[routerl]

It seems to be an article about all those "harmless" lies we tell students.

The vast majority of people think mathematics is about numbers, when it is actually about relations, and numbers are just some of the entities whose relations mathematics studies.

Nobody is born with this misconception; we teach it, and test it, and thereby ingrain it in the minds of every student, most of whom will never study mathematics at a level that makes them go "wait, what?". The overwhelming majority of people never get to this level.

I suspect this is also why statistics feels so counterintuitive to so many people, including me. The Monty Hall problem is only a problem to those who are naive about probability, which is most people, because most of us don't learn any of this stuff early enough to form long lasting, correct instincts.

It's not fair to students to bake "harmless" lies into their early education, as a way to simplify the topic such that it becomes more easily teachable. We've only done this because teaching is hard, and thus expensive. Education is expensive, at every step. It's not fair or productive to build a gate around proper education that makes it available only to those who can afford it at the level where the early misconceptions get corrected. Even those people end up spending a lot of cognitive capital on all those "wait, what?" moments, when their cognitive capital would be better spent elsewhere.

[rich_sasha]

> The vast majority of people think mathematics is about numbers, when it is actually about relations

It is somewhat unfortunate that mathematics is two different things, simultaneously very closely related and very different. One is the abstract study of relationships between axiomatic entities, and the other is arithmetic.

Vast majority of people out there need only arithmetic, and boy they really need it. Calculating tax, taxi fares, shopping bills, splitting bills etc. And to some extent, you need the abstract maths to understand arithmetic.

We have one curriculum for that vast majority of people and for the few who move on to academic maths. Simplifying ideas like integers to number lines doesn't seem like a high price to pay.

[atoav]

I recently thought a student who "always was bad at math" how to use it to make his marionette puppet servo controlled. On the way we encountered a lot of scaling ranges, working with angles, trigonometric functions for motion synthesis, all kinds of complex remappings with saturation functions, random walks, linear interpolation functions and so on.

He was absolutely stunned and asked me why mathematics wasn't thought that way all the time. Instead of a bunch of things he had to do, he came to see it as a toolbox with things you can use.

And I myself wonder why the hell my maths teachers failed at making this easier as well. I distinctly remember my math teacher wbo failed to answer me when I asked after months of solving integrals why we need those. I had to figure that out myself, pre-internet.

[fenomas]

> when it is actually about relations

I think of it more as, math is ultimately about symbols. Like, if a mathematician says that "2 = 2" is a true statement, a reasonable onlooker might ask "Does that mean that all twos are interchangeable? Or that there's a unique concept called two and it equals itself?" And the mathematician replies, "Neither! It means that the string of symbols '2 = 2' is reducible to the symbol 'true', given certain axiomatic symbolic transformations. Nothing more, nothing less!".

And obviously we can project concepts onto the symbols, like "integer" and "real number", and talk usefully about them, but those are the map and the symbols are the terrain, as it were. At the edge cases where we're not sure what to think, we have to discard the concepts and consult the symbols.

[bheadmaster]

> math is ultimately about symbols. [...] Neither! It means that the string of symbols '2 = 2' is reducible to the symbol 'true', given certain axiomatic symbolic transformations. Nothing more, nothing less!

If that were true, math would be useless, and nothing more than an esoteric artform.

The true power of math comes from the correspondence between those symbolic transformations and observation from the real world. Two objects that look alike can be placed in juxtaposition with any other (different) two objects that look alike, and no matter how much we move them around, as long as we don't add or remove any objects, they can still be placed in the same juxtaposition as before (while this description may seem verbose and clumsy, in the real world it does not need a description - it is a much more primitive sensory perception, learned at an early age).

> obviously we can project concepts onto the symbols, like "integer" and "real number", and talk usefully about them, but those are the map and the symbols are the terrain

It wouldn't be "obvious" that we can project concepts onto symbols, if we didn't discover that symbols correspond to concepts and that symbolic transformations can help us predict the future. Thus I'd say it's the other way around: symbols are the map that we know how to read - of the terrain that we can't traverse easily.

[fenomas]

> The true power of math comes from the correspondence between those symbolic transformations and observation from the real world.

Not at all, think it through further. Obviously it's true that mathematics is more practically useful in cases where its symbolically-proved claims have some kind of relation to real-world observations, but if that relationship were a requirement, math would be useless - you could prove a theory on paper symbolically, but you wouldn't know whether the thing you proved was "really true" until you found a way to check whether the result is also true in the real world. And if you found it was true of apples, it might still not be true for electrons, etc etc.

Rather, math's power stems from the fact that it emphatically does not expect or require the symbols to have any connection to real world observations. If you prove something on paper, it's proved and that's that. If the thing you proved also happens to be useful for describing apples or electrons, that's great - and the fact that this often happens is why the whole "unreasonable effectiveness of mathematics" is a thing. But if there's no relation to the real world, that doesn't in any way affect the truth of the symbolically proved claim, or its usefulness or interest to mathematicians.

[bheadmaster]

> Rather, math's power stems from the fact that it emphatically does not expect or require the symbols to have any connection to real world observations.

What exactly do you mean by "power" here, if not the ability to predict real-world phenomena? In absence of it, what exactly would make it anything more than an exotic artform?

[fenomas]

I meant the fact that math works at all, as a symbolic framework with consistency and the power to prove some statements and disprove others, etc. Math only has those features because it examines symbols abstractly regardless of any connection to the real world.

