/math/ - Mathematics

>>202

Yeah. And I learned eigenvalues and vectors before multivariable calculus, with linear algebra.

But my educational system obviously differs from yours, since most of you are yanks.

>>211

For me too. I'm going into compsci, tho, and from there to maths it's just a little bit.

>>247

Same here. I imagine there are a few other topologists on here, but as it seems, we only post once every two months.

>>219

Lmao look at the comp sci faggot thinking he does ""math""

shit

I'll do it since I'm bored enough.

sum (n^2-k^2)^1/2*(n^2-(k-1)^2)^1/2 < sum (n^2-(k-1)^2)^1/2*(n^2-(k-1)^2)^1/2

so, if you prove that sum n^2-(k-1)^2 < (2n^3 + n)/3, you have also proved that sum (n^2-k^2)^1/2*(n^2-(k-1)^2)^1/2 < (2n^3 + n)/3. It's not a hard thing to do from there.

>>278

fucking formatting.

sum (n^2-k^2)^1/2*(n^2-(k-1)^2)^1/2 < sum (n^2-(k-1)^2)^1/2*(n^2-(k-1)^2)^1/2

so, if you prove that sum n^2-(k-1)^2 < (2n^3 + n)/3, you have also proved that sum (n^2-k^2)^1/2*(n^2-(k-1)^2)^1/2 < (2n^3 + n)/3. It's not a hard thing to do from there.

One cool cheat is that you can perform operations on ONE side of an inequality, provided that you can justify that it wouldn't change the inequality.

ie:

a < b

a - 1 < b

UPDATE:

my math was wrong. if im right now, it should be

m(t)=4(t log_(1/2) 1)

Why don't you post something useful?

I only started coming here again these past week

>>>/tv/res/117721

>>>/res/117721

>>>/tv/117721

>284

>2 * 142

>2^2 * 71

From wikipedia:

>Divisible by 1, 2, 4, 71, and 142

1 + 2 + 4 + 71 + 142 = 220

>220 is divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

>Therefore they are amicable numbers.

>>250

>Sure you can. Truth is what we choose to be true.

I'd have to disagree slightly: truth should be, at least in a foundational sense, something which is "intuitively correct" (see: Brower); then you can create a hierarchy such as Tarski's which shows how the truth of a logical proposition is determined from the truth of it's logical components, but that is far from arbitrary.

>In higher level mathematics there is plenty of debate about which axioms are the right ones

only half correct: no one (for instance) disputes the axiom of union, or the axiom of replacement (well, the schema in first set theory). only powerful axioms (such as constuctibility L=V) or the global axiom of choice, and this is because the consequences for assuming such axioms become quite strong (non-existence of strongly inaccessible cardinals in the case of L=V etc etc).

>>257

Foundationally, truth is exclusively a model-theoretical concept. Recall that a formula is true iff it is valid in the particular model. For instance, in theories that aren't categorical you can have many different and interesting models of the same cardinality, so that what is true in one may fail spectacularly in another even for countable models.

What is true depends on which model you choose. Even in logic there are models with universes of different cardinalities, despite the theory with no nonlogical symbols or nonlogical axioms being m-categorical for every infinite cardinal m, for each m you have a different universe and thus the potential for different valid formulas.

What we find to be intuitively correct should only be an aspect of what we choose to be true, intuition being a guide towards the rigorous elucidation of what is formally true in each framework. To limit truth to that which is intuitive would be irresponsible.

As far as your statement about powerful axioms, I agree that not many people talk about the less "popular" axioms, but the ecumenical stance of today towards strengthening and weakening various theories means that looking into variations of, say, replacement, which you might consider uninteresting, could be both fruitful and interesting. Why suggest that talking about replacement is silly when it with the axiom of infinity provides the basis for transfinite recursion, one of the most important tools in set theory?

http://math.bu.edu/people/aki/20.pdf

Here's a good paper if you want an interesting discussion on replacement, I think in section 5 he mentions variations and equivalences of replacement in some obscure theories. I feel that a similar discussion could be had on a smaller scale for the other axioms, and an investigation into what happens when you omit axioms that are usually assumed without question could be a good learning exercise. If it is interestingPost too long. Click here to view the full text.

>>260

please note: i'm italian, i'm trying to translate the terminology i've learnt during my studies in english (from italian) as best as i can, if there are major fuckups, please let me know.

>Why suggest that talking about replacement is silly when it with the axiom of infinity provides the basis for transfinite recursion, one of the most important tools in set theory?

