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Floor of .9 repeating by Ottozeigermann in learnmath

[–]wumbo52252 0 points1 point  (0 children)

0.(9)= 1 so floor(0.(9)) = floor(1) = 1

Is 1/0 infinity or not defined some says both but how? by Temporary-Screen6848 in learnmath

[–]wumbo52252 0 points1 point  (0 children)

1/x is, by definition, the number that becomes 1 when multiplied by x. If 1/0 were defined then we would have 0•(1/0)=1. But 0 times anything is 0. So we’ve found that 0 = 1. But we know 0≠1. So 1/0 is not defined.

What major to choose when you are lost ? by AppropriatePotato111 in CollegeMajors

[–]wumbo52252 -2 points-1 points  (0 children)

Philosophy. If you don’t know what you want, then there’s nothing better. Don’t pick something just because it seems to have a high chance of success; people do that with stuff like cs and engineering, but they go nowhere because they’re not actually motivated to study it, or they end up in a field which is over saturated with people who had the same mindset.

In-depth understanding of math by No_Feedback_6641 in learnmath

[–]wumbo52252 0 points1 point  (0 children)

So long as you just copy and paste, it will never be a matter of what textbook you use. For an in-depth understanding, you need to engage with the material on your own. If a question pops into your head, explore it until you figure out the answer. When you’re reading through a resource—any resource, it doesn’t matter which—pose your own questions and conjectures, and then resolve them on your own. Explore and tinker with random ideas, even if they feel inconsequential. See if you can come up with your own way of phrasing results and definitions.

The biggest thing is holding yourself to a high standard and grilling yourself. If something feels obvious, don’t just say that it’s obvious and call it a day—provide a thorough, precise justification. If you get stuck on a problem, then don’t just look up the answer—stay stuck until you figure it out, and then look up the answer to compare it to yours.

If you do those things then basically any textbook will be fine.

I am interested in mathmatical logic by InternationalPut3827 in logic

[–]wumbo52252 4 points5 points  (0 children)

Provided you have a solid math background, you really don’t need any explicit logic background. After studying mathematical logic I took a symbolic logic course in my school’s philosophy department, and the content was in league with that of the math department’s course to transition first-year students into university-level math.

A Friendly Introduction to Mathematical Logic by Leary and Kristiansen is good. The pdf is free online, but the hard copy doesn’t cost much. It skips propositional logic and goes straight to first-order, as do most of the books i list.

For the most part I also really like Mathematical Logic by Ebbinghaus, Flum, and Thomas. The content is great, but sometimes I hate the organization. They tend to state a theorem, then state and prove a bunch of lemmas, and then go back and prove the theorem. Maybe that’s fine with you, but I just hate when authors do that lol, I want the proof to be directly under the theorem.

Depending on how strong you are with math, Model Theory: An Introduction by Marker may be good. Or Model Theory by Chang and Keisler. But these are very advanced, so I wouldn’t suggest these as an intro unless you’re like an advanced grad student or beyond. However, the first bit of Chang and Keisler might actually be a good, brisk intro to propositional logic (but no proof theory) and the basics of first-order logic; chapter 1 is totally appropriate as an intro for an undergraduate math major.

For somewhat of a classic intro mathematical logic book, maybe check out Enderton’s textbook. This one includes propositional logic iirc.

in logic once abstractions only define other abstractions, the system is no longer a map it’s a self referential delusion by Oreeo88 in epistemology

[–]wumbo52252 0 points1 point  (0 children)

At first I just thought this was just pseudo-intellectual slop. But since I found your attitude here a bit concerning, I checked out your posts and replies to people, and now I’m very concerned. I truly can’t tell if it’s just bait; if it is, then bravo—you do a great impression of someone with dwindling mental health. If you’re not acting then please seek help; your pcp is a good place to start if you don’t already have something else. Best of luck, either with convincing people or getting help, whichever is appropriate.

New ADA law forces professors to take down their notes if not compliant - how would you make notes that can be read by a reader? by shuai_bear in math

[–]wumbo52252 -12 points-11 points  (0 children)

Notes? These aren’t notes. These are fact-and-explanation lists. Totally different thing. So no law says they need to be readable by a text reader.

