Great video! Moving from place to place as you narrate is unusual, but I liked it. The theme of trade-offs between generality and applicability of the result is rather interesting.

@DaylenThimmMath

10 ай бұрын

Yes this was a bit of an experiment. I deliberately tried to do "unusual" but that brings some uncertainty of how it will be received. I'm glad you like it!

@raphaelfrey9061

9 ай бұрын

Its like veritasium

19:00 "careful jill there's a wild mathematician over on the rocks"

Wow what a crime this only has 2.3k views. Incredibly accessible presentation of a topic of a frontier of academia.

@DaylenThimmMath

9 ай бұрын

Glad you liked it! :)

17:26 I'm genuinely curious how you physically made this circle of sand with a bunch of holes. It's quite impressive.

@DaylenThimmMath

9 ай бұрын

I used a pair of compasses to draw the circles and took away the sand with a putty knife. The smaller holes I was able to cut out like cookies (the sand was a bit wet) by taking aluminum or plastic strips and taping them together to the correct circle sizes. The tiny circles I was able to make using straws of different sizes.

@maxfierro3493

8 ай бұрын

​@@DaylenThimmMath Quick question, am I right in thinking that the resulting figure is determined by the center and radius of the first disk you choose to remove from the initial disk? Is the algorithm to make such a figure picking a first disk to remove from another disk, then choosing the biggest remaining contiguous disk next for removal? It reminds me a bit of Cantor's set, but on a plane. I wonder if there is anything that can be said about the elements of the underlying disk that can never be removed by this algorithm (would there be anything left after an infinite amount of steps of the above algorithm if we removed closed disks?).

@kikivoorburg

6 ай бұрын

@@maxfierro3493 I believe what you describe is close to an Appollonian Circle Packing! There it's not required that the 2nd circle is the biggest possible, just that it's cotangent to both the outer (containing) circle and the 1st circle. I don't know enough to say whether removing closed discs would remove all points, but I assume someone's figured that out for the Appollonian circle packing given it's an ancient problem edit: it's "Appollonian" not "Appolonian", oops. Fixed it now

Loved the tribute to Archimedes 😂

"Don't disturb my circles!" Better be careful; that's how Archimedes died.

@DaylenThimmMath

8 ай бұрын

haha... well luckily soccer balls are less dangerous than roman swords :)

Solid work. Proving that there are non-expansive functions which have no fixed points is not really a "loss"; it's progress, unless your goal is subsequent work that relies on fixed points.

You have a great talent. By changing places in nature it seems that you somehow you invoke spatial cognitive abilities as it is very easy to comprehend the abstract concepts. I did my PhD from Uniwien and visited Innsbruck as well. Reminds me of old good days. Subscribed your channel. Please add more videos. Stay blessed.

@DaylenThimmMath

9 ай бұрын

Thanks for your very kind message! Glad you liked the video!

Absolutely love the physical graphs!!!! Was just thinking it would be cool to see more of this in the burgeoning math YT scene! Also the result and justification are very cool!! The explanation is quite elegant - most functions have a region with a fixed point. Extremely intuitive in the end! Well done and thanks. Oh and the don’t disturb my circles was amazing lol

@DaylenThimmMath

9 ай бұрын

Glad you liked it! Yeah I think I spent like 50€ just on different wires haha

Omg this video has such high quality editing. I honestly can't believe it doesn't have more views. Keep up the great work, you've earned a subscriber!

@DaylenThimmMath

9 ай бұрын

Thanks! Glad you liked it! People subscribing definitely helps in getting more views :)

I like the out door freshness here with the math

This is phenomenal and incredibly inspiring. Please continue to make videos; I would love to see more!

@DaylenThimmMath

10 ай бұрын

Thank you so much! It being my first video, I'm really glad to hear you like it!

Brilliant. I encourage you to make more …. Really great quality. I loved that you kept re setting the complex ideas in practical examples. Using nature / varying settings genuinely added something helping me understand. I have subscribed !!

