/sci/ - Science & Math

[Anonymous No.6540434]

Why is wild hamburger such a nutjob?

>https://www.youtube.com/watch?v=pdYe4BKcm74
>"the fundamental theorem of algebra is not logically very solid in fact"

Is this what happens to all unaccomplished math profs, that they come up with shit nobody cares about?

[Anonymous No.6540455]

>>6540434
I have often heard that there is no "purely algebraic" proof of the fundamental theorem, meaning you need analysis.

[Anonymous No.6540468]

The "fundamental theorem of algebra" isn't fundamental to algebra anyway. Don't sweat it.

You have to bring in more axioms from outside of algebra to prove it. That's what he's talking about. If you have to add axioms to prove a result, you didn't really prove it, you decided to assume what was necessary for it to be true.

Selection axioms is an extralogical process.

[noko No.6540474]

he also doesn't believe in real numbers or infinite sets, so basically a constructivist or whatever they're called

[Anonymous No.6540483]

FTA: you can extend fields infinitely and solve everything!!!
NJW: orly
OP: I literally cannot stop sucking dicks

[Anonymous No.6540511]

>>6540434
>being this pleb

[Anonymous No.6540614]

His views on the foundations of mathematics are anything but mainstream and sometimes he says things that reflect that. I love how you've assumed he doesn't know what he's talking about, as if he isn't vastly more educated, knowledgeable, and accomplished as a mathematician than you will ever be.

[Anonymous No.6540646]

>>6540455
Yeah it's true, so what? The problem with wild hamburger is, that he can't stop talking about his caveman views of maths, like some sort of political activist. That's the definition of a nutjob.

>>6540614
njwildberger go pls.

[Anonymous No.6540692]

yeah for I while, I thought wildhotdog was a kook too... "Rational Trigonometry"? WTF?!?...

But his mathematics foundation series is the best on the whole interwebs. It's a GODTIER intro to mathematics for kids. The more I went thru his shit and thought about it, I realized he is actually profoundly right: the foundations of the real numbers are mostly swept under the rug in the contemporary teaching of mathematics. No uni courses do it right, mainly because there is nowhere near enough time in the standard courses to do it right. we all just want to make some hasty assertions about the reals and skip to the stuffs needed for modern (ie. publishable) research.

Dedekind cuts vs equivalence classes of Cauchy sequences vs infinite sequences of (non-computable) decimal digits are all just fig leafs to hide our shame. The fact is, no mathematics dept wants to "waste" months actually rigorously proving statements like "a complete ordered field exists and is unique up to isomorphism," or the convolute product of an infinite sequence of uncomputable decimal digits is actually associative/commutative and corresponds to the appropriate equivalence class of cauchy sequences of rationals, blah blah blah.

yes, we call wildhotdog a heretic, but really it's only because he embarrasses us by pointing out our hypocrisy and exposes our lack of rigor. The fact is, there are still some open issues in this area that nobody wants to acknowledge, so instead let's just lampoon the messenger.

[Anonymous No.6540700]

>>6540692
>heretic
Alternative views of foundations aren't that controversial lol.

[Anonymous No.6540713]

>>6540434
>all unaccomplished math profs come up with shit nobody cares about

lels... okic, so you're not interested in the novel theorems of hyperbolic geometry that NJW comes up with.

Do you have any appreciation that the vast overwhelming majority of mathematicians today do not care about the vast overwhelming majority of theorems proven today? Go to any major mathematics conference and you will witness the disjoint union of hundreds of small topical conferences. Most mathematicians nowadays cannot substantially converse with others even slightly outside their own subject areas. You exhibit no clue about the actual state of fractionalization and specialization in modern mathematics.

Hardly any mathematicians care about MOST of the new mathematics done today, so forgive NJW for not having his name printed in your Calc 1 textbook or for not being named Terence Tao.

[Anonymous No.6540716]

>>6540700
>Alternative views of foundations aren't that controversial lol.

>alternative logical/set-theoretic foundations are needed
yawn

>transcendental numbers shouldn't used in trig or uni-level mathematics
burn the fucking witch

[Anonymous No.6540741]

He has some dumb "philosophy" about finitism that isn't even valid

so he says downright false things which must be very misleading to everyone he is teaching

I pressed him on some of his false claims on the construction of real numbers and he was unable to even reply

[Anonymous No.6540747]

>>6540692
> the foundations of the real numbers are mostly swept under the rug in the contemporary teaching of mathematics

it's such a shame that it's impossible to find a book and teach yourself how numbers are made

furthermore see if any of your classmates can prove associativity of addition for integers or something. I bet they can't define integers either, the problem isn't with real numbers.

