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Stream: community: general

Topic: Categorical Thermodynamics

Ellis D. Cooper (Jul 16 2020 at 15:42):

Towards categorical thermodynamics, with your help, I hope to identify the categorical structure(s) naturally inherent in the following circle of ideas. For sets $A_1,\ldots,A_n$ the set of finite sets of distinct elements one from each of the $A_i$ is denoted by $A_1\rtimes\cdot \rtimes A_n$. Three-dimensional Euclidean space is denoted by $\mathbf{E}$. A core is by definition a structure of the form $(J,S,P)$ where $J\subset \mathbf{E}\rtimes\cdots\rtimes\mathbf{E}$ with $n$ factors, so $J$ is a set of $n$ distinct points in space. These are called the joints of the core. $S\subset J\rtimes J$ is a set of pairs of distinct joints, and these are the struts of the core. $P\subset J\rtimes J \rtimes J$ is a set of triples of distinct joints, and these are the plates. (Set-theoretically, a core is nothing but a 2-dimensional (un-oriented) simplicial complex.) The shape (a.k.a. "conformation") of a core is the set of lengths of the struts, the set of angles between contiguous (shared joint) struts, and the set of angles between contiguous (shared strut) plates. In terms from chemistry, joints are locations of atoms in a molecule, lengths of struts are bond-lengths, strut angles are bond-angles, and angles between plates are "dihedral-" or "torsion-" angles. A spatial-thing is a core with a shape. For every strut there is a line segment of points strictly between its joints, and for every plate there is a triangle of points bounded by its struts (think positive barycentric coordinates). By definition, the joints, line segments, and triangles of a spatial-thing are all disjoint sets of points in space. Therefore, a spatial-thing occupies a subset of space. (Think adjunction space.) By assumption, there is a variable set of spatial-things, that is to say, over different intervals of physical-time there may be more or less spatial-things. Also by assumption, at any particular time, any two spatial-things occupy disjoint subsets of space. Two spatial-things are isomorphic if there exists a bijection between their joints which preserves struts, plates, strut lengths, and strut and plate angles. There are weaker forms of equivalence between spatial-things. For example, two spatial-things may have isomorphic cores, but arbitrarily different shapes. Or, they may have isomorphic cores, and conformations that differ within specified constraints on lengths and angles. A spatial-thing is connected if for any two distinct joints there is a path of struts between them. A species is an equivalence class of connected spatial-things (for some specified equivalence relation). The connected spatial-things of a spatial-thing are called its molecules, and they are partitioned by a choice of equivalence relation. Each part of the partition is a subset of a distinct species. The ratios of the cardinalities of the parts is called a material-substance. The total number of molecules in the spatial-thing is its amount of material-substance. So, two spatial-things may have different amounts of the same substance. If there is just one species then the spatial-thing is an amount of a pure material-substance. This concludes the " statics" of spatial-things. Next up, ``dynamics."
NotQuiteSpatialThing.jpg
This is not quite a spatial-thing because some struts are not disjoint from some plates.

xavier (mathematical artist) (Jul 17 2020 at 01:13):

@Ellis D. Cooper, am I correct in interpreting your formalization of spatial-things as a preparation for defining a control volume?

Ellis D. Cooper (Jul 17 2020 at 14:44):

@xavier (mathematical artist)
Yes, Sir. I should say that "control volume" as explained in its wikipedia article is, however, a special case of spatial-thing. As I see it, a spatial-thing is nothing but a more or less arbitrary collection of generalized "molecules." If, however, the plates of a connected spatial-thing form a combinatorial manifold, a.k.a., a closed two-dimensional surface, then the spatial-thing is a container and so the three-dimensional Brower-Jordan theorem applies. In other words, a container separates inside from outside. Like the membrane of a biological cell.

xavier (mathematical artist) (Jul 17 2020 at 17:16):

@Ellis D. Cooper , Excellent! Thank you.

Ellis D. Cooper (Jul 22 2020 at 20:06):

A (spatial-)thing has a core $(J,S,P)$ and a shape $(\beta,\alpha,\delta)$. No two things overlap in space. Two things may match along a common sub-thing. New struts may joint two previously unconnected things. A thing may lose plates, struts, or joints. A thing may also change shape while maintaining core. A thing may move in space. So, things undergo transformation and/or transportation. A thing may be a container of material-substance, both of which are sub-things of it. A physical-process is a combination of transportation between, and transformation within, of material-substances in containers. Transportation is motion in physical-space; transformation is motion in substance-space. Every kind of motion is the result of a "conjugate" kind of drive. Energy is the product of conjugate drive and motion. There are just a couple of dozen different kinds of conjugate (drive, motion) pairs. These are the basic physical-processes. Each thing has energy, and the energy of a thing may change only due to a process involving the thing. Dynamics is the scientific study of (physical-)processes. Processes occur over a wide range of spatial and temporal scales, roughly categorized as macro-, micro-, and nano-dynamics. Different mathematical disciplines with different types of equations are tools in each range. Biology involves every range, so there is no single comprehensive mathematical discipline appropriate for it. Whether "categorical thermodynamics" could be created to express a mental model of biological autonomy, variation, and motility, very much remains to be seen.

Steve Huntsman (Jul 23 2020 at 01:25):

Ellis D. Cooper said:

A (spatial-)thing has a core $(J,S,P)$ and a shape $(\beta,\alpha,\delta)$. No two things overlap in space. Two things may match along a common sub-thing. New struts may joint two previously unconnected things. A thing may lose plates, struts, or joints. A thing may also change shape while maintaining core. A thing may move in space. So, things undergo transformation and/or transportation. A thing may be a container of material-substance, both of which are sub-things of it. A physical-process is a combination of transportation between, and transformation within, of material-substances in containers. Transportation is motion in physical-space; transformation is motion in substance-space. Every kind of motion is the result of a "conjugate" kind of drive. Energy is the product of conjugate drive and motion. There are just a couple of dozen different kinds of conjugate (drive, motion) pairs. These are the basic physical-processes. Each thing has energy, and the energy of a thing may change only due to a process involving the thing. Dynamics is the scientific study of (physical-)processes. Processes occur over a wide range of spatial and temporal scales, roughly categorized as macro-, micro-, and nano-dynamics. Different mathematical disciplines with different types of equations are tools in each range. Biology involves every range, so there is no single comprehensive mathematical discipline appropriate for it. Whether "categorical thermodynamics" could be created to express a mental model of biological autonomy, variation, and motility, very much remains to be seen.

A productive way to pull on this thread would be to look at coupled Glauber spins versus uncoupled ones from the POV of Pollard’s Markov process formalism. One would hope to get the real-space renormalization group as an operad or some such

Ellis D. Cooper (Jul 24 2020 at 12:23):

@Steve Huntsman Sir, the leap from what I wrote to your references is too far for me. Could you maybe spell out more on how you bridge one to the other?

Steve Huntsman (Jul 24 2020 at 16:12):

I think it’s a pretty good PhD thesis, not an easy thing, but the first moving pieces are https://arxiv.org/abs/1709.09743 and https://aip.scitation.org/doi/abs/10.1063/1.1703954. Take the former and apply it to the latter in a way that lets you do block spins a la https://en.wikipedia.org/wiki/Renormalization_group.

Ellis D. Cooper (Jul 26 2020 at 14:11):

@Steve Huntsman The Pollard thesis constructs functors for abstracting open Markov processes in terms of semi-algebraic relations. Open Markov processes are probabilistic structures strongly pertinent to non-equilibrium thermodynamics. The virtue of Pollard's functors is the virtue of all arguments for compositionality, namely the reduction of behavior of complicated systems in terms of behavior of simpler ones.

My approach to any thesis (doctoral or otherwise) is to ask, what is missing? In this instance, there are no references to Onsager, Fourier, Hess, constitutive equations, the First Law of Thermodynamics, let alone the Gibbs Fundamental Equation. To be fair, Pollard mentions early work in mathematical thermodynamics by Oster, Perelson, Katchalsky, Hill, Prigogine, and very important more recent work by Hong Qian.

Most important to me, what is missing from Pollard, and from all of these scientists, is an explicit mathematical theory of physical-things (and fields) occupying and moving through regions of space. It seems to me that theoretical systems biology calls for this. My reason for saying this is that there exists a community of theoretical biologists, including Longo, Soto, Sonnenschein, Moreno, Montevil, Mossio, and others, who are "In Search of Principles for a Theory of Organisms." This group is analogous to Lawvere's group at Dalhousie University in the 1970s. My dream is to bring my own tiny little ladder to the orchard and try to pick some of the low-hanging fruit.

