Can math be learned backwards?

Question

In C++, we can reverse engineer and performance binary analysis to know exactly what a piece of binary will do, even without seeing the original source code. In math, can this be done?

Basically, can math be reverse engineered so one could, say, find the end and reach the start?

Like in computer programming, we can look at something like:

if(X == Y) - > MOV DX, X - > 0001 1001 0110 0011 - > connect latch A to register X

See? Even without seeing representative little zeroes and ones, we can still track back to what computer code is doing by inspecting a microprocessor's control store/microcode/RTL/etc.

In math, can, instead of learning all the "high" stuff, we learn the end and find the beginning backwards or reverse engineer math and take it on in a totally different approach?

Answer 1

this is more of a comment than an answer but i have a lot to say.

there are people who believe in a new wave of math education, where we could teach "higher" math (meaning calc, maybe basic linear algebra and differential equations) to students as early as elementary school or middle school.

people in support of this argue that schools (at least in the US) spend FAR too much time focusing on the basics, such as calculation, and not enough time on the actual mathematics. this new instruction would sort of "admit" that we can let computers do the computing, and use this to understand more complicated subjects.

the reality is that we already do this in most sciences. in physics these days, at least in my experience, it is growing less and less common to do difficult integrals "by hand" when it saves so much time to use a program like mathematica. the argument is that, once we know how to do a calculation, there isn't a ton of reason to spend hours and hours of busy work.

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personally, i believe that the answer is not one sided. of course students need to learn the basics in order to fully understand higher topics. that said, as a product of the US public school system, i also understand how much time is wasted on, for example, calculating the derivative of 50 polynomials on a page.

i think that, to some extent, lots of higher COULD be taught to younger students. i don't think abstract algebra and real analysis are really as hard as most people think they are, in the sense that we could teach them, partially, as long as we let go of the rigorous understanding of them until those students are older.

Answer 2

I'm still an undergraduate student but I would like to show my point of view.

The construction of mathematics is based on axioms, that is the lowest level possible you can reach; once logic bases are established it is possible building things at higher level using theorems. I think of theorems in mathematical sense as functions in programming: you have some conditions and from these you can show a result in a new, different form through a chain of logical steps of axioms and results of other theorems; on the other hand functions in programming work the same: they get keywords, variables and other functions to compute the needed result.

While learning the mathematics we don't learn its low-level structure, while it could have sense seeing from how logically the mathematics is built but we start from something close to our common experience and then build the knowledge over the knowledge we already have; on the other hand we cannot start learning programming from assembly even if its performance are the best and we can know exactly what is happening inside the program, it would be an overkill!

Making an example would be useful. Take for example the definition of indefinite integral for function $f: \mathbb{R} \to \mathbb{R}$ (function $f$ that inputs a number and returns a number): $$\int f(x)\, dx$$ Was is it? The inverse operator of derivation, so: $$\int f(x)\, dx = F(x) + c \text{ so that } \frac{dF(x)}{dx} = f(x)$$ but was the derivation operator (for $f: \mathbb{R} \to \mathbb{R}$) is? $$\frac{dF(x_0)}{dx} = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}$$ for $x_0 \in \mathbb{R}$. Then extend it for $\forall x_0 \in \mathbb{R}$ where to limit exists.

But we are using limits: what are they? $$\lim_{x \to c} \, f(x) = l$$ is $$\forall \epsilon > 0, \exists \delta > 0, \text{ such that } |x-c|<\delta \Rightarrow |f(x)-l| < \epsilon$$

But we are using inequalities, that is a characteristic of $\mathbb{R}$ (real numbers), so we can describe $\mathbb{R}$ is a field that has a total order ($<$ operator).

• field is a ring with some properties (won't describe in depth);
• a ring is are two groups with some interesting properties too (won't describe in depth too);
• a group is couple (for example $(\mathbb{R}, +)$, $(\mathbb{R}, \cdot)$) for which some properties are true (won't describe in depth too again);

We could ask what $+$ and $\cdot$ are: they are functions that do some stuff (examples: $2.1 + 5.4 = 7.5$, $\pi \cdot \pi = \pi^2$) from a set ($\mathbb{R} \times \mathbb{R}$) to another set ($\mathbb{R}$).

Continuing, a fuction is a collection of all couples (for example for $+$ is $((\mathbb{R} \times \mathbb{R}) \times \mathbb{R})$ such that the relationship given by the function between members of the couples is true (example: $4+6 = 10$ so $((4, 6), 10) \in A$ where $A$ is set of all couples that make $+$ function true).

And what $\mathbb{R}$, $(\mathbb{R} \times \mathbb{R})$, $((\mathbb{R} \times \mathbb{R}) \times \mathbb{R})$ are? Talking broadly they are sets that have special properties (for example $(\mathbb{R}, \mathbb{R})$ is a set that cointains all couples real numbers, such as $(\pi, 3)$, $(0,0)$, $\dots$).

It's also worthy to say that $\mathbb{R}$ set is a "generalization" of the natural numbers set $\mathbb{N}$ and natural numbers are the starting point.

But there are still questions: what is a set and how natural numbers can be constructed? A simple answer to explain a set can be done by saying it's a container that can contain everything with no distinction of exactly same elements. To define natural number set we can use set theory definition (see also): \begin{align} 0 &= \{\} \\ 1 &= \{0\} = \{\{\}\} \\ 2 &= \{0,1\} = \{\{\}, \{0\}\} = \{\{\},\{\{\}\}\} \\ 3 &= \{0, 1, 2\} = \dots \end{align}

We still go further and ask how could we define a set better or how can we logically explain demonstrations and its forms (for example induction proofs)... In this case things like ZFC, Gödel and other interesting things emerge.

In this point have probably arrived to "assembly" of math starting just a single from a "high-level function" (math-programming comparison intended).