Andrew Kwon
For most people who do not study philosophical issues, the idea of meshing the study of physical sciences with the debate of philosophy is tantamount to mixing feathers and tar; one is only asking for trouble in combining two seemingly independent worlds. However, keeping the quantifiable and the pontificated separate denies a fairly sizable intersection of domain for both. Indeed, scientific philosophy is called in for solving many of the physical world’s problems. For instance, statistical measurements combined with evidential probability give a way for scientists to reduce the margin of error in data measurements. Theory construction plays a part in creating a precise hypothesis, which is a component of a well-thought experiment, as well as analysis and extrapolation.
Of course, only select topics in philosophy relate, and even fewer play the important roles in scientific discovery. Though somewhat popular currently, the notion of postmodernistic thought does not provide much useful insight for the scientific community. However, as such a philosophy becomes more popular, it begins to affect the scientific community, suiting it towards demonstrating how even a trivial notion can affect science.
A simple definition of postmodernism is provided by Stenger in that postmodernism’s claim is:
"
…all statements, whether in science or literature, are simply narratives -- stories and myths that do nothing more than articulate the cultural prejudices of the narrator. In this view, one narrative is as good as another… texts have no intrinsic meaning… no narrative can have universal validity and that ‘Western’ science is no exception."Stenger provides a terse rebuttal to the claim itself, but for the moment consider the consequences of postmodernistic thought. As the saying goes, the law of gravity cannot be repealed. However, in light of such considerations, the many indelible laws of science are merely just interpreted as such and can vary with the next person. Such a thought implies that one does not have to fall if one merely re-interprets the law of gravity.
Though Stenger clearly disputes the idea of postmodernism, he nevertheless issues a warning directed to the general scientific community. This signifies that despite the readily apparent rebuttals to it, it has become something to be contended with. Though perhaps an experienced scientist would pay no attention to it, as Stenger claims, its ramifications on the less inclined audience would be visible. Such is the significance of philosophy in scientific study; so much that even a less than reliable ideology could possibly threaten the entire base of scientific knowledge if unchecked.
The idea of measuring (whether it be counting marbles, the length of a plank of wood, the area of a yard) does not seem likely to be the ground of much debate; superficially, a meter is a meter is a meter. Though only three prominent expositors are mentioned, their opinions appear to be shared with many of the leading experts in the filed. However, notable variants and dissenters will also be discussed.
Some shared concepts are at the core of many interpretations; however, even these are open to dispute. Most agree that there exist two types of measurements one can conduct; direct (or fundamental) and indirect. Direct measurements are generally determinable without the aid of an instrument, device or technique. (eg counting a small number of pencils) The bulk of scientific data lies in indirect measurements, or that which must be determined using other than simple observation (eg the exact length of one pencil at any given time).
The concept of error in measurements (direct or indirect) is also widely acknowledged. Though determination and rectification methods vary, error is agreed upon as being a certain degree (ratio, percentage, etc) of inaccuracy in the reported measurement(s) versus the actual data. Other statements would defeat the generalization. The majority consensus on the issue of quantity lies in considering it as a relation more than an individual datum. Quantities are often compared to each other; quantity x may only be larger than y, equal than y, or smaller than y. Such relations are defined to be transitive (though its symmetry is debated) and mutually exclusive; no quantity x may be larger and smaller than some y.*
II(a). What is a quantity?
To begin a cursory investigation into the question of measurements, a meaningful starting point is in defining exactly what a quantity is. After all, measurements are commonly made with the intent of obtaining quantitative results. Shouldn’t one know what measurements are precisely doing?
One view that is popular is that a quantity is as much a property of an object (or a set of objects) as any other given property. (Ellis, 24) It is intrinsic into a pencil that it is exactly one pencil. By extension, a dozen pencils share the property of each having a quantity of one, resulting in a combined property of twelve.*
On a superficial level, this would bode well with most people. It makes sense. However, delving into it would lead one to notice that in fact, such an interpretation virtually ignores the relational properties of a quantity mentioned above. (Ibid, 25) By assigning the quantity to particular object(s), it does not become possible to compare quantities.
