[iammisc]

You are being downvoted but you're correct.

Consider the natural numbers. There is no 'why' behind them. There are axioms behind them, and -- given those axioms -- there are statements about the natural numbers that can be logically reduced to the axioms, but the axioms have no why.

Moreover, it is provable the axioms have no why, because they cannot have a why, because the axioms cannot be proven except in relation to themselves. If you're so convinced the axioms have a why, please prove me and Godel wrong.

The 'why' behind natural numbers is a social one, and one of convenience. The natural numbers make it easy to solve and communicate about certain problems, but they are not the only way to solve those problem nor are they the only way to communicate about these problems.

For example, another way to deal with basic arithmetic, is to talk about numbers as sets. Now you can define certain operations on them, and completely ignore the axioms of the natural numbers. This model is way better than others for certain problems. However, you now have a new set of axioms.. and oh yeah, actually the most obvious ones are completely self-contradictory, so you'll need to choose Zermelo-Frankel or something else.

Or if you want to be even more general, you can simply talk about the lambda calculus, but good luck trying to 'prove' the lamba calculus theorems in itself, because you'll quickly hit the halting problem.

Of course you can then say... well let's get rid of that and use the typed lambda calculus, but then oh yeah you can't do anything interesting. Why are these choices made? Can the choices be justified in the systems themselves? No of course not. The idea that you can use 'logic' to derive these systems is also ridiculous because formal logic is itself a system with axiom (and a very controversial system at that).

But if you look at the lambda calculus, ZF set theory, and the natural numbers as simply models and systems that are sometimes useful, then it makes sense as to 'why'. But the 'why' exists independent of them and is not provable in them and is social and cultural in nature. It is certainly not mathematical as in order to 'do mathematics' (symbolic manipulations) you first need axioms.

Mathematics education in this country has been replaced by rote dogmatism which is why many Americans cannot handle this ambiguity.

What must be explained is that mathematics is a language and in order to communicate with other educated humans about these abstract concepts it behooves everyone to speak the same language. It is the same reason the word for 'dog' in English is taught as being spelled D-O-G. There is no why behind it. It's just the result of thousands of years of culture. Except mathematics is a more global language and more useful for different kinds of manipulations.

[coolgeek]

> Consider the natural numbers. There is no 'why' behind them.

In a universe with more than one object, cardinality exists. Natural numbers are how we can discuss cardinality.

Natural numbers are also how we discuss ordinality, because ordinality exists in any universe having at least one dimension.

Axioms are how we discuss natural numbers rigorously. But natural numbers exist independent of any axioms. That's why they're called natural numbers

[horsawlarway]

Again - correct but misguided in the general sense.

> It is the same reason the word for 'dog' in English is taught as being spelled D-O-G. There is no why behind it.

I agree with you completely, but I think you're guilty of speaking the wrong language in response to the question (and it would behoove you to consider it from the perspective of someone outside the field).

The question "Why" in maths almost always gets asked by someone new to the field, and they are not asking from a mathematical perspective - They are not asking you for a formal/provable "why", they're asking you what is the utility of learning this thing.

So lets go back to D-O-G. The utility is clear - I have this hairy, 4 legged animal that keeps licking me that I'd like to discuss with you. We can agree that D-O-G (or perro, or 개) refers to it.

But with math, SO MANY PEOPLE (especially those established in the field) jump right into the "Here are the rules of this system of math" without ever taking the time to talk about why someone might give a flying fuck.

It would be like me going and making up my own language and forcing you to learn it. No one else speaks it, it's got no books/literature/history, there are no works of art that reference it - it's literally the language this random person made up that serves ZERO purpose except for talking to that person.

No wonder so many kids don't like math!

Instead you need to explicitly start with the utility of math - ideally in ways that are entertaining and fun. Once a person has an application for some of the rules, they become SO MUCH MORE INTERESTING! Suddenly I care about why this rule might impact that rule over there, or why A and Z are related, or what sin/cos/tan mean.

Basically - sell me on the value proposition of your fucked up whacky language - That's what "why" is asking. Once you know those rules do something useful, it becomes a much more engaging field of study.

[iammisc]

You are correct, but this explanation is cultural and sociological, not logical.

We must motivate math, absolutely. And to do so in my opinion starts with socratic questioning. You must convince the student that such an inquiry is even worthwhile.

One thing I'll point out is that we're not just seeing this in math. We see it in every field. More and more kids every year are insisting ( and their teachers are agreeing) that we can do away with inquiries into the English language and the humanities as well. There is a small, but continuing, effort to remove the knowledge of English masters like shakespeare and classic philosophers and treatises from the curriculum.

As a whole, American schooling fails to motivate learning of any kind. Math was the first victim, but the other subjects are also failing.

[horsawlarway]

Sure, but we're in a thread talking about why some folks are choosing to send kids to "Russian" style maths teachers.

The whole discussion is from the perspective of the cultural and sociological.

---

As an aside, I generally agree with you about american schooling. I think it's less a concerted effort, and more a sad reality of the fact that modern schools have essentially become federally funded child care in the US.

[iammisc]

> federally funded child care in the US.

Indeed... As my mother was told by her principal in her inner city school for poor minority kids... "We're just here to watch them until they go to prison".