🧵 Rigorous Basic Math
Anonymous at Thu, 28 Nov 2024 00:21:41 UTC No. 16495659
I have been looking for a book that explained in a rigorous but down-to-earth way basic math, beginning with arithmetic and up to (and including) calculus.
By searching online on my own, I stumbled on "Understanding Numbers in Elementary School Mathematics" by Hung-Hsi Wu.
This seems to be the closest I could find to what I was searching for, the only issue being that the target audience is schoolteachers so it seems to waste some time talking about pedagogical stuff (in which I don't have much interest) even though most of the book just talks about math from what I have seen thus far.
I wanted to ask whether anyone has read Wu's book (and his other books as well, which seem to cover everything up to and including (single-variable) calculus) and whether anyone has any better recommendation for someone who is NOT a teacher but is still interested in learning more about basic math in a reasonable rigorous way.
I'm liking Wu's approach thus far since he remains very concrete and doesn't stray too far into abstraction. And he also explains how to carry out calculations in practice with the standard algorithms and he explains why the algorithms are correct. Which is something you won't find in books with a more advanced outlook texts like "Foundations of Analysis" by Edmund Landau. Landau does explain how the number system works starting from Peano's axioms but the whole development is very formal and detached from the concrete way in which numbers are used in everyday life.
At the other extreme, books like "Basic Mathematics" by Serge Lang aren't basic at all and they already assume a considerable degree of familiarity with numbers (and rigor is somewhat optional anyway, especially in the chapters about geometry).
Anonymous at Thu, 28 Nov 2024 00:35:31 UTC No. 16495673
>>16495659
>a rigorous but down-to-earth way basic math
nigga, the down-to-earth way to explain basic math is how you teach it to kids in elementary school
>rigor
>he remains very concrete and doesn't stray too far into abstraction
rigor is pretty much pure abstraction.
if you want rigorous mathematics concerning numbers and calculus, pick up textbooks on abstract algebra and real analysis.
however, unless you understand some basic applications and examples already, going straight to abstraction will melt your brain.
there is no golden road to mathematics
Anonymous at Thu, 28 Nov 2024 00:59:17 UTC No. 16495695
>>16495659
rigour isnt defined in mathematics
Anonymous at Thu, 28 Nov 2024 01:14:54 UTC No. 16495706
>>16495673
>nigga, the down-to-earth way to explain basic math is how you teach it to kids in elementary school
It's not down-to-earth at all.
They just teach kids the procedure (e.g., long division) without explaining them why they are supposed to do the steps they are told to do or why these steps will lead to correct results.
This is not a down-to-earth explanation: it isn't an explanation in the first place.
>rigor is pretty much pure abstraction.
Nevermind, I guess you are just retarded.
There are levels of abstraction, and not all math closer to the concrete has to be unrigorous.
Take something like Apostol's Calculus, for instance. It is fairly rigorous (the first volume at least; the part about multiple integrals in the second volume can get a little iffy) but it absolutely isn't a book that relies on abstraction. You deal with fairly concrete objects all the time (namely real numbers and Euclidean spaces) and the proofs are "ad hoc", they rarely if ever invoke a more general and abstract result. Which isn't the case in modern mathematical analysis textbooks that deal with more general spaces than the Euclidean ones.
Conversely, there are many books about abstract mathematics that aren't rigorous at all.
So your notion that "abstraction = rigor" is pants-on-head retarded.
>>16495695
Point taken, but good luck passing your exams by drawing pictures and making analogical and plausible arguments when writing a proof instead of abiding by what is generally deemed to be the "rigorous" way of proving something.
By your logic, both Mochizuki and I could claim to have proven the abc conjecture and you would have no way to criticize our "proofs".
Anonymous at Thu, 28 Nov 2024 02:11:04 UTC No. 16495763
>>16495659
Why do you think there would be a rigorous book at preuniversity level? Rigour equals undergraduate level and above and russian or european approaches to high-school rigour try to downgrade some university subject, no the opposite of upgrading elementary school mathematics. If you want some kind of rigour at a very basic level maybe you would need to look for older textbooks, like 1930 or older, before the advent of Bourbaki. Going to some extreme, try reading Leonhard Euler's Elements of Algebra, which was recently edited but goes as far back as 1765.
