/sci/ - Science & Math

>>12430186
Based

>>12430186
>Assume certain set of rules
>Get unintuitive results
Bah, this math is wrong. The assumptions are nonsense.
>Assume certain set of rules
>Get unintuitive results
That's fine. It is not necessarily the case that everything will be intuitive.

>>12430186
>objects only have length if that length would be rational, otherwise they have pseudo-length
lmao

>>12430652
This.
The cope is palpable.

Is intelligence an attribute of integer science man?
No, no and no!

>>12430652
>>12430654
How ironic.

>>12431123
There is no irony. Stop projecting and stop coping.

>>12431123
Let me define two lines.
The first is from (0,0) to (1,0), the second is from (1,0) to (1,1). Any reasonable person would say the length of the first is 1, and the length of the second is 1.
Now we define a third line connecting (0,0) to (1,1) to form a triangle.

Is it more reasonable to assume the third line has length or does not have length?

>>12431134
We've tried to make sense of irrational "lengths" for years. We haven't managed to come up with a coherent theory. As desirable it might be to say "all line segments have a length", maybe, after hundreds of years of attempts, it's time to admit the problems inherent to incommensurability.

>>12431157
>We haven't managed to come up with a coherent theory
Sure we have.

>>12431168
Present it.

>>12431151
>:(

>>12431178
whats wrong with R? :(

>>12432837
It's a fake arithmetic.

>>12434090
Why?

>>12434093
There is not even an algorithm for adding two real numbers together. How pathetic.

>>12434118
Let [math] A,B\in\mathbb{R} [/math] so we may write [math] A=\{a_n\}_{n\geq1}, B=\{b_n\}_{n\geq1}. [/math] Then [math] A+B=\{a_n+b_n\}_{n\geq1}. [/math] There's your algorithm.

>>12434118
Oh boy, you actually believed that retarded lie?

>>12431151
>>Is it more reasonable to assume the third line has length or does not have length?
does not have length

but to be fair, the first length is a mental constructs, just like all maths, which is why I support classical maths over finitism.

>>12434552
It is decisively not more reasonable to assume that the third line does not have a length.
>If I can't compute it it doesn't exist!
If you genuinely believe this, you are retarded.

>>12430186
>A line segment does not necessarily have a length.

>>12434269
There is no algorithm to compare whether two real numbers are equal.

>>12434093
Because it relies on undefined notions that are too vague to be meaningful. Also you can't compute with them.

ITT: Undergrads who just passed calc 2 and diff eq pretend to understand reals

>>12435111
Which ones are the undergrads?

>>12435094
>>12435104
>t. filtered by analysis, salvaged by wildberger's videos

>>12435123
I bet I know more analysis than you do.
Here's a puzzle for you: let f be a function [0,infinity] ->R.
Prove that if for all x, f(nx) converges to 0 as n->infinity then f(x)->0 as x->infinity.

>>12435094
Let [math] A [/math] and [math] B [/math] be as above. Then [math] A=B \iff \forall \epsilon>0,\, \exists N\in\mathbb{N}, a_n-b_m<\epsilon [/math] whenever [math] n>N,\, m>N. [/math]

>>12435149
That's not an algorithm, retard. It's a formula in first order logic.

>>12435134
what's R?

>>12435154

>>12435162
The real numbers.

>>12435184
There is no algorithm. Keep coping.

>>12435190
so they do exist

>>12435201
No.

>>12435194
I showed you >>12435149
You called it a formula.
Formula is synonymous with algorithm.
By your own admission, it is an algorithm.
You have provided nothing to suggest the contrary.

>>12435208
>Formula is synonymous with algorithm.
No it's not, retard. Theyre completely different things.
>source:thesaurus
Get the fuck out

>>12435134
That's the problem you come up with to show that you know more analysis than him? That's barely even something you'd find in early calculus; it's just the basic concept of a limit.
Let A=(1/2,1] and nA=(n/2,n]. Pass n to the subsequence 2^n and it follows from assumption that f(nA) \to 0.

>>12435218
Fine, define formula and define algorithm.
>inb4 define define

>>12435204
why do you use stuff that doesn't exist in your exercises then, are you a schizo??

>>12435235
Not even that guy but an algorithm is an iterative usage of some equation or mathematical transformation in a proscribed order.

A formula is in simple terms just something you plug a value into once and output another value.

>>12435243
Because we were talking about analysis. Im not the schizo because I admit they dont exist, youre the schizo who thinks theyre real.

>>12435293
Then using what I had written, >>12435149 just run through all the epsilon = 1/n, for integers n, and find the N that satisfies what is needed. Continue in this way, and if it holds for all integers n, then you are equal. That is an algorithm as you described it (as far as I can tell).

>>12435311
>and find the N that satisfies what is needed
How do you do that?

>>12435316
Run through all the integers until you find one.

>>12435321
How do I check whether or not I found one? How do I know if N is the right one?

>>12430186
This is like saying "rocks don't change over time, therefore evolution isn't true" in that:
1) It's a non-sequitur and a fallacious analogy and
2) It's completely wrong in the first place. Rocks DO change over time (erosion is one example), and all lines have a length.
3) It makes so little fucking sense as to be completely fucking useless. Are you telling me that if I have a line of length 5, and then segment into into lines A and B with |A|=sqrt(2) and |B|=5-sqrt(2), where did the length go? Did the lengths take a fucking vacation?

