I grew up programming (from 10 or so) and couldn’t agree more. I have almost no formal education yet I’ve picked up enough math to do ML and (mediocre) cryptography and keep up with people who have scary sounding degrees. in the process I’ve fallen in love with math and think as programmers we haven’t been meeting mathematicians half way.

Mathematics is not just functions and calculus.

And it seems to me that you learned the concept of functions through programming first but that there's not evidence that you couldn't have learned it from mathematics at first. Functions are a pretty easy concept, so I think it's pretty easy to introduce it from a variety of points of view. So I'm not sure the anecdote backs up any argument about learning programming makes mathematics easy.

I am of the opinion that programming can be used to explore and learn mathematical ideas and am a big proponent of that, but that is something different than "I know how to program so mathematics will be easy now".

Always wonder about this. Other than foundation of maths, the function abd calculus view … my wonder is about the data part of the maths. Not the process. Yes one can view matrix as a function but could it be data. One can real number line, complex or integer as data?

functions in traditional programming and functions in math are two different things entirely. They should have been called procedures (I think there's historical debate about this but I can't find the reference). Obviously, functional programming is an attempt to address this.

For most mathematicians, calculus as is often taught in typical undergrads is not "true" mathematics. It's just a tool for computation. For them, calculus is analysis (theorems/proofs that are used to build up calculus).

So my question is: Did you study analysis and would you credit programming in helping you get good at it?

+1 for this comment. Calculus, i.e., Calculating with mostly finite numbers and (usually) a known set of well defined rules is only a part of mathematics.

As a student that currently learns Analysis and linear algebra, it is far more complex and abstract than calculus. It is not neccessarily harder to learn but different. And adapting to this paradigma takes time (and effort).

It is similiar to learning Assembley as a Python developer. Knowing Python will help with Assembley. But the levels of abstraction are obviously different and will require a lot of learning.

Not them but I def would. I built a fintech/econometric-lite system in python years back and it was mostly just taking in a ton of “obvious” knowledge and realizing the cool stuff you could do in practice

I don't understand this comment. Calculus as taught to mathematics undergrads is analysis, right?

Where is calculus "often taught" without "theorems/proofs that are used to build up calculus"?

Not US, but for GCSE A's further math equivalent, calculus primarily involved applying intermediate rules (chain rules, trig identities, etc.) to evaluate differentiation/integrals on elementary functions, and a bit on geometric and arithmetic series.

Analysis discusses less well behaved functions and spaces than these.

You can do the first part without even relying on analysis in a mathematical sense. You simply define a differential algebra, by introducing a derivation function that just happens to respect the correct rules. Then "calculus" is the topic of how to perform computations in such an algebra. Note however that you do need analysis to rigorously address other parts of a typical "calculus course", especially those dealing with infinities, sequences and etc.

In most US universities, the courses called "calculus" are mostly about computing integrals and derivatives. Yes, they'll have some theorems (Fundamental Theorem, Mean Value Theorem, etc), but most of the problems are related to computation than proving.

Unfortunately, since engineering students outnumber math majors by a large margin, the departments cater to them and not the math majors. The latter study analysis in later years - usually third or fourth.[1] Universities with strong math programs may offer it in the first or second year.

[1] I just checked my undergrad's program. They begin taking analysis in their 4th year, and it's offered only one semester a year!

That's just sad.

I was taught Calculus in high school, years 11 and 12 (two optional senior years for those that want more education prior to university, etc rather than transfer to a trade apprenticeship or leave school altogether).

That was a mixture of basic theory and proof and some computation (eg: derive an expression for the volume of intersection of two pipes at right angles, etc).

University went straight in to heavy analysis and foundations (for Math 100 - math for the serious (math, physics, hard chemistry)), with seperate streams for "casual math" - engineering, business, law, etc.

Engineering math prepped people for calculating dynamics and kinetics with vary loads, masses, thrusts, harmonic forces, network propagation, mesh computations, etc.

This was Australia in 1980.

> They begin taking analysis in their 4th year, and it's offered only one semester a year!

This is insane to me. In the UK about ten years ago, on literally the first day of my degree, my first class was real analysis. Yes, it was the easy stuff like proving sequences and series converge, various things about continuous functions, but we learnt how to prove it all and the exam was all about proving various things. And we built up to harder stuff as the year went on.

What is even happening if math majors aren't studying analysis until their fourth year?

Sorry I'm so incredulous, it's just that I literally don't know what I would've been studying if analysis had been delayed so much.

> get the impression we might be overloading the term "mathematics".

We are; the challenge is there is a near infinite difficulty in both "programming" and "mathematics" and at a certain point, the skill sets diverge greatly.