, sacrifice, and accomplishment. Humanity has arrived at a point now, however, at which a certain obstinate, perverse, and disingenuous resistance to self-understanding has gone too far.

Although the mathematical sciences excel in refining the precision of their tools, longstanding but largely untapped resources for improving the meaningfulness and value of moral and political knowledge have been available for decades. “The obscurity is much oftener in the passions and prejudices of the reasoner than in the subject.” Methods for putting passions on the table for sorting out take advantage of the lessons beauty teaches about meaning and thereby support each of the three levels of complexity in communication.

At this point we encounter the special relevance of those three levels of complexity to the separation and balance of powers in government. The concrete denotative factuality of data is the concern of the executive branch, as befits its orientation to matters of practical application. The abstract metalinguistic instrumentation of words is the concern of the legislative branch, in accord with its focus on the enactment of laws and measures. And formal metacommunicative explanatory theories are the concern of the judicial branch, as is appropriate to its focus on constitutional issues.

For each of us to give our own individual understandings fair play in ways that do not give free rein to unfettered prejudices entangled in words and subtle confusions, we need to be able to communicate in terms that, so far as possible, function equally well within and across each of these levels of complexity. It is only to state the obvious to say that we lack the language needed for communication of this kind. Our moral and political sciences have not yet systematically focused on creating such languages. Outside of a few scattered works, they have not even yet consciously hypothesized the possibility of creating these languages. It is nonetheless demonstrably the case that these languages are feasible, viable, and desirable.

Though good will towards all and a desire to refrain so far as possible from overt exclusionary prejudices for or against one or another group cannot always be assumed, these are the conditions necessary for a social contract and are taken as the established basis for what follows. The choice between discourse and violence includes careful attention to avoiding the violence of the premature conclusion. If we are ever to achieve improved communication and a fuller realization of both individual liberties and social progress, the care we invest in supports for life, liberty, and the pursuit of happiness must flow from this deep source.

Given the discontinuities between language’s levels of complexity, avoiding premature conclusions means needing individualized uncertainty estimates and an associated tolerance for departures from expectations set up by established fact-word-concept associations. For example, we cannot allow a three-legged horse to alter our definition of horses as four-legged animals. Neither should we allow a careless error or lucky guess to lead to immediate and unqualified judgments of learning in education. Setting up the context in which individual data points can be understood and explained is the challenge we face. Information infrastructures supporting this kind of contextualization have been in development for years.

To meet the need for new communicative capacities, features of these information infrastructures will have to include individualized behavioral feedback mechanisms, minimal encroachments on private affairs, managability, modifiability, and opportunities for simultaneously enhancing one’s own interests and the greater good.

It is in this latter area that our interests are now especially focused. Our audacious but not implausible goal is to find ways of enhancing communication and the quality of information infrastructures by extending beauty’s lessons for meaning into new areas. In the same way that geometry facilitates leaps from concrete figures to abstract constructions and from there to formal ideals, so, too, must we learn, learn about that learning, and develop theories of learning in other less well materialized areas, such as student-centered education, and patient-centered health care. Doing so will set the stage for new classes of human, social, and natural capital property rights that are just as defensible in courts of law and financially fungible as real estate.

When that language is created, when those rights are assigned, and when that legal defensibility and financial fungibility are obtained, a new construction of government will follow. In it, the separation and balance of executive, legislative, and judicial powers will be applied with equal regularity and precision down to the within-individual micro level, as well as at the between-individual meso level, and at the social macro level. This distribution of freedom and responsibility across levels and domains will feed into new educational, market, health, and governmental institutions of markedly different character than we have at present.

A wide range of research publications appearing over the last several decades documents unfolding developments in this regard, and so those themes will not be repeated here. Some of these publications are listed below for those interested. Far more remains to be done in this area than has yet been accomplished, to say the least.

 

 

Sources consulted or implied

Andrich, D. (2010). Sufficiency and conditional estimation of person parameters in the polytomous Rasch model. Psychometrika, 75(2), 292-308.

