Ep 0: "Allll the Numbers??" The Case of the Missing Numbers & Much Ado About Vectors
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- Опубликовано: 14 ноя 2022
- If the surreal numbers are so {great, great^great, great^great^great, … | } then why don't they actually capture allll of the numbers? …and much ado about vectors, to be resolved only in coming episodes.
Next episode teaser: "This was all just 2-Complex, you promised Quaternions and Octonions, and oh my!" Patience attains Episode -1 ( • Ep -1: The Axiom of Ch... ), and lo, a blooming rose by Episode i.
Acknowledgements:
Deepest gratitude again to my amazing family, and their grace, who help me through all of my trials and tribulations. Surreal Physics has been a balm for me in these challenging times.
Thanks also to friends and strangers alike for spending a few precious moments of downtime watching.
And thanks again to @TheoriesofEverything for spurring this channel.
Thank you, Blender and Manim communities (including of course @3blue1brown) for being open, free, flexible, and ever-so helpful.
Additional information:
See description of Episode Null for more links and resources on Conway's Surreal Numbers ( • Ep Null: In the beginn... ), as well as read-aheads for next time and sundry related items.
As mentioned @upandatom has a fantastic video on Russell and Whitehead's Principia Mathematica:
• The 379 page proof tha...
Also as mentioned @veritasium has outstandingly told the history of Imaginary i: • How Imaginary Numbers ...
And @WelchLabsVideo has also created a truly excellent account of i that goes into depth we’ve had No time for here:
• Imaginary Numbers Are ...
For a quick read (for those not quite ready to dig into the scientific literature) check out this lighter article by Charlie Wood (I'm sure there are even more recent ones that are also good, as the evidence keeps mounting): www.quantamagazine.org/imagin... ...such as this one by Alessio Avella: physics.aps.org/articles/v15/7 (but check out the literature references too if you have the time!)
Throughout this video you see a couple of formulations of the Standard Model Lagrangian when I refer to the latest standard formulation of the laws of physics. @pbsspacetime has a wonderful explanation of what this is, and actually many other videos that go more in depth on the elements: • The Equation That Expl... (which also better explains the coffee mug reference!)
If you haven't heard much about Navier-Stokes, @numberphile has a great mini-series on the equations: • Navier-Stokes Equation...
Backwards in time?? Check out this popular-style article by Alfredo Carpineti and technical references therein (and several more from recent years can be found in literature searches): www.iflscience.com/experiment...
Yes, you correctly noted references to The Lord of the Rings, Star Wars, Back to the Future, Star Trek, 2001: A Space Odyssey, Hitchhiker's, and it will not end there. I can't help myself. Thanks for putting up with it.
Humans aren't the only ones who have invented and reinvented the concept of one-third, over and over again. Check out the all too rare trillium, who, like us, seeks impossible beauty in the shade of trees: en.m.wikipedia.org/wiki/Trillium
Some wheels are just too perfectly simple yet unattainable to not keep reinventing.
Stay tuned for more on Clifford's, Noether's, and a whole swirl of others' brilliant works as we try to put this all together in a @surrealphysics.
BTW, of course I still love you, vectors and tensors.
Original soundtrack by Surreal Physics. 
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im guessing the next video will involve the surcomplex numbers? awsome!!!
Yes! But it will not end there!! 🙃
@@surrealphysics SURQUATERNIONS!?!?!?
Amazing work! Excellent balance of explanation, audio, visuals and satire. ^.^
Thank you so much!! And I promise the *-ions are on their way!! ;-)
I found your channel from episode null and I absolutely love these, and from how you mentioned clifford it really looked like you were going to go into geometric algebra even going as far as to mention an i.
Thank you! Oh just wait... we'll get there. You will like Episode -1, which is recorded already, still animating. And actually also Episodes *, 1, and so on... geometric algebra takes awhile, and so do these renders. Now if only continua actually did exist, so they'd be done already! Thanks again!
@@surrealphysics I only recently heard about geometric algebra and thought it might be very powerful if used in conjunction with the surreal number system but from what I have seen so far both of them are still not very prominent, although it seems that surreals are less prominent than geometric algebra as of currently since I can't seem to find much on how the equations for addition and multiplication were come up with and how less than and greater than are defined, only really have some not strong base intuition as to how you would come up with such things.
Yes, I think these ideas are worth more prominence, hence this series. For more about what you are mentioning, check out Conway's book On Numbers and Games, if you're near a library... Or maybe they could order it if they don't have it already. Or if you have the funds, I'm very glad I got my own copy.
Oh I should have mentioned also Don Knuth's book, which is even more readable, where the characters struggle through ideas along with you. He said he was struggling through it himself, so a fun journey along with some great mathematicians!
Oh if you meant the geometrized version of the surreals, then you're right, there's essentially nothing I've been able to find, so I'm working on a paper, but I'd love if anyone finds and shared any existing work!! It's easy to overlook good work.
"i" can't wait for the next video. lol
coolest video ever ty!! 1 0 🎉 that explains everything!
Thank you so much, my friend! ♡
I am inspired to think of the left and right side in Conway's construction of the subject and object, completely dependent on one another, such that the absence of one collapses the other into the void, or the empty set.
