Sean Carroll: Hilbert Space and Infinity
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- čas přidán 16. 05. 2024
- This is a clip from a conversation with Sean Carroll from Nov 2019. Check out Sean's new book on quantum mechanics titled Something Deeply Hidden: amzn.to/2C6aCaf New full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days. You can watch the full conversation here: • Sean Carroll: Quantum ...
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Note: I select clips with insights from these much longer conversation with the hope of helping make these ideas more accessible and discoverable. Ultimately, this podcast is a small side hobby for me with the goal of sharing and discussing ideas. I did a poll and 92% of people either liked or loved the posting of daily clips, 2% were indifferent, and 6% hated it, some suggesting that I post them on a separate CZcams channel. I hear the 6% and partially agree, so am torn about the whole thing. I tried creating a separate clips channel but the CZcams algorithm makes it very difficult for that channel to grow. So for a little while, I'll keep posting clips on this channel. I ask for your patience and to see these clips as supporting the dissemination of knowledge contained in nuanced discussion. If you enjoy it, consider subscribing, sharing, and commenting.
Sean Carroll is a theoretical physicist at Caltech and Santa Fe Institute specializing in quantum mechanics, arrow of time, cosmology, and gravitation. He is the author of several popular books and is a host of a great podcast called Mindscape.
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This is a clip from a conversation with Sean Carroll from Nov 2019. Check out Sean's new book on quantum mechanics titled Something Deeply Hidden: amzn.to/2C6aCaf New full episodes are released once or twice a week and 1-2 new clips or a new non-podcast video is released on all other days. If you enjoy it, subscribe, comment, and share. You can watch the full conversation here: czcams.com/video/iNqqOLscOBY/video.html
(more links below)
Podcast full episodes playlist:
czcams.com/play/PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4.html
Podcasts clips playlist:
czcams.com/play/PLrAXtmErZgOeciFP3CBCIEElOJeitOr41.html
Podcast website:
lexfridman.com/ai
Podcast on Apple Podcasts (iTunes):
apple.co/2lwqZIr
Podcast on Spotify:
spoti.fi/2nEwCF8
Podcast RSS:
lexfridman.com/category/ai/feed/
"What is euclidian space?"
So nice of Lex to force Sean to take it from the start of all starts.
Having someone in this space representing engineering and computer science is the greatest gift you could have ever given humanity besides your actual work. The questions you ask, the standards you hold people to, the neutrality you maintain and the way that you respond are so different from other platforms that I have stopped listening to many others for conversations about random topics. Even three years from now you're still consistent. Thank you.
Glad Carroll made the point about "comfort" as distinguished from objectivity.
This is awesome! Was hooked immediately by the math theory and thoroughly enjoyed Sean’s ability to meaningfully comment on complex ideas in plain and simple english
In pure math, infinites often reveal details about finite things which were otherwise. Like generating functions for integer partitions of n into distinct parts and odd parts, or the notion of a finite limit of an infinite sequence of real numbers, which is the fundamental idea in all of calculus.
Finally... finally, I have a description of Hilbert space that I can understand. Thanks!
Simple explanation. I needed that clarification, not being a scientist or mathematician myself. Musca gratitude and fist bumps to the host and to Sean Carroll!
Deeply absorbing discussion with Sean Carroll. It helps to have some knowledge of physics to understand his very cogent explanations and analogies, but it also helps to have Lex prompt him with real or feigned somewhat naive questions. I believe Sean and Richard Feynman to be very different people, but I also see parallels in their abilities to make the complex understandable.
I love discussions about the infinite as well as unique and interesting mathematical concepts. I am thoroughly math stupid, but this still is extremely thought provoking.
5:13 SEAN A REAL ONE FOR THIS STATEMENT
AS EASY IT IS TO THINK OF THAT AND SAY IT
NOT MANY PEOPLE CAN ACTUALLY REALLY BE ABOUT IT
I am nearly finished reading Max Tegmark’s Our Mathematical Universe and am seeking a bit more understanding of Hilbert Space. I found this and am seeking to understand the ideas. Thank you for what you are contributing to me and our fellow primates. I do have Sean’s book!