Bateson, G. (1972). Steps to an ecology of mind: Collected essays in anthropology, psychiatry, evolution, and epistemology. Chicago: University of Chicago Press.

Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21, 5-31.

Black, P., Wilson, M., & Yao, S. (2011). Road maps for learning: A guide to the navigation of learning progressions. Measurement: Interdisciplinary Research & Perspectives, 9, 1-52.

Fisher, W. P., Jr. (2002, Spring). “The Mystery of Capital” and the human sciences. Rasch Measurement Transactions, 15(4), 854 [http://www.rasch.org/rmt/rmt154j.htm].

Fisher, W. P., Jr. (2005, August 1-3). Data standards for living human, social, and natural capital. In Session G: Concluding Discussion, Future Plans, Policy, etc. Conference on Entrepreneurship and Human Rights [http://www.fordham.edu/economics/vinod/ehr05.htm], Pope Auditorium, Lowenstein Bldg, Fordham University.

Fisher, W. P., Jr. (2007, Summer). Living capital metrics. Rasch Measurement Transactions, 21(1), 1092-1093 [http://www.rasch.org/rmt/rmt211.pdf].

Fisher, W. P., Jr. (2009, November 19). Draft legislation on development and adoption of an intangible assets metric system. Retrieved 6 January 2011, from Living Capital Metrics blog: https://livingcapitalmetrics.wordpress.com/2009/11/19/draft-legislation/

Fisher, W. P., Jr. (2009, November). Invariance and traceability for measures of human, social, and natural capital: Theory and application. Measurement: Concerning Foundational Concepts of Measurement Special Issue Section, 42(9), 1278-1287.

Fisher, W. P., Jr. (2009). NIST Critical national need idea White Paper: metrological infrastructure for human, social, and natural capital (Tech. Rep. No. http://www.nist.gov/tip/wp/pswp/upload/202_metrological_infrastructure_for_human_social_natural.pdf). Washington, DC:. National Institute for Standards and Technology.

Fisher, W. P., Jr. (2010). Measurement, reduced transaction costs, and the ethics of efficient markets for human, social, and natural capital, Bridge to Business Postdoctoral Certification, Freeman School of Business, Tulane University (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2340674).

Fisher, W. P., Jr. (2010). The standard model in the history of the natural sciences, econometrics, and the social sciences. Journal of Physics Conference Series, 238(1), 012016.

Fisher, W. P., Jr. (2011). Bringing human, social, and natural capital to life: Practical consequences and opportunities. Journal of Applied Measurement, 12(1), 49-66.

Fisher, W. P., Jr. (2011). Stochastic and historical resonances of the unit in physics and psychometrics. Measurement: Interdisciplinary Research & Perspectives, 9, 46-50.

Fisher, W. P., Jr. (2012). Measure and manage: Intangible assets metric standards for sustainability. In J. Marques, S. Dhiman & S. Holt (Eds.), Business administration education: Changes in management and leadership strategies (pp. 43-63). New York: Palgrave Macmillan.

Fisher, W. P., Jr. (2012, May/June). What the world needs now: A bold plan for new standards [Third place, 2011 NIST/SES World Standards Day paper competition]. Standards Engineering, 64(3), 1 & 3-5 [http://ssrn.com/abstract=2083975].

Fisher, W. P., Jr. (2015). A probabilistic model of the law of supply and demand. Rasch Measurement Transactions, 29(1), 1508-1511  [http://www.rasch.org/rmt/rmt291.pdf].

Fisher, W. P., Jr. (2018). How beauty teaches us to understand meaning. Educational Philosophy and Theory, in review.

Fisher, W. P., Jr. (2018). A nondualist social ethic: Fusing subject and object horizons in measurement. TMQ–Techniques, Methodologies, and Quality, in review.

Fisher, W. P., Jr., Oon, E. P.-T., & Benson, S. (2018). Applying Design Thinking to systemic problems in educational assessment information management. Journal of Physics Conference Series, 1044, 012012.