That's a great way to put it, especially if you mindfully recognize that the subject and object as L and R respectively can be relativistically inverted, this begins to generate some symmetries for us with very cool implications. But in the very next episode I hope to get out soon, you'll see a fittingly surreal take on the latter part of your comment here. It will be great to know what you think when I can finally get it uploaded.
Why am I only seeing this now??
I feel cheated.
This profound beauty is simply marvelous and the way it is presented embraces it. If only I had words that could rival the beauty of contents of the Video oh I would leave them here as they would be well earned
Ooh thank you so much! And thanks also for effectively telling the algorithm you'd have liked this presented to you earlier. 😉 I deeply appreciate your kind words ❣️ thanks again
It seems that with these videos that you just proved that all of mathematics can be inductively derived in our minds!
Maybe not proved, but I am suggesting that the collection of minds that is the universe may contain many more minds than a human-centric default might be led to believe. We don't know what it takes to be conscious yet, but the answer to this question is increasingly urgent, and thus what structures have the capacity to be conscious is TBD. But a mind, not necessarily conscious, could be very simple indeed.
Mixing geometric algebra with surreal numbers is a very interesting concept - the geometric product allows for anti commutativity so eventually the idea that a multi vector can square to negative 1 - the Conway numbers acting as the field make it interesting - and the space time version of geometric algebra allows for special relativity and em without a problem - i understand that conformal geometric spacetime algebra is needed for general relativity but that there is no need for tensors just multivectors - i really look forward to seeing developments.
Thank you for sharing in the excitement of these ideas! This story is ultimately going to fold and unfold over quite a few episodes, but yes, what the games make possible is not unlike a space from multivectors, but if you can believe it, even more structured (and superposed!), which are extremely resonant and exciting for physics. They evoke these gorgeous mixes of algebra, geometry, and topology... and time! And I just wish I had more of it that the storytelling would be done already! Alas! We seem to have the inductive and coherent universe we do. Anyway, the promise of pulling all this together increasingly bears out the proof, so it will be wonderful to see what you think, both from these more artistic RUclips expressions as well as the technical papers to be posted with the next episodes.
Fantastic subject
Fantastic music and videos
Could the surreal belong in a Hyperbolic space
What a sharp question! Seriously bravo! Said at a whisper, I actually believe the Surreals can build hyperbolic space, not just play within it. But for that we will need more than what Conway calls numbers, so I will say no more to not spoil what unfolds over the next two episodes and beyond. :-) Again, awesome question and many thanks for liking not just the visuals, but the music as well!! I spend perhaps too much time on the latter but can't help myself.
"squeeze it all into a coffee mug" - its this a subtle reference to topology?
Such a great question, because originally it was just a joking reference to how we love to try squeezing the standard model Lagrangian in various forms on tshirts and coffee mugs. But then this turns out to be a great example of how the slow process of art makes me think things and ask questions I wouldn't have in the standard business physics. As I was creating the mug in blender, I realized the funnest way to do so was as a torus, and that got me thinking about "topological frames" with respect to the standard model Lagrangian in a way I don't think I'd have entertained otherwise. I guess you noticed that this mug is even closer to a donut than normal? ☺️ And now I am just over the donut 🌒 that you thought to ask that question--- maybe this is just me, but it feels like an art loop has just made a full turn, if that makes sense. Lol, and sorry if you were after a simpler yes or no! 💗
Cant believe you guys are still under 400 subs! Criminal!
Yes! More episodes coming very soon, so hopefully then the algorithm will show us a bit more love and grace, as it's so clearly very cautious with a newcomer like me. I get it, and it makes me all the more appreciative of comments like yours...I thank you so very much!!! 💕
Literally had been trying to figure out the "i" cliffhanger you left us off on with the last video. Great video and ending! I couldnt figure it out but thought maybe it was related to the continuum hypothesis in some strange way (i got no ide lol. This series is amazing though and keeps getting better. Can't wait for the next one again haha
Thank you!!! And maybe I'm not supposed to answer, since it's a literal cliffhanger, but I'll say that the CH is actually a slightly shorter path to what's next than the average path of the rest of mathematics, given it's all connected, but not directly the resolution...I do hope you like what's on it's way though! :-) Thanks again, and it's nice to hear it has you apondering!
I think I'm caught up now. The wordplay is wonderful, especially in this episode.
Thank you for watching/listening and with a careful ear!
Even more generally than the tensor, the vector is a line drawn on another structure underneath: the bivector, a quantity also indexed by 3 numbers.
This increased generality is quantifiable, as it generalizes to nD vectorspaces.
Edit: Curiously, you mention Clifford right after the tensor.
Yes, we have just begun with the vector-space here. I know these amazing tools aren't going anywhere, nor should they. Clifford's and many others' brilliant ideas I'd like to lift up to the extent possible.
Pure awe.
Thanks!!
Ok, you made me go out and find a used copy of Conway's book. I guess I might as well read along :^) I'm taking linear algebra II this semester, so re-watching this one for fun.
The surreal construction of multiplication looks similar to the summation of dimension of vector spaces and also the probability of the union of disjoint events in probability. I guess that's not a coincidence.
