I found a non trivial zero off the critical line, but there is not enough room in these comments to write it out, and they keep deleting my comments...
@@dentonyoung4314 No, it's not, as there is no such thing as a universal, context free Gödel-undecidable sentence. A sentence may be Gödel-undecidable only in the context given by a formal system (strictly, an axiomatic formal deductive system) such as e.g. ZF, and moreover Gödels procedure gives a proof that the sentence is true (in context). Thus, if you could show that the Riemann hypothesis is Gödel-undecidable in ZF, you have a proof that given ZF, then Riemann is true.
@theevilcottonball that could be an interesting problem, actually. someone should make an actual sofa in the shape of gerver's sofa and attempt to get it around a bend in a correspondingly wide hallway.
Actually, I do wonder what happens if you vary the corridor angle, or do a left plus a right turn or do a turn in one plane and a different plane for a 3D sofa.
Please do more videos like this! It's always fascinating to hear from the author about their thought process in their own words, which is often a big part that's missing in the final paper.
Not to mention that legacy media will half-ass an article and not ask the author for feedback or input. I much prefer hearing it from the original source.
Someone really needs to start a "Journal of null and negative results". It would help not only with this sort of thing but also with experimental stuff where it seem publication biases are causing at least part of the reproducibility crisis. (If I ever fund any kind of research, I'm intending to put a publication requirement on the money; "If you take my money, you must publish whatever results you get even if that's just a description of what you did and that the results where not significant.")
The problem is, journals only make money when people read them, and how many people will read most of these negative results if they're not contradicting something already important? Especially if it regards ideas, like here, rather than an experiment or something - "I played around with this that and the other thing, and nothing worked out." Partial leads on problems, maybe, if they're actual leads, but those can also be called "papers I'll finish someday, when I find the rest of the ideas I need"
@@Stirdix A valid point, but that just suggests that it wouldn't function as a "traditional" journal. It would have to be a non-profit and likely digital-only, _not_ a paper publication. Maybe something that a university (or group of them) could run it like with arXiv. Or for that matter, something funded by some government departments and the like on the basis of "if we fund your research with taxes, then you are obligated to publish (and we will ensure there is at least _one_ place that will publish it regardless of result)". As for who would read them, it would be a valuable source for people getting started on a project or even considering funding someone else's project: "what has already failed?" is a very important question for both those situations. Likely the writing style would be rather different, I'm guessing more narrative and a lot more focus on the "future directions" section and an occasional "how I F'ed up and sank the project before I even started" cautionary tale.
@@benjaminshropshire2900 There is another problem with publishing negative results: if information about how my failures occurred can benefit me in achieving success later on (e.g. after I acquire better equipment or better understanding of theoretical approaches), then why would I transmit to others the information about how I failed? In other words, for a "journal for negative results" to become popular, there needs to be a personal (not altruistic) incentive for publishing negative results. I still think such a "journal" can exist, but at the scale of small communities, in which the members would gladly provide each other advantages as tokens of friendship. Or maybe if there can be an agreement that, if you tell me what you did to get negative results then your names will end up in my paper as co-authors. Again, the mutual trust in technical practices is the key to making this work.
@@Stirdix A huge problem with academia is most of the papers don't matter, and there is a huge amount of money put toward them if they affirm some political bias even if they violate the priors that very field rests on like say stereotype threat stuff like Ariely and Gino do or say candidate gene studies where they not only throw tons of money wastefully at such papers but they also seem to like the negative results
I don't think a journal is what is needed here. I think the main problem is people don't see "worth" to publish something if it didn't get them anywhere, and that's a problem. I have thought a long time about this (and it is one of the reasons I ended up abandoning maths as a career) and I feel an asynchronous collection of information (similar to how Wikipedia/StackExchange work) is better. It removes the "publish moral issues" requirements because now it's not a publication. If you work in X conjecture even if it's just for a couple of months, you can just add a small comment "I tried this and this approach, they didn't work because of Y and Z", and anyone checking that specific conjecture can see at any point in time all the attempts and get some ideas of how others have approached already the problem. Public, accessible, easy to edit, easy to add new information, you can filter by information added by professionals, you can have a small moderation team to remove vandalism or you can go the StackExchange route and just rely on users to self-moderate proper answers from false information. It's honestly baffling to me the amount of effort and time people spend researching mathematics and not researching how to research better as a community. Creating better tools, systems, communities, software. It's a massive waste. It's like working in computer science when Git and standard libraries weren't a thing yet, professionals are doing the same work over and over, independently from one another. I feel the funding problem is different from this, and it's too big to handle right now, because it depends on your country, it depends on the contract and it's a bit of a mess everywhere. I don't think it's realistic to solve it in the next following years. While a simple system or software to just add information to open problems is something that can be achieved in just a couple of years by a small team, there is simply no incentive or people working on it at the moment.
That's a good point. Now I'm wondering if we must have either a fishbole or an anti-fishbone. My gut feeling is that this goes the same way as the fishbone theorem, although it being true would be really nice.
2:42 Both our wallets are stuffed full of a mixture of Australian dollars and Turkish lira. Unfortunately we have no idea what the exchange rates between the Aussie dollar and the Turkish lira are. Under what circumstances we be able to tell who has more money?
One of the problems with maths is that only conclusive results tend to get published. In physics it's fine to say "we tried doing X but it didn't work", but in maths most things are "we did Y and this is how", with maybe a comment about all the other ways they tried.
All physics to the best of our knowledge world is finite, therefore publishing methods that have not yielded effective results is still a useful publication. There is realistically or at least approximately an infinite number of ways to do mathematics wrong, does it seriously benefit anyone by publishing it? Likely not.
When coin flipping, the coin decides to go and stay in the horizontal positions, instead of accepting an either heads or tails question of the entscheidungsproblem... the good moments of math.
“This should morally be a counterexample” I have had this exact experience in grad school. My buddy and I were working on an algebra problem with modules and tensor products, and I was trying to come up with a counter example for something. I trying constructing an explicit bilinear map that should work but didn’t because it nulled everything out, until my buddy figured out a smart way to tweak my example to make it work. When I asked how he thought of it, he said basically the same thing! He was like “what you came up with feels like it should work and morally it should,” and that’s what gave him the idea. Anyway, it’s funny how much in math we know what something “ought to be” but it can be hard actually showing that
Provable for finite sets - ok so that means the counterexample isn't finite - save the link to the paper for when I have an unbounded amount of time (after Christmas?).
ha posting this in exam season might have been cruel:D But yes, if it is a finite set then for sure it has a fishbone and the counterexample actually has to be crazy large, not just infinite but infinite with specific weird structures.
I'd never heard of the Fishbone Conjecture before this video, but it's cool that it's a (now disproven) conjecture that can be explained to an undergrad.
its really exciting that you were able to include an interview in this video :D i really hope you’re able to do more interviews with mathematicians for these math news videos in the future!!
