The Great Math Mystery

Last night we watched The Great Math Mystery, a NOVA episode on PBS television. It was excellent and I recommend it. It can be watched on-line here at least temporarily. There is a full transcript, too. The mystery is:  Is math invented by humans, or is it the language of the universe? Reasons are given for both -- some math is invented and some is discovered.  I believe the best answer came near the end. Math concepts such as numbers are abstracted by humans, but then they and their relationships are found to apply beyond their origin and lead to further discoveries.

The topics include the Fibonacci sequence, the number pi, Galileo's mathematics of falling bodies, Maxwell's equations, Marconi's discovery of radio telegraphy, the quantitative intelligence of lemurs, and the difference between pure math and applied math. Regarding the last, pure math is exact and imaginative but becomes much more useful via approximating with short-cuts such as done by engineers.

How We Know #8: Higher Level Concepts

“The intensity of seeing varies in regard to: clarity, acuity, time, attention, and purpose” (HWK, 164).

"Considering the wealth of conceptual subdivisions of "seeing" that have been formed to capture sub-ranges within the above axes of measurement. Here are some, listed in alphabetical order: descry, espy, gawk, glance, glimpse, look, ogle, peak, scan, stare, watch" (165).

Huh? What axes of measurements? There are some measurable differences, e.g. time, between these subdivisions. On the other hand, they are not fully sorted by time. Moreover, there are qualitative differences as well that Binswanger does not acknowledge as qualitative, for example, different purposes. Also, regarding these alleged measurements, what standard unit analogous to an inch and what measurement instrument analogous to a ruler or tape measure apply? For the sake of argument hypothesize such a standard unit. How is it that gawk is N₁ of said units, glance is N₂ of said units, scan is N₃ of said units, and so forth, where the N's are non-ordinal numbers? To echo a frequent comment Ayn Rand made: Blank out. Am I using a different meaning of "measurement" than Binswanger? Yes, one that is more rigorous, objective, and based in perception, which is the ultimate base of all knowledge. It is not some fuzzy or corrupted meaning.

Binswanger does not say what measurements are "omitted” for the concept motion. He only mentions "measurement ranges that were left open in forming" the concept (154). Regardless, the concept motion highlights qualitative differences even more. Varieties of motion include walking, running, crawling, flying, riding, swimming, jumping, rolling, swinging, and dancing. Non-human motions would add many more varieties. Are the differences between all these subdivisions of motion solely a matter of measurements? Clearly not; they differ qualitatively. For example, swimming is in water and the others are not. Riding in a car is different in multiple ways from the others. Running, walking, crawling, jumping and dancing use the legs in qualitatively different ways.

It also follows that, contra Binswanger (p. 166-7) and Rand, teleological measurement is a flimsy metaphor. It is teleological ranking. The differences between authentic measurement as I described above and ranking overwhelm their similarity.

How We Know #6: Concept-Formation

Binswanger says, "Things that are similar differ quantitatively" (HWK, 110). Then noting that a young child may not see a pig and a collie as similar, he says in all cases similar concretes possess varying degrees of the distinguishing characteristic, and those degrees fall within “specified categories of measurement” – which is why they appear as similar. Ayn Rand called this “measurement omission.” (HWK, 115).

In effect he denies that similar things or attributes can have only qualitative differences. Similarly, in the Appendix of ITOE2 Rand assented to “establish the similarity by showing the characteristic is the same and only [emphasis mine] the measurements vary” (ITOE2, p. 221).

Is it true that all differences between units that fall under the same concept are only quantitative? Both Rand and Binswanger say yes. I say absolutely not. There are many, many exceptions. It takes only one to disprove their claim, but I will offer more anyway.

Consider boats. Having some means of locomotion is essential to being a boat. There are oars, sails, outboard motors, water jets, paddle-wheels, air fans, and inboard engines of various kinds – steam, gas, diesel, nuclear, electric, coal. These are qualitative differences, not quantitative ones. One attribute – speed – of locomotion and even other differences being measurable does not imply that every attribute is measurable.

Consider animals. Some live on land, some in water, some both. Some are carnivores, some herbivores, some omnivores. For some respiration uses lungs, others gills, still others skin. Some have fur, some have scales, and some have feathers. I could go with many other kinds, not simply degrees, of differences. These are qualitative differences, not quantitative ones.

Consider different tools -- hammer, screwdriver, wrench, pliers, file, saw, etc. Each has a different purpose, which is qualitative, not quantitative.

I will defer qualitative, non-quantitative, differences of motion to a future post.

Binswanger says nothing about it, but later in ITOE Rand undercut her prior claim of omitting only measurements when she addressed concepts of consciousness.

“For instance, the concept “thought” is formed by retaining the distinguishing characteristics of the psychological action (a purposely directed process of cognition) and by omitting the particular contents as well as the degree of the intellectual effort’s intensity. The concept “emotion” is formed by retaining the distinguishing characteristics of the psychological action (an automatic response proceeding from an evaluation of an existent) and by omitting the particular contents (the existents) as well as the degree of emotional intensity” (ITOE, 32).