Ellis D. Cooper (Jul 28 2020 at 12:40):

Tiny Little Ladder: Life is a thing. What is a thing, and what kind of thing is Life? At the minimum, a (physical-)thing is the set of alternative behaviors of a core (joints, struts,plates) with a shape (strut-lengths,strut-angles,plate-angles), for short, a shaped-core. A shaped-core may have multiple possible-behaviors, but they are the (possible-)behaviors of one thing. The variable-set of things persisting over a specified time interval is a category with finite colimits, at least a join semi-lattice, a.k.a. idempotent commutative monoid. Hence, an arbitrary finite set of things is also a thing. Very importantly for chemistry, two things sharing one sub-thing constitute a "new" thing (e.g., $H_2O$). The morphisms include inclusion-maps of sub-things, but other maps are possible, such as removal or addition of joints, struts or plates. Also, homotopies of shape, and in particular linear homotopies. Note that for a homotopy of shape there is no "homotopy time." It is better to distinguish re-conformation-progress from process-time. Thus, at any particular time, two distinct things may be equivalent via an invertible change of shape, a.k.a., "deformation." (Question: can't "any" invertible deformation be approximated as closely as specified by piece-wise linear homotopies?) Question: Does the forgetful functor from the category of things to the category of cores have a left-adjoint, in other words, does every core somehow determine a "free-thing"? Reply: Not really. But, assume every thing has shape-dependent energy. Different shapes of the same core means different (strut-lengths,strut-angles,plate-angles). There may well be dependencies among these parameters (e.g., the sum of the angles of a plate is $\pi$ radians). Hence, the actual degrees-of-freedom of a thing determine a sub-manifold of a high-dimensional real vector space. Energy is a non-negative real-valued function on this manifold. The graph of the energy of a thing is called its landscape. Assuming the energy function is smooth enough, one imagines hills and valleys, peaks and depths, and passes. There is a set of (local or absolute) minimum-energy shapes with a specified core (think protein-folding). Regarding what kind of thing is Life, profound hints were published in a special issue of "Progress in Biophysics and Molecular Biology," entitled "From the century of the genome to the century of the organism: New theoretical approaches," edited by A. M. Soto and G. Longo. Your thoughts?

Morgan Rogers (he/him) (Jul 28 2020 at 14:06):

In what sense is $H_2 O$ two things sharing a sub-thing?

Morgan Rogers (he/him) (Jul 28 2020 at 14:22):

Ellis D. Cooper said:

By definition, the joints, line segments, and triangles of a spatial-thing are all disjoint sets of points in space. Therefore, a spatial-thing occupies a subset of space. (Think adjunction space.)

When you say "by definition", you mean you consider the points in a strut or plate to be indexed externally? If so, why do you bother taking points in $\mathbf{E}$ in the first place? And either way, what do you mean by a "subset of space"?

Morgan Rogers (he/him) (Jul 28 2020 at 14:26):

I also wonder why you chose the name "(spatial-)thing". These don't seem as generic as the title of "thing" would ordinarily imply.

Ellis D. Cooper (Jul 28 2020 at 15:50):

@[Mod] Morgan Rogers The two covalent bonds in a water molecule are "struts" between two hydrogen nuclei "joints" to one oxygen "joint".
One hydrogen-bond-oxygen shares oxygen with the other hydrogen-bond-oxygen. If you think of the water molecule as approximated by a CW-complex, then it has three 0-cells and two 1-cells. The hydrogen nucleus is common to two subspaces.

By "space" I mean 3-dimensional Euclidean space $\mathbf{E}$ with coordinate system $R^3$. "Joints" are points of space, "struts" are line-segments between points, and "plates" are triangular subsets of space. ("A simplicial complex $K$ is a subspace of some Euclidean space..." (Wallace, "Algebraic Topology")). A core is just a finite 2-dimensional simplicial complex in $\mathbf{E}$, with its 0-cells, 1-cells, and 2-cells given idiosyncratic names joints, struts, and plates. The reason for this idiosyncracy is that I think of macroscopic spatial-things such as the body of an organism as approximated by a core, just as much as a macromolecule is approximated by a core. Also, since a core is a subset of space, the concepts of length and angle come automatically with the inner-product structure. Thus, a spatial-thing is a core with a shape, which is a lot more than just a simplicial complex.

I may be confusing because I tend to abbreviate "spatial-thing" to just "thing," and sometimes I try to be careful and say "(spatial-)thing." The concept of "thing" in general is the subject of philosophy books (e.g., Heidegger). I want to be very concrete, and by "thing" I ALWAYS mean spatial-thing as a shaped-core. Any other kind of thing should be named X-thing, such as imaginary-thing, or supernatural-thing.

Morgan Rogers (he/him) (Jul 28 2020 at 16:02):

Ellis D. Cooper said:

[Mod] Morgan Rogers The two covalent bonds in a water molecule are "struts" between two hydrogen nuclei "joints" to one oxygen "joint".
By "space" I mean 3-dimensional Euclidean space $\mathbf{E}$ with coordinate system $R^3$. "Joints" are points of space, "struts" are line-segments between points, and "plates" are triangular subsets of space. ("A simplicial complex $K$ is a subspace of some Euclidean space..." (Wallace, "Algebraic Topology")).

Right, but the Euclidean space that Wallace is referring to may have more than three dimensions. In particular, it's easy to come up with a core in the sense of your definition where two struts (or rather, the Euclidean line segment joining the two joints of a strut) overlap, or overlap with a plate. I take it you are explicitly excluding these somehow?

Morgan Rogers (he/him) (Jul 28 2020 at 16:03):

Or, since we're talking chemistry, are overlaps actually allowed to some extent (but they don't get realised physically so much because they represent extreme energy values)?

Morgan Rogers (he/him) (Jul 28 2020 at 16:06):

Ellis D. Cooper said:

Also by assumption, at any particular time, any two spatial-things occupy disjoint subsets of space.

I gather that this statement got changed later on, too, since we've discussed sub-spatial-things :wink:

Ellis D. Cooper (Jul 28 2020 at 16:26):

@[Mod] Morgan Rogers First, my objective has to do with biological-organisms, and Life as a whole spatial-thing, in our shared everyday space. Second, you are exactly right that I must explicitly declare that (interiors of) struts shall not intersect (interiors of) plates. I alluded to that with the image appended to my opening post to this topic. Third, right again, I have got to be more careful about sub-spatial-things versus spatial-things in general. Sub-spatial-things of one spatial-thing form a Boolean algebra, which includes possible overlaps (intersections) and unions, and so on. I have yet to get into connected spatial-things, sub-spatial-things of connected spatial-things, sets of unconnected non-intersecting spatial-things, like clouds, spatial-things whose connected sub-spatial-things are isomorphic (in various alternative ways), etc.

I assume it is okay to be thinking out-loud here on zulip, which hopefully leads to more refined and careful definitions, so many thanks for your comments. :grinning:

Morgan Rogers (he/him) (Jul 28 2020 at 18:04):

Of course, I've been guilty of similar imprecisions on many occasions! :grinning: Okay, I'm starting to follow: when you say spatial-things are disjoint in space, you mean we consider these objects in isolation, say in their own copy of $\mathbf{E}$, and in the category the morphisms are some choice of mappings between these (possibly respecting the ambient space in some way, possibly only looking at the data of the spatial-things, depending on preference).

Morgan Rogers (he/him) (Jul 28 2020 at 18:09):

Actually, maybe I should ask in a bit more detail about morphisms, since you mentioned the forgetful functor to cores... Given how you defined cores originally, doesn't a core come with a shape equipped, coming from the locations of the points in Euclidean space?

Ellis D. Cooper (Jul 28 2020 at 19:34):

[Mod] Morgan Rogers said:

Actually, maybe I should ask in a bit more detail about morphisms, since you mentioned the forgetful functor to cores... Given how you defined cores originally, doesn't a core come with a shape equipped, coming from the locations of the points in Euclidean space?

Of course, as defined, a core automatically comes with a shape. So, right, every core is a shaped-core, i.e., a spatial-thing. Calling it "shaped" is merely intended to emphasize that a core is more than a simplicial complex, since, as far as I know, algebraic topology ignores angles (and any other metric information). That said, define two spatial-things to be core-equivalent if there is a strut- and plate-preserving bijection between their joints. A core-equivalence class is called an abstract-core, and the abstract-cores are the objects of a category. At the minimum, inclusion-maps of sub-spatial-things induce inclusion-maps of abstract-cores. That assignment is the "forgetful functor" I had in mind.

