Why doesn't it surprise me that the lichess community knows stuff like this?
i may have once known it. i don't mind revising it if that's not imposing.
I may have been used to be spoiled with working with spaces having it all without needing to know where each property came form atomically (a la bourbaki perhaps. knowing exactly the minimal structure required for a given property, does not mean that one can't get that property from a richer structure already implying the minimalist or parsimonious structure.
But i did forget, because my typical spaces of interest were all rich enough, mostly real euclidian spaces, or function space with doman or images also taken from those rich enough spaces (and spaces like probability spaces, also not minimalist in its basic constituents), that not all topological spaces could be separable. One has to show the existence of a boundary, at least one, would that not be that dense countable subset. Could we not avoid the dense word and just go around it with just using open set coverings. exisence of or absence of?
But you post does replace things. thanks. I just don't remember the focus on the dense word. perhaps it has synonym of one could refer to the same subset by not being able to cover its interior (there i get it, i can't even talk about interior without first having existence of such a D dense and coutable (i still remember internally and consciously what countable meant).
I guess now i have enough motivation, to search by myself if my recollection of separability indeed could have been obtained wihthout needing the intermedate step of dense (while obviously it would amount to that any way).
i did not finish my poiint about rich enough space though.. did i .. getting tired. thanks for humoring me so far.. that is stimulating,. specially the last statement. And the topology from the order, I don't replace quite yet. do you mean natural order and addition?
I think i can be lazy and take a rich enough already fully constructed space. and have all the topological properties i need to build things that my intuitoin from having human experienced evolved in would have me build... I must be an applied mathematician finally. able to understand the foundational ones, but quick to forget where things came from if i can already use them to build other things describing nature phenomena...
But i appreciate being able to go purist again.. and understanding some of what you describe. or understand what i don't understand. that's already something. going to sleep and dream about disconnected topological spaces.
but btw did you not just go into number theory.. (not the tough part or most abstract part but still number stuff). i like saying stuff in a math context..... ticking.
I may have been used to be spoiled with working with spaces having it all without needing to know where each property came form atomically (a la bourbaki perhaps. knowing exactly the minimal structure required for a given property, does not mean that one can't get that property from a richer structure already implying the minimalist or parsimonious structure.
But i did forget, because my typical spaces of interest were all rich enough, mostly real euclidian spaces, or function space with doman or images also taken from those rich enough spaces (and spaces like probability spaces, also not minimalist in its basic constituents), that not all topological spaces could be separable. One has to show the existence of a boundary, at least one, would that not be that dense countable subset. Could we not avoid the dense word and just go around it with just using open set coverings. exisence of or absence of?
But you post does replace things. thanks. I just don't remember the focus on the dense word. perhaps it has synonym of one could refer to the same subset by not being able to cover its interior (there i get it, i can't even talk about interior without first having existence of such a D dense and coutable (i still remember internally and consciously what countable meant).
I guess now i have enough motivation, to search by myself if my recollection of separability indeed could have been obtained wihthout needing the intermedate step of dense (while obviously it would amount to that any way).
i did not finish my poiint about rich enough space though.. did i .. getting tired. thanks for humoring me so far.. that is stimulating,. specially the last statement. And the topology from the order, I don't replace quite yet. do you mean natural order and addition?
I think i can be lazy and take a rich enough already fully constructed space. and have all the topological properties i need to build things that my intuitoin from having human experienced evolved in would have me build... I must be an applied mathematician finally. able to understand the foundational ones, but quick to forget where things came from if i can already use them to build other things describing nature phenomena...
But i appreciate being able to go purist again.. and understanding some of what you describe. or understand what i don't understand. that's already something. going to sleep and dream about disconnected topological spaces.
but btw did you not just go into number theory.. (not the tough part or most abstract part but still number stuff). i like saying stuff in a math context..... ticking.
Hum not sure what you're trying to say or to do.
Every topological space can be covered by open subsaces.
And I'm pretty sure you can't define separability without mentioning density.
Also "big" spaces of functions tend to not be separable, because they are too big to contain a countable space that is dense.
Maube you're confusing with the notion of "separated".
I did not go into number theory at all.
Every topological space can be covered by open subsaces.
And I'm pretty sure you can't define separability without mentioning density.
Also "big" spaces of functions tend to not be separable, because they are too big to contain a countable space that is dense.
Maube you're confusing with the notion of "separated".
I did not go into number theory at all.
@PxJ said in #33:
> Hum not sure what you're trying to say or to do.
>
> Every topological space can be covered by open subsaces.
>
> And I'm pretty sure you can't define separability without mentioning density.
>
> Also "big" spaces of functions tend to not be separable, because they are too big to contain a countable space that is dense.
>
> Maube you're confusing with the notion of "separated".
>
> I did not go into number theory at all.
fine, you win! i have a tendency to confuse definitions when their juggling numbers get too high.... i need intuition to take over then... I am one of those doomed souls lost to foundational mathematics.