Fisher, W. P., Jr., Oon, E. P.-T., & Benson, S. (2018). Rethinking the role of educational assessment in classroom communities: How can design thinking address the problems of coherence and complexity? Measurement, in review.

Fisher, W. P., Jr., & Stenner, A. J. (2013). On the potential for improved measurement in the human and social sciences. In Q. Zhang & H. Yang (Eds.), Pacific Rim Objective Measurement Symposium 2012 Conference Proceedings (pp. 1-11). Berlin, Germany: Springer-Verlag.

Fisher, W. P., Jr., & Stenner, A. J. (2016). Theory-based metrological traceability in education: A reading measurement network. Measurement, 92, 489-496.

Fisher, W. P., Jr., & Stenner, A. J. (2018). Ecologizing vs modernizing in measurement and metrology. Journal of Physics Conference Series, 1044, 012025.

Gadamer, H.-G. (1980). Dialogue and dialectic: Eight hermeneutical studies on Plato (P. C. Smith, Trans.). New Haven: Yale University Press.

Gari, S. R., Newton, A., Icely, J. D., & Delgado-Serrano, M. D. M. (2017). An analysis of the global applicability of Ostrom’s design principles to diagnose the functionality of common-pool resource institutions. Sustainability, 9(7), 1287.

Gelven, M. (1984). Eros and projection: Plato and Heidegger. In R. W. Shahan & J. N. Mohanty (Eds.), Thinking about Being: Aspects of Heidegger’s thought (pp. 125-136). Norman, Oklahoma: Oklahoma University Press.

Hamilton, A. (. (1788, 1 January). Concerning the general power of taxation (continued). The New York Packet. (Rpt. in J. E. Cooke, (Ed.). (1961). The Federalist (Hamilton, Alexander; Madison, James; Jay, John). (pp. No. 31, 193-198). Middletown, Conn: Wesleyan University Press.

Lunz, M. E., Bergstrom, B. A., & Gershon, R. C. (1994). Computer adaptive testing. International Journal of Educational Research, 21(6), 623-634.

Ostrom, E. (2015). Governing the commons: The evolution of institutions for collective action. Cambridge, UK: Cambridge University Press (Original work published 1990).

Pendrill, L., & Fisher, W. P., Jr. (2015). Counting and quantification: Comparing psychometric and metrological perspectives on visual perceptions of number. Measurement, 71, 46-55.

Penuel, W. R. (2015, 22 September). Infrastructuring as a practice for promoting transformation and equity in design-based implementation research. In Keynote. International Society for Design and Development in Education (ISDDE) 2015 Conference, Boulder, CO. Retrieved from http://learndbir.org/resources/ISDDE-Keynote-091815.pdf

Platt, J. R. (1966). The step to man. New York: John Wiley & Sons.

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests (Reprint, with Foreword and Afterword by B. D. Wright, Chicago: University of Chicago Press, 1980). Copenhagen, Denmark: Danmarks Paedogogiske Institut.

Ricoeur, P. (1966). The project of a social ethic. In D. Stewart & J. Bien, (Eds.). (1974). Political and social essays (pp. 160-175). Athens, Ohio: Ohio University Press.

Ricoeur, P. (1970). Freud and philosophy: An essay on interpretation. Evanston, IL: Northwestern University Press.

Ricoeur, P. (1974). Violence and language. In D. Stewart & J. Bien (Eds.), Political and social essays by Paul Ricoeur (pp. 88-101). Athens, Ohio: Ohio University Press.

Ricoeur, P. (1977). The rule of metaphor: Multi-disciplinary studies of the creation of meaning in language (R. Czerny, Trans.). Toronto: University of Toronto Press.

Star, S. L., & Ruhleder, K. (1996, March). Steps toward an ecology of infrastructure: Design and access for large information spaces. Information Systems Research, 7(1), 111-134.