[Anonymous No.6540755]

>>6540483
You don't need to extend the reals infinitely to solve everything.

[Anonymous No.6540756]

>>6540692
I like his wild linear algebra videos.

[Anonymous No.6540759]

>>6540755
Well to solve a particulay equation you only finitely many extensions.. but to create a domain where you can solve all equations (the algebraic closure) you do need to put together infinitely many extensions

[Anonymous No.6540763]

>>6540692
what open areas?
Also, in mathematics, it doesn't matter if it doesn't 'sit right' with you, because you could be dumb or whatever. as soon as assburgers proofs one (1) contradiction in real numbers then nobody would ignore him

[Anonymous No.6540767]

>>6540759
excuse me if i'm wrong, but isn't C a finite extension of R wherein every polynomial has a root?

[Anonymous No.6540775]

>>6540767
But you can't get R from finitely many extensions of Q.

[Anonymous No.6540803]

>>6540692
I don't see any problem with the fact that a university can't teach you all theorems with proofs.
You are supposed to understand the basic tools you'll need to be a mathematician in some field. If you want, once you have some experience, you can go look for the construction of real numbers by yourself.
As long as it is proved and there aren't flaws, it seems perfectly reasonable to me.

[Anonymous No.6540825]

>mfw I though this thread was about murricans being huge /x/ folk
>wild hamburger

[Anonymous No.6540827]

I'm in favor of his line of thought, I don't think there are many people who know a lot about math and find the reals as Dedekind cuts, as Cauchy sequences, etc. to be a nice thing. Same goes for other constructions which build a thing from other, somewhat loosely related building blocks. Let's call it model theory.

But I also don't know why he puts so much effort in this - it doesn't matter. I approached him with this sentiment on his youtube channel but I don't think things will change - with him of with the responses he gets.

[Anonymous No.6540829]

>>6540827
>I don't think there are many people who know a lot about math and find the reals as Dedekind cuts, as Cauchy sequences, etc. to be a nice thing.
What's actually wrong about them?
Is it because of finitism?

[Anonymous No.6540852]

>>6540829
If you're a finitist, then it's terrible, but also if not, the the "setty" models of the reals are repulsive enough that -as a consequnce- nobody would actually want to look at the model.
Nothing is wrong with it, in the sense that if your only questing is "is it possible to find a model of them", but not what it actually looks like. That question should be solved by completeness of first order logic, though (not sure).

I've only seen 2 vids of Wildberger or so, one of them is one where he patiently discusses such models (to point out why they suck). Regarding your question, it's actually instructive to watch this for 6 minutes from 0:41 on:

https://www.youtube.com/watch?v=4DNlEq0ZrTo#t=41m30s

[Anonymous No.6540875]

>>6540852
Ok, so far when it comes to computational problems, with "infinite decimals" it is quite nasty...
Thanks for the link btw.

[Anonymous No.6540892]

>>6540755
Extend the rationals infinitely.

[Anonymous No.6540894]

>>6540875
yeah, and well apart from the "nasty" things, there are also the conceptually troubling.
You know, you can play games like "imagine all possible 10h long taped ultra-high def movies anybody could ever witness"
(this includes things like "the tape of your parents fucking/your creation" from all possible angles, and in all thinkable fictional variations as well as "he tape of what Hitler witnessed on the day 18.5.1942", or the day before, or any fictional variatants, or "he tape of Riemann explaining pi to you for 10 hours", or "Riemann explaning 4 to you for 10 hours", or "Grothendieck explaining the square root of 2 to Riemann for 10 hours". Or for 5 hours, and then it cuts to a recording on what happended on the moon on the independence day 4 years ago.)
Now okay, the problem with this is that these tapes can ALL be encoded as numbers, say their bits in digital format. In particular, all numbers (or other mathematical concepts) anyone could talk about are finite, i.e. their collection is "as big as the rationals". Call the definable numbers D. Then D is some infinite countable subset of the reals.
Then the set (in the sense of set theory) X defined as "the reals R, without D", i.e. X=R\D, is all numbers you can not possibly talk about/describe. But they make the biggest block of the reals, the uncountable part.
Consequently, the largest part of the "real numbers" R are a collection of "numbers" X, where I can actually not even _talk_ about a single member. It's a semantically construction, but hardly a set of things.

