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@whknott

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.

@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

@dymaxion @whknott

When you've wished you knew more about stats what was it that you wanted to do, find out, or know?

@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

@dymaxion @whknott

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.)

@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

@dymaxion @whknott

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/

@dymaxion @whknott

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

@futurebird @dymaxion @whknott I didn't do any statistics courses as an undergrad. Many years later I did a graduate course in Geostatistics (spatial statistics) where the lecturer went from absolute basics to advanced concepts step by step, seamlessly, and we were spellbound. In a statistics course. It was amazing.

@louisffourie @dymaxion @whknott

I hope you have your notes. Do you remember what book or books you used?

Such treatments are kind of rare. Statistics is not as well formed as a "course" as calculus and other mathematics.

@futurebird @dymaxion @whknott He had meticulous notes and only gave textbook references. He was a textbook author himself. He also interspersed the theory with real examples as well as anecdotes. The course was geared to industry professionals.

And yes, I do have the notes!

@futurebird @dymaxion @whknott My stats courses (intro, for psychology students) are definitely not "real math." We do zero proofs, zero calculus, etc. I've thought of renaming the course from "statistics" to "applied data analysis with a little theory". This fits our students' needs, and since we don't have a functioning university any more (i.e., we are structurally prevented from requiring, say, another couple of math courses for our majors), it's what we can do. I find it immensely valuable but I often wish we could require and therefore teach a more rigorous sequence of actual-statistics courses.

@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.

🧵‍>

@JorgeStolfi @dymaxion @whknott

This is what it is supposed to be, but it's VERY easy to loose the plot.

And that produces people alienated from mathematics. "Why did I have to learn all of that???"

@futurebird A problem I noticed at school that seemed worse with maths than with other subjects: one bad teacher, or a teacher who's having to read one chapter ahead of the class while juggling a bunch of other jobs, can destroy the spirit of enquiry and experimentation for life.

@RogerBW @futurebird
In my last couple years of high school I had a terrible physics teacher then a terrible calculus teacher.
Took the joy out of learning the cool stuff, the math I'd been waiting *years* to start learning at school.
My uncle noticed and started slipping me his college math textbooks. Had to teach myself trig and statistics and what calculus I worked out.

@silvermoon82 @RogerBW @futurebird I feel like this was me too - had GREAT early math experiences, had some real dud teachers in high school and ended up dropping it after high school

I don't think either of my pre-calc/calc teachers had any idea about WHY they were teaching us what they were teaching, they just knew it was the required course material. If someone doesn't love their subject, it leaves students up to themselves to find their own way to love and have curiosity about the subject.

@silvermoon82 @RogerBW @futurebird serendipitously, I happen to be just now reading Kyne Santos's *Math in Drag* - it showed up in the library's "what's new" list and I checked it out. It's so good and reconnecting! onlinekyne.substack.com/p/i-al

Reading it has been a great reminder about how effective teaching about any complex area requires strong communication techniques in addition to subject matter knowledge.

The Math Queen Digest · I Almost Didn’t Write This BookBy Kyne Santos

@silvermoon82 @RogerBW @futurebird I get pretty good mileage out of trig and elementary calculus. Sometimes the rough sketch of differential equations or modeling is useful. Can’t tell you a single time I’ve needed the Bessel equations outside of class.
(I mean, sure, if I actually went through with modeling a spherical turkey for determining cooking time for Thanksgiving I think I’d need it, but there’s a thumb rule on the package)

@futurebird @JorgeStolfi @dymaxion @whknott

I am going to write The Old Man's Math Book. Starting with Sirius and the stars rotating in the night sky, Babylonians, Hipparchus, the math of the ancients.

Then algebra,taught properly, calculating the payout for everyone involved in the spice trade, shares of a commodity with a yet-unknown price.

"The train leaves the station at 10 mph."

No it doesn't. Takes five minutes to accelerate the train to 10 mph.

Real world math

@futurebird @JorgeStolfi @dymaxion @whknott

May I add in passing, most calculus books should be burnt with fire.

@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.

🧵‍>

@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.🧵‍>

@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. 🧵‍>

@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.

@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?🧵‍>

@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.

@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."

@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?