Excellent work, it was well worth the time you've spent on it! Tiny note: If you find time to make more videos, I'd really appreciate it if you could make the audio a bit louder.

@DaylenThimmMath

10 ай бұрын

Glad you like it! And thanks for the advice! I know that is an issue, there are several other things I would have liked to improve, but unfortunately I was dealing with a lot of software issues, my inexperience in making videos, and a lot of other setbacks so that I wasn't able to do this fine tuning at the end before the deadline. Next time then :)

I am a simple entity. I see gremlin, I click

Man, great pacing and production and excellent use of props! I'd love to see you make more math explainers like this. Grüße aus Graz :)

Great video. I especially like how you state main points in several levels of formality/complexity. Listener can get both intuitive feel of what facts mean and try to understand precise descriptions. This is very important for me - I like to write down main points, but many authors present either only formal statement (which I don't get at all at times) or math-related hand-waving (which you can't write down).

@DaylenThimmMath

9 ай бұрын

Glad you liked my explanations! And thanks for taking the time to write the comment!

Great video! Your presentation with physical demos and models was awesome! I'd love to see more of it.

@DaylenThimmMath

9 ай бұрын

Glad you liked it! :)

Thoroughly enjoyed this video. Both the math as well as the production were top notch.

@DaylenThimmMath

9 ай бұрын

Glad you enjoyed it! :)

Amazing concept for presenting math! It was a great pleasure. Thank you.

@DaylenThimmMath

9 ай бұрын

Glad you liked it! :)

Explaining something as esoteric as an infinite-dimensional space, with *physical* props, is quite an achievement. This was really interesting!

@DaylenThimmMath

9 ай бұрын

Thanks! Although I bag to differ about this being esoteric. haha Glad you liked it

Good edited video. And very informative!

I really like your editing style! Hope to see a series of videos from you soon!

@DaylenThimmMath

9 ай бұрын

Glad you liked it! :) I too hope I can find the time and some good topics for more videos.

great video, I love the 'outdoor math' theme!

This is a really good video. Keep making more in this same format.

@DaylenThimmMath

9 ай бұрын

Thanks! :)

Continue doing this, you are bound to get big (and even if that isn't your goal, continue doing it anyway because it's great!) One thing though, I think you need to equalise your audio a bit better. Your voice is very quiet (I had to put my speakers on maximum), but that made the music in the transitions extremely loud.

@DaylenThimmMath

9 ай бұрын

Thank you so much! Yes I know about the audio problem. Will take care of this next time :)

Thanks man! Your explanations are even more beautiful than the setting … and Innsbruck looks gorgeous 🎉

@DaylenThimmMath

9 ай бұрын

Really glad you liked it! :)

Great video! Can't wait for another one

I enjoyed watching this, great video.

Great video!

I really liked the style! It' almost classic greek math :)

Interesting video!! Please keep up!!! I just subscribed!!

Interesting video, thanks for making it.

It was a fun math adventure loved it!

Great video! Looking forward to more on the channel :)

@DaylenThimmMath

9 ай бұрын

Glad you enjoyed it! :)

This was very lovely and inspiring

Awesome video, it deserves way more views

@DaylenThimmMath

10 ай бұрын

Thanks! :)

It's really fun, You will make really cool youtube channel like a vsauce because you are trying make an interesting picture. Content it's hard for me now, but i will watch this video again, thanks for work

Aha! An Archimedes reference! “Don’t disturb my circles!” 😂

Really good video, and i can't but respect the ad for living in innsbruck!

@DaylenThimmMath

9 ай бұрын

Thanks! Up until now the city hasn't called yet to offer a sponsorship ;-) haha

Amazing video. I'm from Brazil, and I really loved your way of explain

@DaylenThimmMath

9 ай бұрын

Thanks! Glad you liked it! :)

Really really good video, thanks for the hard work!

@DaylenThimmMath

10 ай бұрын

Glad you like it! It really turned out so be so much more work than I expected.

congratz! das ist schönes video!

