/sci/ - Science & Math

You do know this is highschool level maths dont you? infinite sums

>>9943094
the idea is that their is actually an infinite number of points in time that have to pass

>>9943099
yeah. and you can add em up.

>>9943094
>>9943111
you realized you got memed into oblivion by your 'education'?

>it's an infinity isn't real thread
It's all so tiresome

>>9943113
holy shit theres a formula that objectively works you brainlet. a/(1-r) as long as r= -1<r<1. finish school you stupid hick

>hurr infinity dont real
>hurr what is calculus

>>9943152
>calculus dont real either
>nooton was a virgin i dont trust him

>>9943113h
hurr durr calculus was a conspirricy. fuck off deepak

>>9943155
he was actually a virgin tho surely you don't trust him right anon?

>>9943099
>an infinite number of points in time can't happen because that must mean it takes infinite time, according to my naive intuition r-r-right g-g-guys?
kys

This paradox is a great way of finding people who don't actually understand calculus.

Infinite sums do not resolve this paradox. When you have an infinite sum of positive numbers "equalling" a number, all that means is that you can make your sum as close as you like to that number by having sufficiently many terms in your sum - but no matter how many terms in your sum you have, your sum won't actually equal that number. Just like how Zeno gets arbitrarily close to the tortoise but never quite reaches it. Writing down an infinite sum is just rewriting the paradox in symbols.

The resolution lies in physics, not mathematics. Think quantum.

>>9943214
>but no matter how many terms in your sum you have, your sum won't actually equal that number.
Wrong, if you have infinite terms it will equal that number, that's the entire point.

>Just like how Zeno gets arbitrarily close to the tortoise but never quite reaches it.
He reaches it in finite time, brainlet.

Nice Dunning-Kruger.

If infinity isn't real, then why do these threads not end?

>>9943206
There's no evidence to suggest that Newton was a virgin though, there's just no evidence to suggest that he wasn't.

>>9943214
Arbitrarily close to the tortoise?

>>9943304
>if you have infinite terms
Lol okay.
Learn how to think rigorously, brainlet. Maybe start by reading the definition of series. Then come back.

>>9943352
Yeah I read it and it says you're a brainlet talking out of your ass with no argument.

>implying completeness

>>9943214 holy shit its not calculus you dumb shit. its simple convergent series summation. this is highschool stuff. the sum of the shrinking segments is about 1.7 something. its fucking EASY. quantum my ass you pretentious shitskin

>>9943214
>quantum
Ever seen Parseval's Theorem used in quantum mechanics, brainlet?

>>9943214
>you can make your sum as close as you like to that number by having sufficiently many terms in your sum
Yes.
>but no matter how many terms in your sum you have, your sum won't actually equal that number
No, that's the entire point of limits, you cannot identify any specific number of Zeno distance intervals that go beyond what is covered by some specific number of Zeno time intervals. If you don't like infinities because you fell for the Wildberger meme that's still fine because the definition of the limit establishes there is no finite number of these distances you could ever come up with no matter how great which can't still be covered by a finite number of time intervals. There is no need to ever even appeal to infinity, it's enough to show the limit exists and your distances will always be covered no matter how many distances you invoke.
>Think quantum.
Please never post a sentence this pseud ever again.

this is a bit pedantic, but a problem with this paradox is that the universe is not infinitely divisable. There is a smallest distance with meaning (Planck length) and a smallest time with meaning (Planck time)

>>9943485
>but a problem with this paradox is that the universe is not infinitely divisible
it's really as simple as this

>>9943485
>planck length
>epistemological meme
try using your head

>>9943403
The argument is the definition of series. You're not arguing with me, you're arguing with a definition. Sorry you can't understand something to simple, brainlet.

>>9943450
Series is defined in terms of a limit. It's taught in calculus courses. Simple stuff.

>>9943468

Yep, this is what I explained. No, I don't dislike infinity, I'm literally making an argument involving limits and merely explaining the basic definition of series. I'm saying the fact that the limit exists does not resolve the paradox.
Also, I said "think quantum" because I didn't want to spoonfeed any more. I was pointing people in the right direction. Don't take yourself so seriously.

this paradox truly does reveal the overconfident brainlet reddit sam harris types as the frauds they are. go back to reading bill bryson and daniel dennett

>>9943515
let me put it simply.at the limit, achilles reaches the turtle.in terms of physics; the summation of planck lengths over planck time lets you determine when and where achilles reaches the turtle, and amazingly; both summations reach the same answer. 1.7 something

>>9943515
>I'm saying the fact that the limit exists does not resolve the paradox.
It does resolve it though. You cannot come up with a number of distance intervals that won't be covered by some other number of time intervals.
>quantum flapdoodle
>the right direction
No.

>>9943515
>The argument is the definition of series.
How does it lead to what you conclude? It doesn't, which is why instead of making an argument you are just claiming it exists.

>>9943528
>>9943529
>>9943533
embarassing

>>9943561
>The "think quantum" anon unironically believes he's in a position to call other people embarrassing

>>9943565
im not that anon you utter brainlet

>>9943071

Given the law of nature, physics and quantum mechanics, there would be a point in which the soldier would reach a point in which it would not be possible to move closer to the turtle and forever would be "just a lil far away" from the turtle. "just a lil far away"is the planck length.

>>9943573
>plack length
you can't be serious

>>9943071
>math doesn't work because I haven't learned what a geometric series is yet
not gonna make it

>>9943577
its an actual measurement you uneducated shitskin

>>9943607
Planck length isn't a limit on how small a distance can be, that's a pop sci misconception.

so did everyone on /sci/ just listen to a feynman audiobook and now think theyre a scientist? the lack of mental horsepower on display in this thread is hilarious

>>9943528

Yep, it's indisputable that Achilles reaches the tortoise in reality. In addition, we can use the limit to find out when he should reach the tortoise and it's the correct physical prediction. But the limit still does not resolve the paradox. The paradox is the incorrect conclusion of Achilles never reaching the tortoise and what incorrect reasoning leads us to that conclusion. Mathematically, the limit shows us that he'll get arbitrarily close to the tortoise, not that he'll reach it. So this isn't a resolution, it's a restatement of the paradox. However, your reasoning involves the planck length, so you don't have a series - you have a very long finite sum. This would be correct reasoning, assuming the premise that space is discrete is correct, which is disputable among physicists.