Awesome! I've re-read a few of his books at least a dozen times, some parts many more. They definitely have a richness you'll appreciate. 🎉 My humble opinion is that linear algebra suddenly makes so much more sense with Surreals in the back of my mind, as well as some other geometric algebra / algebraic geometry ideas we'll be folding in with some topology before too long. Enjoy!!
"i" is as physical as the notion of an x-direction in physical space. The (unit) complex numbers put a coordinate system on a circle, which is useful for computation. The circle might be a real feature of reality, but the complex numbers are only a tool. If you are not convinced, substitute all i by -i in every equation of physics simultaneously, and note that nothing changes about our predictions.
Yes, this story is further unpacked in the first part of Episode 2, out already, but in a playful combinatorial game telling that will take several more episodes to more fully unfold. Ultimate judgements of what precisely are "real" physically are suspended ☺️
Rewatching this today. Think I managed to unpack more of what you said this time around. I got reminded of something Kant said in the first Critique; when we try to grasp the rainbow with our senses alone, we are not able to grasp the transcendent object. Another thing he said, in the Prolegomena, is that the absolute in experience is not itself an experience.
Oh how great: my reaction to the first would be I love the imagery of the rainbow Kant invokes here, given the way we process certain qualia, especially drawing insight from those with synesthesias. But noting also a strange renormalization that happens where a rainbow is uncountably infinitely parametrizable (in theory) around a color wheel, in hue, and also infinitely mixable in saturation and brightness levels--- however quantum observations and optics would say physically, no, that such emissions are always quantized, and thus strangely countable, so at most countably infinite in their energetics. But then we have this strange loop that is our qualia (that purple is a mix of red and blue in the color wheel, but at the edge of the wavelength spectrum visible to us). Conversely, we may also think of an infinite cycling of octaves in our perceptions of notes that strangely repeat in kind, including their resonances, which are curiously also both quantized and infinitely scaling. However those "transcendent objects" beyond our sense systems that we might be attempting to grasp, while possibly finite in and of themselves, are still not ultimately completely graspable to our minds, due to those lossy projective signals by which we come to sense them in time. This is possibly where the different meanings of transcendence, in different philosophical versions as well as the more formal mathematical definitions, get confused. I actually think the Surreals, together with the division algebras, have a lot to say about this, and we'll be getting into it more and more here. The jury is out on the reality of uncountable infinities.
As for the second quote from Prolegomena, which I also really like (for lack of a better verb), this bubbles up for me the thought that I often believe: that we might live in a strange and wonderful semi-conscious monism -- a broad kind of physicalism, inclusive of mind and experience, but not so flat and reductionist as we often hear physicalism described -- one that makes room for semi-determined, semi-free dances (a surreal physics?) -- that our universe is made out of something truly magnificently complex enough to give us all this remarkable structure that in parts, experiences all we dream in ways mathematical as well as less precise visions, no less beautiful (of which there may still yet be latent math, e.g., in our visual cortices, and other parts of the brain interpreting its processed signals, math that we are merely not fully conscious of thus far, but may one day better formalize) -- but that this substance of the universe is also still consistent enough to persist those patterns we call both matter and mind that makes up ourselves and others. Some flavor of monism is the simplest explanation for consistency. However paradoxically (and perhaps as an ultimate source/explanation of paradox itself) this substance cannot be simple, mathematically, meaning not just totally-ordered, or else there is not enough room for decisions and ultimately, qualia and conscious experience atop the simplest forms that the substance takes, which we would identify as matter and light. Again, I often believe this may be the case, but ultimately both the math and the physical observations could point me to a related, but different, set of beliefs.
For now, we explore a kind of collapse of the dualism problem where matter and mind are just two ways to look at the same universe coin (a la what Russell and Chalmers point to, without knowing quite what to specifically point to). But now trying here to get much more specific about the maths and physics, which I suspect are simultaneously Surreal and Complex, finitely implementable, cycling in octaves, but infinite in their possibilities. And what I'm working towards proposing should be testable and falsifiable, at least in large part, as much as anything like this can be. Thank you so much as always for sharing any thoughts, Kine! (re. the edit: apparently when you put too many of what you think are em dashes into a comment here, it thinks you want to cross out that segment of the text... some sort of mark down... good to know!)
@@surrealphysics Wow, thank you for the detailed reply! I am looking forward to the complexifying :))
The finite and the infinite, yes, this is such a peculiar feature of our exsistence. Last summer I had this wild thought, that it feels like we view the universe «inside out». Not that I believe we are inside anything, really, but more rationally, that we may be confused about what we really see. It dawned on me that the only thing we see (feel, hear etc.) is what is gauged, right? But as you point out, in a totally relativized system, it is difficult to understand how anything could be measured… because one needs to have one stable frame (maybe self-referential). Then it dawned on me: is it really things measuring things though?! The sensed thing is just a change, can a change measure a change? And the renormalized border, where is it? Then it hit me, we know it, but it is not an object for the senses. We think we see ourselves, but we only know ourselves because change is lawlike. whatever is looking out of me, determines me but cannot itself be determined (from my point of view). it is the homogeneous in the manifold, that we know, but cannot give a representation. We get to know that undetermined aspect with our understanding when it glimpses through via symbols which patterns cannot be described a lawlike behaviour.