@DaylenThimmMath

9 ай бұрын

Danke :)

Loved it

6:00 That's not true. Stirring is discontinuous. The spoon cuts thru the coffee, and when the coffee rejoins on the other side of the spoon, it will in general be different molecules that end up next to each other. For that matter, even without stirring, the molecules move in a way such that the "fabric" of the coffee doesn't stay continuous from moment to moment anyway. 12:15 What the fixed point is depends on where the map is. If you happened to be standing at the north end of that bridge, then *that* would be the fixed point. You have some lovely landscape around you. However, I don't watch math videos for landscapes. Additionally, the music playing during some of your interludes is louder than your speech in the other parts. Also, some of the very quiet parts have a *lot* of white noise. No doubt part of this is the outdoor filming, but there are filters for this.

This is beautifully made. How long did it take to make the circular patterns in the sand?

@DaylenThimmMath

10 ай бұрын

Glad you liked it! haha well... took me about 2 hours...doing it once. But I ended up doing it three times, one time for testing, once for filming, and a third time for reshoots after I was able to borrow a better camera and figure out a decent way to film from above😂

What an incredible video. Very James Burke vibes in your presentation.

@DaylenThimmMath

9 ай бұрын

Wow, thank you!

good educating method

Very cool 😎

"You can't always get what you want. But if you try sometime you just might find you get what you need."

"Don't disturb my circles!" -- Archimedes. "How I wish I could recollect of circle round, The exact relation Archimede unwound." (Count the letters in the words.)

@DaylenThimmMath

8 ай бұрын

haha, love it!

Actually diffusion breaks continuity assumption, so there is little chance that there will be one particle in the same place as before.

I wonder if these fixed point theorems are related to gravity. These shrink functions that describe Lipschitz contractions are vaguely similar to time dilation or length dilation from general relativity. The Rokatch contractions is vaguely similar to Newton’s law of gravity where it is proportional to the distance between two points. The Euler-Lagrangian equation which describes the equation of motions are fixed points. More generally from the principle of least action via fixed points the equation of motions are described. For general relativity of gravity it is the Einstein-Hilbert action that has fixed points. The Einstein-Hilbert action can used to derive Einstein’s field equations which predict time and length dilations and under some restrictions are equivalent to Newton’s law of gravity.

We have a Carl Sagan of math right here.

@DaylenThimmMath

9 ай бұрын

You're the second person saying that. I had to look up who that is, but I feel honored being compared to him. :)

Imagine this video with beautiful smooth computer animation. 😻

For anyone who really vibed with this video, you should look into the ideas of Daoism and the Dao. They have some fun connections with each other.

Im a Math student at TU Wien who grew up in Innsbruck. I just got this recommended and after the first 3 Minutens Im really looking forward to watching this video on a tablett with a hot kakao right after I finished dinner

@DaylenThimmMath

9 ай бұрын

Have fun! :) Do we know each other? The math community in Innsbruck isn't that big...

@nudelsuppe2090

9 ай бұрын

@@DaylenThimmMath no we dont unfortunately, im still in my bachelors so not a mathematician

@DaylenThimmMath

9 ай бұрын

I think being in your bachelor's counts as being a mathematician. :) After all, being a student I guess is your occupation.

@nudelsuppe2090

9 ай бұрын

@@DaylenThimmMath thx

1) Banach Th

@DaylenThimmMath

9 ай бұрын

Glad you liked the video! Yes, there are a bunch of fixed point theorems out there. Nice to have such a compactly written list of implications! :)

As far as I can tell all of this concerns a stable fixed point. What about unstable ones? I.e. for the x² case, 1 would not be such a contraction, right? 0 certainly would be, but with 1, everything would *expand away from* it, some towards 0 and the rest towards infinity. (or if you involve the complex numbers, then the entire unit circle would stay within itself, but going even slightly off the unit circle would mean you spiral towards infinity or 0) The two types of contraction on infinite sets you mentioned both only seemed to involve contraction. However, I'm guessing you could simply use the inverse function where available, to find the unstable fixed points of the original function? For instance, sqrt(x) also sorta has the same fixed points as x², but the roles are swapped: At every application you get closer and closer to the unit circle (and if we consider the entire map, it's technically closer to twice as many points on each iteration), unless you start precisely at 0, in which case you stay at 0. So I take it that's basically the answer for the two types of contraction as well? Or is there some subtlety why unstable fixed points simply don't generalize to the infinite dimensional case?

