| Mathematically inclined philosophers often bemoan the restrictions of a natural language such as English; it is not precise enough to express mathematical concepts with clarity. One problem is an inherent vagueness in the way expressions refer to the natural world. Another is that natural language allows a sometimes uncomfortable level of ambiguity. A third problem is redundancy, which may be either intentional or unintentional. It turns out that redundancy in the normal circumstances of communication is actually an asset. Ambiguity is due in part to the fact that natural languages tend to be very flexible. Vagueness or imprecision is a part of the natural world. The philosopher may try to reduce any negative effects these problems might produce by being very careful in one’s expression. In any case natural languages are creatures of convention. It is this aspect that seems to prevent wholesale modification. And so we have tremendous variation between different language users. It seems that everyone differs in disposition toward correctness. It seems that everyone has a different concept of what should be the standard of correctness and understandability. |
An algebraic expression.A certain logician in illustrating the drawbacks of expressing mathematical concepts in the expressions of a natural language drew up the following algebraic expression for his example (Lemmon 1965:3): |
| This logician then attempted the following formulation of this equation in English: |
| (1) | The result of subtracting the square of one number from the square of a second gives the same number as is obtained by adding the two numbers, subtracting the first from the second, and then multiplying the results of these two calculations. |
“Correcting” the grammar.As an English speaker this logician probably didn’t realize nor care that his sentence fails to meet some of the standards of a strict English grammar. Or maybe his point is that it doesn’t need to. Perhaps it is the sheer size of the sentence that has made it difficult for the author to maintain grammaticality. Nevertheless he expects the reader to be able to infer its meaning. Note that the verb of the main clause gives would have been appropriate for the subject if the subject had been subtracting. The actual grammatical subject was result. We will take the liberty of “correcting” his expression to the following, as it would be grammatically more proper and in a sense less redundant. In any event it will be easier to analyze. |
| (1) | The result of subtracting the square of one number from the square of a second is the same number as is obtained by adding the two numbers, subtracting the first from the second, and then multiplying the results of these two calculations. |
Paraphrasing clauses and phrases.With this adjustment the analyst is in a better position to proceed with a paratactic paraphrase. We will study these structures later, suffice it here to point out that we have a degree clause in which there is a series of three conjoined gerund phrases. |
| (1P) | a. | A number1 is obtained by adding one number2 and a second number3. | |
| b. | A number4 is obtained by subtracting one number3 from a second number2. | ||
| c. | A number5 is obtained by then multiplying the results1,4 of these two calculations. | ||
| d. | The result6 of subtracting the square of one number3 from the square of a second2 is the same number5. |
| Despite the fact that the analyst has been very careful to be clear in (1P), the anaphora makes this version somewhat convoluted. The proper interpretation of the anaphora has become the source of ambiguity. To understand the sentences together as a single concept the reader has to be able to infer which number is which. In straightening out the syntax, references may become confused. The biggest culprit seems to be the reversal of arguments when we use the word subtract. |
Atomic analysis.Let’s straighten out the syntax even more. The four component sentences may multiply. These then paraphrase into sixteen sentences further compounding the difficulty that the reader may have in keeping straight all the various references to the different numbers and intermediate results. With the tools developed for logical analysis, however, it should not be too difficult to translate these sentences into the underlying logical propositions. |
| (1P') | a. | One number is the first number. | |
| b. | One number is the second number. | ||
| c. | Someone may add these two numbers. | ||
| d. | This is a calculation. | ||
| e. | It has a result. | ||
| f. | Someone may subtract these two numbers. | ||
| g. | This is a calculation. | ||
| h. | It has a result. | ||
| i. | Someone may then multiply these results. | ||
| j. | Someone obtains a number thereby. | ||
| k. | The first number has a square. | ||
| l. | The second number has a square. | ||
| m. | Someone may subtract these two squares. | ||
| n. | This has a result. | ||
| o. | This is a number. | ||
| p. | This number is the same number as that. |
Mathematics and grammaticality.This example shows that mathematics, in particular algebra, is the language more suited to expressing the concepts of numbers, their operations and relationships. Without some standard of grammaticality, however, mathematics could be as confused as the English of the logician. While paraphrase can clarify certain relationships, it does not succeed in matching the precision of reference that is built into mathematical convention. Many fields of endeavor remain obdurate to attempts at mathematization. Part of the problem is the vagueness of their concepts. Description by means of a natural language allows the student to clarify the relevant concepts to progressively higher levels of understanding. It is possible to view grammatical expressions as one step closer to mathematization. |