Maybe exsistence, without looking at itself (choosing a frame), really is lawless. Then, a look at itself could mean a move from frequentism to bayenism. As soon as you have something determined you have a law right? It is lik God decided to compute and here we are 😅
I cannot wait to hear about the testable aspect. That is seriously cool!!!
@Surreal Physics I just had to share. After this comment I started reading a lot of papers on Newtons epistemology. And it dawned on me that what I had conjured up here actually was something Newton wrote in the preface of Principia. "Relative quantities, therefore, are not the actual
quantities whose names they bear but are those sensible measures of them (whether true
or erroneous) that are commonly used instead of the quantities being measured" Just thought that was a bit funny. Of course form-invariance would do the trick these days hehe
Yes, there are many ideas here: that moving away from a single universal "Newtonian" frame of reference, we found stability in the few discoverable invariants of relative relationships is a true paradigm shift, including in the operative math and its logical translation. Gauge theories in physics have uncovered various ways to formally do this beyond the classical picture, even still with no slam dunk on quantum gravity. We also seem to be doing quite a bit of discrete time signal processing, biased to those frequencies evolution has entangled us with. Better understanding the constancy of light's propagation was great progress.
But even classically, the fact that a mirror shows flipped what would be otherwise from the eyes of another in the same physical locality is a fascinating hint to physicality. That what we get through the senses is not merely a reflection of our shell, is more profound than we usually credit, as much as we affect those signals received. I believe there are indeed several layers of interest to our inside out perspectives, as you say, quite literally at the end of the day, which is actually something I explain in my own way at length in this very next episode to come soon. I describe it in different terms and concepts, but you'll perhaps see some resonances and homomorphisms. This family of ideas feels very Penrose to me, among others, but the conceptual inheritance goes way back. Even more interesting, I'd love to know any non-congruences from your perspective and what you think is interesting but needs work. You'll see soon, I'm working up a scheme to better incorporate peer-review into these episodes, to which I will acknowledge any corrections or improvements in the final version.
That we need each other to gain the fullest and deepest understanding of not just something external to us, but our very selves, is a truly beguiling experience and gives the universe this poetic beauty.
You can think of games as highlighting what can be realest about any interaction, elevating what rules could possibly matter in an existence which means survival in explored possibilities. We normally think of rules making a game, but the full Combinatorial view seems better explained (to my mind at least) that meaningful laws emerge from play. That we could sit there and list a bunch of degenerate, redundant rules, but we try not to, as we like minimal energy surfaces and forms more generally for good reasons.
Intuitively in a game, we cannot escape our given (set of) frame(s), but we can choose to see them differently as they engage with others. We can even choose to understand the rules differently to others, or simply be confused or apathetic to rules not essential to the invariants of play. Basically we don't have to care about laws for them to minimally emerge irrefutably in games. If we find ourselves fortunate with broad agency through many play options in a sum of games, which could be thought of as an ensemble of frames or a "metaframe" (my exploratory language, not convention), then the order of moves chosen across frames controlled/influenced is where limited free will can be found. Those are our determinations, played out in time, which offers a not so "random" source to descriptively probabilistic outcomes.
The Bayesian approach can certainly apply to games like this, to help model how a metaframe would perhaps protect some of its controlled "board space" to model predicted outcomes and learn to make better moves in a "sub-sum" with other players. But here winning is a coarse grain on prediction, and not necessarily taken in the narrow and competitive "zero-sum" sense (to mix game theories).
To be explicit, I'm referring for example to the sense that we have more than one frame of reference swimming around in our brains and perhaps many trillions across the full human body to potentially tap or influence, not to mention the multitudes of our resident flora. Relatedly, I'm fascinated by the research that shows the states of our gut bacteria affecting our mental states.
@@surrealphysics Extremely interesting! I will need some time to digest the reply. With that said, I think you will find the following article interesting, (maybe you know of it): There’s Plenty of Room Right Here: Biological Systems as Evolved, Overloaded, Multi-Scale Machines - Bongard and Levin. All of Levins work I think will blow your mind 🫢🤩 Looking forward to the next video! And yeah wow Penrose, he is something!
I may or may not have gotten irrationally excited when William Clifford was mentioned.
I was wondering if he or his algebra would come up again (probably in later videos). When i was brought up, my main reaction was "which sqrt(-1)? I can name, like 10." Of those 10, one in particular behaves very much like i in many quantum physics equations, while 3 others are effectively the quaternions i, j, and k, and another 3 form the spatial dimensions of 1+3D spacetime.
I happen to hang out in a Clifford Algebra discord server and one of the first things I saw this morning was elastostatics reduced to a 1-liner in Python. Given the genericity of the math involved, the same one-liner can compute elastostatics in an arbitrary number of spatial dimensions and with either Minkowski spacetime _or Galilean_ spacetime. Just to give an idea of what's possible.
Ha! Yes, the Clifford story will unfold over many episodes, and while much more ground will be planted in the next episode, we won't have time to resolve fruit until later. But it sounds like you're already well familiar with where we are headed... but warning, it will not be of a standard flavor :-)
And you ask the same question Conway asks at the top on ONAG about sqrt(-1), and to not spoil, I will say what I have to say about that in Episode -1, which has been recorded for a couple weeks, I'm "just" animating and rendering, which takes many birthdays.
