February 23, 2015

Concepts of Sameness (Part 1)

Posted by John Baez

Elaine Landry is a philosopher at U. C. Davis, and she’s editing a book called Categories for the Working Philosopher. Tentatively, at least, it’s supposed to have chapters by these folks

• Colin McLarty (on set theory)
• David Corfield (on geometry)
• Michael Shulman (on univalent foundations)
• Steve Awodey (on structuralism, invariance, and univalence)
• Michael Ernst (on foundations)
• Jean-Pierre Marquis (on first-order logic with dependent sorts)
• John Bell (on logic and model theory)
• Kohei Kishida (on modal logic)
• Robin Cockett and Robert Seely (on proof theory and linear logic)
• Samson Abramsky (on computer science)
• Michael Moortgat (on linguistics and computational semantics)
• Bob Coecke and Aleks Kissinger (on quantum mechanics and ontology)
• James Weatherall (on spacetime theories)
• Jim Lambek (on special relativity)
• John Baez (on concepts of sameness)
• David Spivak (on mathematical modeling)
• Hans Halvorson (on the structure of physical theories)
• Elaine Landry (on structural realism)
• Andrée Ehresmann (on a topic to be announced)

We’re supposed to have our chapters done by April. To make writing my part more fun, I thought I’d draft some portions here on the $n$-Café.

Looking at the heavy emphasis on topics connected to logic, I’m sort of wishing I’d gone off in some other direction, like David Corfield, or Bob Coecke and Aleks Kissinger. I’d originally been going to write about my current love: category theory in applied mathematics and electrical engineering. But I decided that’s still research in progress, not something that’s ready to offer for the delectation of philosophers.

Anyway, my chosen topic is ‘concepts of sameness’ — meaning equality, isomorphism, equivalence and other related notions — and how they get re-examined in $n$-category theory, homotopy theory, and homotopy type theory. But I don’t want to merely explain piles of mathematics: I also want to think about the question what does it mean to be the same, in a somewhat philosophical way.

So, I might start with a bit of ancient philosophy. Something like this:

In classical Greece and China, philosophers were very concerned about the concept of sameness — and its flip side, the concept of change. Their questions may seem naive today, because we’ve developed ways of talking to sidestep the issues they found puzzling. We’ve certainly made progress over the centuries. But we’re not done understanding these issues — indeed, mathematics is in the middle of a big shift in its attitude toward ‘equality’. So it pays to look back at the history.

Indeed, progress in mathematics and philosophy often starts by revisiting issues that seemed settled. When we regain a sense of childlike wonder at things we’d learned to take for granted, a space for new thoughts opens up.

With this in mind, and no pretense at good classical scholarship, let us look at a fragment of Heraclitus and Gongsun Long’s “white horse paradox”.

Heraclitus

Heraclitus lived roughly from 535 to 475 BC. Only fragments of his writings remain. Most of what we know about him comes from Diogenes Laertius, a notoriously unreliable biographer who lived six hundred years later, and Aristotle, who was concerned not with explaining Heraclitus but demolishing his ideas on physics. Among later Greeks Heraclitus was famous for his obscurity, nicknamed “the riddler” and “the dark one”. Nonetheless a certain remark of his has always excited people interested in the concepts of sameness and change.

In a famous passage of the Cratylus (402d), Plato has Socrates say:

Heraclitus is supposed to say that all things are in motion and nothing at rest; he compares them to the stream of a river, and says that you cannot go into the same water twice.

This is often read as saying that all is in flux; nothing stays the same. But a somewhat more reliable quote passed down through Cleanthes says:

On those stepping into rivers staying the same other and other waters flow.

Here it seems that while the river stays the same, the water does not. To me, this poses the great mystery of time: we can only say an entity changes if it is also the same in some way — because if it were completely different, we could not speak of an entity. Of course we can mentally separate the aspect that stays the same and the aspect that changes. But these two aspects must be bound together, if we are to say that ‘the same thing is changing’.