An example is in order; suppose one would have twelve pencils in his possession. As each pencil has the property of being a pencil, one could say that the combined quantity property of the pencils is twelve. However, the quantity of twelve (pencils) does not become a quantity property of the person who has them. The person remains a quantity of one human. Nor is it the quantity of the group of pencils, despite intuition; a group of pencils itself is one group of pencils, although it is twelve.
One could argue against the act of assigning clustering of objects their own quantity property, in hopes of dissolving the group of twelve pencils into just twelve pencils. However, the problem then transmutes into that of uncountability. It is not difficult to see that, under such a proposition, one skyscraper must be reduced to a quantity of 75 floors or 2450 windowpanes or any arbitrary quantity, which in turn must be reduced itself (eg floors to tiles or windowpanes to volume of glass), which itself must be reduced, and so on. The principle would demand a universal smallest unit, which is not even known (subatomic particles are currently the domain of quantum physicists). The sheer impracticality of such a principle prevents it from resolving the problem of the set quantity.
Ellis further presses the concept of a quantity by relating it to a cluster concept. (35) The underlying idea is that no one set order of relationships belongs essentially to a cluster/quantity, but any one of those orders can be used to identify a cluster/quantity. Ellis makes an excellent example out of a fictional Mr. Tom Jones. Empirical statements such as "Mr. Jones stands 6’3" tall" and "Mr. Jones has blonde hair" can be used to help the receiving end of the statements identify what particular Mr. Tom Jones is in question. But, even having a complete set of empirical statements describing Mr. Tom Jones (if that were possible) do not necessarily define Mr. Tom Jones to someone who does not know him. Such is the idea with equating a quantity with a cluster; one can talk of "a dozen pencils" and another may understand what dozen pencils are being referred to, but the statement "a dozen pencils" and related do not define the quantity.
Furthermore, evidence exists that a quantity is not, in fact, necessarily a property of an object per se. (Campbell, 273) Consider the quantity of velocity. Velocity may be considered a quantity of the power of wind, and wind is commonly considered a force instead of an object. However, the fact that we do associate a value (mph) to it implies that inherently, we do not understand quantity as such.
Some take the somewhat questionable view that quantity does not even exist. Dingle relegates the idea to metaphysics, claiming that all we obtain from measurements are simply coherent results. (Ellis, 24) This appears to be a less radical form of the postmodern thought aforementioned. Even though we know that the results of the measurements are consistent, anything the measurements report are implied to have no inherent meaning and thus to have already been interpreted by the viewer to be so.* Ellis quietly refutes him in pointing out that distinct criteria exist for the existence and identity of quantity have been defined.*
II(b). What is a measurement?
The next logical question to ask has two main schools of thought, both of which will be naturally presented here. It is well known that many things, sharing weak (if none at all) relationships (eg mass and volume), are all expressed in terms of a quantity via the process of measuring. But what precisely is a measurement to accomplish?
By rephrasing the question of "what is a measurement?" into "what is the purpose of a measurement?," two propositions claim to answer the question. Ellis presents that the act of measurement is merely an assignment of numerals according to some non-degenerate deterministic rule. (41) This interpretation, though appearing arbitrary almost to the point of uselessness in the world of science, excels in its simplicity. Consider Huffman codes, a simple process for the compression of files. (Cormen, et al., 337) It reduces longer strings of bits/characters by examining the frequency of occurrence within a file and assigns replacement strings according to a rule.* One can argue that the Huffman process is a measurement, since it does assign numerals (binary) by a specific rule (the aforementioned rule). Intuitively, it is not difficult to view the process as a quantitative measurement of occurrences of patterns.
