Starred posts

user21820

Feb 14, 2024 13:22

The following course, if one chooses to study it, is guaranteed to give you crystal-clear understanding of logical reasoning, which includes all mathematical reasoning. Although mathematical content and insight still must be gained separately, there is completely no obstacle to that once one has fully grasped basic FOL (first-order logic), and it would become mostly a matter of time and effort.

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user21820

Sep 21, 2023 04:34

@PrithuBiswas I believe the author wanted the reader to wave with both hands vigorously rather than write with one hand rigorously.

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Martin Sleziak

May 13, 2024 08:25

This room is now six years old. Happy anniversary!

Prithu Biswas

Oct 20, 2023 15:54

@soupless I don't think you should do something in life out of spite. Do it for your love and interest in it.

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user21820

May 14, 2021 10:07

Learning is not only about learning what is correct and why it is correct but also learning about what is wrong and why it is wrong.

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user21820

Feb 21, 2024 12:48

Right-to-left for everything where the order does not matter. Left-to-right for arithmetic operations (+,−,·,/). In all other cases, brackets should be used.

user21820

Mar 22, 2022 15:48

Anyway, in my opinion the only way to learn a real deductive system is via exercises. Every student I've taught have had to go through this stage in order to really understand logical reasoning.

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user21820

Oct 26, 2020 10:08

@Stupidquestioninc Thank you, I put in a lot of effort into my posts, especially those that I use to teach others. =)

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user21820

Jan 24, 2023 08:34

Although there is no notion of truth in the formal system itself, the goal of the system is to make sure that you can only write true statements (in their context).

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user21820

Mar 13, 2021 09:09

@eryceriousmatherfunker You need to first learn basic FOL (first-order logic), before anything else. For that, you can read and work through "Language, Proof and Logic". Feel free to ask specific questions about that text.

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user21820

Mar 7, 2021 09:37

This is an instance of the canonicalization technique, which you should learn to employ whenever possible; canonicalization just means to restrict your attention to some kind of canonical form that you can reduce every other case to. Of course, you have to figure out what is suitable to be the canonical form, but the point is to have the correct mindset in the first place.

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user21820

Feb 6, 2022 19:43

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A: Predicate logic: How do you self-check the logical structure of your own arguments?

Truth tables are not enough to capture first-order logic (with quantifiers), so we use inference rules instead. Each inference rule is chosen to be sound, meaning that if you start with true statements and use the rule you will deduce only true statements. We say that these rules are truth-preser...

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user21820

Jun 15, 2018 05:29

Recommended by logicians: Excellent logical puzzles.

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user21820

Dec 17, 2019 15:51

I personally recommend a textbook by Spivak called "Calculus".

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user21820

Aug 30, 2021 09:18

One can be very logical even in plain English.

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user21820

Jul 21, 2020 14:13

This is not a trivial theorem to prove, though it is now in every introductory undergraduate course on logic. In Godel's time, it was groundbreaking.

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F. Zer

Apr 10, 2021 15:53

@user21820 I will close my eyes :-)

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user21820

Jul 3, 2018 08:50

Viewing induction as a game.

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user21820

Jul 17, 2020 06:15

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A: Is this a new method for finding powers?

For a little bit more, see the answer "General method for indefinite summation" which explains how exactly this representation using forward differences allows you to easily find the formula for indefinite summation of powers. Applied to your case you get: 0, 1, 8, 27 1, 7, 19 6, 12 ...

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user21820

Jul 13, 2020 11:40

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A: Is the real number structure unique?

To paint a more complete picture, you are right in that an axiomatization may very well have no model. An axiomatization is meaningless if nothing satisfies it. But if we can prove that there is a model, and the axioms are the only properties we care about, then we can happily work within the axi...

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user21820

Nov 19, 2021 13:01

And yes it's a good idea to experiment as much as you can. Trying to break a system is a good way to understand how it works if you fail to break it. =)

user21820

Oct 17, 2021 07:24

And of course I'm not interested in teaching a single thing to people who call others "*****".

user21820

Sep 21, 2019 13:30

If you are guessing, then you're learning in the wrong way. Follow what I told you to do and see if you can figure out the right way yourself, and be sure that you are right.

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Prithu biswas

Sep 19, 2021 16:34

It's not wise to learn set theory from a book written over 70 years ago. — Asaf Karagila ♦ Jun 18 at 17:38

Prithu biswas

Sep 7, 2021 17:59

@Peter I think the problem is not about trying to prove an open-problem and expecting it to be easy. But , after someone finds a proof of a open-problem , they dont verify there proofs. And they probably can't verify there proofs because they dont have a grasp in basic logic.

user21820

Aug 8, 2021 16:14

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A: How to calculate the series $-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}...$?