>>12435328
You run through all the n>N and m>N and check that for each n,m, you have a_n-b_m<epsilon. If such a natural number N does not exist, then A and B are not equal.

>>12435346
But what if for each n,m>N it satisfies the inequality. Then you never stop checking and so never figure out that N actually works.

>>12435235
"First-order formula" and "algorithm" are technical terms. Fuck off with your thesaurus.

Construction is not required for a proof of existence. Constructivists get the fuck off the board.
The entire weird constructivist desire to force math to be some pseudo empirical tradition is just sleazy atheist cope.
Reminder that Platonism is true and God exists. Stop coping and stop seething about it.

>>12435630
>Construction is not required for a proof of existence. Constructivists get the fuck off the board.
>The entire weird constructivist desire to force math to be some pseudo empirical tradition is just sleazy atheist cope.
>Reminder that Platonism is true and God exists.
Ok and how is any of that relevant towards existence of the real numbers? Being a platonist is consistent with doubting the existence of reals (because they're left undefined).

>>12435736
stop saying that reals are undefined. they're defined perfectly well, you just don't like the fact that they can't be "algorithmized" in general. if you understand that there's no largest natural number, because you can always add 1 to it, you understand infinity and sequences in particular. no point in playing dumb.

>>12435736
>they're left undefined
The reals are the smallest extension field of the rationals, that has the completeness property. This definition is provably precise enough to characterize them up to isomorphism.

>>12435571
>They are technical terms
That is not a definition, retard.
>>12435381
In general, that is true. But usually there are techniques that allow you to figure it out in a finite amount of time. Also, why restrict yourself to objects which generally only have a finite algorithm to compute? There is nothing (that we have found) in objects such as those I've described which causes any inconsistencies, so for what reason should we abolish them?

>>12435888
>This definition is provably precise enough to characterize them
Not him, but it isn't. You need to show that such a field exists (or that every field has a Cauchy-complete extension).

>>12434552
>>>Is it more reasonable to assume the third line has length or does not have length?
>does not have length
>but to be fair, the first length is a mental constructs, just like all maths, which is why I support classical maths over finitism.

I'd agree if the first and second lines don't have length, then the third doesn't have length. You might say, such lines don't exist in reality, or there is no way to construct such a triangle in reality, or whatever. That is an okay case to make.

But to say the first and second lines have length, but the third line doesn't, seems retarded to me. Though I'm glad we have anons memeing about this stuff than yet another 0.99... thread

>>12435340
>Did the lengths take a fucking vacation?
kek

>>12435340
How do you go about doing this segmentation?

>>12436023
Assuming you can measure out a line of length 2, then draw a circle of radius 2. It is known that you can construct a line 45 degrees to the original line, and then you take the intersection of that line with the circle. Drop a perpendicular from this point to the original line, and cut the line at this intersection.

>>12436023
Take two lines of length 1, so that they form a triangle (0,0)to(1,0) and (1,0)to(1,1).
Take a line of length 5 with one end at (0,0) and which passes through (1,1).
segment the line of length 5 at (1,1)

reminder to all inf*nityfag brainlets ITT
https://plato.stanford.edu/entries/wittgenstein-mathematics

>Since we invent mathematics in its entirety, we do not discover pre-existing mathematical objects or facts or that mathematical objects have certain properties

>If, first, we examine what we have invented, we see that we have invented formal calculi consisting of finite extensions and intensional rules. If, more importantly, we endeavour to determine why we believe that infinite mathematical extensions exist (e.g., why we believe that the actual infinite is intrinsic to mathematics), we find that we conflate mathematical intensions and mathematical extensions, erroneously thinking that there is “a dualism” of “the law and the infinite series obeying it” (PR §180). For instance, we think that because a real number “endlessly yields the places of a decimal fraction” (PR §186), it is “a totality” (WVC 81–82, note 1), when, in reality, “[a]n irrational number isn’t the extension of an infinite decimal fraction,… it’s a law” (PR §181) which “yields extensions” (PR §186). A law and a list are fundamentally different;

>Given that a mathematical extension is a symbol (‘sign’) or a finite concatenation of symbols extended in space, there is a categorical difference between mathematical intensions and (finite) mathematical extensions, from which it follows that “the mathematical infinite” resides only in recursive rules (i.e., intensions). An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms

>>12436058
How are you sure there is an intersection?

>>12436077
Can you restate that in your own words?
I'm not going to engage you in conversation if you can't demonstrate that you understand what you pasted

>>12436084
Essentially, I assume so. Why shouldn't I?

I agree that not all line segments have lengths defined but if you define the real-valued lengths of the legs of a right triangle, that unavoidably determines the length of the hypotenuse by Pythagoras' theorem. Does Wildberger think the proof of Pythagoras' theorem has an error in it?

>no meaning
This is the worst, stupidest, most ignorant criticism. All words have meaning. All properly formed sentence have meaning which exceeds the sum of the meaning of the words. For instance, consider the idea that saying something has no meaning has no meaning. It does have meaning, it means "devoid of anything meant." It's not a meaningless thing to say, it's just stupid and wrong.

