Algebraic Calculus I: Points and Lines in the Affine Plane

Good work, that is a nice summary of the first lecture!

>>9053874
No problem. I do have a question though: given that you have already started the series, have you already the solved the issues of continuity, integrability and differentiability for a geometric space in the rational numbers?

Have you found a notion equivalent to complete metric spaces but applicable to [math] \mathbb{Q} [/math]?

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>>9053853
I like my calc with epsilons and deltas and sequences of open balls. Its the way Leibniz would have wanted it

>>9053888
my "888" prove it!

>>9053888
True geniuses use epsilon-neighbourhoods and delta-neighbourhoods.

Perhaps this series is not suitable for someone of your... lesser talent.

>>9053853
I took this guy's algebraic geometry class and it was actually pretty good, there was no down with infinity lectures.

>>9053896
It must be nice to be taught by the mad man in person. How does it feel? Do you now understand the secrets of the universe?

>>9053916
Should have clarified I watched the lecture vids, followed the course notes using Paolo Aluffi's Homology book, and did the class assignments. The vids are on jewtube https://www.youtube.com/watch?v=Ap2c1dPyIVo&list=PL6763F57A61FE6FE8

However he gives free seminars at UNSW, if I ever end up in that god forsaken land of straya for sure I will time it to attend one of these lectures where infinity is denied

I like how the approach subtly circumvents the the use of functions for lines. It seems really interesting from a computational efficiency point of view. Will replacing functions with structures be a comon theme in this series?

>>9053896
>>9053927
Algebraic Geometry is a very different subject from Algebraic Topology.

>>9053853
You know, this guy gets meme'd pretty hard here, but I kind of like him. He offers unique perspectives on things, and is a very good lecturer to boot.

>>9054057
Ditto. There's something about his pace and teaching style that I love. I don't mind the whole fear of infinity thing either, if anything it's kinda refreshing.

>>9054121
> if anything it's kinda refreshing.
I don't like it personally, but I wholeheartedly approve of any challenge to the idea that math is completely objective and that its axioms are set in stone.
Basically Voltaire's belief on disagreeing with what you say but supporting your right to say it.

>>9053887
All these hinges on the definition of a limit, right? And as far as I know (though I haven't been bothered to make it precise) the limit can be described for a completely arbitrary topological space -- which includes metric spaces as a special case -- using standard concepts in category theory.
Good luck making [math]that[/math] accessible to the masses though.

>>9054213
>All these hinges on the definition of a limit, right?

Yeah, they all depend on the definition of a limit for sequences. We can't currently do Calculus in the rational numbers because in Rat there exist Cauchy sequences that do not converge and that is really bad because then definitions would be kinda inconsistent. Functions that look like they should have limits or look like they should be continuous won't be because the pertinent sequences do not converge. And then also we would be unable to use powerful tools like the Cauchy Criterion for convergence.

>>9053853
those are good lectures

i dont usually subscribe to anything, but i can tell this'll be very useful for formulating calculus and I like the way he teaches it

>>9054057
>>9054121
he explains the concepts well so they can be easily understood, without leaving anything out or treating you like a retard, and you can tell hes actually very interested in what hes teaching

its nice

Bump.

Daily reminder than in 5 years or so Algebraic Calculus will be the only accepted Calculus so better learn it early.

>>9053853
He is a genius, no doubt. Even the best mathematicians do not understand his clear and methodical arguments. The point is, modern mathematics focuses almost entirely on approaches other than constructivism. For no particular reason, modern mathematics has rejected the completely valid philosophical position of constructivism, simply because it does not include the idea of "infinity". Incredibly, even in the 2015, many (I would even say most) mathematicians will literally get angry if you start to take a constructivist viewpoint. Modern students are specifically taught that this idea of "infinity" is "real". Never are they taught the truth, which is that "infinity" is a philosophical belief. Frankly, I am strongly inclined to think that modern mathematics is deeply flawed in its ideas regarding "infinity". Disregarding constructivism to have a random imaginary concept called "infinity" was a major step in the wrong direction for modern mathematics. At the very least I think we should bring back constructivism and at least treat it with the same dignity that we treat the rest of mathematics. There is no room for philosophical bias in mathematics.

Also, I think it is possible to rigorously prove that non-constructive approaches are logically flawed, however, you basically have to be a genius to understand the proof. Most human brains have not evolved to the point where they are aware that things are meaningless unless they are defined. We as a species are "just waking up" to reality and the idea that reality and computation are synonymous. It will be another few hundred years before people understand what Dr. Wildberger is saying.