Campbell proposes an alternate thought for the purpose of measurement. It becomes a process by which relations are established between properties and numbers. (Campbell, 275) Although strikingly familiar to the proposition Ellis favors, it is slightly less restrictive. The insistence of a conditioned rule is noticeably absent, which allows for more types of measurements. But what stands out is what the measurement does. Ellis’ take returns something representative of the datum in question; Campbell’s results may be more than just that. Though a numeral is related to the data it represents, this presented view allows for greater relational possibilities.*
II(c). What can be measured quantitatively?
Again, two schools of thought exist to solve this question. There exist properties of objects which can be quantified (length, mass, etc.), but all are familiar with properties in which numbers are not (and can not) be assigned (beauty, fun, etc). How do we know what we can measure and that we can’t measure what we do not think we can?
The thought proposed by Kyburg is that a direct comparative judgment may allow for a direct measurement. (109) The condition inherent in the "may" is that not all direct comparative judgments are equal; one may consider justice more proper in one locale as opposed to another, whereas another may consider quite the converse. In such a case where there is such a pronounced difference in individuality, quantitative judgment fails.
One might consider if the condition was to simply restrict acceptable judgments to those which already have a standard, but this leads to profoundly circular reasoning. If we wish to derive a new standard for a concept (say, politeness), we would need to obtain data which could be assigned to numeric symbols; however, in order to do so, we would need a standard to obey. A similar argument applies to a condition which is expanded to include things which may or may not be quantifiable; we will not be able to tell what is what.
If we accept temporarily that indirect measurements can be obtained via direct measurements,* it is inferable that indirect measurements should be obtainable from comparative judgments of direct measurements. A commonly agreed upon indirect measurement is that of density, being the ratio of mass to volume, both direct measurements.* Since we may obtain direct measurements, it is simple to transform them into an indirect measurement.
Campbell applies a similar train of thought. In order for a property to be quantitatively measurable, it is necessary for systems (worlds, possibly) that differ in the said property to be related by some transitive asymmetrical relation R. (273) If R is in a class C of systems, and X is in C, X must have either a relation or the converse of said relation to every other system in C..
What may appear somewhat confusing is that the phrasing implies that there may only exist one relation or its converse which applies to every system in C != X. Viewed as such, this causes a problem of specificity. For such a system to work within the bounds of quantity, the only acceptable solution would be to set the relation such that X >= Y (or X <= Y); then the relation, or its converse, would be true for every other quantity. Yet, by doing so, we cannot judge when X = Y directly; in fact, when X = Y it creates a distinct problem not unlike the mutual exclusivity problem. Is X >= or <= Y?
In fact, it is probably more likely that Campbell meant a relation set R instead of merely one relation. This appeals much more to people, in that it safely allows the relations defined for quantity under its wing.
III(a). How do we represent quantities? How do numerals and numbers differ?
The typical representation of a quantity is in numerals. For instance, we refer to the value 53.76 in the understanding of the metalanguage value being the real number 53.76 (however the brain understands such a number). However, 53.76 is not the only way to state the value of 53.76.* To clarify the vocabulary: a numeral and a number parallel the structure of the object language and the metalanguage. The numeral is used to represent the number.
Campbell makes a case for the distinction and somewhat unreliable nature of representation of numbers via numerals. (274) He proceeds to define two classes of quantifiable properties: fixed properties and arbitrary. Fixed are those in which the representation accurately follows the ratio of the number in whatever object language. (For instance, length. If something measures 4 feet, and another measures 2 feet, the ratio is 2:1 in both the numeral and the number, which qualifies it as a fixed ratio.) There is not much that is argued about such properties; they behave as expected.
Arbitrary properties is where Campbell’s case rests, as he notes that this phenomena occurs primarily in derived (or indirect) measurements. (282). One example which is so is volume amplitude determination (the decibel scale). As it currently stands, a volume which has a value of 800 as opposed to a value of 400 on the scale does not vary by a ratio of 2:1, but instead by a power of 2, as the scale operates logarithmically. Campbell makes use of this scale and many others (Mohs’ scale for hardness, for instance) to show that ultimately, the numeral chosen to represent the number needs not follow the expected ratio; indeed, Campbell goes so far as to claim that numeric representations can be entirely arbitrary.