Morgan Rogers (he/him) (Jul 29 2020 at 09:41):

So abstract-cores are the 2-dimensional simplicial sets whose geometric realisation is embeddable into 3-space. So eg. any graph (no plates) but not a simplicial Klein bottle. And indeed, at first sight there is no canonical functor to act as an adjoint to that forgetful functor. However, if all shaped-cores with the same abstract core are isomorphic/equivalent, then any functorial/systematic/natural assignment of a shape to an abstract-core yields an equivalence between the categories of abstract and shaped cores..!

Morgan Rogers (he/him) (Jul 29 2020 at 09:47):

In order to distinguish shaped-cores, we could equip them with energies in some functorial way, producing the landscapes you mentioned. That's fine, but we still have the problem that energy can go up or down, and so there is still no obvious direction that we can apply to the transformations. Also, there is still no way to guarantee that minimum-energy shapes are attained (maybe the minimum occurs at an overlap, or at the limit where two joints coincide or tend to infinity).

Morgan Rogers (he/him) (Jul 29 2020 at 09:49):

Since we're in the realm of thermodynamics, what's needed is entropy! A clear direction, that incorporates energy in a natural way. Is that where you're taking these ideas?

Ellis D. Cooper (Jul 29 2020 at 12:26):

@[Mod] Morgan Rogers Yes to everything you said. Especially about 2-dimensional simplicial sets, and especially about disallowing a simplicial Klein bottle. And yes, we are in the realm of thermodynamics, and yes, entropy is needed. Down the road, the Second Law of Thermodynamics "guarantees that minimum-energy shapes are attained." But, first things first :smile: !

A spatial-thing is connected if for any two joints there is a path of struts from one to the other. A species is the abstract-core of a connected spatial-thing, together with a range of variation for each shape-parameter (strut-length, strut-angle, plate-angle). For example, a human organism has a lot of flexibility at knees and elbows and shoulders, but there are limits. (Sport and dance work within those limits.) An amount- of-species is a finite set of disjoint spatial-things of the same species, which may be called the particles of the amount. The cardinality of an amount of a species is the quantity of the amount. A substance is a finite set of distinct species together with natural-number ratios, its proportions. A pure-substance is just a species. A mixed-substance has more than one species. An amount-of-substance is a finite set of amounts of its species in proportion. The density of an amount of substance at a point in space may (?) be defined by considering the radius of the smallest sphere centered at the point which contains particles of the amount of substance. The density in that case would be the number of particles on the sphere, divided by the cube of the radius. The big picture, now, is a vast number of spatial-things of various species spread out in space, divided among various substances with varying density. Nothing is moving. Yet.

Morgan Rogers (he/him) (Jul 29 2020 at 12:35):

Given the discrete setting, it seems artificial to want there to be a pointwise defined density, especially as the definition you just gave isn't continuous. What will that be needed for?

Morgan Rogers (he/him) (Jul 29 2020 at 12:37):

In any case, in spite of the ad hoc nature of the terminology, you've successfully conveyed the chemical picture of matter as spatially distributed molecules. What next? :smiley:

Ellis D. Cooper (Jul 29 2020 at 17:27):

@[Mod] Morgan Rogers Last question first: Take time $\mathbf{T}$ to be the (algebraic and topological) structure of non-negative real numbers. A temporal-spatial-thing is an ordered pair (time,spatial-thing)$=(t,\Sigma)=(t,((J,S,P,),(\beta,\alpha,\delta)))$. Since the region of space occupied by $\Sigma$ is determined by $J$, if there are $n$ joints then $3n+1$ parameters determine the position of $\Sigma$ at $t$. Define category $\mathbf{W}$ with objects the temporal-spatial-things. Any morphism $f:(t,\Sigma)\to (t',\Sigma')$ must satisfy $t\le t'$. The spatial component of $f$ comes in different flavors.

1. If $t=t'$ and $\Sigma=\Sigma'$, then the identity-map on joints defines the identity-map of $(t,\Sigma)$;
2. If $t=t'$ and $\Sigma\ne \Sigma'$, and $\Sigma'$ is the result of removing or adding joints, struts, or plates from or to $\Sigma$, then $f$ is a transform morphism (if a joint is removed, all struts and plates to which it belongs must be removed as well);
3. If $t<t'$ and $\Sigma'$ has the same abstract-core as $\Sigma$ but a different shape, then $f$ is a transport morphism if it is of the form $[t,t']\times J\to \mathbf{E}$ which maintains the same abstract-core for all times between $t$ and $t'$.

In general, morphisms are compositions (the obvious ways) of identities, transforms, and transports. Think about the movements of dancers in a dance, or about changing conformation of enzyme and substrates, or about simple harmonic oscillations of atoms in crystals. In other words, $\mathbf{W}$ is the category of motions of spatial-things. Why spatial-things move at all is a separate question (the answer will be that every motion is due to a drive).

First question last: the need for a concept of density is that density of one spatial-thing may constrain the motion of another spatial-thing (e.g., the wall of the vascular system constrain blood flow). "The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system." Herbert B. Callen, "Thermodynamics and an Introduction to Thermostatics, Second Edition." Wikipedia: "In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure."

Morgan Rogers (he/him) (Jul 30 2020 at 14:15):

Ellis D. Cooper said:

In general, morphisms are compositions (the obvious ways) of identities, transforms, and transports. In other words, $\mathbf{W}$ is the category of motions of spatial-things.

I can see where you're coming from conceptually, but the unconstrained description of transform morphisms seems to mean that anything can spontaneously happen anywhere (I can have one morphism which deletes everything, a trivial transport morphism to move time forwards, and then another morphism that creates any form of spatial-thing whatsoever). Which is technically fine, but I do want to check that's intentional!

Morgan Rogers (he/him) (Jul 30 2020 at 14:27):

Ellis D. Cooper said:

First question last: the need for a concept of density is that density of one spatial-thing may constrain the motion of another spatial-thing (e.g., the wall of the vascular system constrain blood flow). "The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system." Herbert B. Callen, "Thermodynamics and an Introduction to Thermostatics, Second Edition." Wikipedia: "In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure."

The density values that are used in this contexts are statistical approximations, though, taken by integrating the mass in some relatively large region about a point, rather than by integrating at the scale of individual molecules. When we discuss the density of pure diamond, for example, we take the density to be uniform, since the structure of diamond is symmetric and we are interested in its density at a macroscopic scale. If you're working directly with the constituent atoms or molecules of a substance, and you're assuming that you have precise information about relative positions (as in the case in the formalism you're describing), then that information should be directly applied to understanding processes, rather than taking an arbitrary/artificial measure of density first.
In your cell wall example, say, we could understand how blood flow in constrained by looking at the actual gaps between struts and plates... Although to be honest this all seems rather classical-physicsy so far.

Ellis D. Cooper (Jul 30 2020 at 18:48):

[Mod] Morgan Rogers said:

I do want to check that's intentional!

It is intentional since I am adopting a strategy that starts with great generality and should spiral down to more and more realistic scenarios.
A forthcoming step is to distinguish the macro-, micro-, meso-, and nano-scales of space, time, and energy, since biology operates at all scales. Therefore, thermodynamics should be built to fit accordingly. In particular, the spatial scale of "actual gaps" of struts and plates of vascular cell walls are a great deal smaller than the sizes of, say, white blood cells. Of course you are right about the suggested "arbitrary/artificial measure of density." Maybe it will be satisfactory to just deal with distributions of strut dimensions and spacing between spatial-things (in motion). But, I must say the real reason behind my use of the word "density" is that I will need to formalize the idea that spatial-things of all scales occupy regions of space with great variety of "density". A piston in a cylinder is a LOT more dense than the gas around it. More generally, "containers" (TBD) of substances are more dense than the substances within. As yet there is no urgent need to introduce quantum field theory into the story :smile: .

Morgan Rogers (he/him) (Jul 30 2020 at 19:36):

Again, the density of a piston is not measured at the scale of its constituent atoms, but at a sufficiently large scale that the discrete atoms can be treated as a continuum, and we treat that matter as being contained within the piston's surface (which at a large scale appears uniform and so can be precisely defined). The density measure you proposed would have, beyond the piston's surface, a contribution to the density of the gas from the presence of the piston, which seems undesirable. I would expect the density to be a property of a substance rather than of a particular arrangement of components.