>>9053888
What's wildberger's problem with delta-epsilon approach? In Tao's Analysis he formulated delta-epsilon definitions using rational numbers

>>9056727
Thx for copypasta

>>9053853

I wish he would do things that aren't so trivial

> Spends 10 minutes describing n^2

>>9053853
I'm interested in seeing what he does with this (mainly how he will define the derivative for things that are not polynomials) but so far, none of this is new or has anything to do with calculus.

>>9056798
He doesn't believe in infinity. Tao's construction (which is essentially defining R to be the metric completion of Q) requires not only infinite sequences, but infinite sets of infinite sequences.

Too many examples, not abstract enough

>>9057411
But he defines reals after he constructs limits of rational sequences

>>9056727
Defining naturals as strokes on the whiteboard and addition as "just adding them" is not good approach to the rigorous foundations of maths

This old geezer is a brainlet.

>>9056727
Constructive analysis isn't exactly new and people don't take Wildberger seriously because he just reinvents the wheel.

In the end no one is denying different formulations of some branch of modern mathematics are possible, but let's see how he uses it to solve open problems, that's the whole point of creating new math anyway.

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>>9058885
>>9058885
>Defining naturals as brackets on the whiteboard and addition as gluing them inside a bigger bracket is not good approach to the rigorous foundations of maths

>>9056727
Kinda this, but Wildberger is an ultrafinitist. Constructive mathematics is way wider and more "consistent" (in common sense).

t. have one paper on constructive analysis

P.S. It is incredibly hard to publish papers on constructive math due to the editor's and reviewers' blatant arrogance. Most of the comments aren't even related to the math itself, but bitch about philosophy. This is bullshit. Any mathematics is worth studying, but the politics is unfortunately strong in the community

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>>9059001
shit i have to agree with you

>>9054121
>There's something about his pace and teaching style that I love.
I know what it is: he does math for middle schoolers, and you're a brainlet, so you're happy that you can follow along.
No, seriously: I learned the stuff in these two videos in 5th grade!
And the lack of rigour is appalling. Why is it that, say, multiplying the line equation by a constant does not change the lines? Wildberger's "explanation": we agree that it doesn't. Of course, he just wants to avoid the more complicated math, which the explanation would require, since he knows that his audience is made up of a bunch of brainlets.

>>9053853
can someone tell me what is supposed to be this guy's target audience ?

>>9059166
The Khan Academy crowd and younger.
Brainlets in general.

>>9059154
>the transitive property of quality is "complicated math"
I get that you're pretending to be smart on the internet but you should really try harder

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>>9059166

people who couldn't make it through rudin and were so traumatized by the experience they rejected the reals altogether in a vain attempt to obfuscate their own failures.

>>9059171
>the transitive property of quality
This has nothing to do with the transitivity of [math]e[/math]quality.

>>9059186
oh shit i made a typo, thus completely invalidating my snarky response
how will i ever recover

>>9059188
I am not talking about the typo dumbass. I am talking about what you meant: transitivity of equality is irrelevant.

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>>9059154
>I know nothing about affine geometry and projective geometry

>>9059212
>brainlet trying to point out implications that aren't there

It seems that this wild mad man is causing quite a furor once again.
Understandably, as all of your knowledge about the mythical [math]``\mathbb{R}\text{''}[/math] will soon be obsolete.

>>9053853
If you are not allowed infinite processes, then you can not prove there are infinitely many primes, as those proofs rely on infinite processes.

In this man's version of math, there is only a finite number of primes.

>>9060157
not according to this guy https://math.stackexchange.com/questions/30127/is-there-an-intuitionist-i-e-constructive-proof-of-the-infinitude-of-primes

>>9061270
Wrong, the constructive proof demonstrates that the set of primes is at least hereditarily finite, not that it is infinite.

More precisely, the constructive proof demonstrates that for any finite set of primes you can find a prime number not in that set. At no point is the notion of infinity mentioned, so the proof works just as well in a model of PA (or ZFC with the axiom of infinity replaced by its negation).

>>9059154
I was more referring to his passion of the material, stop projecting, brainlet

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>>9060157
>>9061270

>>9053853
>intuitionist
dropped

>tl;dr: [math]\mathbf Q[/math]-vector spaces exist
Not watching this waste of time.