>>12436117
Fuck off back to your RH thread tooker

HIDE SCHIZO POSTERS
IGNORE SCHIZOPOSTERS
DO NOT REPLY TO SCHIZOPOSTERS

>>12436106
>Let's just ASSUME that the intersection exists even though we can hardly make any sense of the numbers involved
What other flights of fancy do you wish to just ASSUME?

>>12436084
hey, how does this work in your world? there is an intersection, we just don't have the number to describe it, or is there actually a blank place in your geometry?

>>12436633
We can make perfect sense of the numbers involved. They can be thought of as the set almost homomorphisms of the integers, where we identify any two which are almost equal. It can be seen that this has a total order, has the least upper bound property, has the addition and multiplication which we would expect, and the rationals can be imbedded into this set, and this imbedding is dense.
Tell me specifically what is wrong with this?

>>12436744
>there's no finite algorithm which would decide whether two general cauchy sequences are equivalent, therefore real numbers are impossible to work with

>>12434269
>>12435149
>his algorithm isn't even guaranteed to terminate in finite time
Any real (as in practical, not as in """real""" numbers) application of your algorithm will be on *rational* approximations of your """real""" numbers.
>his algorithm assumes he already knows an infinite, non-repeating decimal expansion to his """real""" numbers.
For example, I can tell you right now that 1/3 = 0.(3). It is a completely finite expression that tells you precisely what the decimal expansion of 1/3 is. In fact if I want to multiply 0.(1) and 0.(3), I can do so precisely by just converting these numbers to fractions 1/3 * 1/9 = 1/27 = 0.(037) and get the corresponding unending decimal expansion.
Can you correspondingly give me the full decimal expansion of pi? What about e? So can you give me the full decimal expansion of pi * e? The answer is you can't, unless you give me a more complicated expression in terms of infinite sums, exponentials, etc. Therefore, when wildberger says that there is no general algorithm for adding or determining whether two real numbers are equal, he is of course referring to richardson's theorem, since most computable """real""" numbers will be expressed in a non-trivial fashion and it is undecidable to determine whether two real numbers are equal or what their sum is.

On a less facetious note, wildberger is really just another foot soldier of a war spanning millennia in western philosophy of Plato vs Aristotle. Your algorithms are of course correct in a platonic sense, since you can carry out the infinite sum or some other infinite amount of operations in theory, so of course in your philosophical framework, it is 100% okay to assume you have access to the decimal expansions. However, wildberger is of course firmly in Aristotle's camp, so this is nonsensical to him. As a result, when viewing his (or anyone's) claims, one must also be sure to view them from a correct philosophical lens to understand what is being said.

Shit, you guys still shitpost with this man?

>>12436796
Did I mention Cauchy sequences anywhere?

>>12435299
Well at least you know your analysis problems. Whether you know the solutions are still up in the air. Glad we could help with your homework though. Good luck on finals bud.

>>12436807
>will be on *rational* approximations of your """real""" numbers.
exactly. that's how we defined real numbers, basically as rational approximations of hypothetical lenghts which cannot be constructed from the unit length by multiplication and division, only approximated. obviously you get only rational approximation when you want anything exact, that's what real numbers are. where's the problem?

>>12436122
Why are we bullying based took? Are you as new as the day you were born? Just let him be.

>>12430186

What the fuck is Wildberger smoking?

If a segment is comeasurable with a unit length semgent, then it has a definable length, but if the segment is not comeasurable with a unit length segment, then it only has an approximate length?

I could fucking take ANY segment to be the unit length, so any segment could have a definable length. Why would some lengths be undefinable just because I pick some segment to be the unit length?

>>12435225
Idk if you're retarded or what's going on with you but your "proof" doesn't work at all. I'm honestly amazed you would type that up and think it solves anything.
>and it follows from assumption that f(nA) \to 0
Where did I make such an assumption? I never did.

>>12435914
>But usually there are techniques that allow you to figure it out in a finite amount of time
That's not accurate. There are techniques only in extremely specific, contrived cases. Usually there is no way to work out the answer.
>Also, why restrict yourself to objects which generally only have a finite algorithm to compute?
Because you want to know what you're talking about. You don't want some fairy tale numbers.
>There is nothing (that we have found) in objects such as those I've described which causes any inconsistencies, so for what reason should we abolish them
You don't know this.

>>12431157

Of course we can say "all line segments have a length" because we can pick any line segment to have unit length.

>>12436894
You literally did retard. You said for all x, so it's safe to assume a subinterval also has the same limit.

>>12436872
>that's how we defined real numbers
Who defined them? I've yet to see anyone provide a proper definition.

>>12436959
obviously I mean the definition via Cauchy sequences

>>12436939
I said that for all x, f(nx) tends to zero as n goes to infinity. That's vastly different from saying f(nA) tends to 0 for all intervals A.
Retard.

>>12436982
But this definition doesn't work because all it does is to push the ambiguity and imprecision to the concept of a sequence and an equivalence class.

>>12436985
For all x in nA, x is in R, so f(nx) goes to 0. Then f(nA) goes to zero. Can I spell it out any simpler for you? Hello? Earth to anon?

>>12437016
>Then f(nA) goes to zero
What do you even mean by this and why would it be the case?