I don’t think he did a great job making the distinction between mathematical Hilbert space and our actual world. He mentions it in the beginning, but near the end Lex keeps bringing up this skepticism about the infinity of dimensions, as if it were a cop out or something. I think Sean could have easily pointed out that Lex may be feeling uncomfortable because he is mistaking a function space for a physical space. Needing an infinite number of basis functions to describe a function in another base is totally common and well founded. Has nothing to do with how many dimensions we live in
Greetings. Max, I think you are right here about Lex's confusion. There is all the difference in the world between abstraction and physical reality, even though we may successfully use the former in describing the latter. (It's one of the ways in which reality is intelligible.) I suspect, though, that Lex was not asking quite the right question. It is not that infinity is 'unintuitive' or 'uncomfortable' as an abstraction. This would be mere unfamiliarity. (I suffer from it myself.) It is that it is hardly clear how infinity applies to physical reality. That's where the mess gets made and where the conceptual difficulties lurk.
For all that, I think this is one of Carroll's finest CZcams appearances. Instead of getting himself messed up with philosophies he only half understands (which he often does) he simply explains a very tricky abstract notion. Being no physicist, I am thoroughly glad to know what is meant by 'Hilbert Space' before I die. Now, in my lamentable ignorance (I am a philosopher, after all - and grieve over it during the long winter nights) all I need from Carroll is a similar clip but clear excursus on what exactly is meant by 'wave function' and the difference between the abstraction and the reality there. I must be asking for much!
Hi, it seems what the host is trying to say is that if a particle, or a a syatem or the whole universe has infinite parametres to describe it, that is pretty mind-boggling. Although, I'd like to think it's true)
In all fairness, Lex is in deep waters here. What could Caroll do?
I was thinking the same! Hilbert space is where the functions that describe a quantum system live; it's not the physical space where stuff really exist or happens. You can extract information about the system in physical space from its representative function in Hilbert space.
I think you could best describe it as there's always an infinite potential in projective space which is a general coherent sphere around its compressed focus. Now the finite is always presupposed, therefore to relate that construct to some higher potential mind you only need a general sphere to magnify its potential for its limited capacity for alternate transformations. Therefore to contain an infinity within reality would be completely redundant and therefore infinity is limited to the necessary space in which is needed to compute all possible states of a living system. Though the Infinity also expands through the resonance to each and every brother/sister star in a reflective capacity. Therefore even though it's spatially limited to its own general spear it's reflectively multiplicative within all spheres within space itself. You can think of this as the wave function of any reality has a nucleus focused in its infinite string of potentiality of which only has a developed interface that extends to a certain degree along that string, whereas the only infinite is the potentiality of the string itself. Though in reality you only need so much as the information can develop and that's why when you look at a wave function you only look at a constant frame of view in a relative frame of time. This is on top of the fact that we see everything through a lens so what we see in the world is already distorted to some limited potential and that lens of our eye represents the crystallized material form of the magnetic field of coherency of which is that higher electromagnetic field of the mind and the lens of which it views all things within ourself.
Therefore you can think infinity is the negative image of the positive interface of which is developed into constructive form within that infinite potential of space from its finite reliable state. Therefore you need an infinite potential to define some limited potential within itself to begin with, otherwise you would be limited in what you could create and you would eventually fill up the whole system and it would collapse or crystallize and never grow or develop or allow for any new creation again.
You ask great /deep questions man and in a humble way, and the fact that you are educated in these fields allows you to press more deeply on these issues so some of these smart men can't "squirm out " as it were.
Most people with a basic college level background in a mathematically intensive field know what a Hilbert space is. It is not that deep and shouldn’t even require a very good mathematician/physicist to explain.
But agreed in general Lex is the perfect man to interview scientists and technologists
@@VedJoshi.. well joe rogan wouldnt be able to ask these questions
such extraordinary explanations
a brilliant brief definition of a space
I usually don’t comment on videos, but as someone that genuinely respects these two scientists, I feel I have to say something.
I don’t particularly feel the question of “What is Hilbert Space?” is a good one.
1st. As I’ve seen in some comments below, there are many types of Hilbert Spaces. The one that is discussed here is L2 over some domain (usually over all real numbers) which is used in Quantum Mechanics since they care about square integrable (lebesgue) functions (which means they can be turned into probability distribution functions). Again, one particular type of a Hilbert Space.
2nd. A Hilbert Space is a complete normed vector space whose norm is naturally derived from an inner product. There are so many inner product spaces that are Hilbert Spaces (and in turn Banach Spaces). As a mathematician, I understand the notion of a Hilbert Space to a physicist is in its utility, so I understand why they bypass the formalisms and go straight to the particular Hilbert Space which helps formalize their mathematics.