Morgan Rogers (he/him) (Jul 30 2020 at 19:37):

Ellis D. Cooper said:

As yet there is no urgent need to introduce quantum field theory into the story :smile: .

I'll reserve judgement if you say so :stuck_out_tongue_wink: So what next..?

Ellis D. Cooper (Jul 31 2020 at 19:27):

@[Mod] Morgan Rogers @[Mod] Morgan Rogers You wrote, " the unconstrained description of transform morphisms seems to mean that anything can spontaneously happen anywhere." Yes, that is why quantum mechanics must be invoked to select the full sub-category $\mathbf{V}\subset \mathbf{W}$ determined by Molecular-Geometry, which is at the nanometer-scale of spatial dimensions and the picosecond-scale of time (Schrodinger's Equation, Born-Oppenheimer Approximation, Atomic Orbitals, Valence-Bond Theory, Lewis Structure, and the VSEPR Model).

The "density measure" I proposed should be abandoned with prejudice. I had failed to distinguish (a) distances between disjoint connected-spatial-things, and (b) strut-lengths within connected-spatial-things.

The theoretical framework called Biological Autonomy routinely refers to "closure of constraints" along with "thermodynamic processes." Categorical Thermodynamics is supposed to encompass macroscopic equilibrium thermodynamics -- which is atemporal -- and mesoscopic, open, dissipative, far-from-equilibrium, stochastic thermodynamics. For me, the starting point is to distinguish spatial-things called containers among the amounts of substances in $\mathbf{V}$. First Try: a container is a connected-spatial-thing obtained from a polygonal decomposition of a topologically connected 2-dimensional orientable surface in space. The boundaries of the polygons are simple closed strut-paths. Some of the polygons may be triangles, and some of those may be covered by plates. The shapes of the polygons, and the ratio of the total non-plate area to the total area is a measure of porosity of the boundary relative to flux density of substances.

By a basic result of algebraic topology, space is partitioned into the container exterior, boundary, and interior. The interior is a region of space that may or may not be occupied by some amount of some (possibly mixed-)substance. The Fundamental Balance Postulate of thermodynamics equates any change in that amount of substance to the sum of substance transport through the boundary, or to transform of substance in the interior. These are morphisms of $\mathbf{V}$. As yet, nothing much has been said about why anything should move or change.

Morgan Rogers (he/him) (Aug 01 2020 at 13:41):

Ellis D. Cooper said:

Yes, that is why quantum mechanics must be invoked to select the full sub-category $\mathbf{V}\subset \mathbf{W}$ determined by Molecular-Geometry, which is at the nanometer-scale of spatial dimensions and the picosecond-scale of time.

It's not entirely clear what this means, but I'm happy to accept a generic statement like "morphisms satisfying certain constraints" for now while I'm getting on top of the ideas being discussed.

First Try: a container is a connected-spatial-thing obtained from a polygonal decomposition of a topologically connected 2-dimensional orientable surface in space...

This seems like a sensible thing to take an interest in, sure.

The Fundamental Balance Postulate of thermodynamics equates any change in that amount of substance to the sum of substance transport through the boundary, or to transform of substance in the interior. These are morphisms of $\mathbf{V}$. As yet, nothing much has been said about why anything should move or change.

So this is saying "any (constrained) morphism of $\mathbf{V}$ can be integrated over the surface of a container". First potential problem: if containers are physical, they can be changed by morphisms, in particular in ways that may stop them from being containers. Second, so far in this formalism, all of the various pieces are identical (there is only one kind of joint etc). The unoriginal physics answer to your "why" question is "there are forces", but in order for that to produce interesting effects from simple rules, it would make sense to introduce some different flavours to our stuff beyond the structural properties.

Ellis D. Cooper (Aug 01 2020 at 20:30):

@[Mod] Morgan Rogers

So this is saying "any (constrained) morphism of $\mathbf{V}$ can be integrated over the surface of a container".

In a textbook, the differential form of the balance equation is

$$\frac{\partial y}{\partial t}+\nabla\cdot\mathbf{J}_Y = P(Y)$$

where $P(Y)$ is the amount of substance $Y$ produced per unit time per unit volume; $y$ is the density of $Y$; and $\mathbf{J}_Y$ is the current density of $Y$ at a point of the boundary. In my framework,

$$dX_{\mathbf{A}}=\sum_{\mathbf{C}}dX^{\mathbf{C}}_{\mathbf{A}}+\sum_Z d\mathbf{A}^Z_X$$

says that incremental change in amount of substance $X$ in container $\mathbf{A}$ is equal to the sum of all incremental transports from other containers $\mathbf{C}$ to or from $\mathbf{A}$ through its boundary, plus the sum of all incremental transforms of substances $Z$ into or out of $X$ in the interior of $\mathbf{A}$.

First potential problem: if containers are physical, they can be changed by morphisms, in particular in ways that may stop them from being containers.

Yes, of course, spatial-things including containers do change in time.

Second, so far in this formalism, all of the various pieces are identical (there is only one kind of joint etc).

Spatial-things called physical-atoms are the ur-joints. Surjections in the category of abstract-cores "analyze" elements of structure in the codomain in terms of abstract-cores which are parts of the domain. Analysis bottoms-out in terms of physical-atoms, possibly through a sequence of analyses.

The unoriginal physics answer to your "why" question is "there are forces", but in order for that to produce interesting effects from simple rules, it would make sense to introduce some different flavours to our stuff beyond the structural properties.

How and why the energy of a spatial-thing changes according to "simple rules," a.k.a. a small collection of profoundly analogous physical-processes, is the subject of forthcoming discussion.

Morgan Rogers (he/him) (Aug 03 2020 at 08:40):

Ellis D. Cooper said:

First potential problem: if containers are physical, they can be changed by morphisms, in particular in ways that may stop them from being containers.

Yes, of course, spatial-things including containers do change in time.

The problem I'm pointing to is that if a container changes dramatically (resulting in a discontinuous change in some topological invariant, say, such a squashed sphere collapsing into a torus, or just self-intersecting) there could be problems applying this reasoning. When a bubble pops, has its gaseous contents passed through its surface? This doesn't seem insurmountable, it's just a bit messy and requires some care.

Second, so far in this formalism, all of the various pieces are identical (there is only one kind of joint etc).

Spatial-things called physical-atoms are the ur-joints. Surjections in the category of abstract-cores "analyze" elements of structure in the codomain in terms of abstract-cores which are parts of the domain. Analysis bottoms-out in terms of physical-atoms, possibly through a sequence of analyses.

I don't quite understand what this means. Are you saying there is a smaller scale of spatial-thing that acts as a joint at a larger scale? As in you intend for this formalism to be nested somehow? Or are you hinting at some categorical machinery you have in mind which will distinguish certain kinds of joint?

How and why the energy of a spatial-thing changes according to "simple rules," a.k.a. a small collection of profoundly analogous physical-processes, is the subject of forthcoming discussion.

Okay..! Go ahead!

Ellis D. Cooper (Aug 04 2020 at 17:08):

[Mod] Morgan Rogers

When a bubble pops, has its gaseous contents passed through its surface? This doesn't seem insurmountable, it's just a bit messy and requires some care.

This is a standard basic example in thermodynamics: "pop of the bubble" is removal of a constraint, and assuming a difference of pressure of ideal gas within and outside the bubble, i.e., a difference of (pneumatic-)potential, there will be a flow of chemical-substance with increasing entropy and reduction of potential difference.

I wrote, "Analysis bottoms-out in terms of physical-atoms, possibly through a sequence of analyses."

I don't quite understand what this means. Are you saying there is a smaller scale of spatial-thing that acts as a joint at a larger scale? As in you intend for this formalism to be nested somehow? Or are you hinting at some categorical machinery you have in mind which will distinguish certain kinds of joint?