@futurebird @JorgeStolfi @dymaxion @whknott Because it was the last hope for those who wanted fundamental certainty in their worldview. Religion struck out, physics was finding fundamental limits. If you yearned for completeness and certainty, it was distressing.

@futurebird making me want to build my mathematics skills and reread Alan Watts’s Zen writings.

Stuck in my head: science is systematic doubt, but has cornerstones ultimately taken on faith and consensus. It weakens your understanding if you pretend you can eliminate them.

(Aimed mainly at mid-century rationalists but handy today)

@randomgeek @futurebird Right?

Also, Alan Watts must be among the most sampled philosopher of all time! There are dozens of songs with bits of his talks.

One of my favorites is Golden Light, by STRFKR

open.spotify.com/track/1rPYEWQ

SpotifyGolden LightSTRFKR · Miracle Mile · Song · 2013

@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.

@JorgeStolfi @futurebird @dymaxion @whknott They can and have been stated precisely without contradiction. This is a problem solved nearly a century ago.

@JorgeStolfi @futurebird @dymaxion @whknott This sounds like a grounding problem. The axioms are perfectly consistent (or automated proof checking wouldn't work) but their relation to reality is what's messy.

Probability theory shows this very clearly. The same set of axioms is accepted by frequentists and Bayesians alike with very different interpretations.

So long as you treat the whole thing as an ungrounded string rewriting exercise with no relation to reality, there is no mess.

@pbloem @JorgeStolfi @futurebird @dymaxion @whknott Well let's see what statistics can be if it is grounded in reality.

@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.🧵‍>

@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"

@JorgeStolfi @futurebird @dymaxion @whknott Are you claiming these are problems in the foundations of set theory right now? My understanding is that the ZF axioms are actually not vulnerable to either - the Russell set simply cannot be defined in the language of ZF, and neither can absolute complements (but relative complements can be defined, and are not problematic).

I am pretty certain there aren't any known contradictions implied by ZF. That would be pretty huge news if discovered.

@danielittlewood @JorgeStolfi @futurebird @dymaxion @whknott It's not that contradictions can be derived from ZF, just that it's so unclear what should be taken as axiomatic. Mathematicians generally use not just ZF, but ZFC — with the axiom of choice. In practice, it isn't controversial, and it gets VERY awkward to try to do math without it. But it's logically independent of ZF, and has some highly counterintuitive consequences.

@dedicto @danielittlewood @JorgeStolfi @dymaxion @whknott

Thanks Douglas. This was a much more clear way to put it than I could have managed at the moment.

@futurebird @danielittlewood @JorgeStolfi @dymaxion @whknott For a real walk on the wild side of set theory, check out the Wikipedia article on the axiom of limitation of size, a particularly bold way of handling set theory with proper classes:

en.wikipedia.org/wiki/Axiom_of

en.wikipedia.orgAxiom of limitation of size - Wikipedia

@danielittlewood @JorgeStolfi @futurebird @dymaxion @whknott There are also versions of set theory that have "proper classes" that can HAVE sets as members, but can't BE members of sets (or of each other). Standardization on ZFC is taken for granted in math outside of set theory itself, but the justification for that particular choice is purely pragmatic.

@JorgeStolfi @futurebird i mean typically we just fix that by only taking complements within a particular superset as the "scope". complements are well-defined within a scope and most of mathematics is about working within defined scopes like the integers and real numbers, etc.

@JorgeStolfi
Ok, Russel and Gödel spent a lot of time and founded mathematics on sets. Last time I checked it all worked and was consistent. A set of sets is different to a set of oranges or numbers for that matter.
@futurebird @dymaxion @whknott

@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...

@JorgeStolfi @dymaxion @whknott

And isn't it amazing that we can isolate this "obvious" thing as an assumption and then ask: what happens if we do not assume it's true?

Sometimes that is fruitful, sometimes it just breaks everything.

To me that's very powerful.

@futurebird @JorgeStolfi @dymaxion @whknott This is tangential to your current discussion, but I thought you might be interested. Raymond Smullyan in The Tao is Silent has an essay where via a Socratic argument discusses whether or not systems of morals are finitely axiomatizable. The conclusion one the the speakers draws is that not only are systems of morals finitely axiomatizable, but that they can be reduced to precisely one axiom: "everyone has the right to do as they please."
At any rate, it's a delightful book and the irony of a logician examining the Tao (which is illogical) is entertaining. I found it after reading "The Mind's I" where it had been excerpted.
End of sidebar.