Wilson, M. (2005). Constructing measures: An item response modeling approach. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Wright, B. D. (1958, 7). On behalf of a personal approach to learning. The Elementary School Journal, 58, 365-375. (Rpt. in M. Wilson & W. P. Fisher, Jr., (Eds.). (2017). Psychological and social measurement: The career and contributions of Benjamin D. Wright (pp. 221-232). New York: Springer Nature.)

Wright, B. D. (1999). Fundamental measurement for psychology. In S. E. Embretson & S. L. Hershberger (Eds.), The new rules of measurement: What every educator and psychologist should know (pp. 65-104 [http://www.rasch.org/memo64.htm]). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

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Simple ideas, complex possibilities, elegant and beautiful results

February 11, 2011

Possibilities of great subtlety, elegance, and power can follow from the simplest ideas. Leonardo da Vinci is often credited with offering a variation on this theme, but the basic idea is much older. Philosophy, for instance, began with Plato’s distinction between name and concept. This realization that words are not the things they stand for has informed and structured each of several scientific revolutions.

How so? It all begins from the reasons why Plato required his students to have studied geometry. He knew that those familiar with the Pythagorean theorem would understand the difference between any given triangle and the mathematical relationships it represents. No right triangle ever definitively embodies a perfect realization of the assertion that the square of the hypotenuse equals the sum of the squares of the other two sides. The mathematical definition or concept of a triangle is not the same thing as any actual triangle.

The subtlety and power of this distinction became apparent in its repeated application throughout the history of science. In a sense, astronomy is a geometry of the heavens, Newton’s laws are a geometry of gravity, Ohm’s law is a geometry of electromagnetism, and relativity is a geometry of the invariance of mass and energy in relation to the speed of light. Rasch models present a means to geometries of literacy, numeracy, health, trust, and environmental quality.

We are still witnessing the truth, however partial, of Whitehead’s assertion that the entire history of Western culture is a footnote to Plato. As Husserl put it, we’re still struggling with the possibility of creating a geometry of experience, a phenomenology that is not a mere description of data but that achieves a science of living meaning. The work presented in other posts here attests to a basis for optimism that this quest will be fruitful.

Geometrical and algebraic expressions of scientific laws

April 12, 2010

Geometry provides a model of scientific understanding that has repeatedly proven itself over the course of history. Einstein (1922) considered geometry to be “the most ancient branch of physics” (p. 14). He accorded “special importance” to his view that “all linear measurement in physics is practical geometry,” “because without it I should have been unable to formulate the theory of relativity” (p. 14).

Burtt (1954) concurs, pointing out that the essential question for Copernicus was not “Does the earth move?” but, rather, “…what motions should we attribute to the earth in order to obtain the simplest and most harmonious geometry of the heavens that will accord with the facts?” (p. 39). Maxwell similarly employed a geometrical analogy in working out his electromagnetic theory, saying

“By referring everything to the purely geometrical idea of the motion of an imaginary fluid, I hope to attain generality and precision, and to avoid the dangers arising from a premature theory professing to explain the cause of the phenomena. If the results of mere speculation which I have collected are found to be of any use to experimental philosophers, in arranging and interpreting their results, they will have served their purpose, and a mature theory, in which physical facts will be physically explained, will be formed by those who by interrogating Nature herself can obtain the only true solution of the questions which the mathematical theory suggests.” (Maxwell, 1965/1890, p. 159).

Maxwell was known for thinking visually, once as a student offering a concise geometrical solution to a problem that resisted a lecturer’s lengthy algebraic efforts (Forfar, 2002, p. 8). His approach seemed to be one of playing with images with the aim of arriving at simple mathematical representations, instead of thinking linearly through a train of analysis. A similar method is said to have been used by Einstein (Holton, 1988, pp. 385-388).

