Philosophy Of Matematics And Language Essay, Research Paper
Throughout its history mankind has wondered about his place in the universe. In fact, second only to the existence of God, this subject is the most frequent topic of philo-sophical analysis. However, these two questions are very similar, to the point that in some philosophical analyses the questions are synonymous. In these particular philoso-phies, God takes the form of the universe itself or, more accurately, the structure and function of the universe. In any case, rather than conjecturing that God is some omnipo-tent being, supporters of this philosophy expound upon another attribute habitually asso-ciated with the Man Upstairs: His omniscience. That particular word, omniscience, is broken down to semantic components and taken literally: science is the pursuit of knowl-edge, and God is the possession of all knowledge. This interpretation seems very rigor-ous but has some unfortunate side effects, one of them being that any pursuit of knowl-edge is in fact a pursuit to become as God or be a god (lower case ?g?). To avoid this drawback, philosophers frequently say that God is more accurately described as the knowledge itself, rather than the custody of it. According to this model, knowledge is the language of the nature, the ?pure language? that defines the structure and function of the universe.
There are many benefits to this approach. Most superficially, classifying the structure and function of the universe as a language allows us to apply lingual analysis to the philosophy of God. The benefits, however, go beyond the superficial. This subtle modification makes the pursuit of knowledge a function of its usage rather than its pos-session, implying that one who has knowledge sees the universe in its naked truth. Knowledge becomes a form of enlightenment, and the search for it becomes more admi-rable than narcissistic. Another fortunate by-product of this interpretation is its universal applicability: all forms of knowledge short of totality are on the way to becoming spiritu-ally fit. This model of the spiritual universe is in frequent use today because it not only gives legitimacy to science, but it exalts it to the most high. The pedantic becomes the cream of the societal crop and scientists become holy men. It?s completely consistent with the belief that mans ability to attain knowledge promotes him over every other spe-cies on Earth, and it sanctions the stratification of a society based on scholarship, a mold that has been in use for some time.
Now that we?ve defined the structure and function of the universe as knowledge, we must now further analyze our definition by analyzing knowledge itself. If the society is stratified by knowledge, there must be some competent way of measuring the quantity of knowledge an individual possesses, which means one must have a very articulate and rigorous notion of knowledge. At first glance, one would think that knowledge was sim-ply the understanding of the universe through the possession of facts about it. This un-derstanding creates problems, however, because it now becomes necessary to stratify knowledge, to say that this bit of information is inherently ?better? than that one. This question was first answered using utility as a metric, but it became obsolete because util-ity is too relative. A new, more practical answer was eventually found: rather than meas-uring knowledge, we should measure intellect, the ability to attain knowledge. Even though this has the same problem of stratification, it?s overlooked because philosophers believe that they know the best way to pursue knowledge. To them, the language of complete understanding is logical inference. If one can state a set of facts in the simplis-tic linear progression of statements using logical connectors, the information is in its most readily understandable form. The philosophers used this convention to rigorize mathe-matics, the rigorization process became associated with it, and logic suddenly became mathematical logic. The name stuck, as people refer to the process by that name to this day.
The previous analytic development is the essence of the modern understanding of the natural universe. It starts from the fundamental belief in a deity and transforms it into this mathematical logic, a system of communication that according to our summation minimizes the number of justifiable interpretations, therefore standardizing the universe. There are some limitations to this approach, however. The rationale is, by its very nature, a logical development: it constructs a functional model of the pure language that is con-sistent (i.e., free of contradiction). Therefore, the pure language inherits any limitations of logic by definition?in other words, it assumes that the pure language is (a subset of) logic. Secondly, even though it?s very rigorous in its approach, it presents pure language as an inherent truth viewed through the lens of mathematical logic, as opposed to pure language being synonymous with mathematical logic. This is an important but distinc-tion, but its subtle temperaments cause it to be frequently overlooked.
There are many ways to demonstrate the distinction between pure language and mathematical logic, most of which rely on the exhaustive nature of the pure language (as opposed to the restricted nature of mathematical logic). One particularly interesting way is to exploit their language status, and demonstrate a difference by contrasting their dif-ferent responses to a property of all languages: their evolution. The pure language is by definition the structure and function of the universe, i.e., therefore, change is taken into account in the definition (i.e., the ?function? of the universe). Therefore all kinds of lin-gual evolution are subsets of the pure language, and so the pure language is invariant relative to lingual evolution. (For example, assume that the pure language was changed from its original form to a variation of itself by a form of lingual evolution. What is the new variation? Well, since the lingual evolution is under the category of the pure lan-guage, the variation must be under it as well. Therefore no change really took place.) Contrast this with mathematical logic, a body of knowledge that evolves through use just as a spoken language. However, any changes in mathematical logic that develop through use aren?t referred to as such: we call such modifications mathematical discoveries. A mathematical discovery is considered to be ?fitter? than is evolutionary prerequisite, and the former is usually discarded to a text on the history of the subject. Hence, we see mathematical logic as a static body of knowledge that we change from time to time to fit our needs (which happens to be in this case, the need to be more correct)?synonymous with any spoken language.
An example of the evolution of mathematical logic is found in the varied ap-proaches for the approximations of the number p. The number p is a commercial icon in the pure language whose decimal expansion (approximately 3.1415926535?) goes on forever, never repeating, never terminating. The first approximations of this number come from ancient manuscripts, like the Christian Bible. In I Kings 7:23, the authors used a sheer estimation of the circumference of a circular lake, divided by its diameter, to get a crude approximation of p:
p » = 3.