Anonymous at Thu, 28 Nov 2024 02:41:33 UTC No. 16495775
>>16495763
>Why do you think there would be a rigorous book at preuniversity level?
I suppose I am unsatisfied with the fact that nobody tries to teach people basic math in a "rigorous" way.
I find myself having clearer ideas about abstract groups and rings than about the simplest and most practical operations with fractions.
I know how to build the natural numbers in set theory starting from the empty set as zero, and I know how to define the rational numbers as equivalence classes of ordered pairs of integers (whose second element is not zero).
I know how to define sums and products of rational numbers but I haven't been taught exactly why the definitions were so chosen (figuring out the reasoning behind the definition of addition isn't difficult, but multiplication is trickier; at any rate, I would have preferred a proper explanation instead of having to rely on my own guesswork). Such definitions, in university-level mathematics, are usually thrown at you with no motivation whatsoever and seemingly with no connection to numbers in the real world. Yet we do use fractions (especially percentages) all the time in everyday life. Shouldn't we know more about the concrete motivations behind the definitions of their operations?
> try reading Leonhard Euler's Elements of Algebra
I opened it up and the author seems to assume the reader already knows what numbers are.
He liberally uses addition and multiplication without defining them.
This is not what I was looking for.
Anonymous at Thu, 28 Nov 2024 02:41:39 UTC No. 16495776
>>16495706
id accept any proof as long as it can be calculated
Anonymous at Thu, 28 Nov 2024 02:51:02 UTC No. 16495785
>>16495776
Calculation isn't defined in mathematics.
Anonymous at Thu, 28 Nov 2024 03:04:09 UTC No. 16495796
>>16495775
Do you think the addition and multiplication tables are a set of axioms, of theorems, or a definition?
Anonymous at Thu, 28 Nov 2024 03:12:45 UTC No. 16495804
>>16495796
If one wants to mirror Peano's axioms in a more informal and concrete approach to numbers, you should take the existence of a process of "counting" as an axiom, and then define addition and multiplication on the basis of that counting process.
The addition and multiplication tables then become theorems deduced from the counting function and the definition of the operations.
This very same mirroring (in the opposite direction) is probably how Peano came to formulate his well-known axioms.
Anonymous at Thu, 28 Nov 2024 07:41:00 UTC No. 16495995
>>16495804
Well, taking them as axioms or as a definition can also be made rigorous (see https://ncatlab.org/nlab/show/decim
>Probably how Peano came to formulate his well-known axioms
Where did you read that? Dedekind had the concrete idea and what Peano wanted to do was to make Dedekind's idea completely symbolic, see this section
https://plato.stanford.edu/entries/
and the quotation on
https://en.m.wikipedia.org/wiki/For
Anonymous at Thu, 28 Nov 2024 19:23:19 UTC No. 16496460
I've read some pieces of Wu's books. Sometimes they feel a bit tedious (even though I am interested in the pedagogical stuff) but they have good insights. I'd second looking at historical texts too, particularly Euclid for geometry and also the historical context behind why we construct the real numbers the way we do. Some people shit on Euclid's rigor, and there are certainly some obvious errors such as missing cases in proofs or principles that should have been in the postulate list, but it's better than the average modern high school geometry textbook, and since it's free from the modern notion of "you can take whatever arbitrary shit you want as a postulate," all the proofs work as explanations.
>Such definitions, in university-level mathematics, are usually thrown at you with no motivation whatsoever and seemingly with no connection to numbers in the real world.
Another good way to motivate those definitions is to try building your own system, with axioms and definitions you chose yourself. Sometimes the pitfalls you run into when doing this are insightful.
Anonymous at Thu, 28 Nov 2024 21:55:29 UTC No. 16496579
>>16495785
it is though.
Anonymous at Thu, 28 Nov 2024 22:35:35 UTC No. 16496602
>>16496460
No that anon, but i suggest OP to meditate on these definitions of addition and multiplication. He can then prove the addition and multiplication tables using ruler and compass.