And you can make that rigorous by replacing video tapes with any logic (a bunch of finite sentences) and you get

http://en.wikipedia.org/wiki/Definable_real_number

I used the video tape description because a variant is actually manifest in all computational approaches dealing with reals, see e.g.
http://en.wikipedia.org/wiki/Specker_sequence

But yeah, whateva.

[Anonymous No.6540897]

>>6540894
PS: and yeah, even if we drop X, the larges part of R, and act as if we can speak of all real numbers (which in fact only makes up the collection D), then even among those definable numbers, there are many which you can give a name to, but you can't compute what they are.
A famous exmaple is
http://en.wikipedia.org/wiki/Chaitin%27s_constant

[Anonymous No.6540904]

There are some problems with real numbers if we are interested in computational aspects. But the fundamental theorem of algebra is not a real problem. If we will consider real numbers as say computable Cauchy sequences then the fundamental theorem of algebra will still holds and we will not need any infinite objects to prove it.

[Anonymous No.6540908]

>>6540892
I never said anything about the rationals.
I said the complex numbers are an extension of order 2 over the reals.
Going from rational to reals is easy. This guy is a crank.

[Anonymous No.6540920]

>>6540468
Which axioms do you need to add to get that C is algebraically closed?

[Anonymous No.6540935]

>>6540897
I see. I actually watched the entire video eventually and it brought up many rational points really, which I didn't thought about(maybe as a future applied mathematician is in my nature).

It makes sense. I wish I had a broad understanding of formal logic and computation theory.
Thanks you for the input.

[Anonymous No.6540938]

>>6540935
np.

in the applied world, when you do numerical stuff with reals, you actually only deal with the computational reals (numbers which mortals can know the digits of),

http://en.wikipedia.org/wiki/Computable_number

which are a proper subset of the definable reals (numbers which you can give a name to, say "a", "b", "\pi", "\phi"),

which are "the small part" of the reals.

[Anonymous No.6540939]

>>6540894
A union of finite sets is not necessarily countable. Who says that I can't come up with an uncountable number of possible tapes?

[Anonymous No.6540941]

>>6540434
>be called wild hamburguer nut job unaccomplished math profs that come up with shit nobody cares about by internet parasite
>be awesome everyday

Stay superficial, plebeian.

[Anonymous No.6540946]

>>6540908
I said something about the rationals because the topic of conversation was Wildberger, who always works in the rationals.

>going from the rationals to the reals is easy
If you are capable of performing infinite operations, sure. I'm not though.

[Anonymous No.6540951]

>>6540938
Even that is being charitable. I have yet to find a suitable computational model of the reals. Every representation method has its downsides. In a sense every representation has the 0.999... problem in its own way.

Rational functions of reals are not too bad. Algebraics can be somewhat managed. But transcendental functions of irrational arguments are fucking retarded.

[Anonymous No.6540954]

>>6540939
Whatever, choose a max refreshing frequency, max pixels, max colors a human can effectively perceive and the videos will not bu uncountable.
The real point is that logic effectively has an countable alphabet.

[Anonymous No.6540956]

>>6540939
Whatever, choose a max refreshing frequency, max pixels, max colors a human can effectively perceive and the videos will not bu uncountable.
The real point is that logic effectively has an countable alphabet.

>>6540951
I'm a physicist and dropped the ball on those problems some time ago.
What's your motivation to find a nice computational model?

[Anonymous No.6540960]

>>6540956
>What's your motivation to find a nice computational model?
Boredom.

[Anonymous No.6540979]

>>6540938
>computer science kid

Step off, we're talking about real algebra here.

[Anonymous No.6540992]

Eh I was worried about the existence of infinite sets briefly after watching some njw

But then I realised

Nothing in maths exists and it's a meaningless question "does this mathematical concept exist"
Choose some axioms and play the game. Formalist masterrace

[Anonymous No.6540996]

>>6540992
>njw
?

[Anonymous No.6540997]

>>6540996
Norman J Wildburger
The guy this thread is about

[Anonymous No.6541036]

>>6540829
>What's actually wrong about them?
nothing,

they are completely simple and straight forward. the entire construction fits on a couple of pages and is elementary enough to teach in university - for some reason they don't.. it's swept under the rug with all other foundations

[Anonymous No.6541085]

>>6540692
>But his mathematics foundation series is the best on the whole interwebs.

I've looked into
>https://www.youtube.com/watch?v=K4eAyn-oK4M

Half of the vid he is talking about how difficult the epsilon-delta is for students to understand (which it isn't), and then he offers his "new definition" which IMHO doesn't have any practical advantages, maybe even disadvantages.