“These concepts [knowledge, science, idea, etc.] are formed by retaining their distinguishing characteristics and omitting their content. For instance, the concept “knowledge” is formed by retaining its distinguishing characteristics (a mental grasp of a fact(s) of reality, reached either by perceptual observation or by a process of reason based on perceptual observation) and omitting the particular fact(s) involved” (ibid., 35).

Why did she say omitting particular “contents” and “facts”? What happened to omitting only measurements?

Binswanger also confuses counting, e.g. the number of sides of a polygon and atomic numbers, with measurement. “An interesting case of measurement is that of measuring materials qua materials, such as wood, copper, water. Obviously, one can measure the attributes of the objects formed out of various materials, but in what sense is the difference between copper and lead a difference in measurement? On the sensory level, one uses difference in perceptible qualities—the colors differ, the densities differ, the hardness differs, etc. … Modern chemistry, however, goes to a deeper level: copper and lead differ in “atomic number.” Atomic number is a measurement. It refers to the number of protons in the nucleus of the atom: copper has 29 protons, lead has 82” (p. 121).

However, counting and measurement are quite different. Both use numbers, but counting uses only integers and measuring uses both integers and fractions. Authentic measuring as done by scientists, engineers, and others uses a measuring instrument – a ruler, weight scale, thermometer, voltmeter, pressure gauge, etc. Counting does not rely on such instruments.

By the way, what is the atomic number of wood? 😊

Note: Some of the above is repeated from my article 'Omissions and Measurement' in The Journal of Ayn Rand Studies, Vol. 7, No. 2 (Spring 2006). Link. Another related article is my 'The Sim-Dif Model and Comparison' that appeared in The Journal of Ayn Rand Studies, Vol. 11, No. 2 in December, 2011. Link. Either can be read on-line for no money with a free JSTOR account.

How We Know #3: Concept-Formation

Chapter 3 is devoted to the nature of concepts. Binswanger says there are four main theories -- Realism (e.g. Plato), Moderate Realism (e.g. Aristotle), Nominalism (e.g. Wittgenstein), and Objectivist (Ayn Rand).

"According to Realism, a concept is a term that designates a metaphysical universal: a special kind of non-specific element present in all the members of a class, an element that is grasped directly by some sort of non-sensory "intuition" or "insight"" (p.101).

"Moderate Realists count as realists because they hold that abstraction refers to metaphysical universals; the theory is "moderate" in holding that these universals exist as aspects of perceptual concretes, not as separate entities dwelling in another world. In effect, Moderate Realism shatters the Platonic Form and puts a fragment of it inside each concrete" (p.102).

Binswanger elaborates his version of Objectivist epistemology. 

Binswanger says more about measurement omission than Ayn Rand did. He quotes Rand: "If a child considers a match, a pencil, and a stick, he observes that length is the attribute they have in common, but their specific lengths differ. The difference is one of measurement. In order to form the concept "length," the child's mind retains the attribute and omits the measurements. Or, more precisely, if the process were identified in words, it would consist of the following: Length must exist in some quantity, but may exist in any quantity. I shall identify 'length' as that attribute of any existent possessing it which can be quantitatively related to a unit of length, without specifying the quantity." [ITOE, 11]

Rand seems to say that "omitting measurements" is only omitting specific numbers -- of inches or centimeters or whatnot -- not omitting the attribute length. The match, pencil, and stick each have a length in reality, even if the child knows nothing about authentic measurement. In other words, there is length (metaphysical, ontological) and measured length (epistemological). The former can be simply perceived; the latter requires a special effort. The former doesn't require numbers; the latter does. The former is not a comparison; the latter is a comparison of two lengths, one from a measuring instrument (ruler or tape measure or whatnot).

Binswanger does not say what I just did, but he portrays the child's understanding of length somewhat differently than Rand did. He writes: "In speaking of "measurements" I am referring to the subconscious mechanics of the concept-forming process, not to any consciously performed, explicit, process of measuring. A child beginning to conceptualize things is, of course, incapable of explicit measurement. On the conscious level, he is only aware of similarities and differences. But the objective basis of those similarities and differences is the quantitative variation of a commensurable characteristic" (p.118). He does not claim the child implicitly measures.

Binswanger tries to explain that humans don't really omit measurements. More exactly they recognize that measurements vary. He says that things that are similar differ quantitatively. "Similarity is measurement proximity. "Proximity" is a relative term, depending on a contrast with something that is more distant, which can be called "the foil." Similarity is thus contextual, a matter of relative proximity of measurements in contrast to the relatively distant measurement of a foil. In such a set-up, the bigger difference swamps the smaller difference, making the smaller difference appear as similarity. What is experienced as similarity is, at root, lesser difference" (p. 112).

Here at least the smaller differences among similar things are swamped rather than omitted. 

There are also sections of Chapter 3 on integration and unit-economy.