Great video and great work by Lawrence! I think the great outcome from disproving conjectures is in the analysis and breakdown done to clearly define the "rules" for when and why it is true or false. That richer understanding is still a great step forward.
I like to use the parallel symbol “||” for incomparable terms in a partial ordering, since it fits nicely with “=“ for equality. I see the logic for the perpendicular symbol too though!
I believe they are referring to the "bottom", which is used in posets specifically, but also finds its use in logic systems as contradiction and type theory. en.wikipedia.org/wiki/Greatest_element_and_least_element?wprov=sfla1
I am chuffed I can follow this video. Have no background in maths to speak of, but in distributed computing partially ordered sets describe the events that occur on a network of computers. If I follow your terminology correctly - if two events (elements in a set) are an anti-chain, that means they happened concurrently, or completely independently of each other. One event could not have influenced the other. You might see this a computer/phone user when something that syncs your data tells you you have a "conflict" - there's no total order of events, so you have to decide which bit of the anti-chain is valid. EDIT: Ok so when you resolve a conflict when syncing data, what you're doing I think is defining the "spine". There's some algorithms that can define spines for you, but for a lot of data it's best to just ask the user what they wanted (this file edited on this machine, this file on another machine, which version do you want?).
An example of a poset that I've constructed in my work is the org chart of a company where some people report to managers (and there are no cycles--generally there isn't--I suppose it would still be a poset if you have situations where someone has "dotted lines" to multiple managers, though my code didn't handle that). A chain in that poset would be any subset of a literal chain of command, and an antichain would be some set of people who are not in each other's management chain. But, of course, this is a finite graph. Posets involving the time-ordering of events even occur in physics, in the theory of relativity. If one event is in the future or past light-cone of another, then there's an invariant time ordering, but outside of the light-cone, when the interval separating them is a faster-than-light path, there isn't--which order they happened in depends on one's frame of reference. In quantum field theory, it's very important for the preservation of any notion of causality for certain quantum-mechanical operators to *commute* for events outside each other's light-cone; this is why non-local quantum correlations can't actually be used to send faster-than-light messages. And all this is assuming that the space has no closed timelike loops, in which case God knows what happens (but no one has ever found any...) But *that* poset is actually an uncountably infinite one, assuming a continuous classical spacetime, which is another big if. (A fishbone on it would be pretty simple to construct, though, at least for most of the spaces physicists actually think about: if you can put any timelike coordinate on it, you can define a chain, and then the "space" slices are the antichains.)
@@MattMcIrvin My understanding is the partial order of events used in computing is very much derived from physics (Lamport '78 mentions this explicitly). Much like there's no one global time in relativity, there's no one global time with different machines, so deciding the causality between events is key.
Thanks for this comment. I could follow the video, the complexity was not overwhelming, but I had neither the math background nor any technical perspective to put it into. The video felt like reading a hallucinated LLM output, everything followed a consistent logic but meant nothing and I was about to leave mildly annoyed lol
You could have made the Cartesian example more concrete by comparing apples and oranges. Two apples and three oranges are less than three apples and four oranges, but can't be compared to three apples and two oranges.
@@argon7624 this analogy does not build on, but takes away from ≤ and ∧. if I prefer apples over oranges, three apples and two oranges are greater than two apples and three oranges, so the premises are just false.
Hi professor! I would like to ask you why we can't compare all nodes in our ordering? timestamp: 01:19. For example you said that we cannot compare C and G, but I can imagine in a sort of 3blue1brown grabbing the branch where G belongs ( branch e and g ) and kind of stretching it so it were to be inline with the road of B and C. In this way, for it to be a straight line it would be G -> E -> B -> C. Here we have an ordering, no?
You could do that, it's just that you could also NOT do it:D That is, there are many ways to define an ordering. One could be "more to the right" or one could be "more to the right AND on the same road" etc. So some the orderings that people define are partial and some are not.
Consider that the displayed diagram is not just an undirected graph - the left-right relations matter. If you bring C to be lefter than B, it'd be a different ordering!
Different metrics are different qualia of general measurement theory. Is there a generalization in which different contexts of comparing different magnitudes can be compared with each other in coherent manner? Langland's program or analogy of it studies this old question which also at the heart of Euclid's Elementa. Unified theory can't be done from reductionistic perspective, as Gödel showed. So we intuit and study now holistic perspectives.
ata bo 3:49 into the presetation it said that any two points in the chain has the property that one point is less than another point but the ordering was (X1,Y1)
Tiny error, during the 6:05 example, the partition is purely into antichains, the chain isnt part of the partition (since partition sets have to be disjoint) but it meets every antichain in the partition transversally.
Fun fact! Spacetime itself is partially ordered. Causally linked events form ordered chains, but it is meaningless to talk about order when considering “faster than light” connections. Being able to break the speed of light barrier always implies the ability to travel back in time.
@@bestaround3323 FTL travel is most likely impossible however warping space to travel farther then light would be able to in the same time is theoretically possible though completely infeasible. A side note is that this would break the partial order of spacetime and could have unforseen consequences though at this point it is merely conjecture
@nothingnothing1799 Theoretically possible with exotic matter that is unlikely to exist. Just because something works in the math doesn't mean it works in real life. Warping space to go FTL is still FTL travel. Which means it likely doesn't exist.
Only if there is no observer for whom both such “spaelike separated” events lie within their light cone. For such observers they do have a natural order (how far they lie in his past), though neither can be the classical cause of the other. Relativity doesn’t require Space time to be a DAG (directed acyclic graph)- that only occurs if you assume the laws of physics prevent things like closed time like curves, which nothing in known quantum or relativity physics does. To be fair, nothing in known physics would create them either. And quantum entanglement problems complicates this even more.
Great video! Something that confused me: you don't partition the poset into a chain and disjoint anti-chains; rather, you choose a chain, you partition the poset into disjoint anti-chains, and then insist each anti-chain meets the chain.
14:30 Imagine how much time could have been saved by knowing what had already been tried. This is why a good researcher, into any STEM field, will document their failures as readily as their successes. Whether that gets properly published or not is often outside the researcher's control, but they will at least make their best effort to provide the information of what didn't work
The interview with Lawrence Hollom was fascinating. Not being a mathematician myself, the creative process of working toward a paper worth printing was very clearly explained. The fact that Lawrence also has programming skills may be significant and will become increasingly important in the future. Can Lawrence please start working on Hailstone numbers. There has been so much trouble proving that is true, that I expect that is also false and needs a bit of the Lawrence magic.
@@Hotrob_J Hint: zig-zag paths of continued fractions in Stern-Brocot type structures (generating number theory from holistic nesting algorithm) and Gosper arithmetic of continued fractions. Which means basically doing arithmetics of arbitrary length words in higher dimension.