Wow!

At 23:10, what does the left upper arrow marked by "almost all Lipschitz contractions" indicate? Shouldn't all Lipschitz contractions be nonexpansive?

@DaylenThimmMath

9 ай бұрын

Thanks for the question! Yes indeed, all Lipschitz contractions are nonexpansive, at least if L All in all, the Lipschitz contractions are not responsible for almost all nonexpansive functions having a fixed point, because there are almost none. Do you feel this answered your question? :)

@6:05 "So Brouwer's Fixed Point Theorem guarantees that no matter how much you mix, there will always be some molecule in the coffee at the same spot as it was before mixing." I believe this is a mistake. This would only be true if molecules were infinitely small and numerous. This mathematical consequence cannot be applied to macroscopic physics so readily. If you "stir" a single molecule within a container, is it assuredly at the same place as where it began? Most certainly not. Extreme claims often reveal their flaws when taken to their logical extremes.

5:55 I'm having a bit of trouble hearing you over your background music. Maybe try toning it down while you voiceover? Otherwise this is a great video, really fun

@DaylenThimmMath

8 ай бұрын

Thanks for the feedback! You are right, this is a bit of a problem. Glad you liked the video otherwise!

Doesn't the shortsighted metric described consider any two functions which differ but not on the integer rings to have distance 0, making it only a pseudometric instead of a metric? Or is the actual metric used the corresponding integral or whatever?

@DaylenThimmMath

8 ай бұрын

We take filled circles with all integer radii. So infinitely many. They cover the whole space. Therefore, the distance of two functions is zero if and only if they are the same. (on every circle the metric d_n is the supremum metric)

@yaksher

8 ай бұрын

@@DaylenThimmMath Ah, they're _filled_ circles. That makes sense. Don't we usually use the term "disk" for filled circles?

I've seen this structure before when I was playing with making my own c fractal generator... it got kinda out of hand after a year... this looks a lot like the magnet equation as you step through escape size with fixed itterations.

@DaylenThimmMath

8 ай бұрын

So this type of fractal actually has a name. It is called an Apollonian gasket.

@valseedian

8 ай бұрын

@@DaylenThimmMath back down the rabbit hole I go. a new thing to add to the generator... if it can be done with any shape, can maybe use it to alter fielded fractals and make even more fractal structures. I mean, can I generate one with a serpinksi type algo? wonder what 3+d shapes can be made infinitely porous in the same way. luckily I have my own home brew n dimensional structure planarizer. . never finished the rotator but now I might have a reason to. off to research.

@valseedian

8 ай бұрын

just looked it up and I'm pretty sure I can make gasketizer... looks a lot like it'll work in 3 dimensions for some number of tangential bodies... no reason why 4+d shouldn't be possible... imagine slices of that space in 2d now I'm intrigued. the fractal project has been in cold storage for over a year. might be time to resurrect it and finish the nD rotator.

Do you happen to know why people working on PDE are always trying to improve their bad behaved solutions on Sobolev space instead of trying to prove the theorems on the space of C^infinity functions, which is a Frechet space and has a nice complete metric?

Can someone help me know if I'm understanding correctly? I've always struggled differentiating a theoretically perfect example (the impossible "flat surface with no friction" used in many introductory physics calculations, for example) with the real world analogue. When he refers to a "stirred cup of coffee," he's referring to a theoretically perfect fluid which has infinite points contained within it, correct? That would make sense to me. Otherwise, if taken literally in the real world example, both turbulence and the fact that the fixed point may end up in between two adjacent molecules would make this statement false. Likewise, with the crumpled paper example (taken from the wiki article on the Brouwer Fixed Point Theorem), it can be similar to standing one sheet on edge with a "V" crease that shifts the points the sheet overlays - or does that make it not convex any more since two points on the "crumpled" plane would overlap? I know the theorem is right, I'm just trying to understand its rules better. Thanks in advance!