>>9943529

Correct. Meaning he gets arbitrarily close - not that he actually reaches the tortoise. Also, I made no pseudoscientific statements, don't sit on a high horse.

>>9943533

I'm explaining how what we casually call an "infinite sum" is the limit of finite sums. That's the definition of a series and it's in opposition to what you're saying. You have to realise what you're exactly saying when you say that when you have infinitely many terms your sum equals some number.


A better version of the paradox which is more precise in the problem it presents is the one which allows no motion. Zeno wants to travel a metre but first he has to travel half a metre and before that a quarter, and so on. Where's your series now? Zeno can't travel at all to begin with. What's the next number in a dense set? There isn't one. This is a physical problem, not a mathematical one.

>>9943653
>Meaning he gets arbitrarily close
Close to which specific number? Any real number you pick for the count of distance intervals will be fully covered, not almost covered.

>>9943653
>I'm explaining how what we casually call an "infinite sum" is the limit of finite sums.
Yes, and how does that imply that Achilles doesn't reach the tortoise? The finite sums get arbitrarily close to the limit. The infinite sum on the other hand IS the limit. As far as I can tell you are confusing the two, but it's hard to tell since you refuse to clearly state your argument.

>Zeno wants to travel a metre but first he has to travel half a metre and before that a quarter, and so on. Where's your series now? Zeno can't travel at all to begin with.
This is even more fallaciously vague than your first "argument." Motion is simply a change of position over time. Therefore motion only occurs over nonzero intervals. Regressing to an instantaneous point tells us nothing about movement, period.

>i've moved epsilon along a trajectory

>>9943665

Yes, which is what I mean by arbitrarily close (to the limit). There is still always distance to go though.

>>9943672

But the limit is *defined* as that number which the finite sums get arbitrarily close to. You only have finite sums, you look at what they tend to, and then whatever number that is you call "the limit" or "the infinite sum". When you say the limit equals a number, you can think of that as meaning infinitely many terms added together equals that number, but by definition, what you've really said is that the finite sums tend to that number.
So for Achilles, think of those finite sums and his movements side by side. The first partial sum is his first move, the second partial sum is his first and second move, etc. The finite sums (his movements) tend to a number (a position) but never reach it. We just *say* that number (position) is the "final" value (position).

As for the version of the paradox I stated, there isn't a problem with regressing to a single point, this is what derivatives are for. This involves another limit: we see what the rate of change tends to as the time intervals get smaller. But the time interval is never zero, as you correctly point out (just as in a series the number of summands is never "infinite").

>>9943725
There isn't still distance to go.
You must pick an actual real number, and whichever number you pick is a number that's covered, not almost covered.
You can try to argue the number you're picking is "infinity," but then I get to use "infinity" too and everything is still completely covered, not almost covered.

>>9943733

>>9943734
Not an argument little buddy.

>>9943725
>So for Achilles, think of those finite sums and his movements side by side. The first partial sum is his first move, the second partial sum is his first and second move, etc. The finite sums (his movements) tend to a number (a position) but never reach it. We just *say* that number (position) is the "final" value (position).
So you are indeed confusing the finite sums for the infinite sum. The infinite sum is the total of his movements to reach the tortoise. Achilles doesn't stop moving until he reaches the tortoise, so saying some finite sum doesn't reach the limit is irrelevant.

>But the time interval is never zero, as you correctly point out (just as in a series the number of summands is never "infinite").
So what is the problem? How is motion impossible? Any finite distance inside the meter can be traversed in finite time, so it doesn't matter how you split the meter up, motion is always possible.

>>9943733

But any number you choose that gets covered leaves a number uncovered. Sure, that number will also get covered, but when that number gets covered there will still be another number left uncovered. Just as you can say "any number gets covered", I can say "there will always be an uncovered number". Every time you choose one, I'll choose one. In fact, I can choose the limit. That never gets covered with finitely many summands. There's the idea of the limit being covered "at infinity" but that's not rigorous and I've addressed this.

>>9943752

But this ignores the whole point of the paradox. He doesn't reach the tortoise according to the reasoning presented. He keeps moving, yes, but he doesn't reach the tortoise. His movements only amount to finite sums.
You say finite distance can be traversed, but the point of the paradox I stated was about how such traversal seems impossible. It just moves the endless division at the end in OP's paradox to endless division at the start.

>>9943774
>leaves a number uncovered.
Prove it (you can't, which is incidentally why the epsilon delta definition works as a more rigorous alternative to dealing with infinitesimals).

>>9943786

Choose a partial sum and I'll give you a number between that number and the limit.

you're not picking random intervals to move along a literal arc. traversing infinitesimal distance takes an infinitesimal time, which resolves the paradox.

>>9943794
it doesnt, try again

>>9943774
>But this ignores the whole point of the paradox. He doesn't reach the tortoise according to the reasoning presented.
He does, since it takes a finite amount of time to cross all the intervals consecutively.

>He keeps moving, yes, but he doesn't reach the tortoise.
He reaches the tortoise in a finite amount of time. Today that he doesn't reach the tortoise is to say that he stops moving at some time before that. You are not making an argument.

>You say finite distance can be traversed, but the point of the paradox I stated was about how such traversal seems impossible. It just moves the endless division at the end in OP's paradox to endless division at the start.
Again, I'm asking you how it shows traversal to be impossible. You're not making any arguments.

>>9943071

ITT.

Unimaginative shallow thinkers, literally fucking cavemen, think they so smart, and "mah limits!" However maths Chads, who are capable of deep thought, understand that Zeno's paradox remains today unsolvable and simply demonstrates fundamental flaws in the very most basic principles of mathematics and so called "logical" thought.