Any spatial-thing $\Sigma=(J,S,P),(\beta,\alpha,\delta)$ has a geometric-center $ctr(\Sigma)\in\mathbf{E}$ whose coordinates are the means of the coordinates of the joints $J$. If every joint $j\in J$ is the geometric-center $j=ctr(\Sigma_j)$ of a spatial-thing $\Sigma_j$, and every strut $\{\,j,j'\,\}$ of $\Sigma$ is associated with a specified set $S_{j j'}$ of struts between the joints of $\Sigma_j$ and joints of $\Sigma_{j'}$, then there is a new spatial-thing $$\Sigma_1$$defined by $J_1=\bigcup_j J_j$, $S_1=\bigcup_j S_j \cup \bigcup_{j,j'}S_{j j'}$, and $P_1=\bigcup_j P_j$. One assumes strut-lengths, strut-angles and plate-angles are also specified where needed. One also assumes that the spatial-scale of $\Sigma_1$ is much smaller than the spatial-scale of $\Sigma$. In these circumstances, call $\Sigma_1$ an analysis of $\Sigma$. There may be more than one analysis of a spatial-thing, and there may be an analysis of an analysis. (In order to define an abstract-core morphism from $\Sigma_1$ to $\Sigma$, such that $\Sigma_j$ collapses to $j$ and $S_{j j'}$ collapses to $\{\,j,j'\,\}$, it is necessary to revise the definition of strut as an unordered set of two distinct points, so that there may be a degenerate-strut which is just the singleton of a joint.

If there is a systematic way to analyze the spatial-things of some particular kind, then one would expect some kind of naturality with respect maps of spatial-things. TBD.

Morgan Rogers (he/him) (Aug 05 2020 at 10:27):

Ellis D. Cooper said:

[Mod] Morgan Rogers

When a bubble pops, has its gaseous contents passed through its surface? This doesn't seem insurmountable, it's just a bit messy and requires some care.

This is a standard basic example in thermodynamics: "pop of the bubble" is removal of a constraint, and assuming a difference of pressure of ideal gas within and outside the bubble, i.e., a difference of (pneumatic-)potential, there will be a flow of chemical-substance with increasing entropy and reduction of potential difference.

I have a physical understanding of what happens when a bubble pops. What I was asking is, if the bubble it treated as a container, how should your container balance equation be interpreted when that container vanishes or disintegrates. Usually I've seen the balance equation applied to virtual volumes, so the state of the container can be ignored, but you're presenting containers which are physical (eg cells, bubbles), which I think makes such considerations more important.

Any spatial-thing $\Sigma=(J,S,P),(\beta,\alpha,\delta)$ has a geometric-center $ctr(\Sigma)\in\mathbf{E}$ whose coordinates are the means of the coordinates of the joints $J$. If every joint $j\in J$ is the geometric-center $j=ctr(\Sigma_j)$ of a spatial-thing $\Sigma_j$, and every strut $\{\,j,j'\,\}$ of $\Sigma$ is associated with a specified set $S_{j j'}$ of struts between the joints of $\Sigma_j$ and joints of $\Sigma_{j'}$, then there is a new spatial-thing $$\Sigma_1$$defined by $J_1=\bigcup_j J_j$, $S_1=\bigcup_j S_j \cup \bigcup_{j,j'}S_{j j'}$, and $P_1=\bigcup_j P_j$.

Aha! This feels like some kind of enriched Grothendieck construction. So it is a nested construction picture, great. :tada: I think you should probably remove the requirement on geometric centers, given the requirement that joints are not allowed to coincide: imagine a system of one triangulated sphere contained and centered in another, with bonds between them, or something more complicated (I'm thinking eg the magnesium ion nested in clorophyll). It's conceivable that one would want to treat this as an "analysis" of a system with just two joints when one is only interested in its behaviour at a larger scale, but currently that's not allowed because the joints for the two spheres would be forced to coincide.

One also assumes that the spatial-scale of $\Sigma_1$ is much smaller than the spatial-scale of $\Sigma$.

I can see the motivation for this extra assumption, but it seems unnecessary to introduce it at this point: we aren't making any calculations where such scale considerations will matter just yet, so I would recommend putting off including them. We just need to assume that the result of this construction is a valid spatial-core.

In these circumstances, call $\Sigma_1$ an analysis of $\Sigma$. There may be more than one analysis of a spatial-thing, and there may be an analysis of an analysis.

There are surely infinitely many analyses of any non-empty spatial-core!

(In order to define an abstract-core morphism from $\Sigma_1$ to $\Sigma$, such that $\Sigma_j$ collapses to $j$ and $S_{j j'}$ collapses to $\{\,j,j'\,\}$, it is necessary to revise the definition of strut as an unordered set of two distinct points, so that there may be a degenerate-strut which is just the singleton of a joint.)

This seems like a harmless concession. There's a reason identity morphisms are so convenient to keep around in category theory :innocent:

Morgan Rogers (he/him) (Aug 05 2020 at 10:51):

Re my comment about a Grothendieck construction... Despite all of our constraining things to Euclidean space, it seems to me that if we define cores in a sufficiently abstract setting, we could interpret an analysis as a spatial-core/abstract-core in the (pseudo-)space of spatial-cores/abstract-cores; that is, an analysis would become a (pseudo-)mapping of cores from $\Sigma$ to this (pseudo-)space of cores. I'm putting in the prefix pseudo in analogy with pseudo-functors from a small category to the category of small categories, which is the original setting for the Grothendieck construction; I'll drop it for now.

A joint in this space of abstract-cores should just be a core. A strut between $(J,S,P)$ and $(J',S',P')$ might consist of a relation $R: J \to J'$, a set of triples $A \subseteq J \rtimes J \times J'$ and another collection of triples $B \subseteq J \times J' \rtimes J'$ such that for $(x,y,z) \in A$ we have $(x,y) \in S, (x,z) \in R, (y,z) \in R$ and similarly for $B$, so that this defines not only the struts in the corresponding abstract-core, but also any plates on triangles between two of the joint-cores. Finally, a plate will define a collection of plates whose joints lie in distinct joint-cores. To match up with the constraints specified previously, an abstract-abstract-core constructed in this way must have a corresponding abstract core which is embeddable in $\mathbf{E}$; the scale considerations you mentioned could probably be rehashed as geometric and combinatorial sufficient conditions for this to hold.

Note that treating this idea in a common framework with the "Euclidean cores" we started from requires a multitude of conceptual extensions: in this setting there can be multiple distinct struts between joint-cores (even if only one such is allowed in a core on these joint-cores), and similarly for plates. That might seem like a daunting detour to make, but given where we're discussing this, I have no reservations in pushing you to find the right level of generality for this exploration. Feel free to put my comments on the back-burner if it will help you to maintain the momentum that you're building up, though!

Ellis D. Cooper (Aug 05 2020 at 19:12):

@[Mod] Morgan Rogers

What I was asking is, if the bubble it treated as a container, how should your container balance equation be interpreted when that container vanishes or disintegrates.

Every temporal-spatial-thing has a finite lifetime. That is the unique bounded open interval of time during which it persists. The partial-order category of bounded open intervals ordered by inclusion is denoted by $\mathbf{I}$. For each bounded open interval $I$, the assignment to $I$ of the set of sets of bounded open intervals whose union equals $I$ is the set of covers of $I$ and is a Grothendieck topology for $\mathbf{I}$ (correct me if I am wrong!). Assume that the set of all spatial-things is a variable set in the sense of a set-valued sheaf $\mathcal{X}$ on $\mathbf{I}^{op}$, so that $\mathcal{X}_I$ is the set of temporal-spatial-things that persist during $I$. Some of them may persist before and/or after $I$, and some may end when $I$ ends. If a container temporal-spatial-thing stops persisting, then its balance equation evaporates with it.

Morgan Rogers (he/him) (Aug 06 2020 at 10:07):

Ellis D. Cooper said:

Every temporal-spatial-thing has a finite lifetime. That is the unique bounded open interval of time during which it persists. The partial-order category of bounded open intervals ordered by inclusion is denoted by $\mathbf{I}$. For each bounded open interval $I$, the assignment to $I$ of the set of sets of bounded open intervals whose union equals $I$ is the set of covers of $I$ and is a Grothendieck topology for $\mathbf{I}$ (correct me if I am wrong!).

This is correct, but the topos of sheaves that you get from this site is equivalent to the category of sheaves on the real line!

Assume that the set of all spatial-things is a variable set in the sense of a set-valued sheaf $\mathcal{X}$ on $\mathbf{I}^{op}$.

Taking this definition of the collection of all spatial-things being a time-parameterised set, insisting that lifespans are finite is a somewhat artificial condition, and not only because such a sheaf allows for something that persists for all time. For example, if a species is destroyed and then a new instance of it is later created (with the same shape and position, in the case of spatial-things), the nature of species is such that these instances won't be distinguishable, and in particular the sheaf won't be able to distinguish between them, so that if this recurs in an infinite loop, this will look the same as a single spatial-thing having an infinite, segmented lifespan. But the fact that lifespans can be finite seems sensible, given the morphisms you established earlier.

If a container temporal-spatial-thing stops persisting, then its balance equation evaporates with it.