Anonymous at Thu, 28 Nov 2024 22:41:02 UTC No. 16496609
>>16495659
Just any book on set theory no? I was always very confused on where math comes from until I saw how they build out everything from sets
Anonymous at Fri, 29 Nov 2024 01:45:57 UTC No. 16496760
>>16495995
>Well, taking them as axioms or as a definition can also be made rigorous
Anything can be taken as an axiom if one so wishes, as it is often done in introductory textbooks about synthetic geometry or calculus, where any theorem that is too difficult to prove is assumed true without proof.
Just because one can do something it doesn't mean he should.
An autistic purist may wish to reduce the number of assumptions to the bare minimum possible; as for myself, that's not necessarily the case, but I don't think accepting addition and multiplication tables as axiomatic is such a good idea. Though admittedly this may be merely a matter of taste.
>On the other hand, i dont think the concept of "counting" is needed in the formulation of Peano axioms
By "counting" I was referring to Peano's successor function, since starting from zero and going through the natural numbers one, two, three, etc. is exactly what we do when counting objects.
>Where did you read that? Dedekind had the concrete idea and what Peano wanted to do was to make Dedekind's idea completely symbolic
And where did Dedekind get that idea?
The notion that succession is so basic to our conception of the natural numbers is presumably much older than either Peano or Dedekind.
>see this section
>https://plato.stanford.edu/entries
Why are you quoting the Stanford Encyclopedia of Philosophy?
I repeatedly said I was looking for something as simple and down-to-earth as possible (with no corner-cutting though; as the saying goes, make everything as simple as possible but no simpler).
Philosophers are (in)famous for their tendency to problematize everything as much as possible (since their livelihoods depend on it) and in over two thousand years they have managed to reach agreement on precisely nothing.
Anonymous at Fri, 29 Nov 2024 01:49:46 UTC No. 16496763
>>16495659
Not starting from arithmetic, but have you heard of the books by Gelfand? His algebra book looked pretty good when I looked at it.
Anonymous at Fri, 29 Nov 2024 01:52:39 UTC No. 16496766
>>16496460
>I'd second looking at historical texts too, particularly Euclid for geometry
I have already read the thirteen books of The Elements years ago.
And I am more interested in arithmetic right now rather than geometry, and Euclid has precious little to say about it, except for his famous Euclidean algorithm and the surprising perfect-number theorem.
>and also the historical context behind why we construct the real numbers the way we do
I don't think Euclid elucidates much on that except for the theory of proportion in Book 5 being oddly similar to Dedekind cuts (I'm assuming Dedekind used Euclid as inspiration for that?).
Euclid doesn't even explicitly use any number except positive integers (no mention of genuine rationals beyond ratios in proportion, and let alone irrational), and his whole treatment of radicals and surds in Book 10 is extremely unwieldy and outdated in modern eyes.
>Another good way to motivate those definitions is to try building your own system, with axioms and definitions you chose yourself. Sometimes the pitfalls you run into when doing this are insightful.
That's no different than telling me "you are on your own" as most textbooks basically tell you implicitly.
Anonymous at Fri, 29 Nov 2024 01:53:46 UTC No. 16496768
>>16496579
Now prove it.
Anonymous at Fri, 29 Nov 2024 01:56:56 UTC No. 16496769
>>16496602
I don't know how I feel about trying to build arithmetic upon (Euclidean) geometry.
As Gauss put it, arithmetic has a much better claim to being deemed an a-priori science than geometry; geometry has a much larger empirical hinterland. (The hint is in the name.)
One would be putting the cart before the horse if he felt comfortable accepting the assumptions of Euclidean geometry and build arithmetic starting from that.
Anonymous at Fri, 29 Nov 2024 01:59:29 UTC No. 16496770
>>16496609
I'm pretty positive that mathematics already existed before Georg Cantor came up with the explicit notion of sets.
Standards of rigor may be always shifting with the sands of time, but one doesn't necessarily have to start with sets.
Kronecker wouldn't.
Anonymous at Fri, 29 Nov 2024 02:07:28 UTC No. 16496773
>>16496763
I skimmed through that book and it seems a decent enough introduction to the subject, but it definitely isn't what I am looking for.