Rant over. Enjoy your day!
A Hilbert space is just a complete normed vector space with an inner product.
According to the definition, but ask an expert you get this!
Physicists are funny little myopic creatures. Math rules you.
You can even say that a Hilbert space is a special type of Banach space where the norm is induced by the inner product structure 😂 an even more descriptive definition
@@Tony-cm8lg yea, but he is correct is clarifying that what is said in the video is a specific example of what an Hilbert space is.
I love the idea of a point defined by an infinite number of coordinates. Maybe the most fun game of hide and seek one can have.
Very interesting and worthwhile video.
Lex, our intuitions are limited, having developed only through evolution. Yet we've been able to re-purpose cognition to discover highly useful abstractions that enabled our modern world. We all need to be comfortable with the reality of this situation we find ourselves in and be less overwhelmed by knowledge which cannot be interpreted by archaic modules within our mental apparatus.
I wonder if Dr. Carroll knows house similar his voice is to Alan Alda’s. I keep feeling like I’m watching MASH, really, really smart MASH.
Sean Carroll gave me a clear understanding of Hilbert space.
Interesting.... This is definitely a very physic oriented description of a Hilbert space.... Kinda nice to see it explained differently.
One thing that helped me grapple with the real ness of infinity isn’t that things can be infinitely large, but the question “how many discreet values are there between the number 1 and the number 2” and the answer there is also infinity. There’s no limit to how many times you can divide up that continuum into discreet values.
from me was the question in elementary school ''how many times does nothing fit into something?'' ,maybe it helped cause I assumed nothing as an entity , as something that exists
I love thinking about this stuff. Reality has been feeding my mind so much lately.. The amount of synchronicities I experience a day now is almost getting to be to much. I'm fricken addicted to the rabbit hole so bad but it's so hard to stop. Gotta cut myself off
So cool! As an abstract artist, I feel I need to know more about Hilbert Space. Thanks! 😺
great show!
Starting to understand "spaces", ty.
I've never heard entropy explained as the amount we know or don't know about a system. I'm not sure if that is a way of characterizing whether there are many or fewer degrees of freedom?
Yeah, if I know what you’re going to do (microstate level) you have no freedom. If I have partial info (if it rains he’ll grab umbrella) you have less degrees of freedom than a totally random actor who may grab umbrella in sun or leave it in rain. That’s all it is. Check out Shannon information entropy for more.
when you say entropy do you mean state, scope or both?
If you did know the orientation and position of each water molecule in that bottle, what difference would it make? It's still a bottle of water regardless of how many times you think about it.
Brilliant sound quality. Could you tell me the make of the microphones if possible please.
They are Shure SM7B's.
I'm planning on doing some maths tutoring online. If I want top quality mics i'd probably get one of these; thanks.
If only siraj had seen this video first he might have avoided getting caught by not swapping complex hilbert space for complicated hilbert space lol
jason 😓
Amazing
Lex, I really liked your question of infinity and how you phrased it with context of "comfort". I don't think he got it. I think he was in lecture mode and wasn't ready to listen and imagine. I wish that went differently and the topic could be explored more deeply.
What is Chode space?
Fear not, the details are not lost in infinity, they combine to give infinity meaning.
"Infinity does not exist within time/space. But time and space exist within infinity. 🎶🌌
So, Hilbert space is a mathematical tool, like a stage used to represent a system’s information. And it may or may not correspond to actual space?? It has an upper limit of infinity but the systems represented in it are not necessarily infinite dimensional??!!
Infinity... How long will say a software process take to halt...as long as a piece of string. I agree with lex, infinity, even if we divide it up like cantor, feels like a catch all. A way to claim we know something whereas in fact it just makes current math work. Maybe every spiralling incalculable is a unique fractal that deserves a name
very interesting
Life is the anti-entropic force carrier. Like in war, all the chaos and commotion converges to a finish, a treaty, a winner is assembled and the chaos ends.
When?
10/10.
Space isn't always bigger.
Sometimes its smaller and smaller and smaller.....
Sean Carroll: “Infinity is simple”
Set Theorists: Im bout to end this man’s whole career
0 is the fulcrum of the universe. 0 is the balance point between the polarities of infinity.