In category theory, we try to negotiate these deep waters using the concept of ‘isomorphism’. If we have an isomorphism $f : x \to y$, the objects $x$ and $y$ can be unequal and yet ‘the same in a way’. Alternatively, we can have an isomorphism from an object to itself, $f : x \to x$, where clearly $x$ is the same as $x$ yet $f$ describes some sort of change. So, isomorphisms exhibit a subtle interplay between sameness and difference that may begin to do justice to Heraclitus’ thoughts.

In mathematical physics, the passage of time is often described using isomorphisms: most simply, a one-parameter family of automorphisms $f_t : x \to x$, one for each time $t \in \mathbb{R}$. The automorphisms describe how a physical system is the same yet changing. The same idea generalizes to situations where time is not merely a line of real numbers.

In general, given an object $x$ in a category, the automorphisms $f : x \to x$ form a group called the ‘automorphism group’ or ‘symmetry group’ of that object. The automorphisms can be seen as ‘ways for to change the object without changing it’. For example, a square has symmetries, which are ways you can rotate and/or reflect it that don’t change its appearance at all. Symmetry is very important in physics, and it is worth thinking about why, because this takes us back to some of the questions Heraclitus raised.

Notes

I’ll continue next time. I may write more than I wind up using, but that’s okay. Here are some notes from the Stanford Encyclopedia of Philosophy article on Heraclitus:

There are three alleged “river fragments”:

B12. potamoisi toisin autoisin embainousin hetera kai hetera hudata epirrei.

“On those stepping into rivers staying the same other and other waters flow.” (Cleanthes from Arius Didymus from Eusebius)

B49a. potamois tois autois…

“Into the same rivers we step and do not step, we are and are not.” (Heraclitus Homericus)

B91[a]. potamôi… tôi autôi…

“It is not possible to step twice into the same river according to Heraclitus, or to come into contact twice with a mortal being in the same state.” (Plutarch)

Of these only the first has the linguistic density characteristic of Heraclitus’ words. The second starts out with the same three words as B12, but in Attic, not in Heraclitus’ Ionic dialect, and the second clause has no grammatical connection to the first. The third is patently a paraphrase by an author famous for quoting from memory rather than from books. Even it starts out in Greek with the word ‘river,’ but in the singular. There is no evidence that repetitions of phrases with variations are part of Heraclitus’ style (as they are of Empedocles’). To start with the word ‘river(s)’ goes against normal Greek prose style, and on the plausible assumption that all sources are trying to imitate Heraclitus, who does not repeat himself, we would be led to choose B12 as the one and only river fragment, the only actual quotation from Heraclitus’ book. This is the conclusion of Kirk (1954) and Marcovich (1967), based on an interpretation that goes back to Reinhardt (1916). That B12 is genuine is suggested by the features it shares with Heraclitean fragments: syntactic ambiguity (toisin autoisin ‘the same’ [in the dative] can be construed either with ‘rivers’ [“the same rivers”] or with ‘those stepping in’ [“the same people”], with what comes before or after), chiasmus, sound-painting (the first phrase creates the sound of rushing water with its diphthongs and sibilants), rhyme and alliteration.[1]

If B12 is accepted as genuine, it tends to disqualify the other two alleged fragments. The major theoretical connection in the fragment is that between ‘same rivers’ and ‘other waters.’ B12 is, among other things, a statement of the coincidence of opposites. But it specifies the rivers as the same. The statement is, on the surface, paradoxical, but there is no reason to take it as false or contradictory. It makes perfectly good sense: we call a body of water a river precisely because it consists of changing waters; if the waters should cease to flow it would not be a river, but a lake or a dry streambed. There is a sense, then, in which a river is a remarkable kind of existent, one that remains what it is by changing what it contains (cf. Hume Treatise 1.4.6, p. 258 Selby-Bigge). Heraclitus derives a striking insight from an everyday encounter. Further, he supplies, via the ambiguity in the first clause, another reading: on the same people stepping into rivers, other and other waters flow. With this reading it is people who remain the same in contrast to changing waters, as if the encounter with a flowing environment helped to constitute the perceiving subject as the same. (See Kahn 1979.) B49a, by contrast, contradicts the claim that one can step into the same rivers (and also asserts that claim), and B91[a], like Plato in the Cratylus, denies that one can step in twice. Yet if the rivers remain the same, one surely can step in twice—not into the same waters, to be sure, but into the same rivers. Thus the other alleged fragments are incompatible with the one certifiably genuine fragment.