But you are right that when considering function spaces, one might need to ask, can i still use my human intuition born of physical spaces constructs, to guide myself there? Be aware of the possible caveats, but still make hypotheses with intuition... The trade off with the matter of number of critter definition math world, is that one has to spend a lot of time away from richer spaces, to rebuild their intuition, and then the distance for their life time possibility to talk about the more physical space mathematics and build models of things more complex gets reduces.. there are worlds of mathematics than can't talk to each other any more. I am glad that i can still talk to you, though.
I hope you are going to stick around lichess for a while. I might have to check with you at some point about tools I may need but i am too lazy to go foundational about (or too motivated by what I could do with those).
Would Hilbert spaces of the most vanilla variety cause topological problems of the global optimization variety?
how much of what i can use in R3 euclidian spaces goes down the toilet? probability spaces, measure spaces?
please contact me. or we could make a thread for refreshing such notions. To start you with, I would go looking at Markov Decision processes the basic framework of that NN techonologies of A0 and LC0 use, with the space construct used in the A0 science paper. NN are basically embedding chess positions and actions in mutlidimensional euclidian spaces. Where the MDP can be still formalized.. The computer algorithms are approximations in silica of that framework (from point of view where i have been used to do things). Basically probability spaces, but measure spaces might be easier for me for consider (i do like to go from big to just enough smaller to make things work, so I do need to think about some foundations sometimes).
my confusion might be something annoying to someone who has not practiced proof discovery for a long time and only remembers the polished construct at their finger tips.. Me i am mostly always in doubt... but with proper benevolent critical bouncing back i can be put back on some acceptable path....
I might have tangled a few concept that were more active in my memory 20 years ago.. but there are not gone, just some sedimentation that i am sure you could help clean if needed at some point.. I would keep pulling for rich but not richer spaces (by rich i don't mean big, i mean rich with useful properties for models of phenomena under study to work without always looking under our feet for foundational weaknesses).
so we are good.. you are solid with your knowledge... I am not with it.. i don't know what I will need a review might be good at some point..
> Hum not sure what you're trying to say or to do.
>
> Every topological space can be covered by open subsaces.
>
> And I'm pretty sure you can't define separability without mentioning density.
>
> Also "big" spaces of functions tend to not be separable, because they are too big to contain a countable space that is dense.
>
> Maube you're confusing with the notion of "separated".
>
> I did not go into number theory at all.
fine, you win! i have a tendency to confuse definitions when their juggling numbers get too high.... i need intuition to take over then... I am one of those doomed souls lost to foundational mathematics.
But you are right that when considering function spaces, one might need to ask, can i still use my human intuition born of physical spaces constructs, to guide myself there? Be aware of the possible caveats, but still make hypotheses with intuition... The trade off with the matter of number of critter definition math world, is that one has to spend a lot of time away from richer spaces, to rebuild their intuition, and then the distance for their life time possibility to talk about the more physical space mathematics and build models of things more complex gets reduces.. there are worlds of mathematics than can't talk to each other any more. I am glad that i can still talk to you, though.
I hope you are going to stick around lichess for a while. I might have to check with you at some point about tools I may need but i am too lazy to go foundational about (or too motivated by what I could do with those).
Would Hilbert spaces of the most vanilla variety cause topological problems of the global optimization variety?
how much of what i can use in R3 euclidian spaces goes down the toilet? probability spaces, measure spaces?
please contact me. or we could make a thread for refreshing such notions. To start you with, I would go looking at Markov Decision processes the basic framework of that NN techonologies of A0 and LC0 use, with the space construct used in the A0 science paper. NN are basically embedding chess positions and actions in mutlidimensional euclidian spaces. Where the MDP can be still formalized.. The computer algorithms are approximations in silica of that framework (from point of view where i have been used to do things). Basically probability spaces, but measure spaces might be easier for me for consider (i do like to go from big to just enough smaller to make things work, so I do need to think about some foundations sometimes).
my confusion might be something annoying to someone who has not practiced proof discovery for a long time and only remembers the polished construct at their finger tips.. Me i am mostly always in doubt... but with proper benevolent critical bouncing back i can be put back on some acceptable path....
I might have tangled a few concept that were more active in my memory 20 years ago.. but there are not gone, just some sedimentation that i am sure you could help clean if needed at some point.. I would keep pulling for rich but not richer spaces (by rich i don't mean big, i mean rich with useful properties for models of phenomena under study to work without always looking under our feet for foundational weaknesses).
so we are good.. you are solid with your knowledge... I am not with it.. i don't know what I will need a review might be good at some point..
more specific than just Hilbert spaces. they require a scalar product.. but now I am not sure that the expression power of neural network theorem uses that. However i am pretty sure about resnet proof of universal function approximator, needing a function space for its proof. (universal being there). so basis of functions, but not necessarily Hilbert space. Or maybe there are Tensor ways to build such structure. Safe is to assume not, without loss of generality, to the contrary.. if no hole in topological tools in the less rich structure, i would hope that adding a nice Hilbert metric might smooth things even more.