@DaylenThimmMath

9 ай бұрын

Thanks for your question! :) Yes indeed, I am assuming an idealized fluid. As you already have pointed out, in order to describe the real world one will usually assume a model that reflects reality "well enough" for the application at hand. The usual model used for such slow fluid movements is the model of laminar flow, which gives you a continuous function. On a physical level this may be justified by recognizing that molecules that are close to each other tend to stay close due to adhesion and cohesion forces.

@DaylenThimmMath

9 ай бұрын

Maybe I am not correctly understanding what you mean with the paper example, but the important part is only that the paper is convex before being crumpled and that it is not teared but only crumpled or folded. Overlapping points on the crumpled plane are okay, the function only has to be continuous (not injective if you've heard of that). This is a video shows a pretty great experiment for the paper example: kzread.info/dash/bejne/gJehk7dpeKzcfag.htmlsi=Ijw8uhZxnQvHCNQM&t=28

@klpaah

9 ай бұрын

@@DaylenThimmMathThanks for your replies! I'm still not quite understanding the paper example, because she intentionally set the folded paper directly above those four numbers, and kept the orientation the same. It seems arbitrary, but I'm assuming she's following topological rules that I don't understand yet. Can you tell me if I'm understanding correctly: if she offset the folded paper by exactly one unit in the x and y direction and then rotated (or flipped) it in place, there would be no fixed point because that translation is like the "+ 1" in the function, and the rotation would be like applying sine to the function, am I understanding this correctly? At the very least, I believe I'm beginning to understand the relationship of equations to their corresponding graphs significantly better, so this is having knock-on positive effects regardless of my level of understanding of the original concept. Again, I'm not saying you or she are wrong, and the video you linked was well done - I'm merely attempting to understand the "rules" that this theorem has to follow. Thanks for taking the time to help me understand, I greatly appreciate you and your work! :)

There is nothing more disturbing in a video than the sound level jumping willy-nilly higher or lower from a shot to another. Or when the music starts 10 times louder than the previous speech.

Do we know that a Rakotch contraction under the metric used for bounded sets is also one under the metric used for unbounded sets and vice versa?

@DaylenThimmMath

9 ай бұрын

Thanks for your question! So the metric d that "we use on bounded sets" is a metric that we use to determine the distance between two functions d(f,g), when the functions are defined on bounded sets. If we have a Rakotch contraction f then the metric d does not affect at all whether the function is a Rakotch contraction or not, since it only is a metric between functions. But you are right, if we choose some metric space (M, rho) and let f: M -> M be a function from M to M that is a Rakotch contraction, then, yes, choosing a different metric rho_2 on M may change the fact that f is a Rakotch contraction. Does this clear things up? If not, feel free to ask again :)

@be1tube

9 ай бұрын

@@DaylenThimmMath I understand; there are two metrics: one is used to characterize whether the function is a contraction and the other is used to measure the distance between functions to characterize the density of fixed-pointedness.

@DaylenThimmMath

9 ай бұрын

exactly!

I don't think that coffee cup statement is correct. If you stir it (somehow) so the lowest centimeter of molecules end up on stop of the rest, with the rest having shifted down, no molecule will be in the same position.

@DaylenThimmMath

3 ай бұрын

You are right, in this case there is no fixed point. Also, this way of mixing is not continuous that is, it is not the case, that molecules that were close before are close after mixing again. However adhesion and cohesion effects between the molecules guarantee this continuity. So any way you can realistically mix is continuous and this is needed to apply the Brouwer fixed point Theorem.

이런거 좋음.

This is essentially set theory.