I'm at the start of the interview, and if the counterexample turns out to be dependent of the Axiom of Choice I'll be quite annoyed. Let's unpause. EDIT: well, I still have that question and now I have another: is the counterexample countable?
hold on. I'm not sure if i understood your definition correctly, but it's either one way or another way. Either a vertex in the chain can have multiple antichains connecting to it, as the graph you show in the thumbnail and several times throughout the video would suggest, and therefore the example at around 8:30 does have a fishbone, or one vertex in the chain can only connect to one antichain as the example at ~ 8:30 would suggest, and therefore the example in the thumbnail is not a fishbone (the ordering does have a fishbone, sure, but it's not the one shown by that graph).
Nice video! I was left wanting to see the counter example though. I understand that it's not possible to rigorously analyse a complicated construction in a RUclips video but a picture of the thing would've been nice for appreciating its intricacy.
I used to watch your vid, Linear Algebra when i was 2nd year in Med School, at that time, i remember your channel only have like a few hundred of couple thousand subscribers, i learn math from pure love. I really thought your channel is gonna dead soon, but it procceed til now, 457k, Congratulation Funny part is i quit Med, Start up all over again, now i am studying Math, your channel turn out to be long lived than my main career, thank you for the vid
Seeing that graph near the start reminds me of the card game Maskmen, where, during the game, you 'discover' relative rankings with pairs of cards, and end up with branching hierarchies like that. So you might know Red is more powerful than Green, and Blue is also more powerful than Green, but the relative strengths of Red and Blue haven't been decided yet.
@@ZMacZ People keep on saying that, as if Gödel, Church, Turing and others didn't give their undecidability results that make the claim absurd. Or as if Zeno had not prove the issue already before them. It's very simple. No, we can not construct an ideal straight line from projective shadow (point) of a straight line.
"no fishbone. Here's the proof:" *writes down the video address, which had been carefully written on top of their desk beforehand. Sadly, because there's writing on the desk they're copying from, the teacher assumes they're cheating and takes the exam away and grades it a 0. You got the bad ending. Try again?* W. Sorry I passed out there why did I write all that.
Have you seen the video on the moving sofa problem by the channel "Wrath of Math"? (It does not go into Baek's proof in much detail, but it gives an overview of the problem and an outline of the basic proof idea)
@@DrTrefor And yet also expected, in the sense that the shape that turned out to be optimal was proposed in 1992. (Coincidentally, this is the same year that the fishbone conjecture was posed.)
If I make a conjecture called the baseball conjecture and it's proven false and you do a video on it, could you have the video title be "Math News: The Baseball Conjecture has been deballed!!"
wait....so if you have a finite chain, you can either have a finite set (fishbone) or an infinite set as possible side tracks.....how exactly did any of what was said disprove that?
proving -> Nice! I am now more confident in something that I already believed disproving -> Oh wow… Today I learnt To me, disproving a conjecture that was around for some time, sounds more like learning than proving it
So to have a fishbone, we'd need to be able to connect that point to the spine via an antichains. That we could do, but the requirement of a fishbone is all the anti-chains are DISJOINT and so it would overlap with one of the existing antichains.
@@DrTrefor Sharing common top down holistic origin like the root of Stern-Brocot tree is not disjoint at the root of a binary tree decomposed by nesting algorithm, but can look disjoint from the bottom up perspective of additive algorithms and their compositions.
It’s a not-completely-standardized definition, but mathematicians tends a bit more towards including 0 in the naturals, particularly discrete mathematicians.
Hi, I do not have the time to read the paper, so unfortunately I couldn't answer my own question. Do the examples also say something about the truthfulness/wrongness of the conjecture in the case of well partial orders (No infinite antichains+No infinite descending sequences)? This would be interesting to know.
We used diagrams like this for analysing and solving engineering problems, critical paths and stuff. I wish we were taught it with a stronger mathematical underpinning.
I had the same question, the video skips over it really fast so for a non math-expert viewer it's not very clear but I figured it out: for an anti-chain to form it needs to be non-related to both the point on the chain and every other point that connects to said chain-point via an antichain. This goes even BETWEEN anti-chains. This is what tripped me up: the shape of the fishbone graph kinda makes it look like there's two different anti-chains connecting to each point on the spine, but as the video said they need to be "disjoint" aka the two anti-chains can't have points comparable to each other either. I bet that to real math nerds that was obvious, but I didn't realize that is what the video meant by disjoint. The point highlighted at 9:43 is non-comparable to the highest point of the finite spine, but comparable to the point directly below it. Same x, higher y. But that point is connected via anti-chain to the very same point on the spine it would be connecting to itself. two comparable points can't be connected via anti-chain to the same point on the spine, so that's not allowed.
9:05 This isn't really a valid justification, is it? You could easily construct a finite partition of antichains of a poset (infinite) for which a chain is found connecting them (each at one point). e.g. from the given example, also remove the points (x, y) for which y
It just so happens that such a partition is not possible given the triangular structure of the lattice shown. But that is not a necessary fact from the choice of a finite connecting chain (leaving infinitely many points left to form antichains).
@@danielyuan9862 i'm not saying that there is a fishbone for the example given, only that the reason given for why there isn't is not a sufficient condition.
Nice fact. When fractions 2/3 and 3/2 etc. inverses of consecutive Fibonacci fractions a/b and b/a have their denominators multiplied by two and < b, then a+2b gives Fibonacci numbers and b+2a gives Lucas numbers. It is possible also to approach the issue from a more general perspective than just coordinate systems, in which it was now presented, and compare comparability also in other contexts.
sir I am a Computer Science engineering student from India and just watched your videos about probability in discrete maths and just love 💖the way you teach that you just teach the thing from basic and cristal clear.So the reason behind I am writing this comment is that i want guidance about how can i become a good engineer that i love to solve mathematics and not getting that much interest in coding and programming so please if you can.😊
It's....complicated:D Not the kind of thing I can easily motivate to a wide audience, but Lawrence shares a bit of the intuition in the interview and you can read the details in the paper itself.
The problem with "larger infinities" is that these types of combinatorial questions are not well posed for uncountable structures, the answer always depends on what model of set-theory you feel like using today. That's because those larger infinities are not "absolute" in a technical sense, any uncountable set can be shoehorned into a countable one by forcing it to collapse onto the set of whole numbers So this guy is selling himself short, he probably has said the most that can be said about this conjecture for any cardinality..
@@creativenametxt2960 I didn't read his paper yet, I don't even know what he did exactly. I just know that whatever he did, there's no way "going uncountable" is going to add anything new, because it never does.
This is ridiculous, there are tons of theorems about uncountable structures, like the real numbers. Absoluteness is a technical concept, and doesn't mean "every problem has an answer." There isn't even an algorithm for determining which integer polynomials have an integer root, so I don't see how some undecidable questions about uncountable infinities undermines their study.