That guy is really a nutjob, I don't see anything useful to be learned in his vids.

[Anonymous No.6541114]

>>6540692
>The fact is, there are still some open issues in this area that nobody wants to acknowledge,
>[citation needed]

[Anonymous No.6541135]

The basic problem I have with this guy is that he seems to think that rigor is the most important thing, even in practical math for the layman.

If you look at his channel, he starts out with a video on how distance and angle make trigonometry "too hard". And how it's cheating to teach trig before calculus, since "angles require calculus".

Rigor, when it comes down to it, is a matter of taste. Trigonometry was around for millennia before calculus, and calculus was used to great practical advantage for at least a century before it was made "rigorous". What did rigor actually contribute? A sense of smug satisfaction?

Physics (i.e. the math of the real world, the part of math that matters) continues to run out ahead of the rigorphiles, who wail and gnash their teeth and scream that everything must be bogged down in the pursuit of as much rigor as their autism demands.

Any more rigor than you need to get answers that work is masturbatory.

[Anonymous No.6541149]

ITT: Foundational Mathematics.

I didn't even know that I didn't even know this stuff.

It didn't exist for me until today.

What the fuck.

[Anonymous No.6541160]

Lel, I just saw that video yesterday.
I laughed when he said he will be vindicated in his life time or something along those lines.

He's still a cool dude though.

[Anonymous No.6541245]

I haven't watched this guy's vids, but I think his general beef with contemporary mathematics is entirely valid.

Half the point of math is to have solid reasoning behind everything; to create an unbroken chain of infallible logic from beginning to end so that we can accept new results without having to worry about previous results. But ultimately much of that reasoning gets swept under the rug when students are learning math, and even the students that become math majors end up being taught more advanced maths rather than getting to see these fundamental arguments.

I'm about to graduate from college with a degree in theoretical computer science, and I have no fucking clue where real numbers came from. Everything in Number Theory? Properties of Integers? I have a great understanding of those, and the truth of the properties of the Natural numbers is readily apparent in computer science.

I haven't taken Real Analysis (which someone suggested would help) but I don't see why "limits" are a mathematically sound concept. Recently I was thinking about it, I came to the conclusion that Real numbers are only approximations and most of the properties of Reals comes from that fact. But why isn't it taught that Real numbers are fundamentally approximations? Is there any way to logically formulate Real numbers from Natural/Integer/Rational numbers, or do Real numbers require separate axioms? If the latter, why are Real numbers ever mentioned in the same breath as N/Z/Q?

Personally, I think all math having to do with Real numbers needs to be it's own explicit math, and concepts from Real Math would never be taught in the same class as concepts from N/Z/Q Math; during early schooling children would be initially taught that we use Two Great Maths, taught where each comes from, and then they'd never touch ever again.

[Anonymous No.6541269]

>>6541245
Real numbers are actually constructed, in a way, as approximations(you take for examples sequences of rationals which get closer and closer).
The things is, most of the times a real number approximates something which can't even be computable and the use of infinities makes some people uncomfortable(i'm somehow in the middle ground between pure formalism and "soundness").
I mean, you can construct real numbers in such a way that you can't distinguish between two of them by just looking at their definition, because they rely on infinite things.

As far as I'm aware, the construction with equivalence classes of cauchy sequences doesn't require anything besides ZF.

[Anonymous No.6541275]

>>6541245
In other words, "Ew, your "Real" numbers are touching my actual numbers, gross!"

[Anonymous No.6541314]

>>6541275
Being a CS student, yes, that is my opinion, but I see no reason why the converse is not equally valid for someone who primarily works with Reals.

The point is that these two maths use numbers in very different ways, and since it's easy to conflate one formulation of numbers with another, they should never be used in the same context.

As an analogy, imagine if in school there were very important classes on the properties of apples, and people used two types of apples: Granny Smith and McIntosh. These two types of apples have very different properties and uses (they aren't even the same color!). Wouldn't it be confusing if all the apple classes just used the term "apple" when referring to properties of apples, and only rarely bothered to make the distinction between Granny Smith and McIntosh? Wouldn't it also be confusing if one were taught the history and origins of Granny Smith apples, but the history and origins of McIntosh apples were sort of just implied?

[Anonymous No.6541317]

>>6541269
>I mean, you can construct real numbers in such a way that you can't distinguish between two of them by just looking at their definition, because they rely on infinite things.
Example?