I guess the most direct thing i could say. is to consider that type of neural net architecture (or even wider deep nets), when taken in full symbolic paramétrisation, (i.e. an algorithm scheme or familiy if requiring that point of view), a basis spanning the function space we would need.
I guess, we just have to look at the theorem proof.. would the space involved there lack topological machinery for my R3 borne intution modulated by my non intuition math education in the past, but adapted. maybe not as low level truth as you would have gone.. but that is why i ask you.. hoping this makes sense. not urgent, not yet back on my active radar, just making sure you would remember the context if you stick around and that topic is of mutual curiosity.
I assume you are on lichess because of loving chess , the things that move on the board and all the generated possibilities.. sure they don't look like nice connected sets of positions at first... but that is not a problem..
Also please put prefix educated, when i make guesses. by saying i guess I let the reader know that i can be wrong and would welcome constructive confrontation or adjustments by people who don't guess about these things for a living (or daily life activity). I don't pretend to have the truth. more making hypotheses with my limited vocab.
I leave you with that suggestion to go read about resnet universal approximatin theorem, and perhaps the google science paper for how a game with apparently discrete and finite size set of states, can still be transformed conservatively into bigger smoother representations where one might have the tools needed for full topology complexity expression. not the totally disconnect topological space.. remember that while the positions are one move hamming "distance" away from each other in a legal game path, the main story is about the evaluation and how the probabilities given a pair of move generators (with variable taken from move quality distributions) of outcome change. How small could such change be, is a different story than how many square got mutated in a legal position transition.
I guess the most direct thing i could say. is to consider that type of neural net architecture (or even wider deep nets), when taken in full symbolic paramétrisation, (i.e. an algorithm scheme or familiy if requiring that point of view), a basis spanning the function space we would need.
I guess, we just have to look at the theorem proof.. would the space involved there lack topological machinery for my R3 borne intution modulated by my non intuition math education in the past, but adapted. maybe not as low level truth as you would have gone.. but that is why i ask you.. hoping this makes sense. not urgent, not yet back on my active radar, just making sure you would remember the context if you stick around and that topic is of mutual curiosity.
I assume you are on lichess because of loving chess , the things that move on the board and all the generated possibilities.. sure they don't look like nice connected sets of positions at first... but that is not a problem..
Also please put prefix educated, when i make guesses. by saying i guess I let the reader know that i can be wrong and would welcome constructive confrontation or adjustments by people who don't guess about these things for a living (or daily life activity). I don't pretend to have the truth. more making hypotheses with my limited vocab.
I leave you with that suggestion to go read about resnet universal approximatin theorem, and perhaps the google science paper for how a game with apparently discrete and finite size set of states, can still be transformed conservatively into bigger smoother representations where one might have the tools needed for full topology complexity expression. not the totally disconnect topological space.. remember that while the positions are one move hamming "distance" away from each other in a legal game path, the main story is about the evaluation and how the probabilities given a pair of move generators (with variable taken from move quality distributions) of outcome change. How small could such change be, is a different story than how many square got mutated in a legal position transition.
@PxJ said in #28:
> Nah it's topology.
>
>
> Yep but that corresponds to choosing a direction east-west, not an orientation left-right.
>
>
> It means its closure is the whole space (ie every open neighbourhood of every point in your space contains a point from your dens subset). Doesn't matter for the matter at hand anyway. Was just nitpicking on terminology.
>
> Of course I am right. If you look at your friend in the eyes, your left will be his right and vice-versa. Because the notion of left and right is defined with respect to the direction (vector) in which you are looking. It is exactly the same as the notion of orientation in the 2D space.
You are FC-UK, aren't you?
> Nah it's topology.
>
>
> Yep but that corresponds to choosing a direction east-west, not an orientation left-right.
>
>
> It means its closure is the whole space (ie every open neighbourhood of every point in your space contains a point from your dens subset). Doesn't matter for the matter at hand anyway. Was just nitpicking on terminology.
>
> Of course I am right. If you look at your friend in the eyes, your left will be his right and vice-versa. Because the notion of left and right is defined with respect to the direction (vector) in which you are looking. It is exactly the same as the notion of orientation in the 2D space.
You are FC-UK, aren't you?
Well, yes. You are. And FC-UK just got chatbanned. So now you are posting.
@mixK9 said in #38:
> Well, yes. You are. And FC-UK just got chatbanned. So now you are posting.
Oh, that's why I wondered why I haven't debated with him for quite time!
> Well, yes. You are. And FC-UK just got chatbanned. So now you are posting.
Oh, that's why I wondered why I haven't debated with him for quite time!
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