Philosophy Of Matematics And Language

.. #61552; ? = 3. The ancient Egyptian manuscript called the Rhind papyrus gives another approximation: ? ? = 3.1604938. Such approximations represented the standard in mathematical logic of the time period. To the respective members of the cultures, ? was a number not unlike the every numbers they dealt with; the difference was they didnt know its exact value.

The above ap-proximations of ? 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 ?. 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 ?. 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 ?. This process was very tedious (one mathematician did his calculations on a polygon with 262roughly 4,610, 000,000,000,000,000 sidesto find the value of ? cor-rect to 35 decimal places), but it provided a new way of conceptualizing the number ?.

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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 cant 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: ? = 1 + + + + Suddenly, the matter of approximating ? [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 ? was. The above discussion uses the quest for the number ? 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 doesnt 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 ?) 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 isnt 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 arent 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), its 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, its necessary to first define, for example, what ? is exactly, and derive all other mathematical relationships involving ? from that. However, since the development states exact the nature of ?, all other results are not much more than mathematical coincidences; they become part of what is ? 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 ? can be (intrinsically) many things at once, because there is no definition that nails down one interpretation of ?. 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 dont have to scrap the whole idea of logic all together; we must, however, understand that logic isnt neces-sarily the truth, and always is neither the whole truth and nothing but the truth. Mathematics.

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