This guy is making it up as he goes along. What a quizzing
Why is it so easy to like Sean?
Wait till Lex learns there are different kinds of infinity.
He already knows. He acts stupid so that he can let his guests describe and explain things.
@@OM-el6oy We look at Hilbert spaces and inner products in ML, I think Lex questions this way to regain a new perspective on the matter.
@@OM-el6oy Lex is being way too nice and polite to the idiot SC - who in his last comment reveals that he utterly fails to comprehend undecidability of Halting problem and what that means to notion of infinity.
@@santerisatama5409 Can you elaborate?
@@AG-ur1lj Look into transfinite numbers - they're well-defined.
The thing that Lex is uncomfortable about is that infinity does exist. Which means an infinite amount of Lexs exist, including one (for sure) that is not so bothered about infinity. We eventually all get to the end, but what happens when we get there destroys the ego.
Is he talking about the ether?
Is this guy a uni student?
Such a cool generalization. What do Euclidean spaces and Fourier series have in common?
Wondering about Infinity from a psychological point of view is like asking a brick if your bum looks big in your new jeans.
A Hilbert space is a place where Hilbert can do whatever he fucking want to do with no mom or wife bothering him.
Space.... a broader term for what makes up the greater part of the universe.
The universe.... a term I, and others, use interchangeably with the name/idea/concept/tangible physicality of (to our corporeal forms) God.
Just trying to make ya think ;)
I don't get what he's saying at all, these really are some Complicated Hilbert Spaces
Err, this "complicated Hilbert Space" is too hard. I'd better go back to Athene's "Theory of Everything" video and buy a Udemy course from Siraj.
Understanding Hilbert Spaces is not easy to comprehensively understand without understanding of functional analysis, you must at least know linear algebra. They are vector spaces (with usually infinite basis, infinite dimensionality) that are applied with the inner product which has a complete Cauchy sequence (so you need some real analysis as well).
Shubham Dhingra lol. If you really want to understand what’s being spoken about here, I’d highly suggest watching “The Essence or Linear Algebra” on CZcams. It’s very engaging and entertaining
Siraj ftw
Not complicating. In order to describe a physical space you only need 3 dimensions or 3 basis vectors
. In order to describe other systems, such as economics, you need a lot more vectors or "factors". A dimension is just a vector that isn't affected by another vector. If I move vertically in the y direction, it doesn't affect affect how I move in the x direction. So it's not so hard to understand there exists a
system in which several vectors are independent of each other, meaning they don't inherently affect each other but are necessary to describe the system. For those who are hung up on infinite, math is a tool that allows for an infinite amount of these dimensions or basis vectors. Carroll is trying to say that it's a possibility that a physical system has infinite amount of dimensions used to describe a system, although it's not something that is shown in science yet.
TL;DR : infinity isn't a noun, its a verb.
Infinity is not a number and its not real, just like numbers aren't real. It's a sequence/process/step using symbols (what we call numbers) to represent the quantity of things we can observe. It basically just tells you to do the same thing you did before (n+1 times). Or negative infinity is just to do it in the opposite direction (n-1) times. In programming, I can reach infinity by using a loop with its stopping point at one step ahead of my current step, while increasing the stopping point by one step each time I execute the code. Thus, at every step, I have "doing infinity." When people say you will never reach infinity, it implies there is a target value that infinity has, but infinity has no value because it is not a number, its an algorithm that can be used to get a particular value until some condition has been met, or keep going until the power goes out.
Wow. Sean Carroll is sooo smart!
My professor does not agree that people don't need to know about Hilbert Space in their lives
My head is spinning
He doesn't explain simply.
Wait, so what is Hilbert space? Did I miss something?
To put it simply, possibilities of combining and multiplying vectors.
I think what Lex was getting at is this: Is Inifinity ABSOLUTELY necessary to COMPLETELY describe some phenomena about the PHYSICAL universe? Or is it just an extremely useful tool? It's important to note the PHYSICAL part. Sean Carroll is correct in saying that it's possible for some inputs that an algorithm will never stop. HOWEVER, this is an algorithm being run on a theoretical machine. A real machine will stop eventually due to entropy.
Let's take space as an example. Maybe space is quantized at some level that is not possible to measure, and therefore useful to consider space continuous. This can limit the smallest possible wavelength of light and interactions between subatomic particles. I'm not a physicist, but its at least interesting to think of lol.