So these equations are intended to be applied only locally in time and space. I can accept that for now.

Morgan Rogers (he/him) (Aug 06 2020 at 10:08):

Are you elaborating this on the fly, or have you taken some of these ideas further? Is there an explicit goal you have in mind that we might already have the vocabulary to express?

Ellis D. Cooper (Aug 06 2020 at 14:31):

@[Mod] Morgan Rogers

This is correct, but the topos of sheaves that you get from this site is equivalent to the category of sheaves on the real line!

I was thinking that I may postulate existence of a particular sheaf $\mathcal{X}$ with certain specific properties, without regard to all other possible sheaves.

Taking this definition of the collection of all spatial-things being a time-parameterised set, insisting that lifespans are finite is a somewhat artificial condition, and not only because such a sheaf allows for something that persists for all time.

"All time" is not a bounded open interval, so I do not understand something here. Is the Grothendieck topology automatically closed under arbitrary unions?

if a species is destroyed and then a new instance of it is later created (with the same shape and position, in the case of spatial-things), the nature of species is such that these instances won't be distinguishable...

Since a species is an equivalence class of connected-spatial-things, its members are disjoint connected-spatial-things, regardless of time. For a connected temporal-spatial-thing $(t,\Sigma)$, an equivalent connected temporal-spatial-thing $(t',\Sigma')$ where $t'>t$ and $\Sigma'\cong \Sigma$ is a different temporal-spatial-thing than $(t',\Sigma)$, and they are not at the same position in space.

"If a container temporal-spatial-thing stops persisting, then its balance equation evaporates with it."

So these equations are intended to be applied only locally in time and space. I can accept that for now.

An equation applies only insofar as the variables in it are well-defined. According to the "finite lifetime" postulate for all temporal-spatial-things, all variables associated with a temporal-spatial-thing are defined on a bounded open interval and not otherwise.

Are you elaborating this on the fly, or have you taken some of these ideas further? Is there an explicit goal you have in mind that we might already have the vocabulary to express?

I have a stack of notes and writeups (mostly dead ends) from which my posts are excerpted and there is more to come, but certainly not a finished draft of a paper or anything like that.

There is a cadre of theoretical biologists who blithely invoke "thermodynamic processes" in relation to their new "theoretical principles for biology," mostly in the tradition of Varela and autopoiesis and some vague ideas (IMHO) about "closure of constraints" in biological organisms. My general goal is a categorical framework in which their ideas could be precisely articulated, and situated in a larger picture. Specifically, there is, I believe, a single temporal-spatial-thing that encompasses all living organisms past, present, and future. This is called Life, it is unique among all temporal-spatial-things. The goal there is an existence and uniqueness theorem.

Short of achieving that lofty if not pretentious goal, I would be happy to articulate categorical thermodynamics. For that the challenge is to integrate a macroscopic framework on the scale of organisms with a microscopic framework on the scale of molecules. So, for example, this runs into the seemingly endless literature on coordinating Clausius entropy with Boltzmann entropy, and the more well-worked-out coordination of deterministic non-linear mass action differential equations with stochastic chemical master equations.

The basic reason for defining a category of shaped-cores and their motions is that conventional thermodynamics pretty much ignores the shapes of things, but biology cannot do that.

Morgan Rogers (he/him) (Aug 06 2020 at 20:21):

Ellis D. Cooper said:

"All time" is not a bounded open interval, so I do not understand something here. Is the Grothendieck topology automatically closed under arbitrary unions?

A sheaf assigns a set of spatial-things to each open interval, with restriction maps for containments of intervals. As I mentioned, the category of sheaves is equivalent to the category of sheaves on the real line, and indeed it makes sense to talk about arbitrary unions of intervals, because the value of the sheaf on an infinite interval (or more generally any union of open intervals) is well-defined as the limit of the values on the finite open sub-intervals. If something exists in every finite interval of time, it exists for all time.

Since a species is an equivalence class of connected-spatial-things, its members are disjoint connected-spatial-things, regardless of time. For a connected temporal-spatial-thing $(t,\Sigma)$, an equivalent connected temporal-spatial-thing is $(t',\Sigma')$ where $t'>t$ and $\Sigma'\cong \Sigma$ is a different temporal-spatial-thing than $(t',\Sigma)$, and they are not at the same position in space.

If I'm interpreting this right, a temporal-spatial-thing is a real-time-indexed collection of spatial things, hence the sheaf framework. What I'm pointing out is that lifetime is not a well-defined concept in the framework defined so far. If I have the simplest spatial things (joints, say), nothing so far prevents me from bringing two of them continuously closer together at times t<T, until at time T they meet and are replaced by a single joint (just because joints aren't allowed to be at the same point in space..!), and at times t>T there are again two of them and they are moving apart. What is the lifetime of any of these joints as temporal-spatial-things?

My general goal is a categorical framework in which their ideas could be precisely articulated, and situated in a larger picture. Specifically, there is, I believe, a single temporal-spatial-thing that encompasses all living organisms past, present, and future. This is called Life, it is unique among all temporal-spatial-things. The goal there is an existence and uniqueness theorem.

A noble goal, and the right place to be promoting it :heart: In what sense are you expecting this temporal-spatial-thing to be unique?

The basic reason for defining a category of shaped-cores and their motions is that conventional thermodynamics pretty much ignores the shapes of things, but biology cannot do that.

That seems sensible. Do you have ideas or insights into how shape might affect thermodynamics?

Ellis D. Cooper (Aug 07 2020 at 17:29):

@[Mod] Morgan Rogers

If something exists in every finite interval of time, it exists for all time.

Every living thing persists over exactly one bounded open interval of time. The set of living things is a variable set. However best to say that in categorical thermodynamics is okay by me.

In what sense are you expecting this temporal-spatial-thing to be unique?

Up to isomorphism!

Do you have ideas or insights into how shape might affect thermodynamics?

1. MACROSCOPE: Volume change is always due to change in shape of a container. Change in volume (over a period of time) entails an increment of work (hence, power). Work is energy in a physical-process identified by conjugate variables (force,distance). Hence, transport morphisms.
2. NANOSCOPE: In chemical thermodynamics, substances change due to molecular interactions. Molecules are complicated connected shaped-cores which interact mostly by electric forces that "change molecular conformation" (strut-lengths, strut-angles, plate-angles), and that "break or make bonds" (strut removal, strut addition). Hence, transform morphisms.
3. MACROSCOPE: Measurements are repeatable. Volume measurement is repeatable. There may be many duplicates of a container (isomorphic connected shaped-cores). A container may enclose a quantity of a chemical substance. Contact of two non-empty containers may alter their volumes (hence, transform morphisms). Choose a non-empty container as a standard for mensurable change in volume upon contact with non-empty containers. Call it a "thermometer". Observe that two non-empty containers with the same temperature do not change in volume upon contact with one another. There exists a non-empty container which has relatively negligible change in temperature $T$ upon contact with a substantial range of non-empty containers. Call it a "thermal-source" (a.k.a., "heat bath") at $T$. Contact with a thermal-source over a period of time is a physical-process identified by conjugate variables (temperature,entropy), and as for all pairs of conjugate variables, their product is energy.

In these examples, and many others involving other kinds of physical-process (describable in terms of the motions of non-empty containers), textbooks have illustrations exhibiting shapes of containers (e.g., pistons in cylinders, etc.) but the variables in equations represent either (possible results of) measurements, or theoretical connecting-variables or constants. But not shapes per se. However, biological-processes such as mitosis, for example, crucially involve macroscopic and nanoscopic changes in shape. Hence, transport and transform morphisms at wide-ranging scales.

Morgan Rogers (he/him) (Aug 08 2020 at 12:51):

Ellis D. Cooper said:

In what sense are you expecting this temporal-spatial-thing to be unique?

Up to isomorphism!

Okay, I should have asked more precisely what understanding you expected such a uniqueness theorem to provide. We know that we don't live in $\mathbf{E}$, only a space that's locally Euclidean (at least in its habitable regions, to our knowledge). So at best you'll be able to get a local model of life (on the Earth's surface, say) as a temporal-spatial-thing. Even if you do manage to impose enough conditions to define "life", the implication of this definition referring to something unique would surely be an entirely deterministic model of the history of all of the things you consider to be living... although since you're going to the level of molecules, you'll either have to include a lot of shapes that aren't intrinsically alive or have these constituent pieces spontaneously appear at the instant they are incorporated into a living system, neither of which sounds desirable. The determinism alone is dubious, but any kind of uniqueness seems like a strange goal. The variety of living things is vast, and I'm personally of the opinion that things are alive independently of the entire history and future of life (that is, life should be a (semi-)local property rather than a global one). So, again, in what sense do you expect "life" to be unique?

