myrmepropagandist
I teach at a very academically rigorous school. But, there are a few things this school does that might surprise people:
1. Many teachers teach a wide range of grade levels. So you could have a teacher who *could* teach Linear Algebra teaching you in 4th grade math.
2. The school makes time for creative math and CS in addition to the regular class. So I get to work with students without pressure to get them past any particular test or goal posts.
Wyatt H Knott
@futurebird Yabbut do you teach physics with calculus or without?
myrmepropagandist
They know a little calc before they get into physics. And they often tell me about how they used it in my calc class.
But, what I wish we could do is stop treating Statistics like it's... the math class for "weak" students who couldn't do calculus.
Part of the problem is there is still a tendency to classify kids as "math people" and "not math people" although I'm breaking my peers of this notion every chance I get. Part of it is this snobbishness pure math people have about stats.
Eleanor Saitta
@futurebird
I work in tech, not physics or more classical engineering, but I can say that the number of times I've wished staff knew calculus when they didn't? Zero in twenty years. The number of times I've wished they knew basic statistics when they didn't? At least once a month, for twenty years.
@whknott
Jorge Stolfi
@futurebird @dymaxion @whknott
Indeed the emphasis of math teaching, in high school and early college, is in the sort of logic that enables one to gain trust in a mathematical statement C from facts A and B that one trusts are true.
And that is a good thing, because most problems the student will require algorithms or formulas that he did not learn at school.
🧵>
myrmepropagandist
Because what mathematics really is are the *proofs* not the solutions, not the algorithms, but that unbroken chain from a minimal set of assumptions to the solution.
Over and over we work to demonstrate this unbroken chain, though in most undergrad statistics courses this is just something we give up on in favor of getting the students competent enough with the algorithms to mostly apply them correctly. And that's part of why such courses aren't seen as "real math." 2/2
myrmepropagandist
I realize that mathematicians can make everyone very tired because we are more interested in the foundations and shape of the containers than their contents.
We teach students how to complete the square to solve quadratics ... mostly so they can prove the quadratic formula for themselves.
But often these students don't even really understand that if a*b=0 that means a, b or both must be 0 or how that is at all relevant to "finding x"
1/
Eleanor Saitta
@futurebird
Yeah, largely in that kind of space. Also rigor around what it means to measure something, etc. Like, obviously calculus is useful — if nothing else, not having the conceptual tools of first and second derivatives makes looking at a line on a graph or the area under it less intuitively useful — but I think it's more about the things that you learn along the way.
@whknott
Eleanor Saitta
@futurebird
Hmm. Often, really just basic statistical numeracy so they could understand data they seeing in papers, etc. Sometimes, more the ability to reason about what is and isn't good data for quantitative decision-making — folks really love to make up meaningless numbers to let them avoid qualitative decisions when quant data isn't there. Sometimes data analysis to understand things like perf impact from log data for edge cases. So in some ways, not really statistics itself, but all things that I find that folks who made it through at least one real stats course are likely to be better at, if that makes sense?
@whknott
myrmepropagandist
So, more experience with the creation of data visualizations and summary statistics (where do they come from?) Methods to access their quality and predictive value?
I ask because I've had many people say something along these lines "why did I learn so much calculus, everything I need to do with math is statistics?"
This can confuse me a little because understanding distributions is so much easier if you know calculus. (To find maximums and areas under curves.)
myrmepropagandist
When you've wished you knew more about stats what was it that you wanted to do, find out, or know?
Jorge Stolfi
@futurebird @dymaxion @whknott
🧵> But, as you say, math teaching at school indeed tries more: it tries to derive every math statement all the way from a few "fundamental axioms". That is a leftover from the days when the only serious math taught at school was geometry, and geometry meant Euclid's book.
🧵>
Jorge Stolfi
@futurebird @dymaxion @whknott
🧵> I don't see THAT -- going all the way to "fundamental axioms" -- as a good thing. On one hand, the choice of axioms is arbitrary: one could take any sufficiently large set of theorems from Euclid, declare them to be the axioms, and then derive his axioms from these.🧵>
Jorge Stolfi
@futurebird @dymaxion @whknott
🧵> Furthermore, over the last couple of centuries we have realized that those "fundamental axioms" are not as "true" as Euclid assumed. General relativity and the Heisenberg principle make Euclid statements wholly unreal -- like axioms about gods, ghosts or dragons. The foundations of set theory are a mess. 🧵>
Jorge Stolfi
@futurebird @dymaxion @whknott
🧵> And, finally, since Gödel we know that, no matter which axioms we choose, there will be infinitely many math statement that are true but cannot be proved starting from those axioms. Indeed, *practically all* true statements have no proofs -- because proofs are countable, but true statements are uncountable.