Posted at February 23, 2015 12:20 AM UTC

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Re: Concepts of Sameness (Part 1)

Nice! I’ve also been working on drafts of parts of my chapter and planning to post some of them on the blog, but you beat me to it.

Heraclitus’s remark reminds me of the oft-repeated claim that some massive fraction of the atoms in a human body are replaced on an annual basis. A quick google for citations turned up an NPR program and a stackexchange question, and thereby to a wikipedia page about the Ship of Theseus which I guess is a fancy name for the same paradox.

You didn’t say this explicitly, but I guess the point is that this is exactly the sort of problem that is solved by a structural attitude? That is, every time we replace some part of an object, or some bit of water in the river, we obtain an isomorphic object, and from a structural point of view that’s all that matters (e.g. we don’t really care whether the elements of “the” cyclic group with 2 elements actually “are” numbers, letters, automorphisms, or homotopy classes of maps $S_4 \to S_3$).

Could you say a little more about what you have in mind regarding one-parameter families of automorphisms in mathematical physics? It’s been a while since I thought much about physics. In classical physics I’m used to thinking of time evolution as describing a path through phase space, which is maybe the flow of a vector field (Hamiltonian blah?) and thus consists of isomorphisms; is that what you are thinking of? So the isomorphism is an automorphism of the phase space of the system, the system itself being the same over time, but the state of the system changing? I guess this is roughly the same idea as the Schrodinger picture of quantum mechanics? (And looking again at your final sentence, maybe this is the direction you are heading for the next post?)

Posted by: Mike Shulman on February 23, 2015 6:02 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

You didn’t say this explicitly, but I guess the point is that this is exactly the sort of problem that is solved by a structural attitude? That is, every time we replace some part of an object, or some bit of water in the river, we obtain an isomorphic object, and from a structural point of view that’s all that matters.

That sounds right. I hadn’t planned to analyze this “same river, different water” issue in detail, but I suppose I should expand on my point a bit by returning to Heraclitus’ original example and showing how category theory, or “a structural attitude”, can shed some light on his puzzle.

By the way, somewhere Jorge Luis Borges claimed Heraclitus was sneakily making a second point as well: “you can’t step in the same river twice” not only because it’s a different river, but also because it’s a different you. But again, it’s an isomorphic you, at least in some approximation.

I recently learned that “you can’t step in the same river twice” is probably not something Heraclitus originally said. This way of putting it only goes back to Plutarch’s remark:

It is not possible to step twice into the same river according to Heraclitus, or to come into contact twice with a mortal being in the same state.

And here the sneaky second point lies quite close to the surface.

It turns out that the ship of Theseus example, too, was discussed by Plutarch! He wrote:

The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.

I guess Plutarch had these issues on his mind.

In classical physics I’m used to thinking of time evolution as describing a path through phase space, which is maybe the flow of a vector field (Hamiltonian blah?) and thus consists of isomorphisms; is that what you are thinking of? So the isomorphism is an automorphism of the phase space of the system.

That’s right, and in quantum mechanics the isomorphism is an automorphism of the Hilbert space of the system. In both case our physical system has a set of states, or more precisely some sort of “object of states”, and time evolution acts as automorphisms of this.

So actually we should be a bit careful here if we want to analyze Heraclitus’ example using category theory. Even for a falling rock, time evolution is an automorphism of the set of states. The individual states, points in this set, are noticeably different, but time evolution acts to permute them. The river has an extra feature, which is that states can be different yet isomorphic (at least approximately). So the flowing river, unlike the falling rock, can “move while seeming to stay the same” — which is clearly why Heraclitus chose this example of a physical system.

In short, if we really delve into Heraclitus’ example — which I probably won’t in my paper — we’ll see he’s foreshadowing a kind of categorified version of physics! Instead of a mere set of states (or an object in some category, like the category of manifolds or Hilbert spaces) we should think of the river as having a category of states (or an object in some 2-category), where states can be isomorphic!

This is the idea behind gauge theory — and in fact there’s a formalism for fluid mechanics that makes this rather explicit! But I need to think about this more.