As in each Mersenne prime can be written as 6x+1 for some integer x? Try thinking about the remainders of powers of 2 when you divide them by 6, and what kind of powers of 2 give rise to Mersenne primes…
I like counterexamples more than proofs, because counterexamples show new areas of mathematics that we haven't explored. For example, non-Euclidean geometry is interesting, and finding geometry that doesn't follow Euclid's postulates is much more interesting than showing that Euclidean geometry is the only type of consistent geometry would have been.
That was another conjecture I'd never heard of. So it's provably true in most case, but there's this one weird condition where if if something really bizarre happens it can be false.
Not really "most cases", more like the "nice cases". For example, by any sensible measure there are definitely "a lot more" uncountable posets than countable posets, but the proof only applies in the countable case. The other condition also seems to be more on the restrictive side than not, although I could see why it's a "nice" condition.
My issue with your first example is that it can easily be changed to show infinite antichains + fishbone. The written conjecture is fine (no infinite antichain => fishbone), but I don't believe your "exclusive or" presentation makes sense.
The NxN example? That doesn't have infinite antichains. Given a point, anything else on an antichain is either down and to the right OR left and up. But since there is only a finite amount of down and a finite amount of left possible in NxN it has to be finite.
Sorry, I meant the one at 6:37. This example would work even if the y-axis were extended infinitely down (into negatives), no? In which case those antichains become infinite (not just arbitrarily large).
The distinction between discrete and continuous has become very confused in modern mathematics, mainly due to set theory destrying basic mereological intuitions. Discrete and continuous is not an either-or-relation. Computation theory starts to make sense after we realize that analogical and digital is both-and, they necessitate each other. You can't have digital Turing-Head without analogical Turing-Tape, which is also called "quantum time".
Hi professor! I hope you reply to my comment I want to pursue theoretical physics and am a Physics major(3rd semester) currently we're studying Bessel's functions and frobenius method, laplace transforms etc in ODE. Since these problems get pretty Lengthy, how much would you suggest i practice and how much do you suggest I focus on the proofs(there aren't many in this course) and also what areas of Mathematics do i need to look into for studying General Relativity?
I do think practice makes perfect, but the big goal is to UNDERSTAND why you are doing what you are doing. That is don't focus necessarily on all the technical details in a long computation, do you understand the big picture of what say a laplace transform and inverse laplace transform is doing to solve an ODE?
@DrTrefor Coincidentally that is the exact question i was going to ask my professor but he'd left his office early today due to an emergency. I was going to ask him how does someone come up with something that looks so random as if it was a revelation only they knew of and then they were somehow mysteriously guided that this reduced ODEs to algebraic problems! Thank you very much prof
Huge thank you to Lawrence Hollom for talking to us about his cool new paper!. Check out the interview starting at 10:41 in the video.
Are you pronouncing words correctly? Also, did you check your audio volume?
I can’t wait for the holy grail to be found!
can't wait for Hollom to disprove the Reimann Hypothesis, cuz so many of my friends believe it has to be true intuitively.
I found a non trivial zero off the critical line, but there is not enough room in these comments to write it out, and they keep deleting my comments...
If its disproven, physics would be in a pretty tough place...
The Riemann hypothesis is probably Godelian-undecidable -- meaning it's true but we can never prove that fact.
Fermat2.0@@thomashoglund5671
@@dentonyoung4314 No, it's not, as there is no such thing as a universal, context free Gödel-undecidable sentence. A sentence may be Gödel-undecidable only in the context given by a formal system (strictly, an axiomatic formal deductive system) such as e.g. ZF, and moreover Gödels procedure gives a proof that the sentence is true (in context). Thus, if you could show that the Riemann hypothesis is Gödel-undecidable in ZF, you have a proof that given ZF, then Riemann is true.
apparently, the moving sofa problem was just solved. "The Optimality of Gerver's Sofa" by Jineon Baek.
Amazing
It's still pending peer review, but nonetheless exciting
But does it solve the problem of moving the old non-optimal sofa round the corner of the corridor to get it out?
@theevilcottonball that could be an interesting problem, actually. someone should make an actual sofa in the shape of gerver's sofa and attempt to get it around a bend in a correspondingly wide hallway.
Actually, I do wonder what happens if you vary the corridor angle, or do a left plus a right turn or do a turn in one plane and a different plane for a 3D sofa.
Please do more videos like this! It's always fascinating to hear from the author about their thought process in their own words, which is often a big part that's missing in the final paper.
Glad to hear that! I definitely loved doing the interview bit and if people enjoy it I'll definitely do more of that style.
Fr, the narrative aspect helps me visualize the math better 😂
Not to mention that legacy media will half-ass an article and not ask the author for feedback or input. I much prefer hearing it from the original source.
Someone really needs to start a "Journal of null and negative results". It would help not only with this sort of thing but also with experimental stuff where it seem publication biases are causing at least part of the reproducibility crisis. (If I ever fund any kind of research, I'm intending to put a publication requirement on the money; "If you take my money, you must publish whatever results you get even if that's just a description of what you did and that the results where not significant.")
The problem is, journals only make money when people read them, and how many people will read most of these negative results if they're not contradicting something already important? Especially if it regards ideas, like here, rather than an experiment or something - "I played around with this that and the other thing, and nothing worked out."
Partial leads on problems, maybe, if they're actual leads, but those can also be called "papers I'll finish someday, when I find the rest of the ideas I need"
@@Stirdix A valid point, but that just suggests that it wouldn't function as a "traditional" journal. It would have to be a non-profit and likely digital-only, _not_ a paper publication. Maybe something that a university (or group of them) could run it like with arXiv. Or for that matter, something funded by some government departments and the like on the basis of "if we fund your research with taxes, then you are obligated to publish (and we will ensure there is at least _one_ place that will publish it regardless of result)".
As for who would read them, it would be a valuable source for people getting started on a project or even considering funding someone else's project: "what has already failed?" is a very important question for both those situations. Likely the writing style would be rather different, I'm guessing more narrative and a lot more focus on the "future directions" section and an occasional "how I F'ed up and sank the project before I even started" cautionary tale.
@@benjaminshropshire2900 There is another problem with publishing negative results: if information about how my failures occurred can benefit me in achieving success later on (e.g. after I acquire better equipment or better understanding of theoretical approaches), then why would I transmit to others the information about how I failed? In other words, for a "journal for negative results" to become popular, there needs to be a personal (not altruistic) incentive for publishing negative results.
I still think such a "journal" can exist, but at the scale of small communities, in which the members would gladly provide each other advantages as tokens of friendship. Or maybe if there can be an agreement that, if you tell me what you did to get negative results then your names will end up in my paper as co-authors. Again, the mutual trust in technical practices is the key to making this work.