[Anonymous No.6541334]

>>6540747
>le peano axioms faec

[Anonymous No.6541343]

>>6541317
If you define real numbers by equivalence classes of cauchy sequences of rationals, pie for example is something like {3, 3.1,3.14, 3.141, 3.1415, 3.14150, ....}
But if I define a = {-56, 22/689, 89, 88, -3.2, 6, ....} with everything beyond 6 the same, it is the same cauchy sequence and therefore it represents the same real number.
So in general if you take any cauchy sequence and change arbitrarily the first N digits, it is the same equivalence class because after N they are the same. This is because to identify cauchy sequences, you only need the same behavior at infinity.

[Anonymous No.6541356]

>>6541245
The fundamental arguments are a fruitless pit of autism.

It's like when you get a kid who realizes they can just keep asking "Why?" in response to any answer you give them on any question. It doesn't take many repetitions for this to become silly, and most of them soon grow out of it and start asking better questions, and thinking harder for themselves about the answers they receive. But it illustrates that you can take any question and arbitrarily decide that the answer is unsatisfactory, no matter how good or useful it is.

Mathematics is about abstract structure. You can't build every structure from one set of axioms. When we model something new, we often start by making up some axioms that seem reasonable and appropriate. Then we build from those, and maybe we have to change them, if the model ends up not having the features we want.

Then someone else comes along, and proves that the axioms can be constructed from another set of axioms, that you can model one kind of math in another kind of math. This isn't actually important. It's just playing a game.

Some very badly philosophically confused people think that any new math, with its new axioms, isn't valid until it's unified with the old math, and that we should not make new math this way, but only by deriving it from old math (regardless of this being practically infeasible). They'll become obsessed with one set of axioms, and insist that everything valid must be constructed from them, and anything that can't be, should be discarded.

[Anonymous No.6541362]

>>6541343
OK. I thought you meant you wouldn't be able to tell two different reals apart.

[Anonymous No.6541390]

>>6541245
>"Half the point of math is to have solid reasoning behind everything; to create an unbroken chain of infallible logic from beginning to end "
This is not how mathematics has ever developed or ever worked, it is a product of the way mathematics is made to look that way by textbooks. I recommend you read Proofs and Refutations by Lakatos and The Structure of Scientific Revolutions by Kuhn

[Anonymous No.6541429]

>>6541356
>Mathematics is about abstract structure. You can't build every structure from one set of axioms. When we model something new, we often start by making up some axioms that seem reasonable and appropriate. Then we build from those, and maybe we have to change them, if the model ends up not having the features we want.

>The fundamental arguments are a fruitless pit of autism.

You heard it here first on /sci/, Abstract Algebra is merely a fruitless pit of autism! We don't need to worry about underlying structure, we can pull algebras from the infinite and formalize them as needed!

[Anonymous No.6541448]

what a dire thread, well done /sci/

[Anonymous No.6541449]

>>6541343
>it is the same equivalence class
No, it is IN the same equivalence class.

[Anonymous No.6541468]

>>6541449
Yup, my bad.

[Anonymous No.6541480]

>>6541390
>This is not how mathematics has ever developed or ever worked
Yes it is. Just because the criteria for "infallible logic" has changed over time doesn't mean that hasn't always been the goal. If a mathematician comes up with a proof which is initially accepted but then later shown to be false, both the original mathematician and whoever disproves him were still trying to add to the infallible chain of logic-- it's just that the former sought to extend the chain while the latter sought to verify the chain.

[Anonymous No.6541495]

>>6541429
Welcome to the real world. Most of the goals of pure math are just someone's hobby, rather than anything of practical significance. You might as well be doing pottery, or origami.

Get over this feeling that your game is of Earth-shattering importance. Pure mathematicians, insisting on strict rigor the whole way, would never have come up with so much as calculus, and calculus worked fine before anyone bothered about making it "rigorous".

[Anonymous No.6541610]

>>6541495
Good Lord how big of a tool can you be? You're suggesting that a deeper understanding of mathematical concepts is of no practical importance, which is tantamount to believing in superstition. Could you imagine your position in ANY other human endeavor?

Chemisty:
"No guys, we don't need to figure out what makes up these chemicals or why certain chemicals are the way they are, you just mix shit up and use the results! Those who try to discern why helium is helium just enjoy playing with their chemistry sets!"

Medicine:
>Get over this feeling that your game is of Earth-shattering importance. Pure doctors, insisting on strict rigor the whole way, would never have come up with so much as molds, and molds worked fine [to cure disease] before anyone bothered about making them "rigorous".