I am already very familiar with elementary algebra.
I was looking for something more fundamental than that (and yet something that didn't lose itself in too many convoluted and artificial pseudo-philosophical technicalities).
Maybe the kind of book I am looking for doesn't even exist and I should stick to Wu's books after all. They are the best I could find.
Anonymous at Fri, 29 Nov 2024 02:08:11 UTC No. 16496774
https://www.amazon.com/Algebra-Isra
This is the Gelfand algebra book. It is well regarded. And he wrote other books intended for high schoolers.
Anonymous at Fri, 29 Nov 2024 02:50:50 UTC No. 16496786
>>16495659
Maybe there is a book meant for future school teachers that would suffice. I’m guessing here.
🗑️ Anonymous at Fri, 29 Nov 2024 03:41:08 UTC No. 16496830
>>16496760
>Why are you quoting the Stanford Encyclopedia of Philosophy?
Read the section and you won't find that much philosophy, its a summary of Dedekind's contibution mathematics, although some may argue that Dedekind was a mathematician-philosopher, like Frege and Russell. Remember, logic was first and foresmost a philosophical endeavour, namely, that part of philosophy where they (mathematicians and philosophers) have managed to reach precise agreement on something.
Anonymous at Fri, 29 Nov 2024 03:44:11 UTC No. 16496832
>>16496760
>Why are you quoting the Stanford Encyclopedia of Philosophy?
Read the section and you won't find that much philosophy, its a summary of Dedekind's contribution to part of mathematics, although some may argue that Dedekind was a mathematician-philosopher, like Frege and Russell. Remember, logic was first and foremost a philosophical endeavour, namely, that part of philosophy where they (mathematicians and philosophers) have managed to reach precise agreement on something.
Anonymous at Fri, 29 Nov 2024 07:07:34 UTC No. 16496946
>>16496786
Yeah, I was starting to suspect that myself, but thanks for the sarcasm.
>>16496832
>namely, that part of philosophy where they (mathematicians and philosophers) have managed to reach precise agreement on something
Heh, not really.
Bertrand Russell once said that, so far as he knew, the fact that the ontological argument (put forward by Anselm of Aosta, Descartes, and others) was fallacious was the only thing on which nearly all the philosophers of his own era had agreed.
But, a few decades later, Anthony Kenny remarked that nowadays even that much is not agreed upon, since there were some theistic modal logicians who were trying to resurrect the ontological argument and prove its validity.
So no, it seems that philosophers have never managed to agree on anything, not even on matters of logic.
Which isn't to say that mathematics (or any other science) doesn't have its own controversies and disagreements (Norman Wildberger is a famous example and a living meme in this regard), but my impression is that, throughout the centuries, scientists and mathematicians have managed to reach a broad consensus on most matters.
In Western philosophy, on the other hand, there isn't even much agreement on what philosophy really is about and what methods it should follow. Which is why there still is a split between Analytic and Continental Philosophy, with the analytic philosophers building a kind of "cargo cult" as they try to ape the methods of natural science while the continental philosophers just say whatever passes through their mind at any given point, write a book of obscure aphorisms, and get praised by their colleagues for their prose poetry while the world keeps spinning in more or less complete indifference to the endeavors of both groups of "philosophers".
Anonymous at Fri, 29 Nov 2024 09:25:42 UTC No. 16497021
>>16496770
Their math was derived from observations of the natural world. Thinking of negatives as debt, or reals as lengths, however that is pretty limiting as there is no very obvious natural interpretation of "i". Set theory makes all of this very clear and easy to understand.
Anonymous at Fri, 29 Nov 2024 12:17:28 UTC No. 16497113
>>16496946
>Implying philosophers doesn't agree on predicate logic as an useful model for argumet
Then why philosopy students must take courses on mathematical logic or even set theory (see the link at he end for an example)? What do you think the "cargo cult" of analytical philosophy is a cult of, if not of mathematical logic? Are you still dismissing some paragraphs of text because the "stigma" word "philosophy" is attached to the overall website?
https://www.phil.cam.ac.uk/current-
Anonymous at Fri, 29 Nov 2024 13:04:19 UTC No. 16497136
>>16496946
>while the world keeps spinning in more or less complete indifference to the endeavors of both groups of "philosophers".