Please make no mistake, I never said that replacement is uninteresting or that talking about it is silly, just that differently from what happens for (as an example) the axiom of power-set, you can, via model construction, prove that adding it to the axiom set doesn't ruin coherence, whereas in the case of the "strong" axioms (power-set being an example) you cannot prove that adding them doesn't fuck up the theory (at least in the case of ZF \ (foundation)). furthermore, i was talking about accepting them as a basis for mathematics, not in a general set-theory scenario. as you've pointed out there are many other axiom sets, some of which generate extremely exotic theories (i'm beginning to study something which translates to "fragments of set theory" right now, sound a bit like what you were saying in your last paraghaph).

>To limit truth to that which is intuitive would be irresponsible

i agree with the whole paragraph, in fact i never said anything otherwise; what i meant (by paraphrasing Heyting) was simply that basic proposition should be constructive and "intuitive" (intuitive in a Intuitionist sense, not in a "oh, that's obviously correct" sense, that leads to contradiction very fast); this was taken from Heyting's formulation of arithmetic (HA). then there are notions of semantics and syntax which complicate the matter even further, as you've said. (I for one appreciate Godel's proof of his first theorem much more over the Cantor-Godel one...)

>For instance, in theories that aren't categorical you can have many different and interesting models of the same cardinality, so that what is true in one may fail spectacularly in another even for countable models.Post too long. Click here to view the full text.

>>273

Sure they can be contradictory. A model of group theory with 6 elements is not the same as a model of group theory with 4. The same thing happens at infinite levels, for instance ACF of characteristic p (p prime) can have countably infinite models that aren't isomorphic, while is m categorical for uncountable m. If there isn't a model isomorphism, different things are valid in that model.

>>275

sorry, i'm a twat, i thought you meant that a model M could prove T and (notT)...

>>258

Why are you using NBG? Mind posting parts of what you have on here?

>reading math textbook using "<p,q>"

>not sure if ordered pair or sequence number

>>258

hey there Anon

I'm using NBG because i'm studying expansions of ZFC set theory, currently i'm trying to find a new proof for MK proofs Coer(NBG)... no meaningful results thus far unfortunately.

The thesis in the last post was a overview of ZF, NBG and MK, and the proof of MK proofs coer(NBG) done through set theory.

I'd be glad to drop it here, but I'm italian, and the whole work is in italian as well. sorry...

>>272

Ah ok, so you're proving that MK proves consistency of NBG? That's awesome, good luck. MK is "bigger" than ZFC, so if you have a model of MK you can cut it down to ZFC right?

>>276

yes, the "model" is actually a class in MK of the "statements true in NBG", and then you define a predicate for the truth in NBG thru this class.

what i'm trying to do now is avoid model building and do basically a similar thing using only proof theory and sequent calculus. my professor wants to use something which i can translate as "cut-elimination theorem" (i hope it's similar to the actual english name), but as of now, i haven't been able to construct the needed hypothesis for the theorem to work... still going.

>>263

I can't tell because they didn't pack optimally.

3417?

>>270

How did you arrive at that number?

I got 979 somehow.

>What are your favourite non-textbook math books?

This has a canonical answer. It's the Princeton Companion to Mathematics.

Princeton Companion to Mathematics:

>overview of history

>describes some problems

>no actual math

Well, then the Princeton Companion to Music must have

>overview of what music is

>describes some songs

>no actual music

I suggest something like How to Prove It or Godel Escher Bach

>>264

Not sure what book you're reading, or what you consider "actual math", but in my opinion the Companion has lots of actual math in it, meaning mathematical concepts and ideas.

Obviously it can't go into miniscule detail, but if you think that the Companion has no actual math, then you must also concede that review articles don't contain actual math.

Seriously, read it. I guarantee that anyone who has even the slightest bit of interest in math will be enthralled.

>>265

Mostly, I was feeling argumentative last night but what I meant is that there are no proofs in the book from what I found about it. Math isn't math without proof, and the idea of an encyclopedia of concepts without any attempt to link them together seems like it would serve only to confuse more people. An array of disparate theorems with the proofs all swept under the rug isn't really math, for the same reason a dictionary isn't spoken word and a top 100 list of songs isn't music.

>>266

re argumementative: That's understandable, given my own provocative statement of a "canonical answer"^^

To me, math is anything that increases mathematical understanding (ignoring the circularity in the definition ;) )

There is the occasional proof or proof sketch in the book, but it's true that that's not what it's about. But it certainly is far more than an array of theorems. Rather, to each topic covered (topics include "concepts", "branches of mathematics", "famous theorems/conjectures", and "famous mathematicians"), an article by an expert in the field is included. In the case of theorems, these articles provide enough background that a mathematician reading it will be able to grasp the statement of the theorem as well as why it's important.

