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Let's assume our students are actual aspiring mathematicians.

Why would we introduce our students to Calculus rather than Real Analysis?

After all, "Calculus is a subset of Real Analysis". He will have to learn everything that he had already learned in Calculus once again. In addition to this, the student learns about mathematical concepts like axioms or proofs and mathematical working in general.

It's like saying: "We assume you're stupid, so learn Calculus first."

But I think this will make math harder instead of easier. For instance, If we just assume the real numbers as they are but do not show where they come from and how they can be constructed, the student will have a harder time understanding the key concepts of calculus.

Let's take the extreme case, where we just tell the student the differentiation rules. This might do seem to be easier. But the student will ask himself what he's actually doing and why. Because of the missing theory, he can't grasp it. Because of that he will maybe see himself as a failure or even start hating math.

Well, historically we did not have any formalism. But our student aren't people from 2000 B.C.

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why start at real analysis? Let's start with set-theory, then topology, then maybe then real analysis, linear algebra, eventually, calculus. – James S. Cook Feb 25 at 17:15
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Your starting assumption is flawed. Calculus is very clearly geared toward students in the sciences and is a course that "pays the bills" for mathematics departments. – Jon Bannon Feb 25 at 17:19
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@JonBannon indeed, the reality is Calculus is an economic necessity for math departments. I am not at all convinced it is the right logical starting point for students. But, I'm not sold on real analysis either... maybe a proofs course with baby set theory would be the right starting point. – James S. Cook Feb 25 at 17:22
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Your starting assumption describes few of the students who will take either course. It doesn't even necessarily describe those who will become mathematicians; for many, that's a decision made after being exposed to the power of the subject, and it isn't unusual for a calculus class to provide such a revelation. I think it is harder for most people, even those who are far to the abstract-thinking side of the spectrum, to appreciate real analysis than it is to appreciate the applicability of calculus to physics, optimization, etc. If anything, calculus creates the appetite for real analysis. – Austin Feb 25 at 19:19
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(You can get by for a couple of years with no formal concept of a limit. Derivatives are just slopes of curves. If it is uncomfortable for a curve to have a slope - it isn't straight, no "step over tread"! - handwave away that it is the slope of the tangent. Indefinite integrals take you back from gradient function to curve, definite ones are areas. That's enough to cover a lot of practical material, but conceptually there's a substantial hole in a student's knowledge.) – Silverfish Feb 26 at 1:01

You may as well ask: Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first? Why teach high-school Algebra without starting with the basics of groups, rings and fields? Why, for that matter, teach children to read first, instead of starting with the fundamentals of grammar and linguistics?

Everything we know about how learning takes place tells us that complicated ideas are formed and absorbed by building them on a foundation of simpler ones. It is important to realize that "simpler" does not mean here "more fundamental", i.e. logically more primary; rather "simpler" is here used in the sense of "easier".

A human being is not a formal system, and our minds do not develop according to deductive principles. Almost nobody understands anything complicated the first time they encounter it; we master things by revisiting them, over and over again, gaining deeper insight into it each time. We teach people Calculus first because if we taught them Analysis first they would have no experiential or conceptual substrate on which to build.

Edited to add: In addition to these points, I'd like to echo (and somewhat restate) an observation made (by Jon Bannon and by Austin) in the comments below the OP's post. The opening sentence:

Let's assume our students are actual aspiring mathematicians.

describes a situation that I think needs closer scrutiny. In my experience, anybody who is an aspiring mathematician has already taken Calculus. It is extremely rare, I think, for a person to formulate such an aspiration on the basis of nothing more than high school Algebra and Geometry. I am not sure there is data on this, but I expect most people who eventually become mathematicians did not reach that decision until after at least one semester of Calculus, possibly much more. So the question essentially addresses a hypothetical audience that does not exist.

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+1 for "nobody understands anything complicated the first time they encountered" (and note my moving of the focus to "understands"). I believe that the concept of "reification" explains this well - learners need to be able to integrate and hold in their mind clear ideas of the more basic concepts before they can use them as building blocks for more complex or more abstract ideas. – yoniLavi Feb 25 at 23:21
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"He who will learn to fly one day must first learn to stand and walk..." - Nitsche. – Paul Feb 28 at 3:03

In addition to other good answers, I do think that what "calculus" usually refers to is what people did with calculus prior to about 1830, as opposed to the foundations of it. Namely, they solved a lot of physically meaningful/intuitive problems, as well as geometrical problems, problems in mechanics, celestial mechanics, fluid flow, ... and beginning of applications to number theory, as well as the beginnings of complex analysis.