I like all of the examples you've given of how shape interacts with thermodynamics; it's seems like a neat corner of chemistry that is often judged too messy to be usefully tractable. Of course, at that scale there are lots of quantum effects, but maybe those can be strapped on later once there are some geometric hypotheses to investigate.

Ellis D. Cooper (Aug 08 2020 at 18:03):

@[Mod] Morgan Rogers

So, again, in what sense do you expect "life" to be unique?

This topic was stimulated by a collection of papers in certain journals on theoretical biology. There is a particular paper,
"Is a General Theory of Life Possible? Seeking the Nature of Life in the Context of a Single Example." There, Carol E. Cleland writes that "universal biology presupposes a set of core concepts and unifying principles, mechanisms, or processes capable of explaining how living systems differ from nonliving systems." She points out the "N=1 Problem," which is that the only life with which we are familiar represents a single example and there is worrisome evidence that it is unrepresentative of life considered generally." That single example is what I call ``Life," a rather large temporal-spacial-thing, with every living thing past and present and future a connected sub-temporal-spatial thing that persists for exactly one bounded interval of time. To keep things as simple as possible the framework does not (yet) join time and space in relativistic spacetime, nor does it (yet) confine the story to the Earthly biosphere. The hope is to enunciate a formal (category-theoretic) system that accomodates the biological intuitions published in papers with titles such as "Autopoiesis, Biological Autonomy and the Process Vew of Life" (Meincke), "A Universal Definition of Life: Autonomy and Open-Ended Evolution" (Ruiz-Mirazo, Pereto, Moreno), and "Toward a theory of organisms: Three founding principles in search of a useful integration" (Soto, Longo, et al). These authors seem sufficiently careful to define what they mean that I think there is a (long?) shot at formalizing their intuitions.

More specifically, Mossio, Montevil, and Longo "propose a diagrammatic description of closure, which provides a structured understanding of the principle [of biological "organization"]. In this framework, biological organization refers to the mutual dependence (closure) between constraints, exerted on processes occuring in open thermodynamic conditions." Closure is "a global biological property, an overall determination that is conserved through ontogenetic and phylogenetic times." (In my terms, through Life.) As a category-theory inclined human being, I am big on "diagrammatic descriptions," and as a mathematician I am concerned to explain thermodynamics, with its
raucous history, as simply as possible. This is all very tough stuff, but I don't have anything better to do with my time.

Morgan Rogers (he/him) (Aug 09 2020 at 14:13):

Ellis D. Cooper said:

[Mod] Morgan Rogers
That single example is what I call ``Life," a rather large temporal-spacial-thing, with every living thing past and present and future a connected sub-temporal-spatial thing that persists for exactly one bounded interval of time.

1. You expect organisms to be connected spatial-things?
2. You still haven't addressed the problems I was pointing out with lifespan. As a more life-relevant example, when a bacterium self-replicates, what happens at the point of mitosis, when two virtually identical cells are produced? Does the life of the original cell end while the lives of two "new" cells begin? Does the answer change as soon as there is any asymmetry introduced (eg for a mammal giving birth)?

The hope is to enunciate a formal (category-theoretic) system that accommodates the biological intuitions published in papers with titles such as... These authors seem sufficiently careful to define what they mean that I think there is a (long?) shot at formalizing their intuitions.

From skimming through some of these papers, it seems to me that the whole point of universal biology is to find a definition of Life that is not constrained to the instance(s) of life with a single common ancestor that we have direct experiential access to, or even to the material configuration of that life. Even if one avoids specifying the concrete interpretation of the joints, struts and plates in the model you have proposed (which until now I have assumed at the smallest scale will be atoms or molecules and bonds between these), supposing that there should only be one formal structure up to isomorphism that represents (an entire history of) "life" doesn't seem to me to be a positive step towards that goal. If you have some ideas that led you to the intuition that this should be the case, however, I would be very interested to hear them.

On a more abstract note, there could be some hope of identifying one or more "autopoietic units" - instances of temporal-spatial-things which, even if not universal, can be identified as representing life in the abstract, so that any instance of them (where joints, struts, plates get interpreted as physical things) can be identified as life. I'm biased towards this kind of pursuit due to the belief I mentioned earlier that life seems to me to be a locally definable phenomenon, so that life on Mars would still be considered life, independent of its origins.

Morgan Rogers (he/him) (Aug 09 2020 at 14:39):

Incidentally, there is a little piece of the Cleland article that I would like to critically analyse...

One cannot safely generalize to all life, wherever and whenever it may be found, from a sample of one. To do so would be a bit like trying to come up with a theory of mammals based solely on observations of zebras.

On one hand, as any category theorist (or indeed mathematician) can appreciate, it is perfectly possible to generalize starting from a single example. Category theory is all about extracting the key features of a situation and identifying that same combination of features in other contexts - identifying examples of things having some but not all features in common with one we are interested in. We can identify all of the features of a mammal in a zebra, and those would be perfectly adequate to identify or theorize about other mammals.

On the other hand, without a concrete frame of reference consisting of several different examples having key features in common, there is no way to determine preference for one collection of features over another. Studying zebras in isolation, there would be no reason to prioritise the feature of producing milk over, say, that of having stripes. It is improbable that one would come up with a definition of mammal as a super-class to which zebras belong from studying zebras alone. This is the real reason that Cleland's "N=1 problem" is a problem: not because it prevents us from forming reasonable definitions of life (that we can test against all of the organisms at our disposal), but because no matter how abstract we make our definitions, we won't have anything new to label as a living thing, beyond perhaps some complex systems themselves consisting of collections of organisms... at least until some aliens visit.

Ellis D. Cooper (Aug 09 2020 at 21:57):

@[Mod] Morgan Rogers

You expect organisms to be connected spatial-things?

Yes and no, it depends on scale. With a COSMOSCOPE, organisms are mere joints in a vast space. With a MACROSCOPE, organisms may be analyzed in terms of sub-spatial-things connected by bones at ``joints." With a MESOSCOPE, sure, an organism is analyzed in terms of sub-spatial-things that are not connected, like tissue cells and blood cells. With a NANOSCOPE, an organism is a vast assembly of molecules, but any two molecules may be related by some chain of struts at least for just a moment.

what happens at the point of mitosis, when two virtually identical cells are produced? Does the life of the original cell end while the lives of two "new" cells begin?

In mitosis a single cell divides into two cells (final step of M phase). In terms of morphisms, prior to division a cell undergoes changes in core and shape of constituent sub-spatial-things both of which require transport and transform morphisms, and at division there is removal of struts resulting in two distinct unconnected new spatial-things, which is a transform morphism. So, the original cell no longer persists at the moment of division, and the bounded time intervals of persistence of each of the new cells begin.

I respectfully beg to be forgiven for a confusion that I should remedy right away. That is to say, I retract the concept of a "temporal-spatial-thing" because that is contrary to the whole idea of a variable set of spatial-things that persist during bounded open intervals. But the latest end of one bounded open interval may coincide with the earliest end of another bounded open interval, so only in that sense is there a ``moment" when the parent cell ceases to exist and the daughter cells begin to exist. A similar analysis of exocytosis, and of birth, in terms of transport and transform morphisms seem possible. In the case of birth, of course, a woman before conception, then during pregnancy, then after birth is the same person, but she has undergone a very complicated morphism. And then there is a child, a completely new organism. That's different from mitosis.

it seems to me that the whole point of universal biology is to find a definition of Life that is not constrained to the instance(s) of life with a single common ancestor that we have direct experiential access to, or even to the material configuration of that life...(an entire history of) "life" doesn't seem to me to be a positive step towards that goal.

True that, but my goal is not that goal. My goal, infinitely lofty though it may be, is to identify and formalize a recognizable system of principles that characterize Life as it is known, and there is only one example. It is described, for example, in ``The Book of Life, An Illustrated History of the Evolution of Life on Earth," edited by Stephen Jay Gould.

On a more abstract note, there could be some hope of identifying one or more "autopoietic units" - instances of temporal-spatial-things which, even if not universal, can be identified as representing life in the abstract, so that any instance of them (where joints, struts, plates get interpreted as physical things) can be identified as life. I'm biased towards this kind of pursuit due to the belief I mentioned earlier that life seems to me to be a locally definable phenomenon, so that life on Mars would still be considered life, independent of its origins.