>>9053853
How does he define rationals? Does he claim there are finitely many rationals, with [math]10^{200}[/math] being the biggest and [math]\frac{1}{10^{200}[/math] the smallest positive rational? What does he do about the fact we can find a rational between any two other rationals, meaning there are infinitely many rationals in arbitrarily small interval? Does he say that some rationals are close enough to each other and no other rational lies strictly between them? Or does he accept there are infinitely many rationals, and then all of his videos boil down to "modern mathematics sucks because it uses infinities, and the solution is my mathematics, which too involves infinities"

>>9058962
Having naturals defined as braces inside braces put together in a specific manner is made precise by set theory. Union of sets is a valid, rigorously constructed mathematical operation, bunching sticks together isnt. Just look at his proof of addition commutativity, "just flip it, and it looks right, hence proven" (which isn't even a fully correct proof in his theory, because he draw circles, whereas naturals were defined as sticks, and he didn't prove they're equivalent) and say it's not less rigorous than this wrong, evil modern maths approach

>>9059344
>we must forget about most of abstract algebra and field extensions because one guy doesn't like that most of decimals don't terminate

>>9060157
Assume the set of primes is finite, then
[eqn]0<\prod_p \sin\left(\frac{\pi}{p}\right)=\prod_p \sin\left(\frac{\pi\left(1+2\prod_{p'}p' \right)}{p}\right)= 0[/eqn]
no infinite processes here

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>>9062305
>using pi and sin

>>9062296
Anyone who genuinely subscribes to the set-theoretic postulate that everything is a set will define a bunch of sticks as a set of the requisite cardinality, and bootstrap the proof of commutativity of stick addition from the underlying proof of commutativity of cardinal addition.

The advantage of using sticks rather than sets is that the proof remains valid even for the 99.99% of people who don't subscribe to the reductionist ontology of set theory.

>>9063797
So circles and complex exponentiation don't exist?

>>9060157
>as those proofs rely on infinite processes

No, you are misunderstanding Euclid's argument. His argument is not saying that
>look, I can construct a new prime, and then a new one, and then a new one infinitely

It says that if you assume there is a biggest prime, you reach a contradiction. That is all.

>>9053916

How is life in the slow lane, pleb?

>>9064271
Not true, Euclid's proof was constructive, meaning that he can construct new primes infinitely, his original proof wasn't by contradiction

I want to see what he does with the square root of 2.

>>9064304
>meaning that he can construct new primes infinitely

This is wrong on so many levels. You cannot construct new primes infinitely. Even today we do not have prime generating functions that aren't shit.

Euclid's proof says that IF you assume there is a last prime, you can start generating infinitely many primes. But really, that part about generating infinitely many primes is not even necessary to conclude the argument. It is just a neat trick.

>>9064309
I didn't say he gave a method for constructing arbitrarily large primes, but only that it is always possible to find arbitrarily large prime

>>9061503
>the constructive proof demonstrates that for any finite set of primes you can find a prime number not in that set. At no point is the notion of infinity mentioned
How fucking dense are you?
If you say there's N primes I can prove there's at least N+1 primes.
If you say there's N+1 primes I can prove there's at least N+2 primes.
If you say there's N+x primes I can prove there's N+x+1 primes, for ARBITRARILY LARGE x. If you're not a fucking moron you'll understand that this is a proof there's infinitely many prime numbers.

>>9064431
Arbitrarily large doesn't mean infinite

>>9062305
How's that? I don't see where the last equality comes from

>>9053853
Why do point have to be enclosed by square brackets? would its properties change if we used parentheses instead?

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>Grothendieck
>The question you raise "how can such a formulation lead to computations" doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered.
>brainletberg
>pic related

>>9064459
no it's just notation

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>>9064184
>doesn't "subscribe to" set theory
It's not a statement of fact that everything is a set and nothing else, it's just a useful model to consider everything as a set.
Mathematics isn't "believed", it's just internally consistent.
It's like saying you "subscribe to" A being the first letter of the alphabet. We didn't measure the letters and conclude A was the first, we just decided that it was, and therefore, it is.

>>9064541
Can you explain what Groethendick meant here? I don't quite understand what 'formulations' and 'computations' specifically mean. I think I'm wrestling with same issue.

>>9064679
But that's how he defines a point, a pair of rationals separated by a comma, enclosed by square brackets, he doesn't show points enclosed by brackets are isomophic with point enclosed by parens so it's unjustified to say "it's just a notation"
and if his notion of point doesn't rely on the shape of delimiters then why even include it in the definition?

>>9064184
then go and try to prove anything more trivial than 1+1=2 in wildberger's framework
have in mind not even he can do it, look for example at his proof that multiplication is commutative, he uses circles and not sticks and he gives just an example of 3*4=4*3 or something similar, showing commutativity hold only for two naturals (assuming naturals defined as circles are isomorphic to naturals defined as sticks and are invariant under "just flipping them") and not for all of them, be it countably many or [math]10^{200}[/math]
I'd rather stick to conventional maths rather to wildberger's theory, which is both silly and useless

bbup

>>9064456
yes it does

>>9066615
how

>>9053895
those neighbourhoods are just open balls with the usual euclidean metric

Just study [math]{\operatorname{Def} _{\mathbb{A}_\mathbb{Q}^1}}\left( {\mathbb{Q}\left[ \varepsilon \right]/\left\langle {{\varepsilon ^2}} \right\rangle } \right)[/math]