So, for me, and I suspect for many, that notion of "calculus" is very persuasive that it is useful, whatever its foundational interest or challenges may be.

In particular, we could ask why we should care about the axioms for something if some sort of utility... whether to physics or to geometry or to number theory, let's say! :) ... hasn't been demonstrated. Sure, study of axiomatic systems has some intrinsic interest, but perhaps more so after one sees why we'd care to be sure our machinery works as specified.

For that matter, some of those "applications" help us detect "stress points" in the theoretical development.

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Because your core assumption is bad and most students are not aspiring mathematicians

Your opening statement - "Let's assume our students are actual aspiring mathematicians" - is almost certainly a very bad assumption to make about students at the age where calculus is taught. In fact, very few of the students who are taught calculus will go to study mathematics at degree level and fewer still will go on to become mathematicians. So Mathematics (like other subjects) needs to be taught in a way that is beneficial to students who do not go on to specialise in it.

Calculus is used as a tool in a wide range of subjects such as Biology, Physics and Engineering and many more students will go on to study these subjects than will go on to study Mathematics. Calculus, as a skill, is thus much more useful to a larger group of your students than Real Analysis is.

I think the pedagogical arguments made in other answers also hold true but I think the error in your core assumption is more critical.

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In many respects this is what is done in European universities, where the degree is often a three year program. First year students reading maths take analysis. Of course they probably learned calculus as part of their A-levels.

Years ago, as a US citizen I did a first degree as a math major in the United States, and then did a second undergraduate degree in Oxford, reading maths. I remember being amused that many of the maths students, who were very bright and talented, seemed to know little about the Riemann integral since the British university curriculum started with the Lebesgue integral. On the other hand, I was very comfortable with the Riemann integral but had never heard of the Lebesgue integral.

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I think calculus is actually more important for a working analyst than 'real' real analysis. You can (and should) always learn more about why you are allowed to do things and how they can be constructed from simple principles but to get into the flow of figuring things out yourself, you cannot have yourself looking up the divergence theorem, substitution rule, or Cauchy-Schwarz. You have to be at absolute ease (and intuition) with all kinds inequalities, ODE, series, integrals. No matter how 'easy' those are conceptually, and no matter if you do stochastic processes, dynamical systems, modelling, statistics, or whatever other applied mathematics.

Disclaimer: This of course is coming from a child of the German curriculum that never learned Cauchy-Schwarz in an analysis class 'because you did it in linear algebra' (and how we did it. 'Take this miraculous expression, insert it into this sesquilinear form, and there it is'. Duh), that learned the divergence theorem without having the slightest clue what a divergence is, that took two classes in PDE and knew topological vector spaces, negative order Sobolev spaces, real interpolation before learning about, say, the transport equation (what? there are PDE that cannot be solved with Lax-Milgram?? I actually once took a class abroad and in a mid-term exam solved a hyperbolic equation ending with 'all that is left to be done is to show cooercivity') )

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Learning is easiest and the most enjoyable when it answers questions one already had. A student who has played around with functions before, and wondered how he might find their maxima, or wondered about how to express the amount by which something changes when some other thing changes, will be thrilled to learn of the powerful techniques of calculus which solve just that.

If you try to teach someone the rigor of formal real analysis, he will find it boring, pointless and difficult. Unless, of course, he has already grown a healthy appreciation for calculus, so that he can appreciate analysis for being the foundation of calculus.

Since calculus is the foundation of the physical world, you definitely want to teach it as early as humanly possible. In my humble opinion, students - and especially those who are curious, enthusiastic, and mathematically oriented - should be taught calculus quite early. Failure to teach calculus early enough is, IMO, a problem with the curriculum.

So instead of teaching analysis when we're now teaching calculus, we should just teach calculus earlier - when students are still too young and inexperienced to follow rigorous analysis. It's really "We assume you're a kid, so learn Calculus first."

The "He will have to learn everything that he had already learned in Calculus once again" part also makes no sense - if indeed those are the same things, then the second time will be that much easier for having learned this before, so you can spend less time on it. Besides, learning isn't a one-off thing, to truly understand and retain knowledge you have to approach it several times, from different angles.