TLDR = Maths is still in the dark ages.

>>9943793
No, you have to choose a real number first or else you're not even asking a question.
The entire point of the epsilon delta definition is that there's never any need to appeal to infifinitely small non-real quantities. You can instead just prove that for every real nonzero number epsilon that you require the distance between f(x) and the limit to be less than there exists a real nonzero number delta such that for all real number inputs x if delta is greater than the distance between x and the limit point c then epsilon is greater than the distance between f(x) and the limit.
This means there is not one single real number you can ever point to as not covered. The limit is a symbolic representation of the fact all real numbers greater than zero are covered. Your only possible appeal is to use infinitely small as your challenge which is not part of the set of real numbers. And if you're OK with using infinitely small as the challenge then it's OK to use infinitely small as the answer too i.e. If you resort to saying there's an infinite number of distance intervals than the answerer can resort to saying there's an infinite decrease in time required and the distance is covered completely, not partially.

>>9943814
You are so dumb

>>9943798
>you've only come infinitely close to solving it but haven't actually

>>9943800

>it takes a finite amount of time to cross all the intervals consecutively
We can apply the same paradoxical reasoning to time as well as space though. Why should time flow?

>To say that he doesn't reach the tortoise is to say that he stops moving at some time before that
Same question here. You're thinking of an infinite number of steps before a certain time so you're endlessly dividing time. This is another reason why my version of the paradox is better I think. People don't realise the same paradox is happening with time alongside space, my paradox only looks at space.

>I'm asking you how it shows traversal to be impossible
Well it shows traversal to be impossible because there's no clear next position to be. It's like asking for the next real number after 0.

I'm addressing all of this mathematically and making no mention of how at a small enough scale you have quantum uncertainty and all of that because I'm making the argument that the resolution isn't mathematical, it's physical.

>>9943814

Yes, I know the definition of limit, I'm using it as my argument. But I really don't get what you're saying with "This means there is not one single real number you can ever point to as not covered. The limit is a symbolic representation of the fact all real numbers greater than zero are covered". If you graph 1+1/2+1/4+1/8+...+1/2^n against the natural numbers, you'll see a curve reaching 2. I'm saying if you choose an n, I'll choose a point above that graph and below 2.

>>9943152
>it's another finitist trash denies basic properties of analysis episode

>>9943863
>We can apply the same paradoxical reasoning to time as well as space though. Why should time flow?
Why not? You still have not presented an argument for why either is paradoxical.

>Same question here. You're thinking of an infinite number of steps before a certain time so you're endlessly dividing time.
So what?

>This is another reason why my version of the paradox is better I think.
Your paradox is not a paradox, and you can't even explain why you think it is.

>>9943071
What about by inventing a time machine to have my hand against the other guy's ancestors until he becomes weak enough to be guaranteed to fail when we meet in the present? What if, by inventing a time machine and also having an appealing rhetorical proposition regarding justice and equity, the people in the future who use the time machine will use that first bit of math to channel the victory of that proposition through the inventor, not his enemy.
>when they meet
>in the present

>>9943879
he cute

The paradox doesn't take time into account correctly
The paradox is always explained as "so first he goes to where the turtle was, right? but the turtle moved in that time. so then he goes to the turtle's new place. but the turtle moved again!" and so on. This gives the illusion that every instance of Achilles catching up to the turtle's previous position are alike.
Assuming both Achilles and the tortoise move at a constant speed, the time required to pass these points on the line will decrease exponentially as the distance between them also exponentially decreases. These two decreases perfectly counteract all the way up until the points are infinitely dense and crosses infinitely frequently, after which Achilles will have overtaken the tortoise and the distance and frequency will continue to increase to infinity.
Achilles is performing a supertask

>>9943953

Have you ever considered fixing lawn mowers for a living?

>>9943970
I tried but I just mowed my index finger off instead

I don't understand how this is still considered a paradox if you avoid imagining Achilles pausing every time he reaches a prior position of the tortoise, and/or avoid describing every overlap Achilles makes in a way that takes a set amount of time to communicate

>>9943993
you just ignored the metaphysic that ground almost every conception of reality

>>9944014
Would you mind pointing out how I did that?

What's a handful of uranium atoms plus a few slow moving neutrons?

i cant understand how this is a paradox at all

I can't see how this is a paradox at all

>>9944046
The idea is that distance can be divided infinitely, so to travel any distance one must complete an infinite amount of actions, which seems impossible

>>9944049
why is it impossible? it doesn't seem to contradict anything

>>9943071
Suppose this made moving such a distance impossible. This would make any movement impossible, as such logic could be applied to every step of the path, rather than exclusively the path as a whole. But the problem implicitly assumes you CAN move half the distance, and therefore discrete movement forwards is possible.

>>9944122
>But the problem implicitly assumes you CAN move half the distance, and therefore discrete movement forwards is possible.
Dude
Proof by contradiction
I don't even believe that this is a real paradox but come on man

>>9944127
Look, I'm sure I'm mangling my explanation, but Achilles moves from the point where he starts to the halfway point in the first time period. That's part of the 'paradox'. But suppose the turtle was placed half between his original starting point and Achilles' own, instead of where he was originally, and the finish line was moved to where the turtle originally was.
If Achilles doesn't get into the new finish in the first period (and at the same time as the turtle if the logic is to be believed), he wasn't running as fast as he was in the original, which doesn't make sense, as the turtle's position shouldn't change his speed.

The idea of limits just formalizes calculus concepts in a rigorous mathematical framework. The paradox is not resolved by this in any sense or form, it actually shows the weird nature of the continuum. It's still a real possibility that the nature of space is discrete. We just need more energy to go smaller.

>>9944162
How is it not resolved by the fact that we can calculate exactly how much time it takes Achilles to catch the tortoise?