>>12436998
what do you mean "doesn't work"? it gives you rational approximations up to arbitrary precision, is this not good enough for you? does your finite math provide something better?

>>12436872
If we take an Aristotelian point of view, all of you have access to are the rational approximations. You can get that approximation as close as you'd like to the true real number (or as he'd likely say, some unending decimal expansion that can get arbitrarily close to your desired value), but you can never say that the real number itself is actually a thing. Because doing so would require you to be infinitely precise and would require you to fully perform the infinite number of operations to construct the corresponding real number.
Now assuming you are a Platonist, you have no problem with this since a real number is a concept in abstract, so the fact that you can have an infinite sequence of closer and closer approximations implicitly implies that you can also have the true real number.
It's just the philosophical viewpoint that you take. Take a look here: https://en.wikipedia.org/wiki/Actual_infinity#Aristotle's_potential%E2%80%93actual_distinction.. Basically, your infinite sequence of rational numbers is a potential infinity (something that you came up with in your head). The actual infinity does not exist (at least according to Aristotle).

I personally lean towards the Aristotelian camp and welcome a new take on the Platonist modern mathematics, but I have no actual stake in the game with respect to wildberger himself. My only issue is that most people trying to argue with him either implicitly assume the real numbers or implicitly assume you can do an infinite number of operations or have an infinite number of things (e.g. infinite sets). Wildberger rejects both of these and since the you get an infinite set in ZFC with the axiom of infinity, you are basically arguing his choice of axioms. This is fine, but is likely never to reach a conclusion, since the axioms you choose are metamathematical (philosophical).

>>12437035
>what do you mean "doesn't work"?
I explained this literally in the next part of my sentence.
> it gives you rational approximations up to arbitrary precision
It doesn't give you shit.
>does your finite math provide something better?
Yes, because it actually make sense. And also I have nothing against "infinite" math per se, as long as the concepts are properly defined.

>>12437078
>I explained this literally in the next part of my sentence.
no, you've just said that you don't like sequences and equivalence classes for some reason
>It doesn't give you shit.
you know that's not true
>And also I have nothing against "infinite" math per se, as long as the concepts are properly defined.
now you're gonna pretend that you don't understand sequences?

>>12437395
Be so kind as to define what a sequence is for me.

>>12437022
For every x \in A, f(nx) \to 0. This is true from the assumption. Since the nA cover [0,\infty), the assertion holds.

>>12437022
>He claims to have done analysis, but doesn't even know what the image of a set is.

>>12437503
>the assertion holds
Why?
The question asks you to prove f(x) tends to 0 as x tends to infinity.

>>12437549
I know what an image of a set is, retard. I just don't understand what he means by the image going to 0. He's talking about some kind of convergence of sets, which he hasn't defined.

>>12436923
>That's not accurate. There are techniques only in extremely specific, contrived cases. Usually there is no way to work out the answer.
Yeah, fine.
>Because you want to know what you're talking about. You don't want some fairy tale numbers.
In what regard do I not know what I'm talking about. Tell me specifically what I do not know.
Also, suggest an alternative without any of the problems that you seem to have.
>You don't know this.
Actually, I do. Not a single inconsistency has been found, so the statement is true.

>>12436959
Refute my definition >>12436744 then, retard.

>>12437455
a wall full of whiteboards stretching far far in one direction so that you can't see the end. there's a first whiteboard, but there's no last one. for every whiteboard, there's also one which comes after it. certain amount of strokes. is scribbled on every whiteboard. that's a sequence of natural numbers.

>>12437455
A sequence is a map whose domain is some subset of the natural numbers.

>>12437598
>In what regard do I not know what I'm talking about. Tell me specifically what I do not know
You don't know what a "real number" is.
>Also, suggest an alternative without any of the problems that you seem to have.
Alternative to what? Lots of calculus can be done perfectly well with rational numbers.
>Actually, I do. Not a single inconsistency has been found, so the statement is true.
There have been a lot of inconsistencies found in set theory, which have been patched up by now. Sure, in the last few decades we haven't encountered any new inconsistencies but that's not really a reason to think they don't exist.

>>12437620
Explain to me what an "almost homomorphism" is.

>>12437634
What is a map?
>>12437627
Highly interesting and unusual :) I admit to never have encountered such a wall, but please do indulge my curiosity. How do you do do arithmetic with these whiteboards? And what are the rational numbers in terms of these walls?

>>12437554
Since f(nA) \to 0, then for any epsilon>0, there exists a natural number N such that ||f(nA)||<epsilon whenever n>N. It follows immediately that max_{x>N}(|f(x)|) < epsilon, since x>N implies that x is in nA for some n>N. Hence, |f(x)|<epsilon whenever x>N.
Fucking hell, you're dense.

>>12437696
>there exists a natural number N such that ||f(nA)||<epsilon whenever n>N.
Why would this be true and what the fuck do you mean by ||f(nA)||, retard?

>>12437648
A map [math] f:\mathbb{Z}\to\mathbb{Z} [/math] such that [math] \{f(m+n)-f(m)-f(n):n,m\in\mathbb{Z}\} [/math] is finite.

>>12437722
>A map f:ZZf:ZZ
What do you mean by this? What is a "map"?

>>12437661
A relation such that each element in the domain corresponds to exactly one element in the codomain.

