Counting numbers vs Natural numbers; Peano Axioms

Question

I can feel that my question is going to be a somewhat lengthy one, but I will try my best to deliver it in as short a form as I can manage.

So to begin, I've always thought that the numbers such as 1, 2, 3, etc. exist outside of the realm of mathematics, i.e., one does not need to "define" what it means to have ONE apple, TWO apples, etc. Following this line of thought, we can also believe that there is a natural concept of "order" for instance, we can always determine which basket has "more" apples (of course, so long as they hold only finitely many of them). In sum, we have the concept of "unity", which we will represent as the symbol 1, the concept of "quantity immediately follows the unity", which we will write as 2, and so on. So it makes sense to talk about these "numbers" without ever invoking the concept of natural numbers, and we can even develop the method of mathematical induction on this set. I will call this "naive" set as the counting numbers, since they are what you use to "count" things.

Now, there is what is called the Peano Axioms, which in essence declare the qualities that any candidates of "natural numbers" must possess. Clearly, the above mentioned set {1,2,...} meets this criteria if we define 2 to be the successor of 1, 3 to be that of 2, etc.

But here is the question: to me, it seems all fine to just have these "naive" set of "symbols" i.e., $\{1,2,3,\ldots\}$ that we will casually call natural numbers. We can go all fancy and define additional properties such as addition, multiplication, etc., but the fact remains that these are NOT the fruit of human creation, but something that inherently exist in "nature" (hence the name natural numbers, I presume). So, why do we bother in the first about these Peano Axioms? Is there a necessity to include in our system of natural numbers sets as $\{-1,-2,-3,-4,\ldots\}$, $\{\frac12,\frac 32, \frac 52,\ldots\}$,$\{e,e+1,e+2,\ldots\}$, etc.?

Thanks in advance

Answer 1

There are people who will disagree with you. They think that the natural numbers end somewhere. There is a largest natural number.

Can you argue with beliefs? Thousands of years of human history taught us that the answer is more or less negative. If I believe that the natural numbers have an end somewhere, there is nothing you can do to convince me otherwise.

So you might say now, "okay, but suppose that for the sake of proof there is no largest number and now proceed ..." so I'll argue again, I disagree about the definition of addition, and I disagree about the order and about multiplication and so on and so forth. I mean, sure, I am willing to agree with your definition which "clearly" and "obviously" seem true all the way up to $2^{1000000}$. But not a single number above that. After $2^{1000000}$ everything goes haywire!

Can you prove me wrong? Can you convince me that my belief is somewhat mistaken? No, you probably can't, again look at history.

So you want to say now "Okay, fine, but suppose that you agree with me about the basic definitions ..." and then you show me a proof about one thing or another.

What happened here? You essentially wrote me a list of properties which you think are obviously true for the natural numbers, and then you showed that if those properties are indeed true then some proposition is also true.

So what you really did was to write down some axioms and claim that from those axioms you can prove something. The axioms for the natural numbers are no different than that. They help us put aside whatever belief we have and agree that these axioms, at least for sake of argument, are all true in the natural numbers. Then we can start proving all sort of things.

Answer 2

First off, the distinction you're attempting to between "counting numbers" and "natural numbers" doesn't match how the words is usually used in mathematics. To a mathematician, "the natural numbers" in general means the pre-formal concept you're calling "counting numbers". (And the phrasing "counting numbers" generally isn't used at all). In particular, being a model of this or that formal system doesn't qualify anything as being the natural numbers; only the true Platonic natural numbers are natural numbers.

(That's except for people who have learned enough set theory to know how natural numbers are usually modeled there, but haven't yet gained a sufficiently wide horizon on the matter to know that the model is not the actual thing).

Whether the natural numbers are a human invention or not is something of a philosophical morass that I won't go into -- but the mere fact that they were not created by Peano is not a controversial statement.

Now, since we understand the naturals intuitively, at least to a pretty good extent, what do we need axioms for? Well, most primitively, to see how well we can do by reasoning purely mechanically from axioms. For example, it led to a significant improvement in geometrical reasoning when the ancient Greeks started demanding rigorous proofs from axioms rather than just geometrical intuition; it is natural to wish to know whether something similar could be done for arithmetic.

As it happens, it took many centuries after the Greeks before formal mathematical reasoning had progressed enough that it was thinkable that it might be possible to reason about integer arithmetic without depending on intuition. Peano's axioms were not aimed at telling anything new about the integers -- he was simply one of the first to have a sufficiently well developed notion of formal logic that he could hope to make an axiomatic treatment of the integers work.

The goal of the investigation, then, is not primarily to discover something new about the integers themselves, but to see whether a formal system can be made to support a "mechanical" reasoning about them that is rich and strong to conclude what we already know about them intuitively.

The Peano axioms (and their modern first-order successor, Peano Arithmetic) were somewhat successful in that, but nonetheless the crowning achievement of the whole program turned out to be a negative result: Gödel's Incompleteness Theorem tells us that every reasonable formal system for proving things about the integers will be incomplete: there are truths about the natural numbers that it cannot prove.

Nevertheless, it is still interesting in itself to investigate the strength and properties of various such formal systems.

Also, of course, it is pragmatically useful that when something can be proved in a formal system with few axioms to it, then it is very certain to be true about the actual naturals ... there's a much smaller risk of having overlooked something during the proof than if we base our reasoning purely on an intuitive understanding of how the naturals ought to behave. In this way it's kind of a gold standard for a proof that it can (in principle) be reduced to Peano Arithmetic or a similarly bare-bones system. We recognize that there will be things that can't, but these exceptions will be subject to a lot more scrutiny before one considers oneself convinced that they are in fact true.

Answer 3

Just to add a different perspective, counting numbers go with the idea of the first, second, third etc and this idea is formalised in mathematics by the notion of ordinal numbers. You don't need to formalise the idea, but formalisation goes with making things precise so that simple observations can be generalised.

There is a different concept of number associated with measuring - "the same size as four", for example. This is captured in various ways, including in cardinal numbers. But the concept of measuring is different from the concept of counting, for example in the need to measure fractional or continuous amounts. The formalisation of cardinality gives an isomorphic structure for the positive integers, but is mathematically distinct for infinite cardinals.

So even in thinking naïvely about numbers, the concepts of order and measure arise. The first is discrete and the second is continuous even in informal thinking. Making these ideas precise is tricky (it took clever mathematicians centuries to pin the ideas down).

Answer 4

Peano's axioms give us the (usually 5) essential properties of these "counting numbers" from which, it appears, all of their known properties can be derived. There is no axiom that no number is its own successor, for example, but we can prove it using only Peano's axioms and the rules of logic.

As far as we know, all of current number theory, if not all future developments in the field (see Godel), can be derived from Peano's Axioms using the rules of logic and the axioms of set theory. In the most commonly used version of Peano's axioms, there is no mention of addition, multiplication or exponentiation, but you can construct each of them (i.e. prove their existence) using logic and set theory. It is even possible, to construct the real and complex numbers.

Just how "natural" are Peano's axioms? I think each can be easily justified at a very intuitive level. (See "What is a number again?" at my math blog.)

But if you really insist on constructing the natural numbers from some other, simpler structure, you can do so starting with a set $X$ on which we have function $f: X \to X$ such that is $f$ is injective, but not surjective (i.e. $X$ is Dedekind-infinite). From this set, you can extract a subset $n$ that satisfies all of the Peano axioms, including induction, where $f$ is used as the successor function. (See "Daddy, where do numbers come from?" also at my math blog.)