My view is global. To me there is a natural phenomenon called Life (on Earth!) that is ONE spatial-thing that has persisted so far for around $3.4\times 10^{9}$ years, and with luck has a ways to go before the predicted heat death of the Universe. Whatever, that is a bounded interval of time during which Life persists. In this view, there are two very, very different spatial-things, Life, and everything else. I believe that it is not possible to abstractly characterize ``autopoietic units" without considering them in the context of Life as a whole. DNA goes way back, and (very indirectly) represents in an extraordinarily condensed molecule the encounters of Life with itself and everything else.

By the way, extra-terrestrial-Life, artificial-Life, and Life-on-Mars are fun to imagine, but that is all they are, imaginary (so far). As a would-be mathematical scientist I prefer to try and think about something that exists, that is difficult enough.

Ellis D. Cooper (Aug 09 2020 at 23:18):

I just had the pleasure of seeing published a letter I wrote last week to the editor of The Boston Globe:

Rara avis

It is a pleasure to learn of Cassandra Extavour and her achievements at "the highest levels of excellence" in the worlds of art as a singer and in science as a systems biologist ("Solvers," Ideas, Aug. 2). Likewise, not only is Olivia Caramello a world-class research mathematician specializing in topos theory, she has also given piano recitals at Trinity College Chapel, Cambridge, Britain, and at other venues. We should treasure such rare and valuable human beings.

Ellis D. Cooper
Rockport

Morgan Rogers (he/him) (Aug 10 2020 at 11:27):

Ellis D. Cooper said:

what happens at the point of mitosis, when two virtually identical cells are produced? Does the life of the original cell end while the lives of two "new" cells begin?

[...] So, the original cell no longer persists at the moment of division, and the bounded time intervals of persistence of each of the new cells begin. In the case of birth, of course, a woman before conception, then during pregnancy, then after birth is the same person, but she has undergone a very complicated morphism. And then there is a child, a completely new organism. That's different from mitosis.

I still feel that you haven't adequately explained the distinction here. What, fundamentally, is the difference between mitosis and giving birth? In each case an organism undergoes a complex shape change and the result is a pair of organisms. What's special about the latter case that makes one of the organisms involved persistent? I could present to you a more morbid "Ship of Theseus but with organs" thought experiment here if you insist (or on a more sensible note observe that our cells get replaced on a regular basis)... but all I'm really trying to get at is that unless you need the definition of lifespan for something in particular, there is no reason to wrestle with such problems for the sake of it.

I respectfully beg to be forgiven for a confusion that I should remedy right away. That is to say, I retract the concept of a "temporal-spatial-thing" because that is contrary to the whole idea of a variable set of spatial-things that persist during bounded open intervals.

It really isn't contrary, these seem like compatible notions to me: a time-indexed set of spatial-things meets the definition of a temporal-spatial-thing you gave previously. But it's your theory, transform it in whatever way is conceptually useful to you :blush:

My goal, infinitely lofty though it may be, is to identify and formalize a recognizable system of principles that characterize Life as it is known, and there is only one example.

I wouldn't say that's infinitely lofty, given your insistence on mortality :stuck_out_tongue_wink:
On a more serious note... why? If you characterise something, anything that has the same characteristics will definitionally be an instance of it. If you make your theory so specific that its unique instance 'up to isomorphism' is the known instances of recent life on Earth (and its extension into the future), what are you hoping to get out of that? I'm not trying to be dismissive here, I'm genuinely curious what you picture the potential of this line of research to be.

By the way, extra-terrestrial-Life, artificial-Life, and Life-on-Mars are fun to imagine, but that is all they are, imaginary (so far).

What I meant was, if the first person on Mars discovers some bacteria buried in the dust and isn't immediately certain how or when they got there, whether or not that thing is considered alive shouldn't depend on whether or not it turns out to have the same common ancestor as us. Given what you said about your goal being distinct from that of universal biology, that might sound like conflation of "Life" with"being alive", but ultimately your theory will have to draw that line somewhere since you have to decide which things qualify as members of your variable set. (On the subject of artificial life, we could take a tangent on the lab-grown chicken meat that KFC has recently started selling, but I would rather not).

As a would-be mathematical scientist I prefer to try and think about something that exists, that is difficult enough.

Hmm... On one hand, mathematics involves an abstraction, which you've already made, from the realm of things which exist to a realm of ideas. Even if you're in the camp of folks who believe these ideas have a reality of their own, all possibilities live in this realm of ideas, so that the collection of "things which are physically realised" will be much smaller than the collection of "things which the abstract formalism allows" unless your formalism has perfect predictive power for reality (if you manage that in any domain, you'll be remembered until Life ceases at the heat death of the universe!). On the other hand, scientific theories should have predictive power! They should include possibilities that we haven't directly observed, and allow us to make deductions or predictions about hypothetical situations, because otherwise they are untestable and at best provide a new taxonomical organisation of known phenomena.

Putting hypothetical fringe cases of life to one side, I'm curious how you anticipate your formalism will have the potential to predict the future of Earth-based life, since that seems to be the thrust of the predictive power under this theory.

Ellis D. Cooper (Aug 10 2020 at 13:10):

@[Mod] Morgan Rogers
I look forward to getting back to you on those important questions! In the meantime, I have some mathematical questions.

In the framework here, joints are points of space, struts are unordered pairs $ab=ba$ of not necessarily distinct joints, and plates are unordered triples $abc=acb=bac=bca=cab=cba$ of not necessarily distinct joints. Please, add a new geometric structure called tetra (not a fish) which is a quadruple $abcd$ of not necessarily distinct joints. A spatial-thing is a finite set of joints and struts, plates, and tetras among the joints, such that interiors $\overline{ab}$ of struts, interiors $\overline{abc}$ of plates, and interiors $\overline{abcd}$ of tetrahedra are all disjoint. A core is an equivalence class of spatial-things induced by a bijection of joints that preserves the other elements of structure. Among cores there are (partially-)defined algebraic operations: For any two cores, the disjoint unions of their elements of structure form a new core. Also, for any two struts $ab$ and $bc$ there is a new core $ab\underset{b}{+}bc$, for any two plates $abc$ and $bcd$ there is a new core $abc\underset{bc}{+}bcd$. If $B=A\underset{x}{+}xy$ is a core where $x$ is a joint of $A$, then $B\underset{y}{+}yz$ is a new core, etc., etc. A question is, what is the category theory of this algebraic structure?\\

Within the core equivalence class of a spatial-thing, there is variation of shape, namely the lengths of struts $ab$, the angles of cores of the form $ab\underset{b}{+}bc$, and the angles of cores of the form $abc\underset{bc}{+}bcd$. So, there is a refinement of the core equivalence relation in which two spatial-things are equivalent not only if they have equivalent cores, but also the same shape, i.e., equal lengths and angles of corresponding elements of structure. Formally, $\overset{\beta}{ab}$ denotes strut $ab$ with strut-length $\beta$, and

$$\overset{\beta}{ab} \overset{\alpha}{\underset{b}{+}} \overset{\beta'}{bc}$$

denotes core $ab\underset{b}{+}bc$ with strut-angle $\alpha$, and

$$\overset{\alpha}{abc}\overset{\delta}{\underset{bc}{+}} \overset{\alpha'}{bcd}$$

denotes core $abc\underset{bc}{+}bcd$ with plate-angle $\delta$. Again, a question is, what is the category theory generated by these symbols?

Another question is, for a specified core and an assignment of numbers $\beta,\alpha$ and $\delta$ to sub-cores of the forms $ab$, $ab\underset{b}{+}bc$, and $abc\underset{bc}{+}bcd$, what are necessary and sufficient conditions that the assignment is actually the shape of a spatial-thing, bearing in mind the "disjoint interiors of elements of structure" stipulation?\\

Morgan Rogers (he/him) (Aug 10 2020 at 14:10):

Ellis D. Cooper said:

A question is, what is the category theory of this algebraic structure?

To do any category theory, we need to re-establish what the morphisms are. Presuming that they're the morphisms you specified before, the hom-sets are naturally equipped with a partially stratified topological structure (where the strata are determined by the discrete operations of adding or removing components and the continuous motions are topologized in the natural way). An alternative that brings out the structural aspect is to work with what you previously called "abstract-cores", at which point the walking joint, strut etc. generate the category, giving us a generator-based way to study these structures, as your new notation suggests. Some intermediate stage between these might give a combination of properties you're looking for...