@JorgeStolfi @futurebird @dymaxion @whknott No, they do not. If anything, there's an ongoing struggle between physics and philosophy over what would be a good meaning of "real" between them, and whether this is even possible.

GR does not invalidate Euclidean geometry. It just says that we don't live in a perfectly Euclidean space, though local approximations (and, on the large scales, perceived flatness of the cosmos) still work fine.

@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.

@JorgeStolfi @dymaxion @whknott

I think this is analogous to the "foundational problems in animal taxonomy and speciation" there are huge unaddressed problems and questions there (siphonophorae, lichens...) but these questions are exciting and good and don't make the process of classifying animals in clades by last common ancestor any less useful or revolutionary.

The exceptions don't break the system they help to further define it.

It's similar with these paradoxes at the heart of math.

@JorgeStolfi @futurebird @dymaxion @whknott I find the constructions of the reals from sets to be a useful tool in my repertoire, even as a software engineer. We often have to build abstractions up from simpler things and going from the set containing to empty set to the real num/res and continuity etc is a powerful example.

That is teaching how to think logically, which is a good thing.

@JorgeStolfi @futurebird @dymaxion @whknott

@futurebird @dymaxion @whknott Now you got me thinking if there's an algebraic extension in which there's a "reverse number" such as:

x * (reverse x) = 0

... for non-0 x, that is. Sounds pretty useless, but what do I know.

@vriesk It doesn’t get emphasised much in calculus, but by our definitions of what a differential is, it can be non-zero, but its square is zero. ie, dx > 0, (dx)^2 = 0. Because we defined a differential to be the smallest possible non-zero number. This pops up in other contexts as well, other than just calculus.

@futurebird @dymaxion @whknott

@4raylee @futurebird @dymaxion @whknott But how does that even work? I mean, the definition sounds like something that the number theory claims is impossible to exist, so doesn't that lead to really bad discrepancies?

@vriesk What number theory do you think says this is wrong? This is how we define dx in the first place, and one of the natural consequences of that definition.

I remember getting surprised by this while watching a professor do a proof. They set all higher powers of dx to zero and my brain made the record scratch sound.

Anyway, it’s consistent, and useful. It’s just not what we expect from the colloquial idea of non-zero. But differentials are not exactly like everyday numbers.

@futurebird @dymaxion @whknott

@4raylee @futurebird @dymaxion @whknott Not "wrong", just impossible - as both for rational and real numbers there's always infinite numbers to squeeze between any two.

I mean, I understand dx as a concept of "approaching", but I'd expect that concretizing this concept so that it can be subject to regular algebraic operations would lead to contradictions. This feels quite different than extending reals algebra to include infinities and positive/negative zero.

But most likely I'm wrong.

@vriesk Limits are ‘approaching’, differentials are there! In fact, when we first learn the rules about differentials they come hand in hand with the idea of limits. When exploring limits we learn that we can ignore all higher orders of dx.

Have you heard the phrase “The boundary of a boundary has size zero?” This is another aspect of what we’re talking about, in a slightly different mathematical context. (Exterior derivatives and topology.)

@futurebird @dymaxion @whknott

@4raylee @futurebird @dymaxion @whknott No, I haven't. Though I'm unsure what "size zero" means in this concept, "boundary of a boundary" does not strike me as something particularly incorrect, in as much as I can see a boundary being higher-order dependant on some other condition that can be approximated.

It's just that the concept of such described differential does not agree with my intuition of something well-conceptualized. But like I said, I'm wrong, I'm not a maths person in any way. :)

@futurebird @dymaxion @whknott

the one I never got on with was matrices, they made my head hurt.

@Thebratdragon @futurebird @dymaxion @whknott Opinion: if you want to do stats in the way it is done in science and industry in the world today you can get away with not knowing calculus, but you will have to know at least a bit of linear (i.e., matrix) algebra. This is a sad fact many grad students in many fields encounter.