I think you should stop using antibiotics, and treat yourself only with molds you can find naturally. After all, searching for underlying purpose is just wankery, no more useful than pottery, correct? There are plenty of molds out there, you can just try them all to see which ones produce a condition that is consistent with a cure and then use those in the future.

Pic unrelated, just some picture of an Autist I found on the internet.

[Anonymous No.6541612]

>>6541362
In general you cannot tell if two computable reals are equal.

[Anonymous No.6541641]

>>6541610
>a deeper understanding of mathematical concepts
That's not really what most pure math is. There's no "deeper understanding", just arbitrary goals being pursued as a sort of game.

Don't try and conflate science, and its benefits, with pure math. Science is the study of the actual physical universe. Of course it's useful to pursue the goal of a deeper and more complete understanding of how the world works.

The goals in pure math are not about practical results. Mathematicians will tell you so without hesitation. That's what distinguishes pure math from fields like physics, computer science, and applied math, where equally challenging mathematical work is done with some actual concern for usefulness.

While mathematics is of obvious importance, pure math isn't. We don't need and have never needed pure mathematicians.

[Anonymous No.6541707]

>>6541641
>The goals in pure math are not about practical results. Mathematicians will tell you so without hesitation. That's what distinguishes pure math from fields like physics, computer science, and applied math, where equally challenging mathematical work is done with some actual concern for usefulness.

Do you think all scientists do their work with absolute concern for practical results? Do you think every great scientific discovery has been in the quest for results, not merely out of curiosities' sake? Do you imagine that back in the seventeenth century, that there was a pressing practical need for calculus? Or was it really just Newton (and Leibniz) fartin' around with math for the respect and admiration of his peers?

>While mathematics is of obvious importance, pure math isn't. We don't need and have never needed pure mathematicians.

How rude! How arrogant! The ancient Greeks didn't pursue math and logic because of practical reasons, would you throw out the entire foundation of math on the basis that the Greeks had no business inventing such things?

Historically, most of our math comes from people who were simply doing math for maths sake, not because there was some concrete problem that needed solving. It's only in the past couple hundred years that there's really been a need to quantify complex information.

[Anonymous No.6541732]

>>6541390
Wait, if the idea of infallible logic has changed over time then how does it make sense to use the term to describe different mathematical paradigms at different times, if the criteria for infallible logic has changed then the two mathematicians with different criteria were working in different mathematical frameworks. What you are describing, a Hilbert style formalism, is a very new idea in mathematics and by no means the only way mathematics is interpreted or progressed.

[Anonymous No.6541746]

>>6541707
>Newton (and Leibniz) fartin' around with math for the respect and admiration of his peers?
I don't get why this is a thing. No one does math for the respect or admiration of their peers. I mean no one doing math even gets those things.

[Anonymous No.6541780]

>>6541707
>Do you imagine that back in the seventeenth century, that there was a pressing practical need for calculus? Or was it really just Newton (and Leibniz) fartin' around with math for the respect and admiration of his peers?
>The ancient Greeks didn't pursue math and logic because of practical reasons
You live in a particularly strange fantasy world.

The ancient Greeks pursued math and logic, along with other areas of philosophy, because they wanted to understand how the world worked, and thereby, how to act correctly. They used their math for practical purposes and scientific investigation much as we do today.

Greek mathematicians knew they were sharpening tools which could be put to good, practical work.

Newton and Leibniz weren't "fartin' around", they were fully aware of the potential of their inventions. This was the age of artillery, after all. These weren't impractical men by any means. They both invented calculus as a tool to solve physics problems, and they both were active in practical invention and politics.

[Anonymous No.6541796]

>>6541780
>the Greeks inscribed polygons in circles to figure out how to act correctly

[Anonymous No.6541802]

>>6541796
Yes, they were developing geometry as a tool.

It had obvious applications in architecture, art, and engineering.

There was pretty much nothing that the ancient Greeks did which would be classified as "pure math".

[Anonymous No.6541804]

>>6541780
More or less what this guy said. Though it's largely because back in those time periods mathematics was legitimately viewed as just a tool to view the world. If someone created some mathematical arguments that couldn't be applied to the physical world then they were viewed as repugnant to the senses and people believed that they had to be incorrect. This is why so many lifetimes were spent trying to prove Euclid's fifth postulate based using the others and to try to show that alternative postulates led to contradictions. Eventually they gave up and accepted the existence of non-euclidean geometries, though for a long time the very notion was seen as offensive to reason.