The reason why I think it's great is not because of the articles concerning my own field -- I'm not learning anything new there. But I can read about Mirror symmetry or Vertex operator algebras, Ricci flow or Quantum groups etc. and gain at least a glimpse of what these fields are about.

Given the extreme specialization and fragmentation of mathematics today, this is a valuable gift indeed.

(and no, Wikipedia does not provide this. It gives some of the facts and theorems, but none of the "feeling" of a field. As Gowers notes in the introduction, the Companion is not an Encyclopedia)

I think that every student should take a look at this book before choosing a direction to specialize in.

Concerning "proofs from THE BOOK": My favorite one is Monsky's theorem. It's like a mathematical party trick. You can explain it with a piece of paper and 10-15 minutes (given the "audience" knows p-adic valuations), and it will always produce amazement among those who don't know it yet.

I'm not familiar with those books, but Khan academy is a pretty good supplement material. You're right to pick up a textbook, since you really need to be able to do some practice problems to actually learn the stuff.

Skimming through the contents of those books, and it looks like it covers everything a highschool graduate should know (assuming they did pure mathematics instead of general mathematics or remedial mathematics) except trigonometry (including vectors), limits and calculus.

You'll need to go through all the other stuff before you touch limits and calculus.

For extra vector problems, check out some physics problems for forces.

>>231

Yo, forgot about this thread. I haven't worked on this as much as I have tbh. I've been practicing for a certification course I'm set to do tomorrow for my job.

My goal was to brush up on Algebra and Maybe Trig. I'm currently going to community college for a trade and I figured it was in my best interests to take advantage of their advanced math courses even if I have no interest in going for a degree in math or even a bachelor's.

i.e. calculus, so I figure I would learn that their while brushing up on my math.

Thanks for taking the time to review the books btw. I have a few more links, ranging from online notes to videos explanations. I'll survive I think. As for trig, I'll deal with that horse when I get there. I may just touch on it lightly then work on it when I take my precalculus class.

"Calculus" by Michael Spivak & "Concrete mathematics" by Donald Knuth et al. 2 brilliant texts. Both are widely available online for free as a pdf download. They're quite difficult but if you can get through them, you'll know quite a bit of math, it has to be said. On the more applied side of things, try "patrickjmt", a youtube channel, It's brilliant. prior to all this though, for a week or so, find your old high school books & wrestle through them too. Remember, maths is not a spectator sport, you've gotta actually do it (like programming...). I'm a physics major myself, but plan to switch to pure mathematics (possibly topology) & will be devoting this summer to reading maths, I can't wait. Good luck!

>>239

>"Calculus" by Michael Spivak & "Concrete mathematics" by Donald Knuth et al. 2 brilliant texts

will do but i'm nowhere near that level now, I think.

>On the more applied side of things, try "patrickjmt", a youtube channel,

yup, that's one of the links I mentioned.

>Remember, maths is not a spectator sport, you've gotta actually do it

lel, right now I'm on khan acadamy, I think it's great to get your bearings in math. and going through the kent books. well, not recently though. right now I'm doing a week long certification course for corrosion.

>I'm a physics major myself, but plan to switch to pure mathematics (possibly topology) & will be devoting this summer to reading maths, I can't wait. Good luck!

jelly.

I failed out of community college about 3 years ago. I'm going back to another college and restarted for an associate of applied science degree(basically a trade). I wanted to try my hand at math one more time, seeing as how I'm much more interested in the subject now that I've dealt with how it feels to be math illiterate in the real world. Trig especially would make my life a hell of a lot easier and standards in this industry use a ton of math.

The plan was to take all the math my community course offers during and after my associate even if I don't get a degree for them or transfer. I just want to learn. It's the hunger of the mind, son.

>>238

Okay, fair enough. You probably won't need to learn most of that's on that then.

Just learn Pre-algebra and trig. Maybe some basic algebra. If you find algebra interesting, by all means keep going into 'real math'.

>>239

Very overkill for a trade worker. OP does not need to know calculus and discrete mathematics (though, if he's interested, by all means...)

>>20

[*tex]x^2 [*tex]

>>233

$$x^2$$

>>234

[*tex]$$\frac{x}{2}$$[/*tex]

Wait, I'm confused. Are you the admin? Because this board is up for grabs. Nigga, you need to log in.

\frac{ayy}{lmao}