Firstly, your method is wrong as has been pointed out. You cannot rearrange an infinite series, because the definition of its value is the limit of its partial sums, and rearranging it anyhow will change the sequence of partial sums, so there is no guarantee that the limit will be the same. (See ...

user21820

Jul 31, 2021 11:19

I've been called "idealist" by many people, but when illogical reasoning leads to actual harm and deaths, I feel quite disturbed. I mean, what's the point of teaching students how to solve quadratic equations, if it doesn't help them make logical decisions about life-and-death issues, such as social distancing during a pandemic?

user21820

Jun 27, 2021 18:12

@Prithubiswas What's "referential"? I think the right order of learning is: Basic FOL (syntax, semantics, deductive system, all described in as clear natural language as possible) → Practice proving pure FOL theorems → PA (Peano Arithmetic, not only to actually start proving mathematics but also to learn about schemas) → Set Theory (which is described as a FOL system, which can be used as a meta-system) → Mathematics (including analysis of FOL in Set Theory).

user21820

Mar 25, 2021 17:09

@Rover Well, you should definitely study Spivak's "Calculus" (I think there are some freely available older editions online), as it is well-written and rigorous and definitely what you need to learn in undergraduate real analysis. But before that you might want to read Daniel Velleman's "How to Prove It". But if you want to learn absolutely 100% precise mathematics, you must learn basic FOL, like F.Zer has been doing.

user21820

Oct 16, 2018 05:33

Pretty Programming Puzzle: RoboZZle.

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user21820

Jan 2, 2021 07:18

For example, a lot of people who don't really know logic claim that SOL (second-order logic) is more expressive than FOL, but that is crankery or misleading. For example, PA2 (second-order PA) is only categorical under full second-order semantics, which merely means that from the view of the meta-system MS there is only one full-semantics model of PA2 up to isomorphism. So what? MS itself is an FOL theory, and if MS is consistent then there are tons of models of MS that disagree on what ℕ is!

user21820

Dec 20, 2020 18:03

@ParamanandSingh That's a good start! Note that that approach uses an external induction as I described here. However, "and so on" is almost always handwaving, so that needs to be fixed. It turns out that actually we do not need a process here. Let me write out the full proof. Before that, let me give a formal definition of span, just to make sure we are using the same definition.

user21820

Jun 6, 2018 17:05

How to Prove It is a good introductory book to real mathematics, that is, definitions and theorems and proofs.

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user21820

Jun 6, 2018 17:01

@VioAriton Nets, Puzzles and Postmen is a very accessible and interesting introduction to graph theory that has enough detail to make sure you actually understand the mathematics. I was pleasantly surprised to find it in a library.

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user21820

Oct 16, 2020 03:49

@BillyRubina Just for fun, here is a simple example of a limit that cannot be proven to exists unless our foundational system S is inconsistent: Let f(n) = ( the n-th string is provable over S ? 1 : 0). Then we can prove that if S is consistent then f does not have a limit. And if we can prove that f has a limit then we in fact can prove "0=1"...

user21820

Oct 16, 2020 03:23

@BillyRubina As for crazier being more interesting, I think the converse is more accurate; interesting structures are typically crazier than boring ones. Not sure if you have seen this before, but here is a crazy video of the mandelbrot set that your comment reminded me of:

user21820

Sep 26, 2020 01:19

So you should try to prove them to convince yourself that everything works perfectly!

Malice Vidrine

Sep 25, 2020 23:35

Yeah, it's really a technical trick. The element that's a member of both of the sets in the pairs is the first component, and the thing that's only in one is the second component. Unless both components are equal, then you have the singleton case This turns out to be just enough to distinguish the ordering.

user21820

Jul 30, 2020 04:49

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Q: Intuitive motivation for limit computations

This is a Q&A pair concerning intuitive motivation for limit computations. Usually, my standard advice is to use asymptotic expansions to compute limits (especially for harder things like this or this), but if we wish to do it without asymptotic expansions yet in a well-motivated way, we may want...

user21820

Jul 28, 2020 11:44

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Q: Solving Rubik's cube and other permutation puzzles

I've seen two questions on solving the Rubik's cube but none of the answers have given a complete solution using mainly mathematical techniques. Furthermore, I've not seen a good explanation of general techniques for solving permutation puzzles in general, including those of the Rubik's cube fami...

group-theory permutations puzzle rubiks-cube

user21820

Jul 27, 2020 08:57

In their table, both that rule (also called Explosion Principle) and LEM (right at the bottom) are redundant, but they clearly included it because they are convenient to have.

user21820

Jul 3, 2020 13:43

I teach proper mathematics, not nonsense, in this room.

user21820

Jul 20, 2019 11:51

@yh05 Do you mean "Geometry: Seeing, Doing, Understanding"?

user21820

Jun 27, 2019 11:35

2014 NYTimes article on mathematical pedagogy still seems accurate today...

user21820

Jul 3, 2018 08:25

What are mathematical definitions?

user21820

Jun 23, 2018 05:09

For reference see this particular collection of axioms for facts about natural numbers that one could say were empirically observed.

user21820

Jun 15, 2018 07:00

Advanced logical puzzles: Piggy Push, Manufactoria.

user21820

May 7, 2018 15:37

So, do you see how to continue the reasoning to include the other restrictions one by one?

Basic Mathematics

♦ Basic Mathematics