# Skipping elementary calculus and starting at analysis?

• I
• Grasshopper
Prove the Fundamental Theorem of Calculus. 5. Derive theorems of calculus from the the theorems of real analysis.In summary, a person without experience or context would be unable to learn calculus from a textbook. A person with experience or context would be able to learn the basics, but would need more help to understand the more advanced concepts.f

#### Grasshopper

Gold Member
TL;DR Summary
Is it possible? (This is a useless question of course)
I was walking around with my head in the clouds and suddenly I wondered if a smart person, say, a philosopher, could start at the full monster of real analysis instead of elementary calculus.

Would there be any hope for this unfortunate soul? What are your opinions and why?

Or if you feel this is a stupid question then just ignore me.

I'm thinking that the best answer to this question would be to try and read a textbook on it yourself. See if you understand what is being communicated and see if you can have a go at the exercises in the textbook. This website has lots of textbooks on calculus and real analysis.

https://klkuttler.com/

The one in the first row and first column is a book on what you're asking about. The one on the first row and third column contains bits of real analysis and standard elementary calculus.

Learning a certain branch of mathematics is somehow imprinting one's thought structures. E.g. school math is rather biased towards calculus, so students often have difficulties with linear algebra in their first year. Different areas require different schemes. This means that you can start with the full monster easily, or at least as easy as you start with any other new subject. You can learn analysis along the lines of measure theory, which is quite different from what you called elementary calculus. It will teach you a different view. Whether this is better or not is a different question and depends on subjective scales. Furthermore it depends on personal characteristics. Some find group theory easy and functional analysis hard, and for some it is the other way around.

weirdoguy and Klystron
Also one could look at math historically where folks didn’t even have the book and slowly pieced together the theory over generations. Later someone writes a book organizing things into a more natural sequence.

Our courses today are a direct result of that evolution where we start with Algebra from the 12th century and then move onto Geometry from the Greeks and then onto Trigonometry from the Egyptians. Students of the past would have had different starting points and learning sequences.

The wiki article on Algebra gets into the evolution and organization that is ever evolving:

https://en.wikipedia.org/wiki/Algebra

Why would anyone want to? I knew of some engineering departments that taught courses in the theory in one year, where the student did nothing but theory. Then the next year, they did applications of the theory, where they did nothing but applications. It could probably be done, but I can think of nothing that would make mathematics less appetizing than to learn it this way. Is your intention to not take calculus altogether. Suppose you had to teach it someday? I suppose the human mind is plastic enough to learn anything, but this seems like an inefficient way to learn mathematics. I think someone could learn calculus without being able to learn the multiplication tables or how to add fractions too.

It is possible, but it's quite hard. We started with real analysis, first ever lecture ##\varepsilon - \delta##-definition for limits, there were some funny symbols ##\forall , \exists## ..Pretty much nobody understood what was on the black board.

Would not recommend. Get comfortable with calculus and some transition material that deals with propositional calculus, such as discrete math or naive set theory or some such.

My first calculus class was an analysis class that taught just enough calculus to be able to do the standard computations. It was fine, this is totally doable as long as you're actually interested in the math.

Now could a person with no context just take an analysis textbook and crunch through it and at the end understand how incredibly important the results in its pages are? Probably not.

Thanks for the replies. They are enlightening.

berkeman
As an undergrad, I had the option to go for the honors sequence in math, called analysis for two years, and then I would get credit for the engineering sequence in math, (still more demanding mathematically than the life sciences sequence). This sequence was all proof writing and no applications over my first two years of college. The sequence for honors allowed me to get credit for Real Analysis without taking it.
Big deal. I got to write proofs for two years (my freshman and sophomore years), rather than one year or even one semester of Real Analysis (junior year).
Imagine my horror when I was expecting to solve ordinary calculus problems with related rates, optimization and the like and I got a first test like. 1. State and prove Rolle's theorem. 2. State and prove the intermediate value theorem, 3. prove the product of two continuous functions is continuous, etc, and two more years like this. I was helping my roomate and others to get A's and B's in the regular sequence and getting C's in this 'honors" sequence.
I am positive I would have done better taking the regular sequence, and the real analysis separately. As the college was willing to grant me AP credit if I turned down the "honors" sequence, I would have been even further ahead if I just started taking my sophomore year mathematics courses.

The only upside was I had two very prominent research mathematicians teaching that sequence, (although this does not necessarily mean they were the best teachers).

I have seen other injustices at other colleges, and this has caused be to be very suspicious of both honors classes and remedial classes. I have seen students fail remedial classes because they were at the tail end of a curve with 20 students in it. They took the same tests as the "generic" class with up to 1000 students in it. The same performance caused at least one student in the remedial class to fail, athough the student would have passed if put on the same curve as the 1000 student population.

Summary:: Is it possible? (This is a useless question of course)

I was walking around with my head in the clouds and suddenly I wondered if a smart person, say, a philosopher, could start at the full monster of real analysis instead of elementary calculus.

Would there be any hope for this unfortunate soul? What are your opinions and why?
...
NO.

Really, no!

nuuskur, Grasshopper and Delta2
Summary:: Is it possible? (This is a useless question of course)

I was walking around with my head in the clouds and suddenly I wondered if a smart person, say, a philosopher, could start at the full monster of real analysis instead of elementary calculus.

Would there be any hope for this unfortunate soul? What are your opinions and why?

Or if you feel this is a stupid question then just ignore me.
No disrespect, but how can you say "a smart person, say, a philosopher"! Are you asking if a smart person can learn analysis or are you asking if a philosopher can?

Anyway more seriously, why do you ask?

Where I come from there is not such thing as calculus. There is just rigorous analysis from day one.

No disrespect, but how can you say "a smart person, say, a philosopher"! Are you asking if a smart person can learn analysis or are you asking if a philosopher can?

Anyway more seriously, why do you ask?

Where I come from there is not such thing as calculus. There is just rigorous analysis from day one.
No. I was insinuating that philosophers are extremely intelligent. All the ones I’ve met have been unusually sharp. So it was meant as a compliment to the profession.

But anyway, I suppose you are an example that it is quite possible to learn it from the beginning with all the logical justification.

What motivated me to ask the question is that after I finished Calc III (introduction to multivariable calculus), I got permission to sit in a real analysis course for no credit without the prerequisite of the introduction to higher math class, and it was a whirlwind of triangle inequalities and (what seemed to me to be) backwards thinking (like almost knowing the answer first and working backwards), and honestly it seemed way harder. But eventually the class got to many of the tools we learned in elementary calculus. So I figured, why waste three semesters? Why not start here, but maybe go slowly, so that in the same time, the student manages to save a semester of school.

But others have pointed out that maybe you don’t get as much applicability, and I suspect it may actually hamper physics education if a student is still learning the logical basis for derivatives instead of quick short cuts on how to utilize them.

Edited to add: in fact, maybe this situation would be like teaching set theory and things like the division algorithm instead teaching simple elementary arithmetic. Now that I think of it in that light, it does seem a bit absurd. But as you said, you went straight to the rigorous proofs. So it’s clearly not impossible.

Philosophers are likely well acquainted with classical logic. This unusual sharpness might have owed to that acquaintance.. one that did NOT spawn in a few weeks, but years and years.

If you are a physics student, I think rigorous analysis shouldn't be on top of your priority list. Though, I have tutored many physics students in their MSc studies about basics of analysis or linear algebra, hell, we even had to make a detour at propositional logic.

As a sidenote: a difficulty with dealing with physics students is figuring out how to respond to a query of the form "so, is there a formula here I can use to solve that?"