Gadamer (1980) speaks of the mathematical transparency of geometric figures to convey Plato’s reasons for requiring mathematical training of the students in his Academy, saying:

“Geometry requires figures which we draw, but its object is the circle itself…. Even he who has not yet seen all the metaphysical implications of the concept of pure thinking but only grasps something of mathematics—and as we know, Plato assumed that such was the case with his listeners—even he knows that in a manner of speaking one looks right through the drawn circle and keeps the pure thought of the circle in mind.” (p. 101)

But exactly how do geometrical visualizations lend themselves to algebraic formulae? More specifically, is it possible to see the algebraic structure of scientific laws in geometry?

Yes, it is. Here’s how. Starting from the Pythagorean theorem, we know that the square of a right triangle’s hypotenuse is equal to the sum of the squares of the other two sides. For convenience, imagine that the lengths of the sides of the triangle, as shown in Figure 1, are 3, 4, and 5, for sides a, b, and c, respectively. We can count the unit squares within each side’s square and see that the 25 in the square of the hypotenuse equal the sum of the 9 in the square of side a and the 16 in the sum of side b.

That mathematical relationship can, of course, be written as

a2 + b2 = c2

which, for Figure 1, is

32 + 42 = 52 = 9 + 16 = 25

Now, most scientific laws are written in a multiplicative form, like this:

m = f / a

or

f = m * a

which, of course, is how Maxwell presented Newton’s Second Law. So how would the Pythagorean Theorem be written like a physical law?

Since the advent of small, cheap electronic calculators, slide rules have fallen out of fashion. But these eminently useful tools are built to take advantage of the way the natural logarithm and the number e (2.71828…) make division interchangeable with subtraction, and multiplication interchangeable with addition.

That means the Pythagorean Theorem could be written like Newton’s Second Law of Motion, or the Combined Gas Law. Here’s how it works. The Pythagorean Theorem is normally written as

a2 + b2 = c2

but does it make sense to write it as follows?

a2 * b2 = c2

Using the convenient values for a, b, and c from above

32 + 42 = 52

and

9 + 16 = 25

so, plainly, simply changing the plus sign to a multiplication sign will not work, since 9 * 16 is 144. This is where the number e comes in. What happens if e is taken as a base raised to the power of each of the parameters in the equation? Does this equation work?

e9 * e16 = e25

which, substituting a for e9, b for e16, and c for e25, could be represented by

a * b = c

and which could be solved as

8103 * 8,886,015 ≈ 72,003,378,611

Yes, it works, and so it is possible to divide through by e16 and arrive at the form of the law used by Maxwell and Rasch:

8103 ≈ 72,003,378,611 / 8,886,015

or

e9 = e25 / e16

or, again substituting a for e9, b for e16, and c for e25, could be represented by

a = c / b

which, when converted back to the additive form, looks like this:

a = c – b

and this

9 = 25 – 16 .

Rasch wrote his model in the multiplicative form of

εvi = θvσi

and it is often written in the form of

Pr {Xni = 1} = eβnδi / 1 + eβnδi

or

Pni = exp(Bn – Di) / [1 + exp(Bn – Di)]

which is to say that the probability of a correct response from person n on item i is equal to e taken to the power of the difference between the estimate β (or B) of person n‘s ability and the estimate δ (or D) of item i‘s difficulty, divided by one plus e to that same power.

Logit estimates of Rasch model parameters taken straight from software output usually range between ­-3.0 or so and 3.0. So what happens if a couple of arbitrary values are plugged into these equations? If someone has a measure of 2 logits, what is their probability of a correct answer on an item that calibrates at 0.5 logits? The answer should be

e2-0.5 / (1 + e2-0.5).

Now,

e1.5 = 2.718281.5 = 4.481685….

and

4.481685 / (1 + 4.481685) ≈ 0.8176

For a table of the relationships between logit differences, odds, and probabilities, see Table 1.4.1 in Wright & Stone (1979, p. 16), or Table 1 in Wright (1977).

This form of the model

Pni = exp(Bn – Di) / [1 + exp(Bn – Di)]

can be rewritten in an equivalent form as

[Pni / (1 – Pni)] = exp(Bn – Di) .