>>12437661
>How do you do do arithmetic with these whiteboards?
you mean like addition of sequences? that's obvious

>>12437661
>And what are the rational numbers in terms of these walls?
depends on how *you* implement rational numbers in terms of strokes

>>12437731
See >>12437733
The definition is self-explanatory. Please graduate middle school if you don't understand.

>>12437715
||f(nA)|| is the supremum of the |f(nx)| for x \in A. That this is true follows directly from the assumption that f(nx) \to 0 as n \to \infty.

>>12437737
Well I'm asking you to explain what you mean, you can use whatever construction you find comfortable.
So the way I understand it, these walls are the real numbers?
Then pray tell, what would the real number sqrt(2) look like?

>>12437696
you misunderstood the question

>>12437733
Okay and what is a relation? Explain to me like I'm a postgraduate in mathematics (which I could be).

>>12437635
>You don't know what a "real number" is.
I've defined it right here >>12436744 and you still have not refuted it.
>Alternative to what?
Tell me your alternative to the foundations of math, given you have objections with real numbers despite not pointing out any contradictions or inconsistencies with the current framework. So explain your alternative framework.
>There have been a lot of inconsistencies found in set theory, which have been patched up by now.
So there aren't any.
>Sure, in the last few decades we haven't encountered any new inconsistencies but that's not really a reason to think they don't exist.
It certainly isn't a reason to think they do.

>>12437769
What part did I misunderstand. I literally proved the statement.

>>12437773
A relation R from a set A to a set B is a subset of the Cartesian product AxB. If (x,y)\in R, then we write xRy. We call A the domain and B the codomain.
>I could be a postgraduate in mathematics.
You so obviously aren't. You'd not be beyond a freshman.

>>12437786
>Tell me your alternative to the foundations of math, given you have objections with real numbers despite not pointing out any contradictions or inconsistencies with the current framework. So explain your alternative framework.
You sound like a religious zealot.
>You can't point to any contradictions or inconsistencies in our dogma!
People have repeatedly pointed out inconsistencies in the dogma.
>Well yeah but by now we've patched up these issues and these criticisms no longer work.
Ok sure but that's still no reason to believe in the dogma. You don't believe something just because it's internally consistent, it still has to make sense and correspond to something.
>All my friends and neighbors believe in the dogma. If it's good for them, it's good for me.
Sure but how do you know these entities you're talking about actually exist? You admit to not being able to see them, or hear them, or touch them, or knowing anyone who has. You admit to there being no way of interacting with them and you have even proved that it's impossible to interact with them for anyone. So why would you think these entities actually exist?
>That's what my elders taught me. They know best.
Can you at least explain to me what an "interdimensional elf" is? I've never seen anyone properly explain it.
>Sure! It's a quntik that's cuminiforous and ambitian.
Woah woah slow down there. You just introduced a bunch of new terms which I have no idea what they mean. Let's start from quntik. What is a quntik?
>Quntik is a quntik. The notion is left undefined.
Ok so if an interdimensional elf is defined in terms of objects which are undefined and which you don't know what they mean, surely you can't say that it's a meaningful or properly defined concept.
> I just did. An interdimensional elf is a quntik.
...
Alright have a good day sir.

>>12437800
Idiot.
>>12437748
>That this is true follows directly from the assumption that f(nx) \to 0 as n \to \infty.
How. Prove it.

>>12437863
You've yet to provide a more useful alternative to real numbers or show a single inconsistency within them.

>>12437875
You absolute fucking retard. f(nx)\to0 as n\to\infty is an equivalent statement to \forall\epsilon>0\exists N\in\mathbb{N} such that |f(nx)|<\epsilon whenever n>N. Have you not taken a single course in precalculus? How genuinely retarded can you be?

>>12437762
just work through the construction via cauchy sequences, but replace "sequence" with the wall every time you encounter the word. tell me when you run into something which you consider an issue.

>>12437889
You've yet to provide a more useful alternative to interdimensional elves or show a single inconsistency within them.

>>12437903
>You absolute fucking retard. f(nx)\to0 as n\to\infty is an equivalent statement to \forall\epsilon>0\exists N\in\mathbb{N} such that |f(nx)|<\epsilon whenever n>N
Yes that is obviously true, but this is not what I asked about. I asked you to show that
>then for any epsilon>0, there exists a natural number N such that ||f(nA)||<epsilon whenever n>N.

>>12437905
Ok but real numbers are not actually defined as "cauchy sequences" but rather "equivalence classes of cauchy sequences".
So don't you also have to take some kind of equivalence class of walls?
What would such a "class" look like?

>>12435134
f needs to be continuous for this to hold.

>>12437863
>People have repeatedly pointed out inconsistencies in the dogma.
Name one.
>Ok sure but that's still no reason to believe in the dogma. You don't believe something just because it's internally consistent, it still has to make sense and correspond to something.
And it seems to correspond with our reality as we see it. No one has shown that it doesn't thus far.
The rest is just your nonsensical ramblings which have no relevance. I'm willing to listen if you explain the problems with the definition of real numbers which I presented, and/or provide a mathematical framework contrary to the usual one. You have not done either of these things.