Like, consider: parabolas were pretty fully described by the ancient Greeks, purely as a symbolic abstraction. It was only 1500+ years later that anyone realized that they could also predict the motion of cannonballs and planets. But that discovery was completely orthogonal to the math - e.g. symbolic statements about parabolas didn't get any truer just because they now also described real-world phenomena. (And likewise when we later discovered that planetary motion isn't quite parabolic after all, that didn't affect our understanding of parabolas either.)

That's all I was saying here - that the "esoteric artform" part of math where one abstractly examines symbols is the essence of the thing, and the "predict real-world phenomena" aspect is a side effect that sometimes happens and sometimes doesn't.

[bheadmaster]

True, but what I'm pointing out is - imagine parabolas never predicted movement of anything, or imagine that math itself never had any prediction power in the real world. The self-consistency and the aestethics and the ability to prove statement inside itself would all be just a bunch of symbolic games, much like poetry or crossword puzzles.

You may argue that that, in itself, is powerful, in which case fair enough. But that "power" would be comparable to that of poetry or painting, which, in my opinion, does a disservice to the true power that mathematics holds. Mathematics is much more powerful than poetry and painting, because poetry never helped us build nuclear reactors.

[kragen]

Platonist mathematicians do exist.

[dr_dshiv]

The vast majority of mathematicians are platonists. They think math is real.

[Meneth]

And the logical conclusion of that: https://en.wikipedia.org/wiki/Mathematical_universe_hypothes...

[kragen]

They might not admit it if they're not drunk enough.

[fenomas]

I'll edit out the part of my comment where I suggested they don't.

[kragen]

You can't, so you didn't.

[fenomas]

Weirdly aggressive, but fair enough.

[kragen]

No aggression intended; I just wanted to clarify for anyone reading the thread afterwards who thought I might be responding to a different version of your comment than the one that's there now.

[rini17]

And then there are undecidable statements and their ilk.

[mcphage]

> The Monty Hall problem is only a problem to those who are naive about probability, which is most people, because most of us don't learn any of this stuff early enough to form long lasting, correct instincts.

I think it’s more than that… we come with some built-in heuristics for probability, which mostly work pretty well. Until they don’t.

[rbtprograms]

i would argue our built in heuristics for probability are pretty bad, which is why the monty hall problem is so hard for most people to grasp (even though it is a relatively straight forward application of probability). probabilistic thinking comes much less naturally to the human mind than deterministic thinking.

[xboxnolifes]

I find that the bad intuition on the monty hall problem is mostly due to the small delta of going from 1 in 3 doors to 1 in 2 doors, combined with some bad human intuition. If you change it to start with 1000 doors, I find it to be a lot more intuitively convincing.

[stavros]

Same here, although I've talked to people who were equally confused with that formulation.

[gosub100]

> It's not fair to students to bake "harmless" lies into their early education, as a way to simplify the topic such that it becomes more easily teachable

Childrens' brains are not fully developed. I see no gain from telling a 6 year old that "most numbers aren't countable". Especially because most numbers are never used or interacted with in any way shape or form. It's not "lying", it's separating concepts and prioritizing.

[viraptor]

> We've only done this because teaching is hard, and thus expensive.

That's just silly. We've done that to make the math useful and possible to teach. Unless you're saying you're able to start with sets of numbers and defining a ring for kids, before explaining what 1+1 is.

[kamaal]

>>when it is actually about relations, and numbers are just some of the entities whose relations mathematics studies.

Well no. Math is studying anything at their most atomic machinations. That mostly involves.

- Making hypothesis, that is assumptions about start conditions and rules of play.

- Evolving the system you just created. Such that conclusions are consistent with the rules of play.

The real deal is good math involves lots of paper work, to an extent you could almost say Math is a writing skill than a thinking skill.

Think of it like generating a lengthy changelog.

[SyzygyRhythm]

I would say instead that math is a game. A universal game with no predefined rules at all and only one guideline: if the rules you make up lead to a contradiction, then the rules are probably boring. If your rules say that 1+1=3, then you can prove anything and the whole thing becomes uninteresting.

Mathematicians have come up with various rules (axioms) that seem to work pretty well. And they spend a great deal of time figuring out their consequences. But it may still happen that the rules have a contradiction and they need to come up with a different set.

Sometimes mathematicians add extra rules when they run into a roadblock. And part of the meta-game is to come up with the minimum set of extra rules they need to keep going. Sometimes they spend time figuring out if the existing rules aren't needed.

[kamaal]

>>I would say instead that math is a game.

Yup, and as you keep going the level too go up!

But the core ideas are simple though-

1. Start some where where you understand things enough to make sense.

2. Make the smallest possible, atomic change to some aspect of thing you know at point 1.

3. Test if the change sticks- If yes, repeat steps 1 - 3

4. If the change doesn't stick- Go back to step 1. Now either make a different change to the same thing or make a new change to a different thing. Repeat steps 1 - 3.

As you can see you write a lot. Like really a lot. Math is just writing skills.

[krisoft]

> The Monty Hall problem is only a problem to those who are naive about probability, which is most people, because most of us don't learn any of this stuff early enough to form long lasting, correct instincts.

I mean maybe? Depends on what your definition of being naive about probabilities is. The Monty Hall problem has a sordid history of even very learned mathemathicians specialising in probability getting it very wrong. For example Paul Erdős got it wrong[1] (until someone walked him through it)

Now maybe you count Erdős as someone who is naive about probability. In which case I guess you are right. But that puts the bar very high then.

1: https://sites.oxy.edu/lengyel/M372/Vazsonyi2003/vazs30_1.pdf

[sgt101]

Like Art is about making pleasing artifacts whereas it's really about making business connections and laundering money?