And I love that Clifford algebra discord server is a thing. I had no idea, but that makes sense, along with the one-liners, which agreed, is definitely just a slice of the awesome pi.
Yay! And thanks!!
the problem im having is understanding how to 'add' with the empty set, eg
1 + 0 = {0|} + {|} = {0 + 0, 1 + Ø | Ø + 0, 1 + Ø}
i know from experience with normal real arithmetic that 1 + 0 SHOULD be 1, but in order for the surreal numbers above to be equal to that, the empty set needs to be an "additive annihilator" (because the 1+{} and {}+0 terms need to disappear to get {0|}). ie
∀x(x+Ø=Ø)
this seems like a fair assumption to me but not a lot of sources ive looked up (note: online. i havent looked at any books or papers) explicitly state this assumption
addendum: i know adding sets and numbers is kind of notation abuse, but as i understand it, for a set A and a number x,
A+x={a+x|a∈A}
but trying to apply my admittedly rudimentary understanding of set theory to the case of A=Ø, this would be:
Ø+x={a+x|a∈Ø}
and since the empty set has no elements, it would just be Ø+x=x, no? or since there's nothing to add to x you just get a failed sum and so just the empty set i guess?
I had a hard time coming around to this too, and we'll actually keep coming back to how bizarre the empty set is, so I think it's right to not feel settled about any of this.
But the typical references will explain that the intersection of the empty set with anything will always be the empty set. That without any options to operate with, you are left with nothing still, no matter how many times you operate. These operations are working with sets within the composition, so for an operative component, nothing on one side of the operation will result in nothing.
So this might help reveal Conway's "surreal slash" not as an operation as much as what defines the composition of multiple sets. What's so unintuitive is that our normal language equates zero with nothing so it's difficult to see what that Surreal wedge does. But honestly the typical story explaining how this works in standard set theory is similarly unintuitive in my view.
When u were talking how picky and limited and low fps is nature i felt like u were teasing the matrix hhhhhhhhh😂
Btw love the style, as for the question of what is invented and what discovered is this:
Nothing is invented, we are only discovering things that are in the semantic space, possibilities, and why the universe seems to follow them is cuz we are a quantum observation that is entangled with a computing world that is connected and persistent
Ha! Maybe I was teasing the matrix a little bit 😬☺️🙃😉🫠 and thank you so much for your amazing comments! I think I'm getting increasingly fuzzy on the question of invention vs discovery: I continue to believe that individuals (not necessarily humans!) may have a hand in the realization of structures that have never before existed anywhere in the universe (invention?) Early on, perhaps it was easy enough to invent, and even particles could do it with little-to-no awareness, but as complexity was realized, it took increasing capacity and capability of awareness to invent? And impossible forms that are simple are ubiquitously reinvented? But that everything is still in a sense a discovery, because every step of novelty recognizes a new form that spacetime can assume for the first time ever in some locality, and thus we discover our impact on spacetime with a new way we push on its surface. That makes my current answer a very confusing "yes, no, but, and..."
But that makes me extra fascinated by your ontology, which sounds a little platonic (?) in the sense that a semantic space "pre-exists," if I'm parsing you right? I think of sematic space as a product of rules: where we bring values functionally acting on each other into a generating space, which can fit more or less well to all the values generated throughout our universe--- but probably never all of them(?) (Gödel) So as a computational revisionist, I suspect that computing with general values, rather than numbers alone, will bring us into unruly territory. I'm not sure I have the proof yet though so this is a surreal conjecture for now. If that's not making sense, then I'm really hoping I can eventually explain these ideas adequately in the future videos. You're amazing! Thanks!!
@@surrealphysics
Uk i feel like your concept of computing with general values is my same feeling about how language is the thing, language is only the mapping of our internal samantic space to symbolic ids,
Therefore math, logic, philosophy, art, and every creative idea we might come up with, share something in common, and i am very excited about what kind of results would be of calculating such semantic spaces, what the fork does ai imply? An automated infinite convergence towards the factorisation of every concept? Idk really, the first law of the universe is the existence of an observer, otherwise it would be just a superposition of eternal absurdity, a disjoint group, and observation implies the existence of change and connectivity and math and.. . everything we know?? It really becomes tricky when thinking abt such delicious subjects, but i believe we can factor all the prime properties of a thing and suspect they will all come back to the properties of consciousness cuz a computing mind can only be in a computing universe
Thank you for uploading a new video, SurPh. 🤩
Usually, mathematicians don't want to have anything to do with children, because they think they are at a superior level.
But in reality we have to go back to primary grade to understand that we were all taught that the number 1 means something futile as having 'one apple' and that we can count, and say that we can have two apples in our basket, and we have to learn to write these 'numbers' as 1, 2, 3 and so on.
We don't need a 500 page proof of 1+1=2, it is simply an experiance that we have to learn from people that lived before us, it is an essential part of being human.
When we move on to the next level, we can learn that these numbers can get larger and larger behond imagination, and that we can put all of them on a number line.
But now we have to realise that this numberline is a reminder and a tool only.
And some people will invent subtraction, like when I eat one apple, I have 3-1=2 apples, but when we analyze this, we have to conclude that -1 is a function, and not a number.