Ethicists are the only group of philosopher who try to muscle their philosophy into the real world. And even they are too self-absorbed to REALLY do that. Being a philosopher is kind of a mental disease in a way, at least if you are 100% philosophy minded instead of just 50% like someone like me or others in the more "but what is the relevance to the real world?" camp.
I started disliking academic philosophy when ... well, I don't think the individual topic truly plays a role, but my straw-breaking-the-camels back moment was reached during a personal identity debate. It absolutely can have implications for the real world (e.g. composite consciousnesses, which may one day be real due to advances in advanced medicine/synthetic bodies), but mostly it doesn't.
Anonymous at Fri, 29 Nov 2024 15:34:52 UTC No. 16497214
>>16496946
I wasn’t trying to be sarcastic. I am thinking what you want should exist as a textbook for teachers, but I have no idea if it does.
Anonymous at Sat, 30 Nov 2024 04:34:00 UTC No. 16497886
>>16497021
>>16496609
Any specific books you'd can recommend?
Anonymous at Sun, 1 Dec 2024 22:08:39 UTC No. 16499599
Anonymous at Sun, 1 Dec 2024 22:33:44 UTC No. 16499623
>>16497021
i is just a 90 degree rotation
Anonymous at Sun, 1 Dec 2024 23:13:36 UTC No. 16499677
If I was going to try to teach real numbers rigorously at a middle school level, I would use Bachmann's nested interval construction. I like how concrete it is, and how you can practice doing actual computations with the nests which give you meaningful results. (In comparison, you can compute with Cauchy sequences, but knowing the first few terms of a Cauchy sequence doesn't tell you anything for certain about its limit.)
In practice, the main reason not to teach this to middle schoolers is that they'll lose interest. If you haven't seen the paradoxes you can get by treating infinity sloppily, why would you be interested in levels of rigor that otherwise seem pointless and autistic? I do like to show kids how to find the nested intervals converging to [math]\sqrt{2}[/math], but that's usually as far as I go. Sometimes when a student thinks that 0.6 repeating times 4 is 2.4 repeating, I'll show them how to do that calculation properly with the intervals. It's mainly effective as a way of convincing them that yes, sometimes it's easier to leave the number in fraction form.
As I recall, Wu just says that it doesn't make sense to try to teach real numbers rigorously to middle or high school students, and suggests just assuming for the purposes of the class that the field laws which work for the rationals also work for the reals. Although I think he might treat the reals in one of the later books in his series which I haven't read.
🗑️ Anonymous at Mon, 2 Dec 2024 00:42:07 UTC No. 16499748
>>16499677
>how to find the nested intervals converging to [math]\sqrt{2}[\math]
Using the bisection method or something else? Because if it is the former i don't like that it basically is an algorithm, yes it is concrete but for a beginner it doesn't look that different from performing a supertask until you find the actual decimal expansion of the irrational number
Anonymous at Mon, 2 Dec 2024 00:46:29 UTC No. 16499753
>>16499677
>how to find the nested intervals converging to [math]\sqrt{2}[/math]
Using the bisection method or something else? Because if it is the former i don't like that it basically is an algorithm, yes it is concrete but for a beginner it doesn't look that different from performing a supertask until you find the actual decimal expansion of the irrational number
Anonymous at Mon, 2 Dec 2024 02:01:53 UTC No. 16499789
>>16499753
Yes, bisection.
>for a beginner it doesn't look that different from performing a supertask until you find the actual decimal expansion of the irrational number
Perhaps for those willing to hear more details (usually high school rather than middle schoolers), using the intervals and interval arithmetic to find interval sequences around numbers like [math]\sqrt{2} + 1[/math] and [math]2 \sqrt{2}[/math] could help illustrate what's actually going on, although I suppose that could also be misinterpreted.
But I'm not as worried about that supertask misconception as I am about students believing that [math]\sqrt{2}[/math] is exactly the finite-decimal approximation their calculator shows. The Android calculator app's arbitrary-precision calculations are useful for dispelling that one.