A ‘concept of method’ deals with how we deal with numbers. Numbers actually have their basis in reality. 2, 3, 4 each refer to a quantity which we can perceptually grasp. As we accumulate a greater quantity, at some point we can no longer identify the quantity by mere visual means. In a base 10 system, when we get to a quantity of 10, we move the column in which the notation is noted. 10 units is 1 grouping of 10 units, and no groupings of individual units. 10 groupings of 10 units we can express as 100 and so on. We understand the method that each column refers to a group of 10 units of the column to its right hand side. We understand the the method continues each time we accumulate 10 groups of the current column, we can keep track of it by adding an addition column to the left. As a concept of method, we understand how we can continue to add additional columns as required. The columns continues to refer to the relationship of the group relative to one of its members taken as, and serving as a unit. We understand the method of adding a column, and because we are comfortable with that understanding, believe we can drop the relationship to the unit in favor of continuing to add columns without reference to what the column refers to. This leads to the confusion that an infinity of notational columns is available for useage, by dropping the context of specifically what the column refers to. It is when this relationship or context is dropped, that the tie to perceptual reality is dropped, and renders the procedure as a ‘concept of method’ divorced from the referents which gave rise to it in the first place. Again, you can add zeros till you fall asleep or die, and still not exhaust the method, although the result can be so far removed from reality that it would appear to make the unimaginable, imaginable.
It seems that the concept of method claims that in mathematics we are still dealing with quantities of real-world things, and that’s how my claim differs from it. I would agree that, at introductory stages of language education, when we speak of natural numbers, we are in some sense referring to a quantity of things. However, that’s not what we do in mathematics at any stage. For instance, if it turned out that the universe were finite–as far as I know, there is no reason to suspect that it’s finite or infinite in, say, the number of electrons that exist–that would constrain our initial notion of counting because there would be some very large quantity, beyond which no quantity would successfully refer.
Arithmetic would remain oblivious to this fact, since arithmetic assumes an infinity of natural numbers, and so within the system of arithmetic, every number is meaningful and no number attempts to refer at all. Instead, numbers are meant to represent. At the level of arithmetic, this representation is so near to what it attempts to represent, that it’s hard to see the difference between the representation and the thing represented. But in arithmetic, we simply define a primitive element, which we call 0, and then define a function called the successor function, and define our domain as the closure of 0 and the successor function. From there we define notions of addition, multiplication, and limited subtraction and division. This is not meant to refer to the quantity of apples in some basket, or any quantity at all, though this is meant to behave in a way that mirrors counting–so that you may use this system in order to do your counting for you. The reason for having such a system is because, even if there is a largest quantity, we may never learn what it is or even care what it is. It is more desirable to simply have a system which is agnostic about there being a largest quantity, and which is effective under any hypothesis.
So in the end, arithmetic is nothing more than a linguistic practice which allows us to describe the world. When we group things into fives, count the groups, and multiply to obtain the quantity of things, we are using an abstract mathematical construct which behaves the same way that counting does. How we know that the behavior is the same is hard to spell out, but seems beside the point for this particular conversation.
Nobody in their right mind things that numbers are entities. Ontological evasion, as practiced by a non-trivial subset of mathematicians, includes the refusal to recognize the fact that a “mathematical object” is purely the product of a certain algorithm, and that “=” is a technical mathematical symbol which, oddly, is fundamental yet undefined.
“=” in most, if not all, mathematical domains, is defined as the set of all pairs, (x, x), or in logic, as a relationship between elements in a given domain which is reflexive, transitive, and symmetric.
Ah, well, no, an objection could be their anti-concept status (I do suggest reading up on “anti-concept” to understand exactly what that means). Recall that definitions follow identification of fact.
Definitions in mathematics are the stipulation of language rules in particular domain. As argued above, they are meant to mirror certain facts, but in themselves are supposed to retain a generality which is not wedded to any particular facts so that they may be applied to any domain where the facts warrant the use of a particular mathematical structure. Thus we do not talk about the real numbers which measure length versus the real numbers which measure velocity. We merely speak of all real numbers, which we may use to measure anything that is appropriately measurable by real numbers.
To clarify, this is the logical development of mathematics, though it is not the historical development. Naturally, our mathematical systems have been developed with real-world facts as the guiding principle for how we logically develop a given structure, but we often wish to generalize these structures in order to handle whatever comes our way.
BTW, you say “For any two real numbers, a and b, if |a - b| < |e| for arbitrarily small |e|, then a = b.” Some examples are a=2.3 and b=2.31. So |2.3-2.31| = .01 and for e=.02, which is an arbitrarily small value for e, |a - b| < |e|. Therefore 2.3=2.31. You’re trying to give an “intuitive” proof, i.e. one that appeals to common sense understanding of terms, which just aren’t applicable to mathematical concepts like “limits”.
The notion of some element a being arbitrarily small in this context is just that: For all x, a < x. As I said above, the principle that, for any two real numbers, they are the same iff their distance is arbitrarily small, is a consequence of the construction of real numbers by Dedekind cuts. For elaboration, I recommend the appendix of Rudin’s Principles of Mathematical Analysis.
Obviously I am giving an intuitive proof, since the individual to whom I was responding specifically asked not to discuss the issue with too much abstraction, and the full construction by Dedekind cuts definitely qualifies as too much abstraction. So I rely on a willingness to accept something that is not too difficult to see about the real numbers. But I’m doing a little more than just an intuitive proof: I’m promising that, with some added mathematical sophistication, a rigorous version of the same proof exists, which can be built entirely from set theoretic foundations.