>>9944202
Because that's not the paradox. Evidently a shit happens in finite time and we can calculate it, but it shows the problem with doing something an infinite numbers of times. Yes, with limits you don't need to actually compute an infinite number of times to find it, but the problem still stands because what we know is that, no matter how small, there is no "smallest" element in terms of magnitude. This is a problem with infinities and the continuum. What this suggests is that the universe isn't really continuous, though it could run in weirder shit we dont know, but our idealizations are never going to be completely accurate. Hell, you can see that in a modern setting, mathematicians are not talking about motion, but given a vector field, they talk about integral curves, which are continuous functions a priori, and you don't see it's motion, but it's form. You can see this in many other exmaples where motion isn't really formalized in terms of our ideas of something "tracing" a curve, but we construct the whole curve witht the languange of functions.

>>9943325
Someone once told me that he was /r9k/ levels of bad with women, although I've also heard he was personally Knighted by Queen Anne when she visited his college so maybe he was able to score some of that Royal strange at the after party.

>>9944212
>Because that's not the paradox.
It is though, the paradox claims that Achilles never reaches the tortoise. This is a claim about time. This is false since going through all the steps described by the "paradox" only takes a finite amount of time.

>Evidently a shit happens in finite time and we can calculate it, but it shows the problem with doing something an infinite numbers of times.
What problem? It doesn't show anything, it just assumes that infinite operations cannot sum to a finite amount via naive intuition. You keep saying it shows that there is a problem with infinite steps but you can't explain what that problem is. This is because you are expressing your intuition of the "paradox" and not using logic.

>Yes, with limits you don't need to actually compute an infinite number of times to find it, but the problem still stands because what we know is that, no matter how small, there is no "smallest" element in terms of magnitude. This is a problem with infinities and the continuum.
How is it a problem?

>What this suggests is that the universe isn't really continuous, though it could run in weirder shit we dont know, but our idealizations are never going to be completely accurate.
How does it suggest that? Achilles reaches the tortoise in finite time regardless of continuity.

Why can't you just state your argument clearly? Just list the premises and the conclusion in a classical format.

>>9944212

This is a very interesting line of thought from which someone might take to be implying that there is a form of maths that does define the Universe but is also part of that reality, not an abstraction.

Therefore a valid form of maths is one that rejects Infinity as an impossible concept. Yet it still has to deal with the fact that we can conceive of Infinity as an abstraction, so Infinity does exist in some sense, since we are part of the Universe. I think.

>>9944643
The closest thing reality has to infinity is cycles. But alas, mod infinity will return an error in the universe.

>>9944614
Read any intro to analysis book.

>>9944651
Your continuous refusal to make a coherent argument shows you don't have one.

Takes an infinite number of steps != takes an infinite duration

>>9944697
What argument? Mathematically, all youvare saying is wrong, which makes it sound like you don't really understands how limits are defined in math.

>>9943643
basically

Does this help?

or this?

This.

Modelling the observed reality is a difficult task,
if you suck at it, you just end up confusing yourself, that's what was happening with Zenon.

>>9943136
It's not a geometric series
REEEEEEEEE

>>9944844
holy shit guys id this what it feels like to realize you were a brainlet all along?

>>9943136
formula derivation
https://youtu.be/XFDM1ip5HdU?t=5m30s

>>9943071
There is literally a graph that provides an intuitive explanation to the paradox right next to the source image on the wikipedia page
>>9944844
It is, each successive step is a ratio to the last

>>9944922

pfft that might be correct,
but is it as understandable as this?

>>9944767
An argument that the paradox is not resolved by the fact that the time it takes for Achilles to reach the tortoise is finite, you idiot. You keep saying this thought experiment shows "a problem" yet you can't describe what that problem is.

You guys are all missing the point here. Let me summarize it for .

We know Achilles catches the hare, this is a physical fact.

But the point of the paradox is to illustrate that with a system of Maths that includes the existence of Infinity it can be proven that Achilles does not catch the hare. Will never catch the hare.

Obviously something is fundamentally wrong here, not on a level to do with the application of our Maths, but as an application of our Maths as being relevant to the nature of the Universe.

Do you not see the relevance here? Just put aside your engineer hats for one moment and try and look at the problem for what it is, an inquiry into the most fundamental, most basic, principles of our mathematical concepts. The abstraction we call Infinity is incompatible with the physical nature of reality.

Now if you want to ignore that, if you want to bury your heads in the sand like so many people have done, then fine, go off and build a bridge or fix a lawn mower, great, otherwise you are contributing nothing.

>>9944922
>straight line
there are people who don't know calculus right now on this board

>>9945047
>But the point of the paradox is to illustrate that with a system of Maths that includes the existence of Infinity it can be proven that Achilles does not catch the hare. Will never catch the hare.
OK, so do it already.

>>9943071
Either bait or never heard of a divergent serie

>>9945041
I'm saying that the ideas from mathematical analysis don't resolve the paradox as applies to real physical motion you retard. The problem comes from the idea that you can have shit arbitrarily small. Convergence let's us talk about these infinite tasks in a rigorous and without needing to use ill defined shit like the generality of algebra, but the nature of the contiuum of the real number line is still there. That's why when you talk about the solution of a differential equation, which is what all the retards here are saying when showing they know about uniform rectilinear motion, you are bot thinking as a process, but just relating a geometric curve in your 4 dimensional affine space to what you see. But why do you think experimentally we can only plot discretw points? Basically this
>>9945047 Though you don't really prove you never reach it, it just doesn't make any sense to talk about motion that way in a pure mathematical framework. That's why physics=!math.

>>9945047
>But the point of the paradox is to illustrate that with a system of Maths that includes the existence of Infinity it can be proven that Achilles does not catch the hare. Will never catch the hare.
ONLY when some set amount of time and/or space is attributed to each of Achilles' passes over the tortoise's previous positions, because you are multiplying that set amount by infinity. If we describe every passing with a sentence or a picture in a sequence of diagrams like OP's pic, it becomes clear that we will never fully describe Achilles' overtaking of the tortoise. This attribution of "pauses" is a flaw in our intuition of the problem, born out of the impossibility to communicate an infinite amount of things without taking an infinite amount of time or space.