Obviously, infinity is so useful that it would absurd to remove it. No one in their right mind would do that, but I still feel like the question of whether infinities really exist in the physical universe (not the space of ideas or numbers, etc because obviously in that space it does exist) is still a valid question. I am not fully convinced that infinities really do exist in nature.
Tyler Heers
How many points in space exist between here and the moon? Hint: between any two points there are more points. That’s real, at least as real as it gets. There’s an example of actual infinity in reality. Actually there are infinite points between the 1 and 2 on your ruler, so you can see what one type of infinity looks like right in Front of you
@@deanodeboI think the point he's trying to make is that an infinity might not exist when your counting things in reality like planets or particles in the universe or worlds in the many worlds theory. I think lex makes this point also that he thinks people use the word infinite as a cop out instead of doing the math or saying I don't know
Can infinity exist in a finite system?
Does "zero" really exist (i.e. we have never been able to find "nothing" (yet) or I have no apples instead of I have zero apples). A system of nothing is not a system and a system with something doesn't have nothing.
Although useful constructs to help us understand SOME parts of systems but has never lead us to understanding ALL of the system. I believe a math revelation will first need to be found before we can have a better/complete understanding of physics.
Thank you Issac and Gottfried for calculus : ) patiently awaiting the next revelation as it'll likely provide more complete answers to these questions ; )
"...You wrote a five line program-it doesn't halt." What an ironic way to end a video on infinity!
Here: goto Here;
Entropy is what sean caroll does not know and it increases over time, according to thermodynamics. Checks out.
👍
This guy does an excellent Howard Stern impression.
Entropy is not Atrophy
The mean of all kinds of functions is 0
he sounds like alan alda
Moving my hands towards the keyboard i traverse an infinite series of halves of the distance, all in the Now which is infinitely short in duration. My intuitive Euclidean space is conceived as infinite and perhaps cannot be conceived but as infinite. is not infinity all around us.
@@dy_physics9183 sure, convergence is an Infinite process , hence infinity all around us (you cannot metaphysically comprehend motion otherwise. See Zeno et al.). and convergence has very little to do with time or the present which is Infinitely short. convergence has very little to do with the (perhaps necessary) Infinity of space. again, infinity is all around us.
@@dy_physics9183originally i was posing this question to Lex who, if i understand him right, has problems with the notion of infinity being a part of the real physical world. my claim is that infinity as a concept and as a property of objects and changes (motions) in the world is not problematic. (meta)physically there is infinity everywhere. i claim that this hold for (strictly)physically as well.
Is the interview high on something? He seems unsure, slurring his words, maybe confused about his reason for being there, and kind of all over the place. It's difficult to follow the path of questioning. I hope he's ok :(.
He is very intelligent. It is odd to see someone who can use this amount of there cognition. He closes his eyes to ponder and produce the correct wording. Please be respectful to computer programmers and mathematicians and physicists and professors of any degree. You are not on their level of intellect and it is easy to gang up and bully people who are better than you.
Physicalist-formalist notion of 'actual infinity' cancels possibility of motion - Zeno.
Undecidablity of Halting problem is coherent with 'potential infinity'.
Lex is giving Carroll's mumbojumbo infinity too much slack.
Is he kind of talking about the sum of the tenants?
why Lex is talking as if he is on something
There is no room for physicalism aka philosophicalnaturalism , quantum mechanics inevitable shows the ideia of mind as ultimate reality .
If John Mulaney was a physicist instead of a comedian.
There is nothing natural about infinite. There can be no location in space or time in an infinite universe. In essence, if the universe is infinite in age, then it would have taken an infinite amount of time to reach this moment. Infinite cannot be reached by definition so by definition the universe and time itself had a beginning. You can apply the same logic to physical space
Come on.. infinity is not a number.
I'm so interested but I'm too dumb to really comprehend what's being said. Fuck.
I must say the interviewer has no intuivitive sense of how simple the concept of infinity is. Being uncomfortable with Infinity means you just dont understand it. There is no paradox or confusion regarding infinity.
My guy is a research scientist at MIT he has a PHD in computer engineering I absolutely guarantee he has a better understanding of infinity than you do buddy.
@@vicslav4030 maybe he never got to understand the concept of infinity in all this time.
