Division is such a method. A quotient is obtained if and only if the process is completed. However, mathematicians discovered there were division problems that could not be completed. What to do? There are two possibilities. One is to say that division isn’t defined in those situations, just as we say division isn’t defined when the divisor is 0.
This is, in some sense, correct. However, this view suggests that we could, if we wanted, defined division by zero. And that is, in some sense, correct. But then we would not be working with arithmetic, i.e. the mathematical system meant to represent counting. Likewise when we define 5/3, we are no longer working with arithmetic. To extend our mathematical system in this way actually changes the domain in a deep way. Rational numbers do not count things except in the sense that the natural numbers “live inside of” the rationals, or to put it a little more formally, there’s an injective homomorphism from the naturals to the rationals. So the development of the rational numbers from the naturals (or integers) is not one of patching some uncomfortable gaps but of constructing a genuinely new structure.
That is how negative signs came into use, and how the “…”, etc. notation for non-terminating decimals came to be. (It is also how each new type of number came to be defined, Whole, Integers, Rational, etc.)
Particularly on account of the negatives, I challenge this claim. I don’t believe negatives came about in order to allow us to extend the subtraction operation to a larger class of pairs of numbers, but would be much more willing to believe that it came about as an attempt to symbolize debt.
That means that any non-terminating decimal is explicitly not a quotient.
In what way? .333… = 1/3 in the real number system, by the argument I gave earlier. They are two names of the same entity in the structure, just as pi is another name for 3.14… which is another name for the ratio between the circumference and diameter of a circle. They are indistinguishable in their properties as mathematical entities.
Perhaps a large part of the problem here is the idea that the number .333… is somehow created by dividing one by three, approximated to the nearest tenth, then hundredth, then thousandth, and so on, and so the existence of the number is dependent upon how far out we actually go. However, the limit of a sequence is not constructed by us in that sense. For instance, take the sequence of numbers s_n = 1/n. Thus s_1 = 1, s_2 = 1/2, s_3 = 1/3, and so on. This sequence is not a finite sequence, and the sequence does not depend on how long we choose to state examples. Moreover, this sequence of elements approaches 0 in the limit. What I mean when I say that is nothing more than the claim: For every positive real number e, there exists a natural number N such that, for all natural numbers n that are greater than N, s_n < e. So give me any e, there is some rational number p/q which is between 0 and e where p and q have no common factor, then s_q = 1/q, which is less than or equal to p/q < e, thus I have proven my claim. That’s all there is to claiming that some element is the limit of a sequence of elements. One doesn’t have to talk about any infinite process at all, just infinite sets (Here, the sequence is an infinite set of rational numbers, and there is no process of generating the sequence. This is a defined set, regardless of whether anybody writes down or thinks of all the elements in it.). This notion of limit is essentially the same one that we use whenever we talk about any limits in analysis, so really the only objection that one might level against the use of real numbers is talk about infinite sets. But if you object to this, then you must object to the use of infinitely many natural numbers, and I presume nobody objects to that.
They do not qualify as numbers.
You should tell us what you mean by a “number” then, since in mathematics we simply take numbers to be elements in certain structures, usually ones that satisfy commutative unitary field axioms. Hence, complex numbers are numbers.
The argument that says the distance between .999… and 1 on the number line is infinitesimably small, so that they are not separate, is mistaking infinitesimals.
In the real number system, this is false. I think maybe in surreal numbers, this is true. But it is merely a consequence of the definition of real numbers, i.e. merely by the way that real numbers are constructed, that two real numbers are equal if and only if their distance is zero, and again by construction, distance between real numbers is zero iff the distance is arbitrarily small.
Since non-terminating decimals fail to be meaningful designations of quantity, they must be assigned a numerical value. .999… gets assigned 1. But that is an assignment of a quantity to a symbol, it is not a mathematical result. .999… is merely deemed equal to 1, it was not found to be equal to it.
This is just false. That .999… = 1 is not an assumption, it is a result which can be proved by properties of rational numbers and construction.
I am not sure how this relates to my post, but please note the in the case with .999… one never leaves the domain of rational numbers: all terms in the sequence are rational, at least some of the definition of the concept of limit are valid for rationals, the proof that the limit is 1 uses only rational numbers, and the result is also rational.
I speak about reals because the way to construct rationals from integers is just to take them as equivalence classes pairs of integers (where the right-hand part of the pair is non-zero). There’s really no great leap from integers to rationals. There is a tremendous leap from the rationals to the reals, and that’s where infinite decimal notation gets interesting. But yes, the numbers we’ve been discussing are contained in the rationals.
This is a very loose language. Nothing could be equal to an infinite sum, because there are no infinite anything, including sums. There are sequences of partial sums, which can have a limit. Something can be equal to that limit.
First, I’ve never understood this bizarre Objectivist insistence that nothing is infinite.
Second, see the above, when I speak of an infinite sum I’m not speaking of an infinite process–and keep in mind that numbers aren’t things. When speaking about an infinite sum, one merely refers to the limit of a sequence of partial sums. No mathematician means anything else by this, so I can’t see what your objection is.
The initial sense is one of a pragmatic approach.
It depends on what you mean by “pragmatic”. Why do I cook food? Because it is the most pragmatic way for me to get nutrition. Being a practice is not in itself an objection. Mathematics, I claim, is just a practice. It does not, in and of itself, refer to anything.
As the concept of method is the ‘how’ we expand the numbers, and in continuous quantity, it permits the ‘how’ of refinement of resolution. This is done by setting aside for the moment the ‘what’ is being quantified or is being used as the unit.
We might use something like the concept of method to think about the numbers, but that’s not what they are. An infinite set is still an infinite set regardless of whether anybody thinks of some or any of its elements.
The reason to suspect that the universe is finite in the context of quantity of electrons would be the law of identity. Treating arithmetic as oblivious, or as being agnostic seems to suggest a severing of the logical hierarchy between arithmetic and what perceptually gives rise to its necessity.
I never understood why Objectivist think that somehow identity is violated by an infinite quantity of something. It would just have, as part of its identity, an infinity quantity. Unless, of course, this law of identity is actually supposed to have some extra philosophical claim besides just that things behave in the way that they behave; that they are the things they are, and not anything else. Infinite things are infinite. Identity preserved.
And yes, arithmetic is not what historically gave rise to arithmetic. Arithmetic is a formal system with an infinity of elements, regardless of whether there are actually infinitely many objects in existence.
Besides, -5 is also smaller that any positive real number 