>>9943214
Baitlet

>>9944922
Achilles is not running at a constant speed. He is slowing down you fucking brainlet.

>>9945061
>I'm saying that the ideas from mathematical analysis don't resolve the paradox as applies to real physical motion you retard.
I AM ASKING YOU HOW IT FAILS TO RESOLVE THE PARADOX.

>>9945066
Because we have never been able to observe or create arbitrarily small time intervals. The idea is not that zeno thought it was a divergent series you absolute moron, but that arbitrarily small shit has issues, as any notion of scalability or "smallness" if you will, makes no sense in math. If yoy made someone small enough, he would always see achilles "far" from the tuttle.

>>9943071
>he thinks this problem can be formally defined without math

>>9945065
No he isn't because that's not way the paradox is laid out. Brainlet.

>>9945081
Sure it can. Zeno assumes there can be no infinities yet presumes to infinitely fractionate the time/space between the two racers infinitely.

It's not a paradox at all, it's a contradiction.

>>9945073
>Because we have never been able to observe or create arbitrarily small time intervals.
You are a fucking idiot. I've never seen Achilles chasing a tortoise either, doesn't mean jack shit. We can still reason as if they exist. That's the entire point of a thought experiment.

>The idea is not that zeno thought it was a divergent series you absolute moron, but that arbitrarily small shit has issues, as any notion of scalability or "smallness" if you will, makes no sense in math.
No, the idea is that Zeno did not understand that infinite steps could be undertaken in finite time. Zeno was not aware of any of the math that resolves the problem. There is no issue with arbitrarily small units of distance, since they are all traversed in finite time.


Well at least you finally made your argument clear so now everyone can point, laugh, and ignore everything you've said.

>>9945103
>No, the idea that Zeno did not undestand that infinite steps could be undertaken in finite time
O really?, Show me the mathematical proof of this.

>>9945103
>transversed in a finite time
But you can slice time arbitrarily small also you fuckwit. After a second happened, what event happened 1-epsilon seconds? How small can we make that epsilon?

>>9944922
This guy is right. The paradox can be written mathematically as 1 + .5 + .25 ... which is probably the most basic example of a geometric series and this particular series converges absolutely.

>>9945130
And? Does that mean you can not arbitrarily subdivide the real number line?

>>9943214
>>9943071
>>9943099
The idea, when you compute an infinite sum isn't that you're actually summing up an infinite number of terms. Rather the idea is that you're finding the point where the sum converges. This is kind of roundabout but it's a very reasonable thing to do in lieu of not being able to compute an actual infinite sum.

>>9943304
>He reaches it in finite time, brainlet.
In infinite steps, it doesn't take infinite time unless they're each slowing down.

>>9945061
ITT a physics student outs themselves as a brainlet in front of math nerds.

>>9945171
Not really, I'm a math major who actually knows basic analysis. Tedious yes, but It was required for my PhD program.

>>9943094

>>9945190
>Math Major
>PhD. program
>basic analysis

>>9945108
See >>9944922

>But you can slice time arbitrarily small also you fuckwit.
So what, retard?

>After a second happened, what event happened 1-epsilon seconds? How small can we make that epsilon?
It literally doesn't matter, you clearly don't understand basic calculus.

>>9945155
>The idea, when you compute an infinite sum isn't that you're actually summing up an infinite number of terms. Rather the idea is that you're finding the point where the sum converges.
Wrong, the sum of infinite terms is by definition equivalent to the value the partial sums converge to. Welcome to freshman math.

>>9945218
Yes I was first a math major. I then moved on to my PhD.

>>9945169
Try reading that post again.

>>9945245
I'm sorry to tell you this, but you're not going to get one.

>>9945252
Why?

>>9945054

Enjoy the sand sandwich, epictard.

>>9945263
Because you're not intelligent enough.

>>9945062

If you cant understand the problem there is no point even discussing it with you.

There is just no getting through to you dullards is there? It doesn't matter how simply anyone expresses the problem, the point just goes whooshing by, thousands of miles above your head.

Look, just go and fix a lawn mower, or something, okay? You will be a lot happier.

>>9945272
So you have no proof. You are pathetic.

>>9945282
It's amazing how I can be perpetually wrong yet no one will clearly lay out why

>>9945248
It's like the other anon said. The issue is performing infinite steps in finite time.

>>9945243
That's what I said, in less terms.
[eqn]\sum_{n=0}^\infty a_n = \lim_{k\to\infty}\sum_{n=0}^k a_n[/eqn]

>>9945245
Every math major takes a at a minimum two courses in analysis regardless of whether they're going for a PhD or not.

>>9945299
>The issue is performing infinite steps in finite time.
OK, how is it an issue?

>That's what I said, in less terms.
Then there is no issue.

>>9945277
Epic

>>9945309
The issue Zeno put forward, not that it is an actual issue. Are you fucking illiterate?

>>9945299
Yes, and If you actually paid attention in your analysis courses, you would understand how an "infinite sum" is actually defined. Those tedious epsilons and deltas have meaning you know. It has absolutely no say om Zeno's paradox.

>>9945323
No shit, reread >>9945155

>>9945315
I'm saying it's not an actual issue. Are you confusing me with someone else?

>>9945328
>OP thinks Zeno's issue is with infinitesimals
>anons like >>9945103 point out that his issue was with infinite steps in finite time
>retards (you) go apeshit thinking that Zeno's argument is being put forward now, in 2018.
kys yourself, brainlet

>>9943136
Don't you realise that formula is just "fuck it, let assume X as value"

>>9945323
It has absolutely everything to do with Zeno's paradox, since Achilles' infinite steps to reach the tortoise take finite time.

>>9945340
>>anons like >>9945103 point out that his issue was with infinite steps in finite time
Retard do you realize that I'm both >>9945328 and >>9945103 ?