There’s at least one more twist lurking in Heraclitus’ remark, noted by the Stanford Encyclopedia of Philosophy. “Moving while seeming to stay the same” is not merely something a river can do, it’s actually part of what makes a river be a river:

we call a body of water a river precisely because it consists of changing waters; if the waters should cease to flow it would not be a river, but a lake [….]

I don’t really know the meaning of this in terms of modern category-theoretic physics.

Thanks for making me think about this in more detail! I may change my paper to say more about Heraclitus and his river: I hadn’t noticed that he was foreshadowing the idea of a category of states.

Posted by: John Baez on February 23, 2015 6:51 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I wrote:

Instead of a mere set of states (or an object in some category, like the category of manifolds or Hilbert spaces) we should think of the river as having a category of states (or an object in some 2-category), where states can be isomorphic!

Now I’m trying to see if someone has already implicitly described fluid flow using a formalism where there’s a category — probably an infinite-dimensional Lie groupoid — of states.

There’s a nice approach to fluid dynamics based on the group $SDiff(M)$ of volume-preserving diffeomorphisms of the Riemannian manifold $M$ called ‘space’. This group can be made into an infinite-dimensional Lie group with a Riemannian metric. Then Euler’s equation, describing the motion of incompressible fluid of zero viscosity, simply describes geodesic motion on $SDiff(M)$!

All this is well-known, but now I’m wondering if we can think of $SDiff(M)$ as a group of ‘gauge symmetries’ and get a Lie groupoid of states of the fluid. The idea is that applying a volume-preserving diffeomorphism to the positions of the particles in the fluid has no observable effect so counts as an isomorphism between states. However, different velocity vector fields should count as observably different.

I think it could work this way. We start with $G = SDiff(M)$. We get a symplectic manifold $T^* G$ whose points are states of the fluid, describing both ‘position’ and ‘velocity’ (or really ‘momentum’) degrees of freedom. We could then apply symplectic reduction, using the action of $G$ on $T^* G$ by left translations. A point in the reduced phase space just describes the velocity vector field of the fluid!

[Edit: no, symplectic reduction won’t work that way.]

However, we could more subtly take the ‘weak quotient’ of $T^* G$ by the left action of $G$, getting an infinite-dimnsional Lie groupoid. Here we’d count two states of the fluid as isomorphic rather than equal when we permute the particles in the fluid by a volume-preserving diffeomorphism.

Has anyone done any of this in their work on fluid dynamics?

Posted by: John Baez on February 23, 2015 4:29 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

he’s foreshadowing a kind of categorified version of physics!

Haha, neat!

“Moving while seeming to stay the same” is not merely something a river can do, it’s actually part of what makes a river be a river… I don’t really know the meaning of this in terms of modern category-theoretic physics.

It seems to me more of an issue of language than physics. Surely the physics of possibly-flowing water doesn’t care what criteria we use to decide whether to apply the English words “river” or “lake”.

Posted by: Mike Shulman on February 23, 2015 6:50 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Mike wrote:

Surely the physics of possibly-flowing water doesn’t care what criteria we use to decide whether to apply the English words “river” or “lake”.

That’s true, but it does care whether the water is flowing, and how fast! And there’s something interesting about defining the velocity of a fluid when we’ve already decided that the fluid counts as isomorphic after it flows a while!

Its ‘position’ is changing to a new isomorphic position… but we don’t want its ‘velocity’ to be isomorphic to zero.

You might think this would cause trouble, but in fact it seems to work out nicely when we use the usual technique for dealing with these problems: symplectic reduction as described here, or the ‘stacky’ version that we really should be using (which I believe gives an equivalent result in this particular case).

Sorry, that sounds technical. But the idea is cool: in this context, ‘not changing’ shouldn’t be isomorphic to ‘changing to an isomorphic situation’. And it’s not! You just turn the crank of mathematical physics, and this is the result you get!

Posted by: John Baez on February 23, 2015 10:54 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

That is very cool! Maybe one day I’ll understand it.

Posted by: Mike Shulman on February 24, 2015 4:49 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

If I explain it, you can just read the book!