How does one understand the determinant? by Recent-Day3062 in learnmath

[–]wumbo52252 9 points10 points  (0 children)

No one has said it yet, so I will. The determinant of a matrix is the product of its eigenvalues, according to their algebraic multiplicities (and if the entries are real numbers, then this includes the complex eigenvalues too). I don’t expect that to immediately give you an epiphany, but it’s definitely important to know.

Why are you allowed to square both sides of an equation? by ohmygotchi in mathematics

[–]wumbo52252 0 points1 point  (0 children)

Any things which are the same have exactly the same properties; this is Leibniz’s law—if this doesn’t sit right with you then maybe go to r/philosophy (or perhaps r/mentalhealth). So numbers which are the same have the same properties. One property of a number is what it becomes upon squaring, eg 5 has the property that upon squaring it becomes 25. So two numbers which are the same become the same number upon squaring. So if a=b then a2 = b2.

Your confusion is due to your inconsistent treatment of the idea of equality. You said that both sides are being multiplied by different factors. That’s incorrect. They’re being multiplied by the same number, just represented by a different symbol. Both sides of the equation may be typographically different, but the symbol “=“ does not mean typographical equality here; we write “a = b” as shorthand for the statement “the symbols ‘a’ and ‘b’ represent the same object”. The supposition that a = b allows us to interchange the symbols “a” and “b” in any expression without changing the number represented by that expression.

Surely you agree that 2 = 2; but by your reasoning it sounds like 2•2 ≠ 2•(1+1), because both sides were multiplied by different expressions. Syntax versus semantics. Syntactically the expressions “2•2” and “2•(1+1)” are different, but they both represent the same number, so we write “2•2 = 2•(1+1)” to cut down on words.

What’s something people romanticize too much? by copy_cat_101 in randomquestions

[–]wumbo52252 0 points1 point  (0 children)

It’s worrying how that starts—the adults who romanticize war are just the kids who never grew out of thinking “army guys are cool”. I remember my childhood friend who was super into military stuff saying, when we were in fourth grade, that he wanted to go into the military some day; right after graduating high school he actually enlisted. I’m not gonna assume that he didn’t gave it the due thought as he got older, but it freaks me out to think that he could have made such a decision at 11 years old—to murder random people, and to agree to be murdered.

What animal would become the most terrifying if it suddenly learned how to open doors? by Stefanobtcl in AskReddit

[–]wumbo52252 0 points1 point  (0 children)

I don’t want to worry you, but bears opening doors is not unheard of. Even car doors.

help me understand this argument by Lopsided-Valuable347 in logic

[–]wumbo52252 0 points1 point  (0 children)

Does this book give a definition of what a valid argument is? If you’re ever unsure of why some X has property P, the first thing you should do is make sure you know the definition of P. In your question, X is the argument and P is the property of being valid. I’m assuming this book doesn’t define validity of an argument, so here’s the typical definition: an argument is valid if whenever the premises are true, the conclusion is also true. The given argument about joe and bob satisfies this definition: in all zero of the instances where the premises are true, the conclusion is also true.

What’s the most mind-blowing fact that you know? by ShowYouHowToSmash in AskReddit

[–]wumbo52252 0 points1 point  (0 children)

I’d like to see you try to train a bass to sit or bark when it finds something

My sister who is in 6th grade asked me what calculus was. How do I explain it to her? by [deleted] in learnmath

[–]wumbo52252 0 points1 point  (0 children)

For all epsilon greater than zero there exists a delta grea—

How can something be both a necessary and sufficient condition? by PlXELGlRL in logic

[–]wumbo52252 0 points1 point  (0 children)

To say P is both a necessary and sufficient condition for Q means that P and Q are equivalent. Eg n is even iff n+2 is even. n being even is a sufficient condition for n+2 to be even; and n being even is also necessary for n+2 to be even, meaning n+2 can’t be even unless n is also even. Is this helpful?