@@Stirdix A huge problem with academia is most of the papers don't matter, and there is a huge amount of money put toward them if they affirm some political bias even if they violate the priors that very field rests on like say stereotype threat stuff like Ariely and Gino do or say candidate gene studies where they not only throw tons of money wastefully at such papers but they also seem to like the negative results
I don't think a journal is what is needed here. I think the main problem is people don't see "worth" to publish something if it didn't get them anywhere, and that's a problem.
I have thought a long time about this (and it is one of the reasons I ended up abandoning maths as a career) and I feel an asynchronous collection of information (similar to how Wikipedia/StackExchange work) is better. It removes the "publish moral issues" requirements because now it's not a publication. If you work in X conjecture even if it's just for a couple of months, you can just add a small comment "I tried this and this approach, they didn't work because of Y and Z", and anyone checking that specific conjecture can see at any point in time all the attempts and get some ideas of how others have approached already the problem.
Public, accessible, easy to edit, easy to add new information, you can filter by information added by professionals, you can have a small moderation team to remove vandalism or you can go the StackExchange route and just rely on users to self-moderate proper answers from false information.
It's honestly baffling to me the amount of effort and time people spend researching mathematics and not researching how to research better as a community. Creating better tools, systems, communities, software. It's a massive waste. It's like working in computer science when Git and standard libraries weren't a thing yet, professionals are doing the same work over and over, independently from one another.
I feel the funding problem is different from this, and it's too big to handle right now, because it depends on your country, it depends on the contract and it's a bit of a mess everywhere. I don't think it's realistic to solve it in the next following years. While a simple system or software to just add information to open problems is something that can be achieved in just a couple of years by a small team, there is simply no incentive or people working on it at the moment.
10:10 note that we have an... anti-fishbone?
We could have the (0,0), (1,0), (2,0), ..... anti-chain, and all vertical disjoin chains.
An anti fishbone. A fishmeat
That's a good point. Now I'm wondering if we must have either a fishbole or an anti-fishbone. My gut feeling is that this goes the same way as the fishbone theorem, although it being true would be really nice.
@@minamagdy4126An anti fishbone would need an infinite anti-chain a lot of times
2:42 Both our wallets are stuffed full of a mixture of Australian dollars and Turkish lira. Unfortunately we have no idea what the exchange rates between the Aussie dollar and the Turkish lira are. Under what circumstances we be able to tell who has more money?
nice metaphor
took me a second to figure out what you meant! very clever
if you knew the exchange rate for both in a third currency you could get an aproximate value within that third currency
"Lawrence Fishbone" really just makes me think of Morpheus from The Matrix.
you think that's theorem you're proving?
One of the problems with maths is that only conclusive results tend to get published. In physics it's fine to say "we tried doing X but it didn't work", but in maths most things are "we did Y and this is how", with maybe a comment about all the other ways they tried.
All physics to the best of our knowledge world is finite, therefore publishing methods that have not yielded effective results is still a useful publication. There is realistically or at least approximately an infinite number of ways to do mathematics wrong, does it seriously benefit anyone by publishing it? Likely not.
"Ignore the head and the tail for a moment" 4:40 . I'm still waiting for you to get back to the mathematical relevance of the head and tail...
When coin flipping, the coin decides to go and stay in the horizontal positions, instead of accepting an either heads or tails question of the entscheidungsproblem... the good moments of math.
surely that means next video must be on the topic 😎
“This should morally be a counterexample”
I have had this exact experience in grad school. My buddy and I were working on an algebra problem with modules and tensor products, and I was trying to come up with a counter example for something. I trying constructing an explicit bilinear map that should work but didn’t because it nulled everything out, until my buddy figured out a smart way to tweak my example to make it work. When I asked how he thought of it, he said basically the same thing! He was like “what you came up with feels like it should work and morally it should,” and that’s what gave him the idea. Anyway, it’s funny how much in math we know what something “ought to be” but it can be hard actually showing that
Provable for finite sets - ok so that means the counterexample isn't finite - save the link to the paper for when I have an unbounded amount of time (after Christmas?).
ha posting this in exam season might have been cruel:D But yes, if it is a finite set then for sure it has a fishbone and the counterexample actually has to be crazy large, not just infinite but infinite with specific weird structures.
It being provable for finite sets... that means there is always a fishbone, right?
@@minamagdy4126 no because there are infinite sets
@@minamagdy4126yes, since no infinite antichain can be made
@@minamagdy4126 Yes, for finite sets there is always a fishbone.
I got the impression that was regarded as a trivial case!?
I'd never heard of the Fishbone Conjecture before this video, but it's cool that it's a (now disproven) conjecture that can be explained to an undergrad.
its really exciting that you were able to include an interview in this video :D i really hope you’re able to do more interviews with mathematicians for these math news videos in the future!!
This is a fantastic video, loved hearing the intuition behind the new theorem
Great video and great work by Lawrence!
I think the great outcome from disproving conjectures is in the analysis and breakdown done to clearly define the "rules" for when and why it is true or false. That richer understanding is still a great step forward.
15:00
Yes, informing others of what methods do not work for a proof would be very efficient in saving a lot of wasted time.
@@electricpaper269 you learn more about proofs by discovering which methods are ineffective yourself, rather than relying on others.
I like to use the parallel symbol “||” for incomparable terms in a partial ordering, since it fits nicely with “=“ for equality. I see the logic for the perpendicular symbol too though!
I believe they are referring to the "bottom", which is used in posets specifically, but also finds its use in logic systems as contradiction and type theory. en.wikipedia.org/wiki/Greatest_element_and_least_element?wprov=sfla1
In physics, 'II' generally gets used if things are parallel, so it would at least confuse me.
Issue is || can be taken as like “or”
You mean //?
I’ve seen your double vertical bar used in the sense of “just divides”
For example 2 just divides 42 since 2^2 does not.
The interview was the best part. I really liked the author's insight into the creation of the set.
I am chuffed I can follow this video. Have no background in maths to speak of, but in distributed computing partially ordered sets describe the events that occur on a network of computers.
If I follow your terminology correctly - if two events (elements in a set) are an anti-chain, that means they happened concurrently, or completely independently of each other. One event could not have influenced the other. You might see this a computer/phone user when something that syncs your data tells you you have a "conflict" - there's no total order of events, so you have to decide which bit of the anti-chain is valid.
EDIT: Ok so when you resolve a conflict when syncing data, what you're doing I think is defining the "spine". There's some algorithms that can define spines for you, but for a lot of data it's best to just ask the user what they wanted (this file edited on this machine, this file on another machine, which version do you want?).
I actually really love this, building connections to your own area of expertise is awesome
An example of a poset that I've constructed in my work is the org chart of a company where some people report to managers (and there are no cycles--generally there isn't--I suppose it would still be a poset if you have situations where someone has "dotted lines" to multiple managers, though my code didn't handle that). A chain in that poset would be any subset of a literal chain of command, and an antichain would be some set of people who are not in each other's management chain. But, of course, this is a finite graph.