Based on the penmanship, I'm going to go ahead and guess that this man is using Hagoromo chalk. 😮

@DaylenThimmMath

8 ай бұрын

haha almost right: I unfortunately forgot the Hagomoro chalk at home the day I filmed these scenes.

Fractal lotus root

I know that sine and cosine have fixed points, it really just asks what number makes f(x)=f(f(x)) true

@ianweckhorst3200

9 ай бұрын

Tangent has uncountably many fixed points

@ianweckhorst3200

9 ай бұрын

That applies to most of the normal trigonometric functions other than sine and cosine

Im bad at math but good video

Make more videos

sigma-porous = meagre?

@DaylenThimmMath

9 ай бұрын

Well, it is very similar. For meager sets you don't need a metric. But assume you have a metric space: then yes, sigma-porous implies meager. However, meager does not imply sigma-porous, since for porosity you need the linearity between the neighborhood and the size of the holes you can choose. So porosity is a stronger notion than meagerness in metric spaces. The proof sketch in the bonus part does not really go into this distinction, so in the video we indeed only show meager. Actually, I may have lied a little to make the explanations more understandable: We were not able to prove this linearity between the neighborhood and the size of the holes. But we were able to describe the relationship between these two sizes using a function, say phi. What we really were able to show is so called co-sigma-phi-porosity. If you are more interested in this, then you could take a look at the documents I mention in the video description, e.g., my thesis has a summary of these different notions.

Funny thing: I just stumbled across a headline reading "Woman marries herself, cheats on herself and blames men" so the spouse function does indeed have a fixed point, unless she has already decided to get a divorce. ^^

@DaylenThimmMath

8 ай бұрын

Haha Yeah I saw a BBC article about self-merriage. I was thinking about making a short mentioning it.😅

why would you open by quoting Hilbert? he believed that mathematics could be grounded in formal logic using the nonsensical axioms produced by Dedekind and Peano. when it was eventually shown that these axioms don't actually work, Hilbert was pretty much the only person who understood the implication, and gave up on the venture. which means that his vision for a logical foundation of mathematics was misguided, but literally everyone who does math now also views his ultimate rejection of the project as heresy. which means he's the last person you should be quoting when espousing conventional mathematical views, since his career highlights the fact that everything you believe about math is merely cult nonsense.

@sumdumbmick

9 ай бұрын

take for instance the notion that addition is an operator. this is built into Peano Arithmetic, which is directly derived from Peano's Axioms, which is the thing which Godel's Incompleteness Theorems are showing is broken. A little before Godel's demonstration of Incompleteness, however, we get Presburger Arithmetic, which is Peano Arithmetic minus multiplication, because the issues with Incompleteness come from the account of multiplication given by Peano Arithmetic. anyway... so, Peano fancied himself a linguist. he had absolutely no idea wtf he was talking about, but he was a deluded maniac, so he believed that he understood how language fundamentally works. what he tried to do with arithmetic was decompose it into 'nouns' and 'verbs', or 'numbers' and 'operators'. so when looking at something like 6+7 he declared that 6 and 7 are the numbers that the addition operator acts over. and from this is proclaimed that the basic arguments of the addition operator is Natural numbers. but... this is fucking insane nonsense. note, for instance, what happens when we consider a 'subtraction': 4 -3 = -3 +4 why did it turn into an addition? why did the subtraction operator move along with the 3 as if it's part of the 3? well, the answer's really fucking simple. + and - are not operators, they're the sign of the vectors we're processing. that is, 4-3 is not a subtraction of a Natural number 3 from the Natural number 4, rather it's starting with the Integer +4 and moving the Integer distance -3. similarly, -3+4 is just the reverse ordering of this pair of Integers. so this problem just reduces to handling the unordered list of Integers {+4, -3}. and the reason that the + sign disappears on positive terms in initial position is simply to reduce the amount of writing necessary, since we assume a positive polarity by default.