Posted by: John Baez on February 24, 2015 4:59 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Hmm, it’s not working out, and I should have realized that instantly. If you take the cotangent bundle $T^* G$ of a Lie group $G$ and do symplectic reduction with respect to the action of $G$ by left translations, you get a point.

So, I’m a bit mystified about what mathematical process lets us start by describing the time evolution for an incompressible fluid as geodesic flow on $T^* G$ where $G$ is the group of volume-preserving diffeomorphisms of space, and end up with the usual description purely in terms of velocity: namely, the incompressible Euler equations. Somehow it makes all ‘positions’ of the fluid count as the same without making all velocities count as zero. But symplectic reduction would do both.

Posted by: John Baez on February 24, 2015 4:49 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

For a Lie group $G$, the tangent bundle $T G$ (and the cotangent bundle $T^* G$) is parallelizable, in two canonical ways.

Posted by: Jesse C. McKeown on February 24, 2015 6:28 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Yes. As V. I. Arnold pointed out, the dynamics of an incompressible inviscid fluid is formally just like that of a rotating body: geodesic flow on $T^\ast G$ with respect to some left-invariant metric on a Lie group $G$. For the rotating body $G = SO(3)$, while for the fluid it’s the infinite-dimensional Lie group $SDiff(M)$.

The funny thing that happens, apparently, is that because the group is parallelizable by left translations and Hamiltonian is invariant under left translations, we can describe the dynamics of the momentum $p \in T^\ast_q G \cong \mathfrak{g}^\ast$ without ever mentioning the position $q \in G$.

I had momentarily hoped this was an example of symplectic reduction. But that was a burst of stupidity: symplectic reduction discards momentum and position information together.

So I’m left wondering what systematic idea this is an example of, if any.

It reminds me a bit of the ‘classical spin-$j$ particle’, where the phase space is $S^2$. A point in here describes the angular momentum vector of spinning particle. And yet, nothing actually ‘spins’ in this description. So again we have somehow discarded position information without discarding momentum information.

Posted by: John Baez on February 24, 2015 7:17 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

The symplectic reduction of the cotangent bundle of a Lie group $G$ is a coadjoint orbit of $G$. On the other hand the differential geometric quotient of the cotangent bundle by the action of $G$ is the dual of the Lie algebra of $G$ with its canonical Poisson bracket.

John, is that what you were referring to?

Posted by: Eugene on February 25, 2015 3:17 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Eugene wrote:

The symplectic reduction of the cotangent bundle of a Lie group $G$ is a coadjoint orbit of $G$.

That’s when you use the action of $G$ on itself by conjugation, right? When you use the action on itself by left or right translations you get a point, right? That’s the kind of action that’s relevant to me here.

On the other hand the differential geometric quotient of the cotangent bundle by the action of $G$ is the dual of the Lie algebra of $G$ with its canonical Poisson bracket.

That’s what I need to understand Euler’s equation for the incompressible inviscid fluid. What was bugging me is that I didn’t recognize it as an instance of a general pattern. But maybe now I do, thanks to you:

You start with a Poisson manifold $X$ and an action of $G$ on $X$ as Poisson maps. Suppose the quotient $X/G$ is a manifold. Furthermore, suppose we can push the Poisson tensor on $X$ forward along the quotient map

$p: X \to X/G$

to make $X/G$ into a Poisson manifold. (Can we always do this, or are extra conditions required?)

Finally, suppose we have a Hamiltonian $H : X \to \mathbb{R}$ that actually factors through the quotient map $p$. Then the flow on $X$ generated by $H$ descends to a flow on $X/G$.

That’s what’s happening with Euler’s equation, I think! Thanks.

Posted by: John Baez on February 25, 2015 3:40 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

That’s when you use the action of $G$ on itself by conjugation, right?

No, this is when the action is by left (or right) translation. The symplectic quotient of the cotangent bundle is a point only if you are reducing at fixed points of the coadjoint action.

In general (for free actions) symplectic reduced spaces are Poission leaves of the quotient, which is always a Poisson manifold. The easiest way to see it is by thinking of functions on the quotient as invariant functions upstairs. Invariants form a Poisson algebra.