>>6541796
Not really, but there was a lot of ideology that was inexorably tied to greek geometry. I mean you often hear that the greeks didn't believe in the existence of irrationals but it actually goes further than that. They didn't actually believe the rationals were full fledged numbers, rather what we call the rationals they just perceived as a ratio between two numbers. It was laughable to think that the relationship between two things could be as real as the things themselves. All of constructionist geometry was full of stuff like this.


All of that said, modern mathematics doesn't give a single fuck about the real world. If something comes from it then it's kind of neat but not the intention. Of course, if you're doing paid research then you have to give others a reason to pay you for it. Even if you actually spend most of your time working on math shit you're actually interested in.

[Anonymous No.6541812]

>>6541802
>which would be classified as "pure math".
This is wrong, axiomatization was created by Euclid and is considered pure math. It is the basis of pure geometry, which is pure math as fuck.

Pure math doesn't mean that it purposely doesn't have applications in the real world, just that it isn't based on directly on the real world. Axiomatization allows us to make arguments based on axioms without having to rely on the real world.

[Anonymous No.6541863]

>>6541732
What on earth are you on about? All I'm saying is that one of the primary goals of mathematics is to have solid reasoning; although the frameworks for determining what constitutes "solid reasoning" may change, the end goal is still to ensure that new math needn't concern itself with the validity of older math.

[Anonymous No.6541887]

>>6541812
Euclid's geometry is closer to physics than anything. He picked his axioms based on observation of the world, selecting them not as arbitrary rules but as things so obvious that he thought no one could find fault with them, and showed how you could derive important results from them with great certainty.

Pure math pretty much always has its roots in something applied. You can't call those roots pure just because a pure math field grew from it.

Pure math happens when you stop caring about the relation of the math to the real world. A pure mathematician is one who can see the Banach-Tarski paradox and remain content to work with the same axioms that it is a consequence of.

[Anonymous No.6541901]

>>6541887
You're right about that first part, but it's because of all the ideology tied into geometry as I said here. >>6541804

You're wrong about the second part because axiomatization itself is pure. Whether or not the intent is to model something applied. I can see how this is confusing though, here is an example of some different greek geometry that is applied. It's a method for computing the integral dealing with infinitesimals and the Law of the Lever.
http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems

Levers in particular are an applied concept.

>Pure math happens when you stop caring about the relation of the math to the real world. A pure mathematician is one who can see the Banach-Tarski paradox and remain content to work with the same axioms that it is a consequence of.

This is the concept misconception about what pure math is. It's a terrible misconception too. I mean think about it, does pure math stop becoming pure math if someone finds an application for it?

Here is a wiki article on pure geometry. As you can see it's essentially a generalization of Euclid's idea of axiomatization for geometry.
http://en.wikipedia.org/wiki/Synthetic_geometry

[Anonymous No.6541926]

don't care wot u think abt him his vids are focking gr8

[Anonymous No.6542124]

>>6540434
He's a crack-pot.

[Anonymous No.6542215]

>>6542124
Ideologically perhaps, but I can see value in being able to do mathematics using fewer requirements. His wild linear algebra series is great imo.

[Anonymous No.6542238]

>>6540434
speaking of him being a nutjob - does anyone know any better free videos that give you more mainstream algebraic topology? I'm trying to learn it on the side, mostly for finite stuff, and I'm going through Munkres and a few related topoi books. It's going aight, but I learn way faster when I can watch lectures to tie everything together at a faster pace than reading. hamburglar is the only thing that comes up when you search "algebraic topology lectures" that isn't beyond intro level.

[Anonymous No.6542351]

>>6542238
>mostly for finite stuff, and I'm going through Munkres and a few related topoi books
what on earth are you doing with algebraic topology?

[Anonymous No.6542353]

>>6542351
whoops, should mention the catsters youtube channel. i think there was an MIT course online possibly too or not lol.

[Anonymous No.6542372]

>>6541356
I like what you say in the last paragraph here

[Anonymous No.6542391]

>>6541356

You're missing the point though anon. It's not just playing a game or fruitless autism. There is real value in being able to do this kind of nesting. I'll to illustrate the reason.

So, suppose you have two axiomatic systems. Call them A and B. Now suppose that it turns out that axiomatic system B can be modeled inside axiomatic system A. Then, suppose that we later on find out that axiomatic system B is inconsistent, meaning that inside that system we can prove some contradictory statements. Then because axiomatic system B can be modeled in A it means that A is also inconsistent (because in a manner of speaking, B is just a special case of A). Flaws like this cascade upwards. In a way, by modeling axiomatic systems inside one another we're kind of bookkeeping against inconsistencies like this so that in the worst case scenario where an inconsistency is found we can assess the damage.