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The majority of maths students in British universities - even the very most prestigious and world-renowned - only started calculus at age 16 or 17. Issues of mathematical maturity are relevant here but the idea of calculus being something that must be started by 14 is very far removed from many countries' curricula. – Silverfish Feb 26 at 0:51
    
@Silverfish: That's why I said "should". I didn't say that's what's actually common. The OP suggested an optimization of teaching real analysis when we normally teach calculus, I suggested that if anything, calculus should be taught (much) earlier. Also, that's just a single part of my answer which is mostly unrelated to the rest - I wouldn't think it justifies a downvote. Anyway, I don't know how things are done here in mathed.se - maybe correct age<->curriculum correspondence is a big deal. If it offends you that much I'll change it... – Meni Rosenfeld Feb 26 at 10:54
    
@Silverfish: I'll also say your data somewhat surprises me, since 16-17 is the age most students in Israel learn calculus, regardless of mathematical inclination. Myself, I think I started calculus at age 13-14 from a tutor and books. – Meni Rosenfeld Feb 26 at 10:56
    
Not all school systems have much option for acceleration or taking a course early. That's particularly true in systems where post-16 education has separate qualifications and timetabling (and in some countries, is even taught at completely separate institutions to pre-16). In those situations, if a curriculum allocates a topic to post-16 education, then (absent private tuition or - and this remains rare even among strong students - really extensive self-study) even most future universtity maths students just aren't going to see it until they're 16 or 17. – Silverfish Feb 26 at 11:48
    
It's also worth pointing out that at least in the UK, "enrichment" schemes for students identified as "talented", and maths competitions (which target a similar subset of students) don't tend to bring forward next year's material, but rather extend topics students are expected to have already studied (by showing interesting applications, or a focus on creative problem-solving using those methods). Taking more advanced topics earlier isn't the only route to success, so I felt that the "doing it wrong" comment was rather strong compared to the level of evidence put forward in this post. – Silverfish Feb 26 at 12:03

I took an honors calculus course in my first year at the University of Michigan (and my high school didn't have calculus). It was truly an analysis course. It was so far from how I was used to thinking that it felt over my head.

The simple beauty of calculus was lost.

I love teaching calculus, and I often point out to the students where we are leaving out details that they'll encounter in analysis if they go on in math.

I emphasize velocity, slope, rate of change, graphing, and optimization.

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This is one of the most fundamental questions in modern mathematics pedagogy today, and I think that it is high time we seek out new alternatives to old methods. But, surprisingly or not, this question has really been addressed in large part by Goodstein in his HISTORIC text Mathematical Analysis. It is not well known, and that makes me sad, but I do what I can to help people seek it out.

The basic problem in modern Calculus/Analysis is that at the popular conception of Calculus is one without rigor: it's a mix of algebra and geometric intuition. Specifically, the notion of limits, derivatives, and integrals are, in modern treatments of Calculus, tied to the graphs and pictures that we draw. Though this makes it easier (in a sense) to communicate some of the results of Calculus/Analysis to students, it does little to reinforce the most basic principle of mathematics: rigor. The common belief is that the rigor of Real Analysis is inappropriate at the "Calculus Level". I am one such individual: real analysis is so far from the math leading up to what we now conceive of as calculus so as to render it absolutely inaccessible to even some aspiring mathematicians.

What Goodstein does was to take a revolutionary and new perspective on the whole Calculus/Real Analysis problem. I believe that, in time, most calculus/real analysis will be taught using a version of his methods in Mathematical Analysis whether by direct reference or through some circumlocutions route.

Goodstein's approach is as follows: rather than thinking of continuity as the corner stone of mathematical analysis, we can construct a surprisingly familiar theory based entirely on uniform continuity. The result is this, we get a Mathematical Analysis which is not only simpler than standard treatments of Real Analysis, but which makes it possible to present Calculus rigorously without leaping too far from the math commonly learned before taking a calculus class.

There is a further solution to this general problem: the introduction of rigor at an earlier stage in student's education. Rather than presenting algebra and number in a hacked mash of geometric intuition and half-baked formalisms, we must seek out a design of a formal system (specifically one which is not set-theoretic) that is both concrete to students while being fully rigorous. The biggest problem/barrier to such a design is the belief that no such design is possible, or that no such design exists.

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