>>12437935
Correct. I forgot to include the assumption that f is continuous.
>>12437937
>Name one.
Russel's paradox, the paradox due to Forti, König's paradox, Richard's paradox and many others.
>And it seems to correspond with our reality as we see it.
How?
>No one has shown that it doesn't thus far.
So? Also no one has shown that interdimensional elves don't exist.
>I'm willing to listen if you explain the problems with the definition of real numbers which I presented
I already explained that the most obvious issue is that the notion of a "set" is left completely undefined, so the term "real number" is too vague to be meaningful.
>provide a mathematical framework contrary to the usual one
Take a look at Wildberger's youtube channel.

>>12435968
Yes no length exist at all. A length is just a projection of some humans on a rod

>>12437909
Still waiting

>>12437960
>Russel's paradox, the paradox due to Forti, König's paradox, Richard's paradox and many others.
None of those are contradictions in ZFC.
>the notion of a "set" is left completely undefined
The notion of set is perfectly defined by the axioms of ZFC. Stop trying to weasel your way out with pointless rethoric and prove there's a contradiction in them already if you're so sure they're wrong.

>>12438778
From wikipedia:
"In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right." This pushes the definition of a set onto the definition of a collection.
So to avoid some basic paradoxes, there are some restrictions on what sets we can construct. Let's look at just one axiom: the axiom of schema. "Essentially, it says that any definable subclass of a set is a set." What is a subclass? "subclass is a class contained in some other class in the same way that a subset is a set contained in some other set." What is a class? "a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...In work on Zermelo–Fraenkel set theory, the notion of class is informal" So it turns out that we restrict the definition of a set to something that can be "unambiguously" defined (no, there is no info on wikipedia about what unambiguously means here), in an informal manner.
This is not a new objection by wildberger by the way: https://en.m.wikipedia.org/wiki/Set_theory#Objections_to_set_theory_as_a_foundation_for_mathematics (look at poincare's objections specifically).
It is ridiculous to just dismiss wildberger because he dislikes our current definition of sets. They are vaguely defined (probably because in reality they are some meta-mathematical object), so alternatives should definitely be welcomed and carefully examined as well.

>>12430186
PRAISE SAINT WILD BURGER!

Say it with me Brothers and Sisters of the ONE TRUE FINITE FAITH....

PRAISE THE SAINT WILD BURGER!

>>12436889
Verily I sayeth unto thee, SODOMITE! YOU WILL BURN IN THE finitely DEEP PITS OF HELL!

Amen.

And PRAISE THE SAINT BURGER!

>>12439067
>From wikipedia
For fuck's sake. Of course the hand-wavy wishy-washy "definition" for people who don't know formal logic is shitty and inconsistent. That's the whole reason why ZFC was developed in the first place. Here's your rigorous definition: A set is the object described by the axioms of ZFC. Do you have a problem with it? Feel free to come back with a proof of the inconsistency of ZFC.
>It is ridiculous to just dismiss wildberger because he dislikes our current definition of sets
I dismiss Wilderberg because he's dishonest. He acts like there's some glaring error in ZFC that the mathematical community is too dumb to see when, in reality, the only criticism he has is that he doesn't like it. Thats fine but has nothing to do with mathematics. Wilderberg knows this but adamantly refuses to admit it and continues to claim his way is the right way.
>This is not a new objection by wildberger by the way
I know that. In fact, every single mathematician and a bunch of philosophers know that. One of them knew even more than that. Stop acting like you have some kind of secret knowledge; you don't. The choice of axioms is arbitrary, we just use the one we like the most. You are free to use whichever ones you want. In fact, there are tons of finitists who prefer to use something else because they find the results of ZFC to be unintuitive (Banach-Tarski), ugly (Weierstrass functions) or they just happen to like some other axioms better. These are honest people with a passion for knowledge and an eye for beauty. Wilderberg is not one of these people. He's a weasel trying to get some pennies out of impressionable undergrads.

>>12439349
his maths is better than the usual sin, cos maths from HS

>>12439349
>Feel free to come back with a proof of the inconsistency of ZFC.
I'm pretty sure he just dislikes impredicative definitions (see below), which I think is completely reasonable.

>He acts like there's some glaring error in ZFC that the mathematical community is too dumb to see when, in reality, the only criticism he has is that he doesn't like it.
I agree with you here. He is being hyperbolic when he states zfc is "inconsistent" in the traditional sense (and he never shows this from within zfc's axioms, how it is inconsistent; of course if you don't assume the axiom of infinity though, real numbers aren't going to make sense). My personal take on this is that wildberger seems to only want predicative definitions, so the impredicative definitions of zfc are probably what he means by "inconsistent" (https://en.m.wikipedia.org/wiki/Impredicativity).).

>Stop acting like you have some kind of secret knowledge; you don't.
??? I didn't even insinuate that I was talking to you in a condescending manner. I just said that these objects are vaguely defined (in a self-referencing manner), and linked some philosophers. None of that was a statement about you; stop taking this so personally lol.
>He's a weasel trying to get some pennies out of impressionable undergrads.
I mean, he can make money however wants; having a patreon is not an inherently bad thing. I personally will not pay a penny to him, but if a patreon spurs him to make more content, then that's up to him.
My personal take on wildberger is to watch his videos for the new approach to old things (e.g. rational trigonometry). His math is correct, but I usually just tune him out when he rages about zfc.