Posets involving the time-ordering of events even occur in physics, in the theory of relativity. If one event is in the future or past light-cone of another, then there's an invariant time ordering, but outside of the light-cone, when the interval separating them is a faster-than-light path, there isn't--which order they happened in depends on one's frame of reference. In quantum field theory, it's very important for the preservation of any notion of causality for certain quantum-mechanical operators to *commute* for events outside each other's light-cone; this is why non-local quantum correlations can't actually be used to send faster-than-light messages. And all this is assuming that the space has no closed timelike loops, in which case God knows what happens (but no one has ever found any...) But *that* poset is actually an uncountably infinite one, assuming a continuous classical spacetime, which is another big if. (A fishbone on it would be pretty simple to construct, though, at least for most of the spaces physicists actually think about: if you can put any timelike coordinate on it, you can define a chain, and then the "space" slices are the antichains.)
@@MattMcIrvin My understanding is the partial order of events used in computing is very much derived from physics (Lamport '78 mentions this explicitly). Much like there's no one global time in relativity, there's no one global time with different machines, so deciding the causality between events is key.
Thanks for this comment. I could follow the video, the complexity was not overwhelming, but I had neither the math background nor any technical perspective to put it into. The video felt like reading a hallucinated LLM output, everything followed a consistent logic but meant nothing and I was about to leave mildly annoyed lol
Events that are timewise separated are a chain. Events that are spacewise separated are an antichain.
You could have made the Cartesian example more concrete by comparing apples and oranges.
Two apples and three oranges are less than three apples and four oranges, but can't be compared to three apples and two oranges.
I thought this was a math video, not preschool
@ElusiveEel Math always builds upon itself, and never do any parts become wholely irrelevant.
@@ElusiveEel must they be disjoint? The best people are those who could make a single description to fit all people.
@@argon7624 this analogy does not build on, but takes away from ≤ and ∧.
if I prefer apples over oranges, three apples and two oranges are greater than two apples and three oranges, so the premises are just false.
@@xinpingdonohoe3978 preschoolers don't learn math (arithmetic is not math), so they are disjoint.
the rest of what you said is meaningless.
Keep the Math News coming! I love sharing with my friends things like this, which are fairly simple to state but only recently solved.
Congrats for the discovery!
I loved the interview
Very cool! Would be happy to see more on partial orders
SIX YEARS of gradutation that I could not get it, and you explained to me what an antichain is in less the a minute WOW!
Hi professor! I would like to ask you why we can't compare all nodes in our ordering? timestamp: 01:19. For example you said that we cannot compare C and G, but I can imagine in a sort of 3blue1brown grabbing the branch where G belongs ( branch e and g ) and kind of stretching it so it were to be inline with the road of B and C. In this way, for it to be a straight line it would be G -> E -> B -> C. Here we have an ordering, no?
You could do that, it's just that you could also NOT do it:D That is, there are many ways to define an ordering. One could be "more to the right" or one could be "more to the right AND on the same road" etc. So some the orderings that people define are partial and some are not.
Consider that the displayed diagram is not just an undirected graph - the left-right relations matter. If you bring C to be lefter than B, it'd be a different ordering!
Different metrics are different qualia of general measurement theory. Is there a generalization in which different contexts of comparing different magnitudes can be compared with each other in coherent manner?
Langland's program or analogy of it studies this old question which also at the heart of Euclid's Elementa. Unified theory can't be done from reductionistic perspective, as Gödel showed. So we intuit and study now holistic perspectives.
ata bo 3:49 into the presetation it said that any two points in the chain has the property that one point is less than another point but the ordering was (X1,Y1)
Tiny error, during the 6:05 example, the partition is purely into antichains, the chain isnt part of the partition (since partition sets have to be disjoint) but it meets every antichain in the partition transversally.
Fun fact! Spacetime itself is partially ordered. Causally linked events form ordered chains, but it is meaningless to talk about order when considering “faster than light” connections. Being able to break the speed of light barrier always implies the ability to travel back in time.
Yeah that’s cool
Which is why it is highly unlikely that FTL exists.
@@bestaround3323 FTL travel is most likely impossible however warping space to travel farther then light would be able to in the same time is theoretically possible though completely infeasible. A side note is that this would break the partial order of spacetime and could have unforseen consequences though at this point it is merely conjecture
@nothingnothing1799 Theoretically possible with exotic matter that is unlikely to exist. Just because something works in the math doesn't mean it works in real life.
Warping space to go FTL is still FTL travel. Which means it likely doesn't exist.
Only if there is no observer for whom both such “spaelike separated” events lie within their light cone. For such observers they do have a natural order (how far they lie in his past), though neither can be the classical cause of the other. Relativity doesn’t require Space time to be a DAG (directed acyclic graph)- that only occurs if you assume the laws of physics prevent things like closed time like curves, which nothing in known quantum or relativity physics does. To be fair, nothing in known physics would create them either. And quantum entanglement problems complicates this even more.
at 3:31 shouldn't (3,2) itself also be highlighted since x1=x2=3 and y1=y2=2 therefore x1
Oh sure, I was implicitly thinking of what other points could be added to a chain, but absolutely any point is compares to itself.
was just gonna post this
Great video! Something that confused me: you don't partition the poset into a chain and disjoint anti-chains; rather, you choose a chain, you partition the poset into disjoint anti-chains, and then insist each anti-chain meets the chain.
I'm honestly lost with the logic provided in the video 😂
Yes, this is stated almost verbatim in the paper. Sadly the explanation in the video doesn't get it quite right
14:30 Imagine how much time could have been saved by knowing what had already been tried. This is why a good researcher, into any STEM field, will document their failures as readily as their successes. Whether that gets properly published or not is often outside the researcher's control, but they will at least make their best effort to provide the information of what didn't work
pretty nice as recently the moving couch problem was solved, many big news!
The interview with Lawrence Hollom was fascinating. Not being a mathematician myself, the creative process of working toward a paper worth printing was very clearly explained. The fact that Lawrence also has programming skills may be significant and will become increasingly important in the future.
Can Lawrence please start working on Hailstone numbers. There has been so much trouble proving that is true, that I expect that is also false and needs a bit of the Lawrence magic.
Thank you for this very clear explanation
Wonderful video, very nice!
The scientist in me loves disproving more than proving
Thank you! Great video
Lawrence Hollom discovered the counterexample while listening to Everyday Sunshine on repeat for 36 hours straight.
This is unironically super relevant to a sci-fi story I'm writing!
Maybe. After watching it two more times, I'm less sure lol
Yeah not actually relevant (it's branching timelines of multiple countable infinities), but still very interesting, and made me exercise my brain :)
@@Hotrob_J Hint: zig-zag paths of continued fractions in Stern-Brocot type structures (generating number theory from holistic nesting algorithm) and Gosper arithmetic of continued fractions. Which means basically doing arithmetics of arbitrary length words in higher dimension.