So we have to conclude that functions are not numbers, and therefore negative numbers cannot exist in physics.
And superior mathematicians don't like this at all, but they cannot give an example of a 'physical' negative number.
In physics, negative numbers do not exist.
Interesting, this is reminiscent of ancient Greek geometers and their beliefs or framings of negative numbers' nonexistence. We do depart here because I do think of numbers operatively or functionally (hence the proper class of Surreals) and that something like moving "backwards" does exist, even if my backwards is another frame's forward, and no frame is preferred. IMO, relativistic functions are what we need in physics.
Also, just to stand up for mathematicians a bit, the ones I know care for, love, and have quite a bit to do with their children or more generally, and appreciate how profound simple math can be. It is fashionable to do the fancy flashy stuff, if that's what you mean.
@@surrealphysics
I immediately think of negative temperature and negative atmospheric pressure, but second to that, I think your comment smells a bit like going back in time, am I right?
But seriously, negative numbers are the harsh reality in our human lifetime.
My bank for example, subtracts every three months or so, a small amount of money from my balance, as costs for the account itself.
I'm looking forward to your videos about the relativistic functions, and the ones about the backward moving frames, SurPh.
You make very good videos.
@@surrealphysics Mathematicians are super.
I love math, I use it every day.
But we have to remember that we hire a mathematician to help us with a certain problem, independent of our profession.
We all love that, however...
As soon as the mathematicians are among themselves, all hell breaks loose.
So mathematicians have to remember that they work or us, and not the other way around.
The mathematicians are not the crisis in physics, physicists who don't check the work of the mathematicians are.
Not necessarily back in time, but how they project into metric spaces, but I think the squared nature of most measures resolves this. There have however been some experiments hitting science news cycles lately claiming "time reversal" which over the past several years I'm always ultra skeptical of, but actually specific claims technically make sense when it's all at a very small/simple level of quantum systems over quite limited timescales with enormous effort. So I've become only slightly less skeptical thanks to increasingly good evidence. Thanks so much for watching, JW!
What software do you use to make the particle effects? After Effects? Blender? Touch Designer?
Blender alone for the particle effects! They are a mix of hand crafted animation and simulation. No AI. It feels like orchestrating a gravitational symphony, and I highly recommend playing with it in Blender, which is free and open source (sounds like you know that already, but just for anyone else who happens to be reading)! 💕
This is the prettiest, sexyest video that I don't understand at all. Nice work! I'm Impressed.
Thank you!!! We'll be doing more primers on all of this officially complex material 😊 so hopefully it helps with comprehension!
Wow, what a superbly Surreal American @Tebees "whodunnit". The clues are there, almost in passing, but the significance will no doubt be revealed later. The format is great - I also spend a lot of time pausing at each 'clue' and searching the web to find out more, as well as looking up the references.
I was wondering whether, in the invented / discovered table at 22:47, the 1/3 was 'invented' because it is (remarkably!) only produced at birthday ω in the Surreal Number system ? As a result it sounds like a great belief in an actual infinity is needed in order to adopt surreal numbers for physics (speaking as a physicist) ?
Thank you so much on many levels! First, for appreciating and working through the clues. Also for introducing me to @Tebees what a delightful channel, I had never come across! It's actually quite painful to have so much to reveal, and with it all taking so much time--- alas, stuff takes time! I'm also thrilled you're following up on references.
One-third is a fascinating number. It is a simplest of inventions in the sense that a perfect third cannot be quantifiably measured in our experience of the world, but it is very natural to pine for it. Beautiful plants do so in the shade of pine trees, in the formation of their leaves and flowers, and it seems the quark dances of nuclei do so as well. So instantiated numerical infinities can be sought after, in minds and nature alike, but only fully fulfilled in the imagination, as an infinite goal. That's my synthesis anyway.
That might sound like dualism, but it is not meant as such. We don't really understand what minds are yet, nor who all have them, but they seem to be natural phenomena. Somehow they seem to be able to represent an unreachable boundless quantity that cannot instantiate fully in any finite moment, by the structure's own definition (e.g., 1/3). But by definition, multiple frames of reference are sharing in a structure playing out in time that while boundless, is still well defined. That may not be said clearly enough, but I am pining myself for the right descriptions and visualizations for these simultaneously simple yet complex ideas.
@@surrealphysics Very interesting! Have you seen @upandatom, @physicsgirl, @LookingGlassUniverse ? I'm researching Hilbert's open 6th Problem. The associated philosophy is essentially that the universe created humans and therefore the human mind. The human mind invents maths. So as physics models the universe and maths is invented by the created humans who are part of the universe, physics and maths should fuse to become the same thing, in some way. Out of interest I was wondering what is the reasoning behind thinking that fractions, such as 1/3, are not 'finite' and for example "a perfect third cannot be quantifiably measured in our experience". What sort of measurement are you considering and what sort of quantification is being considered? Also what is your view of the number 1 when taking fractions ?
Yes! I do actually subscribe and very much enjoy these amazing creators. (Just somehow Tebees had escaped me!) In fact I referenced/linked/recommended an Up and Atom video in one of mine, and I'm sure I'll get a chance to do the same for the others before long! It makes my job of the story I'm trying to tell so much easier that amazing content like theirs exists.
