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On page of 22 Terry Tao's Analysis I, he writes (after presenting the Peano axioms):

Remark 2.1.14. Note that our definition of the natural numbers is axiomatic rather than constructive. We have not told you what the natural numbers are (so we do not address such questions as what the numbers are made of, are they physical objects, what do they measure, etc.) - we have only listed some things you can do with them (in fact, the only operation we have defined on them right now is the increment one) and some of the properties that they have. This is how mathematics works - it treats its objects abstractly, caring only about what properties the objects have, not what the objects are or what they mean. [...]

Remark 2.1.15. Historically, the realization that numbers could be treated axiomatically is very recent, not much more than a hundred years old. Before then, numbers were generally understood to be inextricably connected to some external concept [...] The great discovery of the late nineteenth century was that numbers can be understood abstractly via axioms, without necessarily needing a concrete model; of course a mathematician can use any of these models when it is convenient, to aid his or her intuition and understanding, but they can also be just as easily discarded when they begin to get in the way.

He then goes on to define sets axiomatically, and then, with the additional axiom that there is a set satisfying the axioms of the natural numbers, he constructs the rational numbers and real numbers.

However, I have often seen the rational numbers and real numbers also defined axiomatically--many times informally, where authors lay out certain properties of these numbers they will take for granted in their development of further results. Add to this list of objects often treated axiomatically (or informally axiomatically): points and lines, knots, symbols (when defining formal languages or formal polynomials, with unstated axioms being that they can be concatenated or exponentiated, etc.), functions, ordered tuples, and matrices, among others.

On the other hand, there are some objects which are typically constructed from the more primitive objects cited above, rather than being defined axiomatically. Among these are the complex numbers (defined as 2-tuples of reals, although axiomatic definitions simply stating, e.g., $i^2 = -1$ are also common), the extension field $\mathbb Q(\sqrt 2)$, or the $p$-adic numbers $\mathbb Q_7$, to name a few.

So, returning to my initial question,

  • How is it determined which objects one should define axiomatically, taking them as primitive objects, and which ones constructively?

In Bill Thurston's essay, On Proof and Progress in Mathematics, he shares in the section entitled "What is a proof?":

Within any field, there are certain theorems and certain techniques that are generally known and generally accepted. When you write a paper, you refer to these without proof. You look at other papers in the field, and you see what facts they quote without proof, and what they cite in their bibliography.

Following along with this narrative, one answer to my question would be that the objects we take as primitive are just those that mathematicians in a given field generally take as primitive. But, even if we accept this as the case, I'm hoping to gain insight into the motivations behind the choices of which objects are primitive--I don't think they're entirely arbitrary.

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  • $\begingroup$ I am not sure about remark 2.1.14. I think axiomatic definition of numbers strictly depends on the construction formed using a primitive called a unit. I can then observe that distance between units is strictly relative. $\endgroup$ – usiro Mar 13 '17 at 2:27
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Short soft answer to a soft question. Others may contribute more.

Where to start is an author's decision. You could teach analysis just with axioms for the real numbers. Tao chooses to start further back - essentially, as far back as we now know we can. He's decided that's a pedagogically sound way to teach real analysis.

My real analysis course in 1957 assumed the real numbers at the start of the first semester and constructed them from set theory and Peano at the start of the second. I remember struggling with integer arithmetic defined inductively when all I'd ever used induction for was proving things.

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A "soft question" maybe, but I think it suggests some interesting issues all the same! Here's just one line of thought.

Suppose you offer me a bunch of axioms for widgets. Why should I care? What's the point of your widgets? Indeed, your theory has got a number of strange-looking features -- I'm beginning to suspect that widget-theory might in fact be inconsistent.

Now, you can at least resolve my last worry if you can construct a model of widget-theory using (for example) ordered sets of wotnots -- where wotnots are mathematical items I'm already happy about -- and where operations on widgets are construed as operations on sets of wotnots. You show me that this construction satisfies your axioms, so I'm at least content that you aren't mired in inconsistency. (I don't need a worldly "concrete" model, an abstract model will quell my anxieties!)

You offer me, for example, an axiomatic theory of "quarternions". I'm a bit dubious and wonder whether this makes any kind of sense. You then say, look, think in terms of ordered pairs of complex numbers, with a "multiplication" relation defined on these pairs in a way analogous to the way in which a multiplication relation was defined on ordered pairs of reals in constructing the complex numbers. We work through a few details, and yes, it all comes out in the wash -- the construction gives a model of your axioms, and ok, I'll accept that your theory is consistent.

So this gives us one important reason we can be interested in constructions as well as in axioms.

But of course, having a construction won't resolve my worries about the point of it all -- why should we care about widget theory? of all the consistent axiomatic theories we could cook up, what's so great about this one?

That's a murky sort of question -- but it hints that we do care, mathematically, about more than just having a bunch of axioms to play with. Would someone who just knows the axioms of pure widget-theory, can prove theorems about widgets using those axioms (and can run you up a nice set-theoretic model of widget-theory, proving it consistent), count as really understanding about widgets? What does it take to really understand about quarternions, get their mathematical point? -- is being able to play around with the axioms enough? We don't want to know "what they are made of" (in Tao's words): but we do seem to want to know something more than just what the axioms say.

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(This was originally included in my question but might be better placed as an answer, though more serving more as a discussion point.)

I think one important factor in the choice of primitive objects is whether or not one can mentally conceive of or "visualize" such objects. I can (or at least think I can) visualize the natural numbers, real numbers, symbols, lines, and knots, but I have a hard time visualizing the $7$-adic numbers without reference to infinite sequences of rationals. Although this should be taken with a grain of salt, since I can't, for example, individually visualize each of the infinitude of natural numbers simultaneously.

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The OP wrote: "I'm hoping to gain insight into the motivations behind the choices of which objects are primitive--I don't think they're entirely arbitrary." This formulation implies that, while the choices of which objects are primitive is not entirely arbitrary, there is some degree of arbitrariness involved.

Such an idea obviously runs counter to the viewpoint of foundationalism, whereby mathematicians have dug deeper and deeper over the centuries until they have reached a satisfactory bottom line (literally and figuratively), upon which the majestic castle of mathematics is built. This view is base upon what is described by philosopher Ian Hacking as the butterfly model of scientific development. Here the evolution of a science is compared to the development of a biological organism like butterfly, which may undergo dramatic and even seemingly unpredictable developments, but it was all encoded in its genetic make-up from the start.

An alternative view is that the castle is floating in midair, that the alleged bottom line is illusory, and that the degree of contingency in the historical development of mathematics is greater than is usually thought. Such a model of develoment of a science is described by Hacking as a Latin model, implying a comparison of the evolution of an exact science (including mathematics) with the contingent development of a language like Latin, dependent as it is on social and cultural factors.

The obvious example is the foundations provided by category theory, which compete with the set-theoretic foundations currently prevalent in a number of fields (though not in all).

A fantastic historical example of different "primitives" as you put it can be found in Cauchy. To Cauchy, a mathematical primitive was the notion of a variable quantity. He exploited it in his textbooks to define other notions, like infinitesimal and limit. For example, to define an infinitesimal, Cauchy declares roughly that a variable quantity that is getting smaller and smaller becomes an infinitesimal (notice that in most cases he does not say that the variable quantity is an infinitesimal but rather becomes an infinitesimal). Cauchy then goes on to point out that the limit of such a variable quantity is $0$. Thus both limits and infinitesimals are defined in terms of the notion of avariable quantity in Cauchy, which is to him the primitive notion.

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