Numerals, as one might suppose, are not necessarily restricted to numbers. In fact, numerals can be in any particular language, natural or otherwise. One can replace [1..9] with [A..I], if one should choose, without affecting the number underneath. One can also rank by presidents, dukes in England, vintage cars, or any other sort of representation which can be ordered. But again, even here, an arbitrary nature be determinate; the requirement is of some sort of order hierarchy.
The inefficiency of numeral representation can manifest itself in nonsensical events. Consider the numeric representation of mass. If you combine 2kg with 2kg, the result is 4kg, which is in line with our expectations of 2+2 = 4. However, consider the numeric representation of density. If you combine an item with a density of 2kg/m3 with another item of density 2kg/m3, the result is an item of density 2kg/m3 , though our arithmetic would have indicated otherwise. No representation can be selected to fix the inconsistency of the latter math, as the trouble lies in that any representation of density is simply inadequate to handle the truth of the situation properly.
III(b). How can direct and indirect measurements overlap?
Something apparently as clear-cut as being direct as opposed to being indirect appears to be devoid of conflict on the surface. As it stands with some properties, being x has little to do with y method of finding it. What obfuscates the matter even more so is that while properties can be found in the opposing fashion, they are not considered both in/direct in any systematic fashion.
Returning to a previously unanswered point, how is it that Campbell can justify claiming weight as a direct measurement when we rely on scales or other measuring devices? (Campbell 277) At the heart of this question lies one of the greater confusions of the matter. Weight may not be precisely identifiable by simply lifting a separate mass in each hand, but what matters in the scenario is that the person is able to distinguish which mass is greater or if they are equal. Finding such a relation is a key part of becoming a measurement as explicated above.
So, what of the scales? This is one such instance of using indirect measuring on a direct property. (Kyburg, 112) Length, mass, and many other properties are quite direct, but yet we often opt for an indirect approach.* Note that this does not allocate any direct properties into the indirect category (although arguments for dual categorization exist). Indeed, many direct measurements can be obtained by the interaction of indirect measurement (for example, obtaining distance traveled = velocity * time). (Ibid, 138)
Is the converse possible? One may recall Archimedes’ discovery of volume by dropping objects in water and measuring displacement. If we extend the experiment to density, we can obtain relative densities of two objects (if both objects share a similar mass and volume) by sinking both into equal amounts of the same fluid. (Campbell, 276) The result (whether the indication is a rising fluid level or amount of overflow) is measurable by simple perception, thus making a density measurement possible direct.
Though the boundaries of what appear to be direct and indirect are not quite solid, many still take the time to make the distinctions. It is most likely that these distinctions remain due to commonality (density is usually derived, not sunk) and just common sense (mass is certainly easier to measure than force).
III(c). How does the concept of error figure into measurement?
Both direct and indirect measurements suffer from imprecision. Though the degrees vary, certainly it would be unrealistic to deny that there is no way to obtain an absolutely error-free measurement.
Direct measurements have the most pronounced flaws originating from human error. Kyburg points out our lack of ability to discriminate at relatively small precision. (114) For instance, if you were to ask someone to stick his/her hand in 50°C water, the response would be that it was hot. Surreptitiously adding more water such that the temperature drops to 49.5° may register a difference of volume, but not temperature.
Experience also tells us that humans are less capable of discrimination as one progresses closer to extremes. (Indeed, experience is everything in direct measurement; a lifetime of observations provide the necessary comparisons needed for cases where only one quantity is being considered. [Ibid, 37]) Certainly, 25°C and 0° C weather is enough for someone to change from less protective clothes to layers. But all a person can sense in a temperature shift from —50°C to —75°C is nothing but the undeniable freezing sensation (shortly before the person loses consciousness).