@sumdumbmick

9 ай бұрын

this then means that the Natural Numbers are not a subset of the Integers at all. the Naturals are in fact numbers, just bare, unsigned, countable magnitudes. the Integers are signed 1-dimensional vectors. and the only way to perform addition or subtraction with Naturals is to implicitly cast them into Integers, because there is no such thing as an addition or subtraction operator. if there were, then math would look really, really fucking different than it does. but literally all of your heroes were too dumb to notice any of this.

@sumdumbmick

9 ай бұрын

for a more in depth look at the issue, consider Peano's successor function: allegedly 0 is a Natural number which is not the successor of anything. then 1 is its successor, and adding 1 from here out yields the Naturals. but... what about 1/2? isn't that a more genuine successor to 0 than 1 is? and now what about 1/3, or 1/4? thus we get an infinite regress (in no small measure due to the fact that the Reals are an Archimedean Group, which by definition means that no value has a successor... good job, morons). further, the assumption all along is that we have arithmetic operators working over numbers. so the claim that 1+1=2 should always hold true, but in truth it virtually never speaks to the truth: 1 dog +1 quail = 6 legs; so 1+1=6 1 half +1 third = 5 sixths; so 1+1=5 1 frog +1 pond = 1 pond; so 1+1=1 1 C water +1 C dirt = some mud; so 1+1=an unknown value the issue this illustrates is that the only things we can 'add' or 'subtract' are vectors. that is, an object which is the combination of a magnitude and a unit. when the units are incommensurate a unit conversion is required, and only after that unit conversion occurs can we add the magnitudes in the way Peano suggests, when he claims that you can add Naturals... but you should note that this makes Peano wrong in the general case, since addition really always acts over vectors, not bare magnitudes like the Naturals, which contradicts the Hilbert quote you opened with. which tells me you really have no idea wtf you're talking about.

@sumdumbmick

9 ай бұрын

what if the 'greedy gremlin' is just a consequence of you using a completely wrong set of foundational axioms in your interpretation of mathematics? a formal logic can be internally consistent and prove all manner of valid things, without any of it being sound. and the entire venture of mathematical proof theory is now based on the assumption that the currently accepted axioms are flawless... which is dumb as hell. and not only because I just demonstrated how some of those foundational axioms are actually just wrong, but because even a cursory understanding of how an axiomatic theory works reveals this basic property of logical frameworks in general.

@adhararadhara4173

9 ай бұрын

u good dog?

Botfarm boosting has ruined social media

I think the coffee analogy is imperfect as molecules in a coffee cup are not continuous.

@DaylenThimmMath

9 ай бұрын

Thanks for your comment! :) Ok, so there is definitely is some truth to your criticism. On a very very small scale one could perceive mixing coffee like mixing a gigantic coffee mug with roughly 8 * 10^24 golf balls, and no this is not continuous. However, in physical fluid models no one thinks of it this way for the the following reasons: 1. dealing with individual molecules is computationally not only infeasible, but impossible 2. due to kohesion and adhesion forces between the particles, molecules that were close together before mixing tend to stay close together after mixing (which justifies a continuous model) This continuous model is usually called the model of laminar flow. It breaks down if the flow speed is really really high. (but I don't see you mixing your coffee at the speed of sound haha) Also, if we'd use this molecule approach, there would be no continuous functions in physics at all. Even a spacial density function would be pointless then (or could only be written as a sum of delta distributions).

@EricDMMiller

9 ай бұрын

@@DaylenThimmMath I agree that the principle holds true for suitably abstract models. But molecules of coffee are just not abstract enough because they occupy physical space in discrete ways and are subject to multiple unique space filling arrangements. Whether space itself is discrete or continuous is a bit of an open question, with some disagreement ongoing (to my knowledge), so it may be possible that it really does have no direct application in any element of reality.

@jakobr_

9 ай бұрын

If we imagine the molecules being attached to some smooth, mathematically ideal structure like some kind of 3D fabric, that fabric would have a fixed point. I think with enough stirring the chance of the guaranteed fixed point lying in the space between molecules instead of on a molecule itself approaches the percent of space not occupied by the molecules.