I'm at the start of the interview, and if the counterexample turns out to be dependent of the Axiom of Choice I'll be quite annoyed. Let's unpause. EDIT: well, I still have that question and now I have another: is the counterexample countable?
The counter example is countable, and um don’t quote me but I think it doesn’t depend on AoC, it’s entirely constructed
hold on. I'm not sure if i understood your definition correctly, but it's either one way or another way. Either a vertex in the chain can have multiple antichains connecting to it, as the graph you show in the thumbnail and several times throughout the video would suggest, and therefore the example at around 8:30 does have a fishbone, or one vertex in the chain can only connect to one antichain as the example at ~ 8:30 would suggest, and therefore the example in the thumbnail is not a fishbone (the ordering does have a fishbone, sure, but it's not the one shown by that graph).
Nice video! I was left wanting to see the counter example though. I understand that it's not possible to rigorously analyse a complicated construction in a RUclips video but a picture of the thing would've been nice for appreciating its intricacy.
I wanted to see the counterexample explained.... though I guess I have to read the paper for that
Yeah me too. That's what motivated me to continue watching. Alas :/
I used to watch your vid, Linear Algebra when i was 2nd year in Med School, at that time, i remember your channel only have like a few hundred of couple thousand subscribers, i learn math from pure love. I really thought your channel is gonna dead soon, but it procceed til now, 457k, Congratulation
Funny part is i quit Med, Start up all over again, now i am studying Math, your channel turn out to be long lived than my main career, thank you for the vid
What is the counterexample??
Probably very complicated from what it sounds like.
8:01 actually you've cut out infinitely less than half ;)
The points across the middle remain
Congrats!!
Seeing that graph near the start reminds me of the card game Maskmen, where, during the game, you 'discover' relative rankings with pairs of cards, and end up with branching hierarchies like that. So you might know Red is more powerful than Green, and Blue is also more powerful than Green, but the relative strengths of Red and Blue haven't been decided yet.
I cant wait for this guy to disprove the collatz conjecture
Appreciate you sticking to the punny naming conventions!
8:02 There is an antichain that is infinite at (0,inf),
which equates the length of the backbone, and thus also is equally infinite.
Ha, that's where my computer screen is so it's how *I* see what is being displayed in the background at any given moment:D
@@DrTrefor It's ok, I'm on gluhwein...
inf isn't a natural number
@@gdclemo If the line of natural numbers is infinite in length, then inf equates the whole line.
@@ZMacZ People keep on saying that, as if Gödel, Church, Turing and others didn't give their undecidability results that make the claim absurd. Or as if Zeno had not prove the issue already before them.
It's very simple. No, we can not construct an ideal straight line from projective shadow (point) of a straight line.
Does the existence of the counterexample rely on the axiom of choice? It sounds like that from the interview.
I don't think so. It is an explicitly constructed ordering on NxNxN.
Good question.
Man this would have been nice for the exam I took 4 hours ago 😭
Ya you would def have gotten an A+ if you just wrote "no fishbone lol"
"no fishbone. Here's the proof:" *writes down the video address, which had been carefully written on top of their desk beforehand. Sadly, because there's writing on the desk they're copying from, the teacher assumes they're cheating and takes the exam away and grades it a 0. You got the bad ending. Try again?*
W. Sorry I passed out there why did I write all that.
Have you seen the video on the moving sofa problem by the channel "Wrath of Math"? (It does not go into Baek's proof in much detail, but it gives an overview of the problem and an outline of the basic proof idea)
Ya that was a cool one for sure, such an u expected shape
@@DrTrefor And yet also expected, in the sense that the shape that turned out to be optimal was proposed in 1992. (Coincidentally, this is the same year that the fishbone conjecture was posed.)
If I make a conjecture called the baseball conjecture and it's proven false and you do a video on it, could you have the video title be "Math News: The Baseball Conjecture has been deballed!!"
wait....so if you have a finite chain, you can either have a finite set (fishbone) or an infinite set as possible side tracks.....how exactly did any of what was said disprove that?
The fishbones can be infinite, and in fact were in the example going up the y axis
all that organic chemistry tutor from learning most of calc 2 in two before my final really brought me to math youtube 😭
I don’t know what’s happening, but I like listening to someone who’s excited about their interest :)
dr. bazett, please try to prove or disprove the Collatz Conjecture.
Brb solving;)
Lawrence Hollom comes off as a wonderful human.
proving -> Nice! I am now more confident in something that I already believed
disproving -> Oh wow… Today I learnt
To me, disproving a conjecture that was around for some time, sounds more like learning than proving it
Cannot C-G create an antichain? 6:23 mins
I don't understand the problem with the (4, 4) point at 9:35. What are the intersections that must be avoided?
So to have a fishbone, we'd need to be able to connect that point to the spine via an antichains. That we could do, but the requirement of a fishbone is all the anti-chains are DISJOINT and so it would overlap with one of the existing antichains.
@@DrTrefor Sharing common top down holistic origin like the root of Stern-Brocot tree is not disjoint at the root of a binary tree decomposed by nesting algorithm, but can look disjoint from the bottom up perspective of additive algorithms and their compositions.
In your cincrete example at 2:40 ish you have dots on every pair of whole numbers. 0 isnt a natural number.
It’s a not-completely-standardized definition, but mathematicians tends a bit more towards including 0 in the naturals, particularly discrete mathematicians.
Hi, I do not have the time to read the paper, so unfortunately I couldn't answer my own question. Do the examples also say something about the truthfulness/wrongness of the conjecture in the case of well partial orders (No infinite antichains+No infinite descending sequences)? This would be interesting to know.
We used diagrams like this for analysing and solving engineering problems, critical paths and stuff. I wish we were taught it with a stronger mathematical underpinning.
Why does 9:43 not work?
I had the same question, the video skips over it really fast so for a non math-expert viewer it's not very clear but I figured it out: for an anti-chain to form it needs to be non-related to both the point on the chain and every other point that connects to said chain-point via an antichain. This goes even BETWEEN anti-chains. This is what tripped me up: the shape of the fishbone graph kinda makes it look like there's two different anti-chains connecting to each point on the spine, but as the video said they need to be "disjoint" aka the two anti-chains can't have points comparable to each other either. I bet that to real math nerds that was obvious, but I didn't realize that is what the video meant by disjoint. The point highlighted at 9:43 is non-comparable to the highest point of the finite spine, but comparable to the point directly below it. Same x, higher y. But that point is connected via anti-chain to the very same point on the spine it would be connecting to itself. two comparable points can't be connected via anti-chain to the same point on the spine, so that's not allowed.
9:05
This isn't really a valid justification, is it?