You are confusing me with someone else.

>>9945346
Yes, that's what we observe. You are just stating a truisim. The problem is that at any instant, you can construct instances arbitrary close and this subdivision is infinite. The idea that two x vs t curves intersect is a static fact from math, but you are never actually talking about motion.

>>9945349
The point is that NO ONE is saying that Zenos paradox is an actual issue in 2018. There are several anons pointing out that Zeno's issue was with performing an infinite number of steps in finite time.
You are arguing with nobody.

>>9945357
>The problem is that at any instant, you can construct instances arbitrary close and this subdivision is infinite.
How is this a problem exactly?

>The idea that two x vs t curves intersect is a static fact from math, but you are never actually talking about motion.
When are we not talking about motion? You're not arguing anything, just stating your conclusion. I'm done with this, either list your premises and show exactly how they lead to your conclusion, or fuck off.

>>9945361
>There are several anons pointing out that Zeno's issue was with performing an infinite number of steps in finite time.

And that's completely wrong. The issue is about scale and the real number line.

>>9945361
There is at least one in this thread, retard. Welcome to /sci/.

>>9945366
You have, in no place here shown your claims. Every retard here is just smugly stating it's about basic calculus. You are the one making the claim that you know the resolution of zenos paradox. Please tell me rigorously how do you prove you reach shit in finite time.

>>9945374
Proven*

>>9945343
Not really. You aren't assuming x has a value you are just defining the value of an infinite sum to be the limit of its partial sums. Obviously when we have a finite amount of terms we have one definition for the idea of equality and that same concept doesn't translate over when we deal with an infinite amount of terms. The idea of infinity and limits has tremendous applications so I wouldn't really discount it to some handwavey bullshit which is what you imply when you talk about infinite sums being "hurr durr lets assume this thing has a value"

>>9945374
>You have, in no place here shown your claims.
I have stated from the start that the amount of time for Achilles to reach the tortoise can be calculated as a convergent sum. What exactly do you need clarified about this? Do you need an example? Do you need a lesson in calculus? Do you need to look at >>9944922 again? Or are you just stalling because you know you have no basis to your position?

>>9943306

Well played , Anon. Well played indeed.

>>9945384
Not him, but you do see that at the end you are talking about the limit of partial sums right?
>>9945390
Again, what the fuck does convergence actually mean? https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit You are only saying that a fucking line is continuous and that there is a common point between these lines. You don't even need a convergent series you just need to solve the equation [math]v_{A}t=v_{T}t+100[/math] to get an explicit expression of the t for which the lines intersect. And well yea you can approximate that number by a Cauchy series of rational numbers, again nothing really that magical. Do you know how that translates to the physical realm? You are ontologically incorrect and that's why your retarded ass can't seem to understand the problem here, as you are already using the fucking idea that the motion of Achilles or the turtle is a continuous straight line. How you came to that construction? Assuming kinematical laws that model space and time as continuous quantities in order to define [math]v(t)=\frac{dr}{dt}[/math] Mathematically, this means you are defining a vector field, and you want to find a curve that fits that field at every point of time, and if you find it, you solved the problem and the make the correspondence with physics and say, whenever I measure values of position and time I should get these pair of values. I fail to see how on earth this tells us anything about Zeno's paradox, because your solution curves are static objects in [math]\mathbb{R}^2[/math] and nothing is moving in this abstract space. Convergence is defined as basically saying you can make the error arbitrarily small, but the problem is no matter how "small" you get THERE ARE ALWAYS SMALLER IN THE REAL NUMBER LINE. A real resolution of the paradox is to postulate the idea that space and time are fundamentally discrete.
>hurr durr show me the proof
I can not talk mathematically about a problem that isn't mathematical.

>>9945422
>I fail to see how on earth this tells us anything about Zeno's paradox, because your solution curves are static objects in [math]\mathbb{R}^2[/math] and nothing is moving in this abstract space
What the fuck are you even trying to say? Why would a curve need to "move" to represent something moving? If this is your objection to movement being represented mathematically then you have no argument, just babble.

>Convergence is defined as basically saying you can make the error arbitrarily small
No, it means that the error of the finite sums is arbitrarily small. The error of the infinite sum is 0. You are still confusing the two after I have explained the difference multiple times.

>but the problem is no matter how "small" you get THERE ARE ALWAYS SMALLER IN THE REAL NUMBER LINE.
So what? For the last time, how is this a problem?

>I can not talk mathematically about a problem that isn't mathematical.
I didn't ask you for a mathematical proof, I asked you for a coherent argument. I know that is too hard for you since you have failed time and time again despite typing several paragraphs.

>>9945438
>Why would a curve need to "move" to represent something moving?
It doesn't, but the correspondence isn't direct as one sees in the physical world. i.e it tells you the representation of the events, but at the end you only measure a discrete number of them, hell, a finite number of them.
>The error of the infinite sum is 0
There is no ifinite sum you absolute twat. Don't act as you are explaining anything if you have no clue about the basic constructions from analysis. You have an infinte set of numbers that aproximate arbitrarily the number you want, but in no formal construction of the real numbers will you ever get a definition of a proper algebraic infinite sum. https://en.wikipedia.org/wiki/Generality_of_algebra
>So what? For the last time, how is this a problem?
Jesus fucking christ, the "last time" makes sense in terms of convergence, but you have an unaccountably infinite amount of events and time that come before your so called "last time". Yes, you have individual points in your line, but to talk about motion in the kinematical sense, you have to talk about derivatives, and so you need to understand the motion in A WHOLE NEIGHBORHOOD of t, which contains an unaccountably infinite number of distinct events. And because the real numbers are not well ordered, its nonsensical to talk about last in your context. The problem here is postulating that motion through space defines an unaccountably infinite amount of events, i.e. a continuous curve, and so if you are precise enough, you should be able to see Achilles always sufficiently away from the turtle, because you can, at least mathematically, always get smaller and smaller. Convergence of a series just tells you that for t close enough to t0 Achilles gets arbitrarily close to the tortoise, but all in a setting where YOU ALREADY HAVE AN INTEGRAL CURVE TRACED OUT.