It's not all bad though, it's also possible for good things to flow back and forth. Like, suppose you have a set of axioms that define Rectangles and another set that define Squares. You can model the axioms for squares inside the axioms for rectangles by thinking of squares as just a special definition (a rectangle + some extra constraint). Then any theorem we prove about rectangles must also be true about squares, because every square is a rectangle. Similarly, if we prove something really powerful about squares then it may be possible to apply the concept in the broader scope of rectangles in some useful way.

Modeling axiomatic systems gives us a lot of very serious power and foundations are kind of an arbitrary way of keeping track of everything. If someone can prove all of the same things using a weaker set of axioms then by all means they should, minimalism never hurt anybody.

[Anonymous No.6542682]

>>6542215
Best linear algebra vids I've seen.

[Anonymous No.6542811]

>>6542351
>>6542353
k i figured that was it. i already follow catsters also it's a secret hehe.

[Anonymous No.6542822]

>>6542353
oh cool i get a cute chinese british woman to teach me category theory?? fuck yeah!!!

[Anonymous No.6542885]

I like how he thinks he invented using similar triangles in place of angles and sine, and we should teach kids trigonometry so it's actually just about triangles, and they don't learn to deal with circular functions at all.

[Anonymous No.6542923]

>>6542885
Sorry anon, but trig functions should only be thought about in terms of their Taylor series. Anything else is handwavy and imprecise

[Anonymous No.6543013]

If he has thought that he discovered the grass, it is only because he has got down off the shoulders of giants.

[Anonymous No.6543022]

I really don't understand why people think he's a nut just because he disagrees with current treatments of analysis. I think you're just mad that he's saying your favorite theorems are wrong.

[Anonymous No.6543125]

>>6542885
So basically you don't understand the aim of rational trig at all.

[Anonymous No.6543158]

>>6543125
What I understand is that he thinks we should teach it to children and surveyors instead of regular trig.

I'm sorry, but "rational trigonometry" is just geometry. We already teach geometry. He didn't invent shit.

He does have a point that it makes sense to start calculus before trig. In a lot of ways, it's easier, especially with computers to illustrate concepts with animations.

[Anonymous No.6543185]

I read his paper 'Infinite Series - should you believe?' up to page 9 or 10.
Either he is joking or retarded. Where he does not have any arguments, he starts to say funny things.

> using a crappy heuristic version of the axiom system, then saying mathematics is crappy because of the version he uses

> no maths prof nor student can tell you how to define numbers or prove simple rules of arithmetics
where i live, you learn this in the very first year

and in general, this paper is everything but scientific or reasonable. There might be serious approaches to this, but Wildberger does not try to make on of them here.

tl;dr
>ishyggydiddy

[Anonymous No.6543186]

>>6541245
>I haven't taken Real Analysis
> I don't see why "limits" are a mathematically sound concept

Well, maybe - and just maybe - you should take the unbearable effort to take this sophomore course before you start babbling about its fundamental concepts.

> Is there any way to logically formulate Real numbers from Natural/Integer/Rational numbers, or do Real numbers require separate axioms? If the latter, why are Real numbers ever mentioned in the same breath as N/Z/Q?
Please buy a book about the foundations of math, or set theory or whatever. Now you're just embarrassing yourself.

> Half the point of math is to have solid reasoning behind everything; to create an unbroken chain of infallible logic from beginning to end so that we can accept new results without having to worry about previous results. But ultimately much of that reasoning gets swept under the rug when students are learning math
Because it is fucking impossible you prepostorous wacko, please try to prove there are infinitely many primes using only Peano's Postulates and the rules of logic, write it all formally down and be surprised that you are an old man.

[Anonymous No.6543300]

he's just really too obsessed about how the current standard model of math is "wrong" or inefficient or unintuitive, and just really wants to propose his alternative ideas

[Anonymous No.6543398]

>>6543300
He wouldn't be so obsessed if it weren't taught to students as if it were fact and the only correct way to think about mathematics (esp. calculus, topology and related fields)

[Anonymous No.6543446]

>>6542811
I asked because it seems heavyhanded to deal with the homotopy of finite topological spaces at the level of topoi.

[Anonymous No.6543464]

>>6543446
Because my focus isn't on homotopy. I'm working on subjects closer related to logic, which is much more straightforward at the level of topoi.

[Anonymous No.6543502]

>>6540434
Do you guys think wild cowsandwich is in this thread right now?

[Anonymous No.6543515]

>>6543502
i hope not