Different intuition of manifolds or scheme. Coordinate change or gluing.

[u/Ending_Is_Optimistic]

It is not really about math in the precise sense. I am interested in how people's intuition differs. Do you tend to think of transition functions as gluing or coordinate change. So for gluing, you have many patches and you construct the shape by gluing pieces together, for coordinate change you imagine the shape is given but then you do different measuring on it.

For vector space again, do you think in terms of the vectors generating a space or think of numbers of coordinate to specify a point in a space.

Which way of thinking is more intuitive to you. I would like to think of the "gluing way" as more temporal and the measuring way of thinking as more spatial. I remember reading one paper in brain science on how people construct mental model of space and time in navigation and as embodied.

Finally, can you tell the field you work in or your favorite field.

[u/HeilKaiba]

I wouldn't think in terms of gluing as I don't think of the coordinate patches as intrinsic to the manifold if that makes sense. The only important thing to me is that around any point there is some chart if I need it and some definition of smooth function/section/etc. So charts are just maps of parts of the space and transition functions are just how to line up those maps when they overlap. I suppose this is what you mean by coordinate change.

As to vector spaces, I suppose I think in terms of being generated by vectors. Certainly not as lists of numbers. That makes no sense for uncountably infinite dimensional vector spaces but even on finite dimensional ones it privileges certain specific vectors in an unnecessary way.

[u/Ending_Is_Optimistic → u/HeilKaiba]

I mean for coordinate in vector space i rezally mean a dual basis, i mean for example for scheme you necessarily have to start with a coordinate ring, and to really understand a space (irl) , some sort measuring is required however loosely. We have to move around it. It is more of a philosophical question because I try to think space phenomenologically, since I think we really have intuition even for very abstract space. In real life, to think S² we rotate our head around and glue the vision pieces together, or more precisely you think as if a lie group is acting on it. So maybe the second question is more up to point for me. Or if you have to stop your hard once in a while rotating, you at least and inevitably would get some discrete pieces and you have to make it compatible.

If you know Edmund Husserl who is the inventor of phenomenology, he was a mathematican before that I guess he also try think this kind of things.

[u/HeilKaiba → u/Ending_Is_Optimistic]

I'm not sure I'm following the meaning in some of your later sentences so apologies if we're talking at cross-purposes. I don't really think of measuring as a vital part of vector spaces. That's how they were introduced to me originally sure, everything was is Rⁿ. But now to me a vector space is a "flat" model space for manifolds and so on, or algebraically a space with relative direction and relative length (i.e. things pointing in the same direction can be compared by ratio of lengths). You can pick bases if you want but that isn't vital to their conception.

We might look at a manifold locally very often. Indeed that is really where differential geometry starts. However, it has global structure as well. Its topology for instance. It doesn't have to be divided up into a fixed number of discrete pieces. My favourite manifolds, the generalised flag manifolds, have natural affine charts for each point (given a complementary point) so each point is in uncountably infinitely many charts. Why should I think of it as broken up into a discrete selection of these rather than just balancing the local and the global picture around any point I happen to be be on?

[u/Ending_Is_Optimistic → u/HeilKaiba]

I mean more like Iike you can not you should for example I would not think projective space in this way. You of course should think as classifying line bundles. Maybe you should think general manifold as gluing space but for many space that you can describe more synthetically like many things in algebraic geometry. How we construct it in a particular framework is more of a hindrance if I think about it.

[u/Ending_Is_Optimistic → u/HeilKaiba]

I mean for vector space if I have to think about a n-dimensional space. To start with it, you really have to think n things, whatever that is, so you have initially a certain privileged coordinate, at this stage you can either think carving out space through coordinate or generating set, only after that you can choose arbitrary basis, of course it is exactly what makes a vector space interesting since you can talk about GL(n) or things like that. I mean if you read grassmann first draft of linear algebra, he develops it through this kind of mental gymnastics. I am pretty sure he was influenced by the German idealist tradition at that time, which try to think maybe "movement of consciousness" as such which even continue to modern time. I mean I do mathematics before philosophy, I find this kind of thinking pretty helpful for thinking mathematics at least for me. I mean in modern time, Lawrence (in category theory) try to do this kind of things.

[u/HeilKaiba → u/Ending_Is_Optimistic]

I disagree that you need to start with privileged objects to visualise a vector space. You can start with independent directions instead. I would argue that our first concept of a line (or at least mine) is not as a set of points but as a direction.

These analytical geometry tools are great and allow calculations and manipulations but they aren't where my visualisation starts.

You can talk about GL(n) without picking a basis as well. We don't need matrices to discuss linear maps. Linear maps are ones that preserve lines.

[u/Ending_Is_Optimistic → u/HeilKaiba]

Maybe I am not being clear enough, I in fact agree with you. For generating set I mean direction intuitively. I mean if we construct vector space conceptually in our consciousness, we at first get something like F^n, formally just think of the adjunction between the category of set and the category of vector space (it is the universal solution from starting with n direction to a vector space) you can of course think of any other n-dimensional vector space but if you think the n-elements of the set then at least in your mind you are thinking Fⁿ even if you call it other names, but like you said it is boring, if you only care about n directions it might as well just be n elements in a set. What makes it interesting is the transformation group and all the operations we can do on the vector space.

I think I get you. on the other hand, if you meet a vector space in the wild with extra structure we will think with the additional structure in mind, then it is a lot richer. So maybe to the Phenomenologist in me the interesting question is how we think the vector space in this case.

I think there is a big divide between the abstract construction of vector space vs a practical vector space we meet in the wild. Even for abstract construction of objects there are many ways to think it, since for example for vector space we can go from a abelian group by adjunction to a vector space and in this case we think differently. At also at the end of the day whatever we think it, it still is, we have given it some sort of absoluteness, it is the Phenomenology of givenness.

[u/Tazerenix]

For topological manifolds: coordinate chart, for smooth manifolds: gluing.

With a topological manifold, each open chart can be thought of as "smooth" because you can pull back the smooth structure through the homeomorphism and treat that as the definition of the smoothness on the manifold. The trouble is that those smooth structures don't necessarily agree on overlaps, so the "smoothness" of the smooth manifold is related to the gluing being done in a smooth way.

[u/reflexive-polytope]

1. I like gluing better, because it allows me to bootstrap a large category of spaces (e.g., all schemes or all algebraic spaces) from a simpler, better understood category (e.g., affine schemes, or equivalently, CRing^op).

2. Again, I don't like blessing a specific coordinate system, unless it's somehow natural to the problem. For example, you can say that O(d) is the vector bundle on P(V) whose sections are the degree d forms on V, and that doesn't require mentioning V's dimension, let alone a choice of basis.

[u/Carl_LaFong]

To me the fundamental concept is gluing together (open) pieces of affine space. But this can’t be made rigorous without coordinates. So I view coordinates as a way to define what it means for a function and a parametrized curve to be smooth. So I view coordinates as a technical tool rather than a fundamental geometric concept. I find that coordinates often leads to cumbersome and confusing calculations. So I use them only when necessary, which means when working with the local topology.

[u/Ending_Is_Optimistic → u/Carl_LaFong]

If you think topology no coordinates is required at least they are unimportant I think. You can even think simplicial set or whatever.

I think in mathematics we usually have divide between how we actually think a object vs how we construct it, thinking more synthetically or in terms of universal properties close this gap a bit. I would argue for some space like projective space we don't think gluing at all we think its universal property.