>>9945464
>It doesn't, but the correspondence isn't direct as one sees in the physical world.
Yeah, that's how math works. Jesus Christ, you're dense.

>i.e it tells you the representation of the events, but at the end you only measure a discrete number of them, hell, a finite number of them.
No, you calculate an infinite number of time steps.

>There is no ifinite sum you absolute twat.
There is, that's what the thought experiment says.

>You have an infinte set of numbers that aproximate arbitrarily the number you want, but in no formal construction of the real numbers will you ever get a definition of a proper algebraic infinite sum.
The infinite sum is equal to the limit by definition, not an approximation. Again, you fail at basic calculus.

>Jesus fucking christ, the "last time" makes sense in terms of convergence, but you have an unaccountably infinite amount of events and time that come before your so called "last time". Yes, you have individual points in your line, but to talk about motion in the kinematical sense, you have to talk about derivatives, and so you need to understand the motion in A WHOLE NEIGHBORHOOD of t, which contains an unaccountably infinite number of distinct events. And because the real numbers are not well ordered, its nonsensical to talk about last in your context. The problem here is postulating that motion through space defines an unaccountably infinite amount of events, i.e. a continuous curve, and so if you are precise enough, you should be able to see Achilles always sufficiently away from the turtle, because you can, at least mathematically, always get smaller and smaller. Convergence of a series just tells you that for t close enough to t0 Achilles gets arbitrarily close to the tortoise, but all in a setting where YOU ALREADY HAVE AN INTEGRAL CURVE TRACED OUT.
This is word salad. I'm done. Enjoy your mental illness.

>>9945478
And you failed basic mathematical analysis, and basic reading comprehension. But it's all ok as long that in your head you feel your right.

>>9945479
>doesn't understand convergent series despite linking the wikipedia article
>claims others failed analysis

>>9945293
>>9945291
>>9945293

Because you are too fucking stupid to even understand an elementary explanation. Its like trying to explain something to someone who doesn't even speak the same language.

>>9945537
EXPLAIN TO ME IN WHATEVER LANGUAGE YOU ARE MOST COMFORTABLE WITH WHAT EXACTLY THE ERROR IS IN MY POST HERE >>9945062

>>9943485
Those are just the smallest measurable quantities.

Off topic but did anyone else think Calculus was the dopest shit when they first learned it? I felt like I was literally a wizard when I learned babby's first integrals and derivations.

>>9945490
Oh I understand them pretty well. And assuming you top understand them, you failed interpreting the paradox anyways, because you don't understand why your claims are ontologically, nonsense

>>9945966
>Oh I understand them pretty well.
No, you don't, since you think an infinite sum approximates a limit instead of equaling it.

>And assuming you top understand them, you failed interpreting the paradox anyways, because you don't understand why your claims are ontologically, nonsense
I have asked you repeatedly to explain how it does not resolve the paradox, but you have failed time and time again. We both know you're lying.

>>9945980
>you think an infinite sum approximates a limit
No you retard, I just understand the basic constructions of series as limits of partial sums, and so there is no actual infinite number of sums. I.e. writting an infinite sum is notation for a limit you retard.
>but you failed
You haven't really tried to refute my points. Jesus I know you brainlets get your dick hard cause you know (poorly it seems) some calculus, but this problem has nothing to do with it. It has to do with the problems of the continuum of real numbers. You understant that if you have a straight line with coordinate t and ordinate x, it's interpreted as giving the position of an object AT EVERY POSSIBLE TIME. If you accept that then by the continuous nature of the line, you can see that at any point you can find a neighborhood arbitrarily close to that, so you have an infinite numbera of events that preceded said event you fixed. And yes the limit exists and the function is continuous, that doesnt mean the set of events tgat had to occurr are infinite, and talking about "reaching" a time makes no sense. That's why the curve is taken as a function that a priori gives every time a position, and doesn't talk in any way about motion occuring in neighborhoods of a point. Physically, if could scale our senses arbitrarily small, we could potentially just go smaller and see the more and more precise steps, and the paradox is still unresolved.

>>9945996
>No you retard, I just understand the basic constructions of series as limits of partial sums, and so there is no actual infinite number of sums. I.e. writting an infinite sum is notation for a limit you retard.
The series that is constructed is an infinite sum, you're contradicting yourself.

>You haven't really tried to refute my points.
You haven't given any points, so there is nothing to refute.

>And yes the limit exists and the function is continuous, that doesnt mean the set of events tgat had to occurr are infinite
The set of events that had to occur is infinite because that is how the thought experiment is presented.

>and talking about "reaching" a time makes no sense.
It is very easy to make sense of it mathematically.

>That's why the curve is taken as a function that a priori gives every time a position, and doesn't talk in any way about motion occuring in neighborhoods of a point.
Motion is simply a change of position over time, the mathematical interpretation agrees with this.

>Physically, if could scale our senses arbitrarily small, we could potentially just go smaller and see the more and more precise steps, and the paradox is still unresolved.
It doesn't matter how "small our senses" are, because a finite distance is always finite and always takes a finite amount of time for Achilles to traverse.

You're not explaining anything, just stating your conclusion in vague language. Nothing you are talking about has anything to do with the paradox, which was and has always been about Zeno's inability to comprehend infinite steps summing up to a finite time.