One third being an invention can be thought of a few ways actually. First off, the Surreal structure of a third is an infinite conversation, you could say. Its construction approaches the one third numerical value in a well behaved way, so we "see" where it's headed, but it takes "Day Omega" to realize, a "day" which I'm suggesting is an invented/imagined day, to "fully construct." However logically and self-consistently, it's actually never "full" or "complete". What is happening in our mind I suspect is a dance of structures in time to point to one third, but they don't manifest that structure perfectly either... But of course that's big speculation.
Another way to think about it is that if you arbitrarily draw a boundary around three identical things in the world and say one is a third of the "whole" you invented, then that is all in your mind, and not really "essential" or "true" to the structure of that part of the world. But that is actually the case for any way of imagining counts as fractions. The Surreals are a guide for me in this kind of structural classification (finite vs infinite), but it's very subtle and ultimately resting on the game(s) that spacetime plays.
So from the above you may already get a taste of my (present) intuition on Hilbert's 6th, much of which will keep unfolding in the main storyline of surreal physics, but I've also written some essays on related topics that I might release here as podcasts (eventually! Time!) But here's a nitpicky question for you in the meantime regarding word choice: there's a lot of talk assuming humans invented math, but what about these countless other minds across time and space? I don't think we give them all enough credit. Clearly cicadas aren't the only ones that can count to double digits (and do much more complex analysis than that)---
I do think we need to keep evolving our mathematical understanding and technological prowess before we can better understand how mind and matter are of the same substance, and this anti-dualist direction is precisely where we are headed on this channel! But not precisely a bland unity either. We all have our own minds, afterall, and it would be a surreal shame to not honor and celebrate these unique experiences we have. Onwards to a true third! 🤓
Will you be going over multiplicative inverses because if you have only one term, (aw^n)*((1/a)w^(-n)) but for something like (1+e)^(-1), long division gets something like 1-e+e^2-e^3... but that isn't close enough with surreals, you might need |No| amount of terms, or is there a way to write it easier. Thanks!
Numbers e+1 and its inverse will be born on day omega+1 and for the inductive surreal definition see here en.m.wikipedia.org/wiki/Surreal_number#Arithmetic under Division however they alter the definition here slightly, equivalently, yet in a way that may be a bit annoying/confusing compared to how Conway writes it up in On Numbers and Games, but hopefully this still gets you on your way. Glad to hear you're making sense of all this! It takes exactly these kinds of exercises (and at least in my case, lots of head scratching). :-)
where are they😫😫😫
Sorry I'm rendering as fast as I can! 😅
-1 comes before +1 for Surreal number order. It matters
Conway would say they are born on the same day. How you draw the tree is like an opinion. If you mean "before" in some spatial sense, that is a matter of perspective and labeling. Everything indeed matters, but how much and for what are some natural follow-up questions.
Check out Geometric Algebra … you don’t need complex numbers. i is about rotation, and you can do it without imaginary numbers!
Ah, you are familiar with where we are headed in geometic algebra, which of course doesn't just include Clifford's work. I agree i is about rotation, and perhaps you noticed i doing quite a bit of rotation as it danced. But where do you think angles, let alone orthogonalities, come from? Conway was no dummy when he invited i to the No party.
@@surrealphysics Your videos are amazing… and the motion graphics ARE impressive.
I appreciate it! Thanks!!
@@surrealphysics ok now “i” get why the i was going crazy, that’s pretty funny
Excellent video, I really enjoyed this. However I don't pretend to take sides in the debate of numbers, formulas, equations, and mathematical functions in general being either invented or discovered. Yes, we have credited various people throughout time with either having invented or discovering them. Yet, I think our initial assumptions may in fact be wrong or taken out of context. Maybe neither of those two choices are valid, maybe there is a third choice that hasn't even been considered... I tend to think that we only rediscover what has already been there all along.
Here's an example, most people today are familiar with the well known Pythagorean Theorem A^2 + B^2 = C^2. This is an equality of the squares of two lengths of the legs of a given geometrical right triangle with the square of the length of its diagonal or hypotenuse. We have credited Pythagoras based on his work at recognizing these patterns and figuring out the calculations and being able to derive this formula.
We know that the properties of right triangles have existed long before Pythagoras ever walked this earth. And since this is a factual realization, we can easily conclude that any and all numbers and mathematical operations have existed long before any of us have walked this earth. What makes it even more intriguing is that numbers themselves don't actually exist in nature as they are abstractions. They are conceptual, a product of the mind. It takes an observer, a consciousness to realize them. Since the properties of geometrical shapes and numbers have already existed long before any human has ever walked this earth and knowing that they are conceptual, a product of the mind the real question then becomes: whose mind do the numbers come from?
It is this kind of correlation that has me fascinated with topics such as information theory, harmonics, symmetry, etc. Just take the Mandelbrot set for example and look at the complexity of the mapping of its outputs and dive into the beauty of its infinite complexity. It's things like fractals that have no bounds, and yet we can see that nature itself mimics these in discrete approximations. Just look at a snowflake, a rams horn, leaves of ferns, trees themselves, the honeycombs of honey bees, ant colonies, the vortexes of tornadoes and hurricanes, and so much more...