Taking the natural logarithm of the response probabilities expresses the model in perhaps its most intuitive form, often written as

ln[Pni / (1 – Pni)] = Bn – Di .

Substituting a for ln[Pni / (1 – Pni)], b for Bn, and c for Di, we have the same equation as we had for the Pythagorean Theorem, above

a = c – b .

Plugging in the same values of 2.0 and 0.5 logits for Bn and Di,

ln[Pni / (1 – Pni)] = 2.0 – 0.5 = 1.5.

The logit value of 1.5 is obtained from response odds [Pni / (1 – Pni)] of about 4.5, making, again, Pni equal to about 0.82.

Rasch wrote the model in working from Maxwell like this:

Avj = Fj / Mv .

So when catapult j’s force F of 50 Newtons (361.65 poundals) is applied to object v’s mass M of 10 kilograms (22.046 pounds), the acceleration of this interaction is 5 meters (16.404 feet) per second, per second. Increases in force relative to the same mass result in proportionate increases in acceleration, etc.

The same consistent and invariant structural relationship is posited and often found in Rasch model applications, such that reasonable matches are found between the expected and observed response probabilities are found for various differences between ability, attitude, or performance measures Bn and the difficulty calibrations Di of the items on the scale, between different measures relative to any given item, and between different calibrations relative to any given person. Of course, any number of parameters may be added, as long as they are included in an initial calibration design in which they are linked together in a common frame of reference.

Model fit statistics, principal components analysis of the standardized residuals, statistical studies of differential item/person functioning, and graphical methods are all applied to the study of departures from the modeled expectations.

I’ve shown here how the additive expression of the Pythagorean theorem, the multiplicative expression of natural laws, and the additive and multiplicative forms of Rasch models all participate in the same simultaneous, conjoint relation of two parameters mediated by a third. For those who think geometrically, perhaps the connections drawn here will be helpful in visualizing the design of experiments testing hypotheses of converging yet separable parameters. For those who think algebraically, perhaps the structure of lawful regularity in question and answer processes will be helpful in focusing attention on how to proceed step by step from one definite idea to another, in the manner so well demonstrated by Maxwell (Forfar, 2002, p. 8). Either way, the geometrical and/or algebraic figures and symbols ought to work together to provide a transparent view on the abstract mathematical relationships that stand independent from whatever local particulars are used as the medium of their representation.

Just as Einstein held that it would have been impossible to formulate the theory of relativity without the concepts, relationships, and images of practical geometry, so, too, may it one day turn out that key advances in the social and human sciences depend on the invariance of measures related to one another in the simple and lawful regularities of geometry.

Figure 1. A geometrical proof of the Pythagorean Theorem

References

Burtt, E. A. (1954). The metaphysical foundations of modern physical science (Rev. ed.) [First edition published in 1924]. Garden City, New York: Doubleday Anchor.

Einstein, A. (1922). Geometry and experience (G. B. Jeffery, W. Perrett, Trans.). In Sidelights on relativity (pp. 12-23). London, England: Methuen & Co. LTD.

Forfar, J. (2002, June). James Clerk Maxwell: His qualities of mind and personality as judged by his contemporaries. Mathematics Today, 38(3), 83.

Gadamer, H.-G. (1980). Dialogue and dialectic: Eight hermeneutical studies on Plato (P. C. Smith, Trans.). New Haven: Yale University Press.

Holton, G. (1988). Thematic origins of scientific thought (Revised ed.). Cambridge, Massachusetts: Harvard University Press.

Maxwell, J. C. (1965/1890). The scientific papers of James Clerk Maxwell (W. D. Niven, Ed.). New York: Dover Publications.

Wright, B. D. (1977). Solving measurement problems with the Rasch model. Journal of Educational Measurement, 14(2), 97-116 [http://www.rasch.org/memo42.htm].

Wright, B. D., & Stone, M. H. (1979). Best test design: Rasch measurement. Chicago, Illinois: MESA Press.

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