>>12437722
Pray tell... How many of these mappings are there? Can you show me how to calculate [math]\pi + e + \sqrt 2[/math] with these almost homomorphisms?

>>12437931
it's one wall. the thing is that it's equivalent to other walls
>how do you algorithmically decide if a general pair of walls is equivalent?
you don't. surprisingly, this is not an issue at all.

>>12439547
>I didn't even insinuate that I was talking to you in a condescending manner.
Then how do you interpret bringing up objections about ZFC that were brought up and resolved almost a hundred years ago as if they were something new? A lot of mathematicians and philosophers have problems with impredicativity; that doesn't make axiomatic systems with impredicative statements vaguely defined. Predicativity, on the other hand, is vaguely defined.
>he can make money however wants
Do you think he would have the same following if he admitted that he just likes finitist predictive axioms but ZFC is just as (in)valid instead of hinting at "errors" and "inconsistencies"?
>watch his videos for the new approach to old things
I did. His maths are shit. He claims some the way ZFC answers some questions is "problematic" but when it's his turn to deal with the same questions, he just says they don't actually have an answer. Pic in the OP is a prime example.
>What's the length of the segment going from (0,0) to (1,1) in your new, better, more precise and consistent maths, Professor Wilderberg?
>Well, it doesn't have a length! Something, something, tax status, something, something, metaphor, something, something, all-seeing leprechaun.
He doesn't prove that some lines don't have lengths. He doesn't prove that assuming every line has a length leads to contradictions. Instead, he reverts to shitty rethoric and ill-conceived metaphors. I can do the same thing he does. Check it out:
>in ZFC we will have a precise axiomatic foundation of math. We will not use problematic concepts such as "string of lines". We don't assume the existence of whiteboards or markers to define the natural numbers but actually build them from a completely logical point of view. We also stay away from any kind of finitism because that implies the existence of a largest number, which is nonsense.
He's Alex Jones with a PhD.

>>12440864
You are like, so an mad try hard.

>>12430186
Why should we assume Euclidean geometry is "right" in any sense of the word.

>>12440864
>Then how do you interpret bringing up objections about ZFC that were brought up and resolved almost a hundred years ago as if they were something new?
I did not bring up as though they were new, retard. Read what I actually said. I repeatedly said that these inconsistencies have been brought to light and by now the theory was patched up so that they no longer work. You asked me to provide examples of these inconsistencies and I gave them to you. Why do you have to pretend I said something I didn't? Is arguing against what I actually said so hard?
Holy shit.

>>12440864
>objections about ZFC that were brought up and resolved
You can't resolve philosophical objections lol. The majority opinion may be that impredicativism is okay, but if someone wants to reject it, that's a meta-mathemathematical choice about how they want to do math and you have no right to stop them from doing it, regardless of what you think about the philosophy itself.
>that doesn't make axiomatic systems with impredicative statements vaguely defined
Impredicativity literally means that a definition is self-referencing. If asked you what "blue" was and you said "not red" and if I asked you what "red" was and you said "not blue," I don't think it would be very far-fetched of me to say your definition of blue is vague. Here is another example: https://mathoverflow.net/a/26233.. Striving for predicativism is not a bad thing at all.
>Predicativity, on the other hand, is vaguely defined
You can go to the wiki page I linked you and take the logical negation of the definition of impredicativity to get the definition of predicativism. Just because you don't like it doesn't mean it is vaguely defined. Now whether it is possible to define mathematics completely predicatively is a whole another question, but predicativism itself is a perfectly valid philosophical stance.
>He doesn't prove that some lines don't
have lengths.
What is the length of the diagonal of a unit square? It clearly is a line segment (two points on the rational plane define its endpoints), and we know its length, if it does exist, is a number x such that x^2 = 2. The Greeks already proved that this number is not rational, so if you don't accept real numbers, this line segment does not actually have a length. This is by the way his motivation for using quadrance over lengths.
(cont.)

>>12440864
>He claims some the way ZFC answers some questions is "problematic" but when it's his turn to deal with the same questions, he just says they don't actually have an answer.
Once again, and this is my personal take, his problem is either the use of infinite sets, infinite operations (e.g. infinite sums), or impredicative definitions.
So when he says something is "problematic," he just means that it has some construct/assumption that he is not willing to accept. So then you ask, what is his answer for real numbers (for example)? His answer (would probably) be that if a real numbers are defined "poorly" (according to him) (e.g. its constructions require infinite sets), you don't need an answer. It's just that real numbers don't exist. You don't need to make something up to cope with that fact. His system doesn't have real numbers and as long as he is consistent with the conclusions that he draws, his reasoning is perfectly fine. Now would his system be useful is a whole another question, but there is nothing wrong with considering his position intellectually.

>Do you think he would have the same following...
Maybe when he was new, I would agree with you. But by now, even everyone on this board probably knows what his philosophical leanings are. The reason (likely) he still has a following a decade later is because he is a professional mathematician doing math without real numbers and people are interested in this new point of view.

>as if they were something new?
I literally said his objections were not something new.

>He's Alex Jones with a PhD.
??? Is it because they both make hyperbolic claims and mention their products in videos? The difference is wildberger isn't marketing his book as a cure for anything, he doesn't run dedicated ad breaks for his products, and his claims, while hyperbolic, are mathematical (as opposed to political), so they can be argued with logic as opposed to hearsay.