Indirect measurements involving the use of instruments generally do not substitute human error, but instead place instrumental error on top of it. Campbell discusses a simple method of determining the maximum bound of error on any given instrument. (447) However, what soon comes into play as well is that we directly measure the output of an instrument.
Then why do we use instruments at all, if all they do is add additional error to our already flawed perception? The answer lies in how the instrument "re-calibrates" error. This may be achieved in two ways: by displacing the interval or by replacing the interval. Continuing the temperature example used above, a person cannot distinguish a .5°C change at that temperature. Inserting a thermometer, however, will present the change of temperature as either a change in volume or a digital display (change of numbers). Both are examples of interval substitution in that they recast the difference in a different light. The other tactic is often used to enhance or enlarge things beyond our normal capacity. A microscope allows us to view things at incredible magnifications, most of which we cannot view visibly. The estimate dimensions of an organic cell, though incapable in the capacity of the unaided eye, suddenly become measurable and reasonable once magnified to a size in which the eye can more reliably make judgments. Thus, though instruments may add a degree of error, they reduce the human degree of error in some method considerably.
Culminating the discussion so far would seem to reduce the question to whether a direct measurement can be as scientifically effective as an indirect measurement. Typically, indirect measurements are considered more precise, often having number values attached to them, whereas direct measurements generally are left in the form of the mutually exclusive relations <, =, >.
Kyburg notes that one should NOT expect any logical relation between what appears to be and what actually is. (62) Indeed, there really are no grounds to accept a statement that "x is really such-and-such" unless one stays in a purely theoretical ground.* Though a statement as such is counterintuitive, a simple example should demonstrate its factuality.
Consider any two lines. Generally, a person is able to determine with good accuracy whether one line is longer than the other or if both are equal lengths. However, as mentioned above, we cannot see smaller than a certain distance, so if the lines differ by less than said length, we are wrong. Furthermore, our perception of the length of a line is often "adjusted" by surrounding information (the perception test which asks if <————> or >————< is a longer line comes to mind), furthering our inaccuracy. The result is that we may be wrong by relying on what "seems" so against what we can tell with greater accuracy. Is this to imply that our perceptions are flawed?*
So far, we have not defined what scientifically effective means. Let us define effectiveness as maximal meaningfulness, and demonstrate by example (and also demonstrate another flaw of human error in direct measurement). Suppose that another planet is discovered close by. Scientists are interested in sending a satellite to monitor it. To find enough information to be useful to the engineering crew (maximally meaningful), scientists will have to measure the distance to it to a specific precision. By human perception (even aided by a telescope), the best one is able to come up with in terms of distance is perhaps an approximation relative to the distance of other planets. In the worst case, "far." Instruments and tools can refine the interval of possible distance drastically, thereby maximizing meaningfulness and thus being more efficient.
Perhaps an accurate model for consideration of the reality of "relative" against "absolute" is to consider the human body and its senses as an instrument in itself. In such a light, other instruments serve to sharpen our interval of confidence in values. Though we are free to assign values to what we perceive (eg this much water as one, that much water as two), it is to our advantage to rely on values derived from measurements with smaller error boundaries. After all, unlike the postmodernist view, reality is not what we perceive it to be. It is what the evidence supports. (Stenger)
Works Cited
Campbell, Norman Robert. Foundations of Science: philosophy of theory and experimentation. Cambridge University Press/Dover, Great Britain: 1957.
Cormen, Thomas H. et al. Introduction to Algorithms. MIT Press/McGraw-Hill, Massachusetts: 1999.
Ellis, Brian. Basic Concepts of Measurement. Cambridge University Press, Great Britain: 1966. Kyburg, Henry E. Jr. Theory and Measurement. Cambridge University Press/University of Rochester Press, Rochester: 1984.
Stenger, Victor J. "’Postmodern’ Attacks on Science and Reality." Internet. URL: http://www.quackwatch.com/ 01QuackeryRelatedTopics/reality.html. 1998.