You could easily construct a finite partition of antichains of a poset (infinite) for which a chain is found connecting them (each at one point).
e.g. from the given example, also remove the points (x, y) for which y
It just so happens that such a partition is not possible given the triangular structure of the lattice shown. But that is not a necessary fact from the choice of a finite connecting chain (leaving infinitely many points left to form antichains).
I'm sorry, but I don't see how your construction qualifies as a fish bone. You've only done it on some subset, but you need to include every point.
@@danielyuan9862 i'm not saying that there is a fishbone for the example given, only that the reason given for why there isn't is not a sufficient condition.
Wouldn’t the two points 2,3 and 3,2 both be greater than eachother?
To be greater both the x AND y coordinates have to be greater, not just one of them.
Nice fact. When fractions 2/3 and 3/2 etc. inverses of consecutive Fibonacci fractions a/b and b/a have their denominators multiplied by two and < b, then a+2b gives Fibonacci numbers and b+2a gives Lucas numbers.
It is possible also to approach the issue from a more general perspective than just coordinate systems, in which it was now presented, and compare comparability also in other contexts.
How cool would it be to study with Dr. Bazett??
how long were you waiting to make this video
That is correct for discrete math.
sir I am a Computer Science engineering student from India and just watched your videos about probability in discrete maths and just love 💖the way you teach that you just teach the thing from basic and cristal clear.So the reason behind I am writing this comment is that i want guidance about how can i become a good engineer that i love to solve mathematics and not getting that much interest in coding and programming so please if you can.😊
You should have shown the actual counterexample!!
It's....complicated:D Not the kind of thing I can easily motivate to a wide audience, but Lawrence shares a bit of the intuition in the interview and you can read the details in the paper itself.
Math teachers are often one small step from being standup comedians.
Love your deboned in lieu of debunked!❤
The problem with "larger infinities" is that these types of combinatorial questions are not well posed for uncountable structures, the answer always depends on what model of set-theory you feel like using today. That's because those larger infinities are not "absolute" in a technical sense, any uncountable set can be shoehorned into a countable one by forcing it to collapse onto the set of whole numbers So this guy is selling himself short, he probably has said the most that can be said about this conjecture for any cardinality..
could you elaborate how this specific problem is not well-posed and how it is different in different models?
@@creativenametxt2960 I didn't read his paper yet, I don't even know what he did exactly. I just know that whatever he did, there's no way "going uncountable" is going to add anything new, because it never does.
Can you explain how there are a countable number of real numbers?
@@thewhitefalcon8539 There aren't.
This is ridiculous, there are tons of theorems about uncountable structures, like the real numbers. Absoluteness is a technical concept, and doesn't mean "every problem has an answer." There isn't even an algorithm for determining which integer polynomials have an integer root, so I don't see how some undecidable questions about uncountable infinities undermines their study.
The title is sheer geniousness 😂😂😂
So b and c are same depth in the network, so equivalence is in the eye of the beholder?
you get to choose the rules
It says he’s a phd student. Is this result sufficient for a PhD dissertation?
My Conjuncture.
For every Mersenne Prime >3, there exists a positive integer x such that 6x+1.
As in each Mersenne prime can be written as 6x+1 for some integer x? Try thinking about the remainders of powers of 2 when you divide them by 6, and what kind of powers of 2 give rise to Mersenne primes…
@malignusvonbottershnike563 what do you want to say.
I like counterexamples more than proofs, because counterexamples show new areas of mathematics that we haven't explored. For example, non-Euclidean geometry is interesting, and finding geometry that doesn't follow Euclid's postulates is much more interesting than showing that Euclidean geometry is the only type of consistent geometry would have been.
Rust users will know all about PartialOrd
SHOW US A PICTURE OF HIS COUNTEREXAMPLE.
The counterexample if quite complicated, but you can check it out in the paper if you like
Does anyone know why he keeps looking down to his lower-right in his videos?
this got anime power scalers mad as hell now
ok, so ... ...so *just where **_IS_* this "counterexample" ??
Visualize it for us pls ?? >=(
.
I…..uh…..it’s too big read the paper:D
That was another conjecture I'd never heard of. So it's provably true in most case, but there's this one weird condition where if if something really bizarre happens it can be false.
That's most mathematics
Not really "most cases", more like the "nice cases". For example, by any sensible measure there are definitely "a lot more" uncountable posets than countable posets, but the proof only applies in the countable case. The other condition also seems to be more on the restrictive side than not, although I could see why it's a "nice" condition.
My issue with your first example is that it can easily be changed to show infinite antichains + fishbone. The written conjecture is fine (no infinite antichain => fishbone), but I don't believe your "exclusive or" presentation makes sense.
The NxN example? That doesn't have infinite antichains. Given a point, anything else on an antichain is either down and to the right OR left and up. But since there is only a finite amount of down and a finite amount of left possible in NxN it has to be finite.
Sorry, I meant the one at 6:37. This example would work even if the y-axis were extended infinitely down (into negatives), no? In which case those antichains become infinite (not just arbitrarily large).
@@Winiumbut it doesn't extend infinitely down
@@thewhitefalcon8539 ...yes thats why I had said "easily changed".
"Roadway?" Is that not a *tree* structure, and {A,B,C,D} and {A,B,C,F} are two *branches*?
This is quite poggers
Conjectures hate this one weird trick.
How about wagon wheel
Finally discrete math comes in useful
lol "finally" what you on about:D
People use discrete math every time they do arithmetic on integers (or integer multiples of some unit).
Discrete math is quite important in computer science!
he says, using discrete math to watch this video
The distinction between discrete and continuous has become very confused in modern mathematics, mainly due to set theory destrying basic mereological intuitions.
Discrete and continuous is not an either-or-relation. Computation theory starts to make sense after we realize that analogical and digital is both-and, they necessitate each other. You can't have digital Turing-Head without analogical Turing-Tape, which is also called "quantum time".
Wow so cool
First the Bunkbed now the fishbone
No conjectures are safe
Hi professor!
I hope you reply to my comment
I want to pursue theoretical physics and am a Physics major(3rd semester) currently we're studying Bessel's functions and frobenius method, laplace transforms etc in ODE.
Since these problems get pretty Lengthy, how much would you suggest i practice and how much do you suggest I focus on the proofs(there aren't many in this course) and also what areas of Mathematics do i need to look into for studying General Relativity?
I do think practice makes perfect, but the big goal is to UNDERSTAND why you are doing what you are doing. That is don't focus necessarily on all the technical details in a long computation, do you understand the big picture of what say a laplace transform and inverse laplace transform is doing to solve an ODE?
@DrTrefor Coincidentally that is the exact question i was going to ask my professor but he'd left his office early today due to an emergency.
I was going to ask him how does someone come up with something that looks so random as if it was a revelation only they knew of and then they were somehow mysteriously guided that this reduced ODEs to algebraic problems!
Thank you very much prof
I remember hoping this was true the first time I learned about anti-chains.
Future Fields Medalist in the making.