>>9945996
[math] \displaystyle
\\ 0 < p < 1
\\
1 = p + (1-p) ~~~~~~\overset{1}{\overbrace{[=====p=====|==(1-p)==]}} \\
= p[x+(1-x)] + (1-p) ~~~~~~\overset{1}{\overbrace{\underset{p}{[\underbrace{=====x=====|==(p-x)==}]} ~~ + ~~ (1-p)}} \\
\\
\dfrac{x}{p-x}=\dfrac{p}{1-p} \Rightarrow x- xp = p^2 - xp \Rightarrow \underline{x=p^2} \Rightarrow \underline{(p-x)=p(1-p)}\\
\overset{1}{\overbrace{\underset{p}{[\underbrace{=====p^2=====|==p(1-p)==}]} ~~ + ~~ (1-p)}} \\
\\
\overset{1}{\overbrace{\underset{p^2}{[\underbrace{=====p^3=====|==p^2(1-p)==}]} ~~+ p(1-p)+(1-p)}} \\
\overset{1}{\overbrace{\underset{p^3}{[\underbrace{=====p^4=====|==p^3(1-p)==}]} ~~+ p^2(1-p)+p(1-p)+(1-p)}} \\
(1-p)+p(1-p)+p^2(1-p)+p^3(1-p)+\cdots =1 \\
1+p+p^2+p^3+\cdots =\dfrac{1}{1-p}
[/math]

>>9946033
>The series constructed is an infinite sum
I mean, this board is full of math autists, I'm surprised no one else has called you out. No, an infinite sum is not the infinite addition of an infinite numbers of terms. Just go read fucking Rudin for this.
>It is very easy to make sense of it mathematically.
Then show me how. Considering you don't understand convergence I find it hard to believe you have any proper mathematical arguments.
> Motion is simply a change of position over time, the mathematical interpretation agrees with this.
So motion is derivative of a curve? The derivative is a local notion defined at each point of a curve, but the curve is already given a priori, it's not the development of something tracing out a curve.
> doesn't matter how "small our senses" are, because a finite distance is always finite
Nice that you are starting to sound dumber, maybe you can realize how retarded your interpretation is. Yes, between to points on a real line, the distance between them is finite, that's a truism. My point is that around any point there are an infinite number of points that are arbitrarily close. As easy as considering the that if we have [math]x\in\mathbb{R}[/math] then we just take [math]x_{n}=x-\frac{1}{2^n}[/math] The limit [math]lim_{n\rightarrow\infty}x_{n}=x[/math] That's no secret and it was not unkown to the thinkers to attempted to answer it, but the problem is that still, no matter wht you do, there is not a single [math]x_{n}[/math] that is equal to [math]x[/math]. You say the point is about doing it in a finite time, but that's not the point, the idea is that of how can you conceive motion if you get any point there is no point "next" to it and zeno is realizing this property of the continuum. Or tell me, what number comes after 0? I also dare you to quote me any text that says that Zeno claimed that the process took infinite time. plato.stanford.edu/entries/paradox-zeno/#FurRea

>>9946039
I don't know what fucking retarded point you are trying to show, but you just need to use the exponential function in the finite geometric series and take a limit.

>>9945996
>It has to do with the problems of the continuum of real numbers

They just don't get it. Its like trying to describe colour to a blind person, one that doesn't even speak the same language as you. I honestly don't know why you are persevering with these mouth breathing autards.

>>9943071
>velocity

>>9946088
>i'm happy with circular logic

>we can sum infinite amount of numbers
>limits are exact numbers
t. this thread

>>9946125
You don't need the exponential function to find the closed expression for a finite geometric sum and you don't need the geometric series to prove anything about such function as it's literally just the inverse of ln(x) which is defined through an integral. That or use any of the equivalent ways to define the function.

>>9946087
>No, an infinite sum is not the infinite addition of an infinite numbers of terms.
Yes it is.

>Then show me how.
I already have. But let's try this again. Let V be Achilles' speed. Let v < V be the tortoise's speed. Let d be the initial distance between Achilles and the tortoise. So Achilles moves distance d in time d/V and the tortoise moves distance dv/V in the same time. This must repeat infinitely as the thought experiment states. So the amount of time for Achilles to reach the tortoise is

d/V+dv/V^2+dv^2/V^3+...

d/V(1+v/V+(v/V)^2+...)

Since v/V < 1 this simplifies to

(d/V)/(1-v/V)

d/(V-v)

Which is finite.

>So motion is derivative of a curve?
No, the derivative of the position over time would be velocity. Motion is a change in position over time. Motion is not defined instantaneously.

>My point is that around any point there are an infinite number of points that are arbitrarily close.
Yes, so what?

>You say the point is about doing it in a finite time, but that's not the point, the idea is that of how can you conceive motion if you get any point there is no point "next" to it and zeno is realizing this property of the continuum.
Motion is coherent for any two different positions, what does the continuum have to do with anything? All points in the continuum are a finite, nonzero distance apart. What is your definition of motion, because it appears you are adding something to it beyond a change in position over time.

>>9946087
>I also dare you to quote me any text that says that Zeno claimed that the process took infinite time.
Try reading your own source, buffoon:

"Thus the series of distances that Atalanta is required to run is: …, then 1/16 of the way, then 1/8 of the way, then 1/4 of the way, and finally 1/2 of the way (for now we are not suggesting that she stops at the end of each segment and then starts running at the beginning of the next—we are thinking of her continuous run being composed of such parts). And now there is a problem, for this description of her run has her travelling an infinite number of finite distances, which, Zeno would have us conclude, must take an infinite time, which is to say it is never completed."

>>9943071
>he thinks this problem is unresolved with high school math

>>9946153
holy low iq
>>9946129
i kno rite

>>9946184
>t. never took calculus

>>9946188
>t. literal mediocre iq

>>9946190
lmao explain how limits arent numbers

>>9946206
i can tell from thousands of miles away that you werent born an ashkenazi bc its coear from the way u think you were never gifted the mental horsepower of a jew-stuck to sam harris brainlet

>>9946221

Spoken like the barely literate peasant you are.

>>9943214

>>9945384
how can you sum infinite numbers idiot

>>9943099
>motion, in reality, is moving through points
no

>>9947785
read a book

>>9947824
>any book will teach you how to sum infinite numbers

>>9947785
by defining equality for infinite sums.

>>9943071
They're never gonna get there. Just letting you guys know.