Numbers are pure abstractions, and we use them to represent nature and reality even though they don't actually exist in nature or reality, however, reality or nature blindly adheres to them and their properties through the laws of applied mathematics, physics, chemistry, etc... The universe is the way it is simply because an intelligent mind imagined it to be that way. I don't, I can not believe everything happened by just mere random chance. My passion and interests for numbers, patterns, mathematics, physics, engineering, etc. is why I promote and support Intelligent Design. Now my own position on the rationale of numbers, mathematics, physics, etc... has nothing to do with this video or is trying to take anything away from it. I just like sharing my position, my views.
Again, what an excellent video with great visualizations, very clear and clean explanations and representations of things, and so much more. All of the hours and hard work invested into this to give us a beautifully well made demonstration is outstanding. And your dedication to your craft is well appreciated.
Thank you for sharing your views! And thank you for the kind words.
Isn't it surreal that the same structures of infinities in possibly uncountably various forms transcend nature and our minds and back again. Mandelbrot starts to cameo next episode, by the way. It does make one think hard about what the word natural should even mean, especially when it comes to an instantiation of a mind and its ideas, not even necessarily human.
I hear you appreciating the invented vs discovered game for what it's worth (of course it's not really in any sense "mine" but rather one of the great dances of the ages), I definitely imply reinvention in invention and rediscovery in discovery, but it sounds like you might prefer the third way in which I initially referred to imaginary i, that something could be so true, that it is both. Which is reminiscent of your Pythagorean example. Or perhaps in yet another way.
Not necessarily here, if you don't want to make such statements publicly, but I'd be fascinated by your response to the "God of the gaps" criticism of many intelligent design arguments given the collection of your previous comments. Actually that phrase takes on a whole new meaning in the surreal context where one could recognize, as Conway seems to in some irreverently devoted sense, though not in any deistic sense, that the infinite gaps of the surreals are the true ineffable and unattainable infinity "beyond." But this is not much like what Bonhoeffer or contemporaries have meant when they speak of not putting God in scientific gaps that they see being continually "closed." Like you said in so many words, this is all at a layer of meaning that is up for the widest class of interpretations, with very little to ultimately constrain it, where experiential evidence is worth its weight in gold, as clearly something many can't help but ponder in the context of the best math and science of our time.
@@surrealphysics Wow, I truly was not expecting to have such a well thought out reply. I'm not exactly "religious", however, I can not deny the possibility that a Divine Creator is real. And for that, I will use some of the scriptures not to "preach", but to quote them to show my reasoning, and how it actually relates to math, and the sciences. All one has to do is carefully read Genesis 1:1-3. The term infinity is right there in verse 2 where it states: "2 And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters."
In the Hebrew language the word for Spirit is Ruach and it also means Wind, Breath. And in other places and contexts it has also been referred to as Sound, Voice, Mind, Person or Being. And this leads me to think that the Spirit of God is the Conscious Speaking Mind of God. What did he speak of? Light. What's the fastest known object to move through space? Light! What is Light? It's Energy. What is Energy? According to Einstein it is another representation of Matter from E=mC^2.
Okay so where does mathematics, physics, chemistry, etc. come in? Well just the word Created implies the ability to Construct, to Plan, to Build, to Engineer. And what was built or made? heaven and earth. We can refer to heaven as space and earth as matter. Yet before he spoke of light, it was described as being empty and void, shapeless... there was no motion, there was no time, there was no energy or heat... it was a Static Universe of his making until he put it into motion. This happened when he said, "Let there be light".
So from this text alone, we have the very first Chemist, Physitist, and Engineer. And all of those fields requires an extensive knowledge of mathematics and numbers. So this is again my position and how I see things. As for others, it's just food for thought, take it or leave. I do believe in a Divine Creator. There's just too much complexity with too many fine tuned variables that all have to be right and synchronized in such a harmonic way just to be able to support life on this earth. That's not happening by random explosions... That's been planned and engineered!
Thank you for sharing your thoughtful beliefs! I hope all feel comfortable sharing here, and like you, not to change anyone's mind, but to share and hear and be heard. Atheist, theist, and agnostic alike. Religious, non-religious... ambivalent or seeking. We're all on this giant gameboard together, where relativistically no definition of winning may be preferred, and thus kindness and generosity in communication, like yours, is appreciated indeed. Thanks again
@@surrealphysics I agree, we don't all have to agree nor believe in the same thing, yet we still ought to be respective of each other. The ability for people to have options or choices is much better than to be driven by a single narrative.
I think there's too much movement with the animations, and I'd like it to be more synthesized, more pauses between the takes. It's very confusing
Thanks for the feedback, again! This episode in particular conveys a disorienting feeling at many points, so I can appreciate it's too much movement for some viewers. The pace of these will probably not slow much so you are welcome to pause or rewatch, or maybe they're just not your cup of tea. We will have a few more disorienting sequences in future episodes, but increasingly fewer, so if you stick with the series hopefully you find them more watchable over time. Thanks again
@@surrealphysics Yes maybe not my cup of tea indeed :/ I'll stick with it though because it does sound interesting
@@raphaelgonzales3481 thanks again for constructive feedback as well as an open mind!