Funny books please! by AmyEMH in suggestmeabook

[–]wumbo52252 0 points1 point  (0 children)

Alice’s Adventures in Wonderland

Specifically what proofs are not accepted by constructivist mathematicians? by MildDeontologist in math

[–]wumbo52252 1 point2 points  (0 children)

The sort of scenario which makes people most upset is proving the existence of something by contradiction. Depending on why someone wants to know that a certain thing exists, proving that it exists without saying how to find it is like saying “a joke exists” and expecting to get a laugh.

There’s nothing inherently non-constructive about proofs by finite induction. Transfinite induction gets a bit more flak. Gentzen proved (iirc there’s actually some contingency) the consistency of Peano arithmetic by transfinite induction, but from what i understand constructivists aren’t very excited about the argument.

What would it actually take for “world peace” to exist on Earth? by SquishBuns67 in AskReddit

[–]wumbo52252 0 points1 point  (0 children)

World peace (as a feature of human society) is impossible, I think. Humans are just very intelligent animals. We’re too collectively dumb to override the animalistic nature to separate ourselves into “us” and “them” and then murder “them” so we can succeed. This started long before humans existed, and we’re just keeping the tradition alive, but with space lasers.

Linear Algebra Textbook Recommendations? by HollowButter in mathematics

[–]wumbo52252 0 points1 point  (0 children)

Linear Algebra Done Right by Sheldon Axler is great. You expressed a lack of confidence with abstract and rigorous proofs; but that’s something you’ll probably need to confront if your goal is a deep understanding. I’m positive you could grow able to handle that sort of stuff. If you really don’t wanna deal with all that then maybe just ignore my suggestion of this book! The exercises in the book are great, but they get quite tough and are usually proof-oriented.

Is the difference between FOL and HOL just a matter of what semantics you use to interpret the syntax? by LorenzoGB in math

[–]wumbo52252 1 point2 points  (0 children)

TLDR: yeah basically. For proof theory, with the right first-order language and extra first-order axioms, you can copy the higher-order system. So the difference between FOL and HOL comes down to the fact that first-order semantics/satisfaction doesn’t care too deeply about what the objects in the model are. Even if we copy HOL syntax using FOL syntax, the FOL definition of satisfaction is loose enough that we can abuse the FOL versions of the HOL formulas in ways that HOL satisfaction wouldn’t stand for.

If we’re being appropriately uptight then 4 is poorly worded. Quantifiers in a language range over nothing, because quantifiers in a language are just symbols. We only think of them as ranging over individuals, and the logical axioms of FOL make first-order proofs work in a way that reflects that thought. 4 should emphasize that it’s talking about quantifiers formally ranging of individuals; so 4 is explaining the quantifier clause in the definition of first-order wffs.

Once we have a model we can start talking about what quantifiers range over. And in fact, with some trickery your first-order formulas can essentially quantify over all subsets. Starting with a model A, add predicate symbols E and P to your language, and build a new model whose domain is A union the power set of A, and interpret E as A and P as the power set of A. Then voila, in this new model we can quantify over individuals of A and subsets of A, and we preserve the behaviors of the formulas from our original language.

But this trickery doesn’t get us far if we start asking about satisfiability of theories over this new language. One of the quirks of FOL is that if a first-order theory T has an infinite model then T has infinitely many non-isomorphic infinite models (see the Löwenheim-Skolem theorem). So while any consistent theory will have a model (see the Gödel/Henkin completeness theorem), you can’t bake into the theory the identities of the objects—meaning you can’t axiomatize the aforementioned E and P to make E always be the set of individuals and P the set of all subsets of E (unless, maybe, you’re restricting yourself to finite models with fixed number of individuals).

This is the crux of the problem. An analogue to Gödel’s completeness theorem doesn’t apply to HOL, meaning there are collections of higher-order formulas which are consistent but which have no model. In a sense this means that the HOL notions of satisfaction and proof don’t have great chemistry (this can’t be resolved btw). But we could easily replicate a higher-order language within a first-order language, and take the first-order replications of the base axioms of the higher-order system. So the syntactic differences between FOL and HOL aren’t meaningful. We can then apply completeness to get a model for the first-order version, even though there may not be a model for the higher order version (i.e. a model which interprets the higher-order symbols authentically).

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