@NumbToons I think the more plausible answer is that you have a very limited understanding of what it means. I mean have you ever tried to write an AI computer program that uses the concept of infinity? No? Then have some respect. I'm sick of naive commenters thinking they know more than any professional. This is the kolch and Doppler effect in action. The stupid think that they are smart and the smart think that they are stupid
@@vicslav4030 I don't have anything against you or the interviewer, i understand what you are saying. Dont worry, im well aware of my lack of knowledge and intelligence. I might be able to convey my point across in a good real life conversation. CZcams comments are just suitable for arguments, not discussions. Have a good day sir.
@NumbToons I'm sorry bro these youtube comments are virtually the only human interaction I have. I'm pissed off cause I have 5 years to live if I'm lucky this disease will rot my brain just like my mother which is terrifying I can't talk to anyone cause it's so sad.
His definition of a Hilbert space is plain wrong from a mathematical point of view... A Hilbert space need not be infinite dimensional at all, and need not be restricted on quantum wave functions.
EDIT: I get that he tries to explain these in layman terms, but he could've done so, without being factually inaccurate.
I argue that you CAN'T do anything mathematically with infinite, and that this is where brilliant people run into trouble. Infinite is not an actual numerical value, and hence cannot be plugged into an equation. "2 times infinite" would hence be as nonsensical as "2 times happiness".
Infinite refers to something never ending, which is the real crazy part. We could swap the word with "eternal", and it becomes more clear why trying to make it a math term is silly.
Infinity and zero point to the same thing
I understand why there is a need from the public opinion to have a good analogue for mathematical objects, but at the end those are mathamtical objects. There is a definition for them and that's it. That is what should be learned in the school, that the mathematical truth is not a part of this material world and every analogue is faint and misleading.
Aren't you triggered by infinity?
I know Dilbert space.....
Theoretical physics seems very redundant, seems just like a bunch of mathematicians circle jrkin ideas and numbers. Like they're trying to get their peers to validate and agree that they're theories are viable. Then they can attempt to ask for more money , grants etc.
You sound bitter.
Search CZcams for "The ELEMENTS in six dimensions, arranged by volume periods of nuclide mass averages"
Well the computer programme is not gonna run infinitely long, computer needs energy to run and computer, funny enough, will disintegrate due to entropy, so no, it is not gonna run for ever .
that's true. it will run until the cpu overheats and possibly catches on fire. lol.
He didn’t actually answer the question correctly
what did he say that was wrong?
Looks and talks like Steve Buscemi.
Imgonnagogetthepapers getthepapers
Shut up.
Any way stroke my long ☝️
But it ain’t really a thing
while(5==5) //sample C++ program that would print out hello word an infinite amount of times
{
cout
Space allows geometric analysis for one thing
The ideal of reality being infinite is so philosophically and logically obvious that it seems almost comical to listen to supposedly intelligent people grapple with the concept. In what thought experiment can one travel to the edge of space and time and run into a brick wall with a sign that says " Universe Ends Here"?🤣
A finite sized universe doesn't mean that you'll get to a point where you can't keep going anymore. If the universe has a finite size, you will eventually come back to the same point if you keep going in one direction. It's like how the Earth has a finite radius, and we found out about this when people travelled around the world and got back to where they set off.
@@rickzegooene2193 So if the finite universe is round, then in what space does the sphere exist?
@@pcpoliceliveleak5735 Our current understanding is that the curvature is intrinsic to our spacetime itself. There isn't some other higher dimensional space that our universe somehow wraps around into a ball in. Or, I should say, we don't know if it exists, but there's really no reason for it to exist. It may be an unintuitive idea, but the geometry of spacetime can be contained within itself.
I knew this was going to be disappointing.
This has very little at all to do with life in reality. They are playing with math, as mathematicians do. But the interviewer seems to want it to be some kind of profound view of reality. Yes math is fascinating, and I studied it for years and that is my academic background. But you will not find a spiritual grounding in it.
The far reaches of specific narrow fields of mathematics are filled with mathematicians that use mathematical analysis and synthesis to find abstract mathematical structures in their research. However, it is a waste of time for a layperson to try to have a discussion about these issues. Instead, it is for colleagues in the field to assist and review each other‘s research purely for the purpose of making progress.
At the point where he is asking about infinity, this discussion might as well be one guy speaking German about bridge building with another guy speaking Swahili about sheep herding. Honestly, we know as much about the number one as we do about the concept of infinity. You could spend an entire life studying the number one, and never run out of genuine research direction