The ancient Egyptian manuscript called the Rhind papyrus gives another approximation:
p » = 3.1604938?.
Such approximations represented the standard in mathematical logic of the time period. To the respective members of the cultures, p was a number not unlike the every numbers they dealt with; the difference was they didn?t know it?s exact value. The above ap-proximations of p were the closest that they could get to capturing the ever-elusive num-ber; therefore, after many years of use in the society, the approximation and the number itself became virtually indistinguishable. The line was blurred between the pure language and the mathematical logic that approximated it and, practically speaking, the number became p.
This was the case until just after the turn of the age, about 150 BC, when the sec-ond phase of approximation began. Fellows like Archimedes and Ptolemy used geomet-ric means to approximate p. They took geometric shapes with equal sides, and calculated the ratio of their perimeter to their diameter to get an estimation of the constant. Then, they doubled the number of sides, and re-calculated the ratio to get a better approxima-tion of p. This process was very tedious (one mathematician did his calculations on a polygon with 262?roughly 4,610, 000,000,000,000,000 sides?to find the value of p cor-rect to 35 decimal places), but it provided a new way of conceptualizing the number p. Rather than thinking of it as a simple number like the rest of the numbers they knew, people now thought of this member of the pure language as the holy grail of a geometric quest that had no end. One could continue to increase the number of sides of various regular polygons to get closer and closer to it, but in the end this geometric limit was un-attainable (because we simply can?t draw a perfect circle).
Then the fifteenth century rolls around, and the famous mathematicians Newton and Leibniz discovered the calculus. When applied to this old problem, they found that if we continually added and subtracted the following fractions, we got closer and closer to the elusive constant:
p = 1 ? + ? + ? + ? + ?
?Suddenly, the matter of approximating p [and therefore this part of the pure language] turned from the geometric problem it had been with Archimedes? regular polygons to a simple arithmetic problem of and adding and subtracting numerical terms. This was a major change in perspective.? (Dunham, p.108) This shift in perspective was a result of the discovery of calculus, and would the new trend in mathematics. Just as in oral lan-guage, use (i.e., the use of logic to produce mathematical discovery) intrinsically changed the conception of what exactly p was.
The above discussion uses the quest for the number p to reveal two forms of evolution apparent in mathematical logic. The first is an ?unofficial? evolution, i.e., practical evolution that results from years of use, while the second is an ?official? evolu-tion, i.e., evolution that is a result of logical deduction. Since the pure language of the universe doesn?t exhibit such change, these demonstrate that mathematical logic is inher-ently different from the structure and function of the universe. There is, however, a re-buttal to the above argument, another modification to the logical construction that seem-ingly makes this difference disappear. If we assume that the pure language is consistent (i.e., contains no contradictions), we can define mathematical logic to be a translation of the pure language, and define our discovery of the language (e.g., our approximations of p) to be the lens we view it through. That way, logic is still the Supreme Being, and the pursuit of it is again legitimized. All our problems are solved.
The problem with such a modification to our definitions is that it isn?t consistent with our practice. Because mathematical logic (or our conception of it at least) is a lan-guage, it has evolved considerably from its definition. Now, math excursions aren?t per-formed through discovery, but through construction: mathematicians state axioms (as-sumptions) and definitions, and logically derive all of mathematical from them. Mathe-maticians believe this process to be more rigorous than any other method of proof in that, aside from the ubiquitous set of axioms (axioms are a necessary part of every construc-tion), it?s logically impeccable. The quest for the truth has become a secondary concern, and the quest for the logically consistent has ran to the top of our list of priorities. For example, in the widely-accepted construction of the field of analysis (one of three ex-haustive subcategories of math), arithmetic involving infinity is defined in such a way that is inconsistent with what we know from other mathematical excursions to be true:
It may seem strange to define 0 ´ ¥ = 0. [According to the pure language, the value of this ex-pression can equal zero as well as any other finite number.] However, one verifies without diffi-culty that with this definition the commutative, associative, and distributive laws hold on [all of the numbers from zero to infinity] without any restriction. (Rudin, p.18)
This reveals a subtle but intrinsic difference between the pure language of the universe (i.e., the truth) and mathematical logic in practice today.
Another aspect of the logical construction that distinguishes it from the pure lan-guage is the linear progression. By its very nature, every logical argument is linear in its development: A implies B, implies C, implies D, etc. But, every line has a beginning, i.e., every logical construction has a beginning, a group of definitions and axioms from which all other results derive. (This seemingly obvious fact was stated earlier and even-tually logically proven.) Therefore, it?s necessary to first define, for example, what p is exactly, and derive all other mathematical relationships involving p from that. However, since the development states exact the nature of p, all other results are not much more than mathematical coincidences; they become part of what is p only in another construc-tion, where these facts are taken into account in the definition. This is not true of the pure language: as has become more and more apparent in science since the 1950s (and the new mathematics that arouse from it), nature is very non-linear. This means that there is no beginning or end to the truth: the number p can be (intrinsically) many things at once, because there is no definition that nails down one interpretation of p. Even though mathematical logic can be used to see the truth, the truth becomes unavoidably biased by it.
There are many shortcomings of logic that keep it from being the pure language, the absolute truth, the Man Upstairs. Yet and still we have embraced this theology whole-heartedly (if not consciously, through societal conditioning). Our desire to com-pletely understand the universe (along with our belief that we can completely understand the universe) has blinded us into accepting falsehoods as facts. We don?t have to scrap the whole idea of logic all together; we must, however, understand that logic isn?t neces-sarily the truth, and always is neither the whole truth and nothing but the truth.