>>12434118

Of course there is

Take line segment 1, extend it, then use a straightedge and compass to move line segment 2 onto the extended line segment 1, joining at one of line segment 1’s end points.

>>12434269
i guess this is why cs filters mathlets

>>12435094
>cstard who thinks computability is the final redpill

>>12435630
But we can construct the real numbers you dopey faggot

>>12436807
Wow, you took intro to philosophy. This has nothing to do with Plato and Aristotle.

>>12435134

There is a counterexample. Let [math] m [/math] be restricted to [math] \mathbb{N} [/math] (this excludes 0).

Define [eqn] f(x) := \left\{\begin{matrix}
\frac{e^{m}}{x}\\
0
\end{matrix}\right. \begin{matrix}
\text{ when } x \text{ is a natural number multiple of }e^{m}\\
\text{ when } x \text{ is not a natural number multiple of any power of } e
\end{matrix} [/eqn]

This function is well-defined because e is transcendental.

>>12443999
Checked.
Actually, his problem is he wants everything to be computable. So either he (a) is too retarded to know that’s a thing or (b) so delusional he thinks he invented the concept.

>>12444954
Very nice.

>>12444945
I mentioned it as an aside, since you can argue that Wildberger's rejection of the actual infinite is very Aristotelian. There's no need to be butthurt about it lmao.

>>12444982
I don't think so. If you look here: https://www.youtube.com/watch?v=lcIbCZR0HbU, he basically claims pi is a metanumber. Of course pi is computable (under the real number system), but he wouldn't even consider that a "number" because to compute it would require an infinite number of operations. So I think he is okay with as many finite approximations of pi as you want up to as many decimal digits as you want, but as soon as you consider the entire decimal expansion as a whole, that's where his problem is. You could make the same argument about sqrt(2), but he is okay with that since you can construct an extension field from the rationals with sqrt(2) in a purely algebraic manner (no infinities). So I think his outlook is more nuanced than just computability.
Regardless, his views are interesting and certainly unorthodox. Ultimately, my take is to tune him out when he starts going on about inadequacies in the real number system and focus on the new methods he introduces. Sure, he may be a "flawed" individual (he makes some overblown claims), but I don't see why everyone has to dismiss everything he has done because of that. Especially when the actual math he introduces is rigorous.
Also he's a professional mathematician with a PhD from Yale and he taught at Stanford. I'm sure he knows what computable numbers are lol and he never claims to have invented the notion.

>>12434093 #
There is not even an algorithm for adding two real numbers together. How pathetic.

>Elaborate.

>>12435134

If f is continuous, then this can be proven pretty easily. Prove the contrapositive.

Because the function does not converge to 0 as x approaches infinity, you can find some [math] \epsilon_{0} > 0 [/math] such that, for all [math] \delta > 0 [/math], there exist some [math] x > \delta [/math] where [math] \left | f(x) \right | \geq \epsilon_{0} [/math].

Because the function is continuous, each [math] x_{0} [/math] such that [math] \left | f(x_{0}) \right | \geq \epsilon_{0} [/math] has a neighborhood around it such that all [math] x [/math] in the neighborhood satisfies [math] \left | f(x) \right | \geq \epsilon_{0} [/math].

For [math] a_{i} > 0 [/math], we can take a closed interval [math] [a_{i}, b_{i}] [/math], so when we find [math] n_{0} [/math] such that [math] n \geq n_{0}[/math] implies [math] \frac{b_{i}}{a_{i}} > \frac{n+1}{n} [/math], we have it that [eqn] \bigcup_{k = n_{0}}^{\infty} [a_{i}k, b_{i}k] \supseteq [an_{0}, \infty) [/eqn]
Taking [math] max[an_{0}, ai] = \delta [/math], we find at least one neighborhood (as mentioned before) covered by [math] \bigcup_{k = n_{0}}^{\infty} [a_{i}k, b_{i}k] [/math]. Take a closed interval subset of the neighborhood such that it has diameter [math] \frac{1}{i} [/math], it is completely covered by [math] [a_{i}k, b_{i}k] [/math], scale it by [math] \frac{1}{k} [/math] and define it to be [math] [a_{i+1},b_{i+1}] [/math]. Note: the ai term is used to make sure that the k scale associated with the closed interval subset increases without bound.

Repeat the aforementioned process as an infinite procedure to create a nested sequence of closed intervals, with diameter approaching 0. The intersection of these intervals contains exactly one real number [math] r [/math]. Therefore, it is not true that [eqn] \lim_{n\rightarrow \infty} f(nr) = 0 [/eqn]
This completes the proof, take the contrapositive to get the corollary you want.

>>12447023

*such that it has at most diameter [math] \frac{1}{i} [/math]

>>12436998
Equivalence classes are not ambiguous at all.

If the difference of two cauchy sequences has limit 0, then they belong to the same equivalence class. If the difference is not 0, then they belong to different equivalence classes.

>>12444945
>>12445149
Wildberger has talked about his Aristotelian influences. Ed Nelson (notorious ultrafinitist) talks about Platonic vs Aristotelian numbers in one of his short papers on predicative arithmetic. Interesting stuff.