The Plane of Existence

Mathematics is the language of science. It, like science, is a collection of procedures with steps. The spoken number “seven”  or the marks we make when we write “7,” are, like the word “slab,” instructions for a procedure. A number is not an entity in am alternate world of forms or an attribute of things. it is instructions for a variety of procedures determined by what we are doing. Just what procedure is, as with the word "slab,"  determined by context. The procedure for counting requires repeating sounds, writing marks on paper or typing them so that they appear on a screen in an exact order. If we notice that the numbers in 3+7 = x are three and seven we can substitute 10 for "x." This expression, "3+7=x", taken as a whole, is an instruction for a counting procedure that results in the number "10" Once we know the procedure  we can simply write down the predicted result. When we “understand,” that is understand how to use, the numbers and the plus sign we can perform the operation.

 

Mathematics involves procedures that require no effort, other than the effort to know what they are. To know what to do is to have already done it. All we have to do is write it down. Mathematics reduces procedures to, so to speak, bare bones. Many experimental procedures require skill beyond merely knowing what to do. In mathematics, stating the answer, “10," the purpose of the operation, is doing the step. 

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The predicted result of an experiment is a mathematical manipulation of several measurements.

 

The simplest operation with numbers is, of course, counting, uttering a collection of distinct sounds, writing a collection of distinct marks on a page or making such marks appear on a screen in a strict order. Counting can be counting something, in particular, token of a type. The word "counting" joins the word to adding something to a pile.

 

Measurements that use numbers but are not directly done by counting, for example weighing a collection of coins, might be checked by counting if what is weighed can be divided into units that, when measured, give the same measurement, in this case individual coins. This is not always convenient.

 

In addition to numbers mathematics also includes mathematical operators like “+.” In 3+7=x , “+” directs us to perform a counting operation with 3 and 7 to produce their sum and replace x with the answer. The plus sign indicates a more complex operation based on counting.  we start counting from 3 and, simultaneously, start counting from one. When the counter starting from one gets to seven the other will get to 10, which is the answer. Of course other elaborations upon the counting games, for example what “-” instructs us to do, quickly proliferate. All kinds of complex procedures, all symbolic, can be done with these little marks on the page. When we have written one expression it allows us to write another, all justified by various procedures with numbers. The student quickly learns that answer and can so skip actually performing the counting routine.

 

Mathematical sentences must unambiguously designate a procedure, but that lack of ambiguity extends only to the mathematics. Since counting can not only be simply counting, but can also be counting something, that procedure, of counting something and not just counting,  can differ depending upon our purpose. If we had a pile of various sized stones and wanted to know its mass it would not do to count the stones. 

 

The minus sign, “-” is an instruction to employ addition operations modified by counting backwards until we get to one, then insert a new number, “0,” and then count forwards again saying “minus” before each number. To do so we have to add zero and minus numbers as mathematical entities. If we start positive counting from a negative number we again count backwards until we get to "0" and then count forwards.

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Multiplication of 3 x 7 requires us to make a rectangle of countable things three on one side and seven on the other and then count the things in the rectangle. And of course there are many new operations that can be done with rectangular arrays of whatever. Mathematics supplies instructions for measurements. Although it sounds strange counting from one to ten is a measurement of the number of numbers between one and ten. And as long as we are just counting we can substitute any kind of token for those numbers.

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