Jorge Stolfi
@futurebird @dymaxion @whknott
"Let X be the set of all elements that don't belong to X" That definition of the set X is *obviously* invalid; but how do you define rigorously what is a valid definition?
The complement C of a set X is supposed to be all the elements that do not belong to X. But since sets can be elements of other sets, the set C is itself an element of C. Is that okay? if not, how do we fix that?🧵>
myrmepropagandist
@JorgeStolfi @dymaxion @whknott
The foundations of set theory are a mess? This is news to me.
There have always been mathematicians interested in pipe dreams like universal axioms or more minimal sets of axioms. But if you need to get things done you *can* ... you just need to be honest about what you are assuming, and it might not be as minimal as some want.
Different assumptions create different mathematics that have their own uses.
Maybe I'm not understanding what you are saying.
Jorge Stolfi
@futurebird @dymaxion @whknott
🧵> Some math theorems depend on the axiom of choice. If I got it right, it says that, given any (finite or infinite) set of non-empty and disjoint sets, there is a set that consists of one element from each of those sets. Seems pretty obvious and lame, but it leads to several counterintuitive results...
Jorge Stolfi
@futurebird @dymaxion @whknott
It is not just that the axioms are arbitrary, but that they cannot be specified precisely without contradiction. If you can't define what a "set" is, how can you expect everybody to agree on whether something is a set of not?
Euclid himself assumed a couple of things that should have been explicitly stated as axioms, such as "If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side."
myrmepropagandist
@JorgeStolfi @dymaxion @whknott
At the base of every axiomatic system are terms that cannot be defined using the system itself. Terms that require consensus and should be recognized as such. And these are always worth re-examining and removing, changing to see what other systems we may develop.
It is still a human endeavor, based on language. And I think that's a feature not a bug.
Jorge Stolfi
@futurebird @dymaxion @whknott
My PhD thesis was about a variant of projective geometry in n dimensions. As a wannabe mathematician, I spent months trying to define that geometry by a set of axioms. In two dimensions, it is basically Euclidean geometry without circles -- no compass, just ruler, so its axioms are straightforward.🧵>
Jorge Stolfi
@futurebird @dymaxion @whknott
🧵> In three dimensions it is more complicated: besides the obvious Euclid-like axioms, it seems one need also an axiom that says "for any eight lines A1,A2,A2,A4 and B1,B2,B3,B4, if 15 of the pairs Ai,Bj intersect, then the 16th will intersect too".
And it only got worse in higher dimensions.
So in the end I gave up on the axiomatic approach. I defined a model with real coordinates, and said "n-dim projective geometry is is any geometry isomorphic to this"
Rich Felker
@JorgeStolfi @futurebird @dymaxion @whknott They can and have been stated precisely without contradiction. This is a problem solved nearly a century ago.
Matt McIrvin
@dymaxion @futurebird @whknott Where I wish people understood calculus is in *political* discussions, particularly involving economics. Not any of the techniques, just the basic idea of a function, its derivative and its integral being different things.
And maybe the idea that if you have a function of multiple variables, its rate of change is going to depend on which specific things you're holding constant.
But that last one is a HARD idea. It doesn't even really show up in AP Calculus, it's a later class. It trips people up when they're studying college-level thermodynamics.
Matt McIrvin
@futurebird @JorgeStolfi @dymaxion @whknott I think that all the way back to at least Leibniz there was this idea that you might be able to automate the general search for truth by reducing it to turn-the-crank derivations. It's the same impulse that leads people to treat ChatGPT as an oracle.
myrmepropagandist
@JorgeStolfi @dymaxion @whknott
Part of understanding these systems and their tremendous power is recognizing that there are undefined terms and knowing exactly what you have assumed through the consensus of language and conventions of meaning.
Do you want to purge subjectivity from mathematics? Good luck with that.
But I think it's also worth asking WHY so much of the mathematics of a century ago was focused on this goal? Why did otherwise brilliant people burn up so much time chasing it?
myrmepropagandist
And where has treating statistics like "the lesser math" gotten us? I blame the messy way LLMs and other tools are being slathered around on everything on the general neglect of statistics by "serious math people"
It's an unsupervised area and it has fallen into chaos. And they have the powerful tools that claim to do the magic that the people want.
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