If the action is not free, the above statement should be true in some derived sense. In differential geometric setting this is, at the moment, a phantasy. It may be rigorously true in derived algebraic geometry.

Posted by: Eugene on February 25, 2015 12:48 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Here’s the last stanza of Borges’ “Arte poética” (The Art of Poetry):

También es como el río interminable
Que pasa y queda y es cristal de un mismo
Heráclito inconstante, que es el mismo
Y es otro, como el río interminable.

The translation by W. S. Merwin in the Selected Poems anthology:

It is also like the river with no end
That flows and remains and is the mirror of one same
Inconstant Heraclitus, who is the same
And is another, like the river with no end.

Posted by: Blake Stacey on February 23, 2015 8:08 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Blake wrote:

Here’s the last stanza of Borges’ “Arte poética” (The Art of Poetry):

It is also like the river with no end
That flows and remains and is the mirror of one same
Inconstant Heraclitus, who is the same
And is another, like the river with no end.

Thanks! That nicely expresses in poetry how both the observed and observer are changing.

But I’m sure I saw Borges state it in plain prose in some essay. I’ve been looking for this online, and I keep finding more poems of his that mention Heraclitus…

Oh, here’s what I want! It’s in his essay A new refutation of time. (Great title, eh?)

… each time I recall fragment 91 of Heraclitus, ‘You cannot step into the same river twice,’ I admire his dialectical skill, for the facility with which we accept the first meaning (“The river is another”) covertly imposes upon us the second meaning (“I am another”) and gives us the illusion of having invented it…

Posted by: John Baez on February 23, 2015 9:58 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I knew there was a prose example, but you found it before I did!

Posted by: Blake Stacey on February 24, 2015 12:26 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I’m a little late to the party, but I found Mike’s comment

You didn’t say this explicitly, but I guess the point is that this is exactly the sort of problem that is solved by a structural attitude? That is, every time we replace some part of an object, or some bit of water in the river, we obtain an isomorphic object, and from a structural point of view that’s all that matters (e.g. we don’t really care whether the elements of “the” cyclic group with 2 elements actually “are” numbers, letters, automorphisms, or homotopy classes of maps $S^4\toS^3$

to be puzzling.

The question, as I understand it, is what it means (for example) for $Nile_0$ (the Nile as it is today) to be the same river as $Nile_1$ (the Nile as it is tomorrow). Mike’s answer, as I understand it, is that there is an isomorphism between them.

I guess I’m confused about what category these isomorphisms are taken to live in. The first possibility that springs to mind, suggested by analogy to mathematical structures like the group with two elements, is “if there is an isometry of spacetime which takes the water molecules of $Nile_0$ to $Nile_1$, then they are the same. This seems wrong for a couple of reasons: first, there’s no reason to expect such an isometry to in fact exist between $Nile_0$ and $Nile_1$ (except in an approximate sense), and second it would seem to entail that if there were a body of water on the other side of the universe in the same shape as the Nile, it would be the same river, which is simply not true – although perhaps from a “structural perspective” one would be tempted to think that it is true?

So the category is something different – its morphisms aren’t structure-preserving maps in the most naive sense. As Mike says, a morphism in this category can be given by the act of replacing some part of the river with a new part, or something like that. But as we try to specify what exactly a morphism in this category is, it seems to me that we’re reduced back to the original question, “what does it mean for $Nile_0$ to be the same river as $Nile_1$?”, and that introducing the term “isomorphism” really hasn’t bought us anything.

I suppose what we gain from a categorical attitude is a way of approaching the possibility that $Nile_0$ and $Nile_1$ could be the same in more than one way, i.e. the groupoid of Niles could fail to be a setoid. Maybe the category of Niles is not even a groupoid – if the course of the river were to split into two branches tomorrow, those two branches would each have some claim to be the same river as $Nile_0$, but maybe it wouldn’t follow that the two branches themselves were the same river (the morphisms of sameness point in the wrong directions to compose them).

Posted by: Tim Campion on February 28, 2015 1:32 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Tim wrote:

I guess I’m confused about what category these isomorphisms are taken to live in.

Deciding this can be thought of as a way of deciding what you mean when you say “This river is the same river that we say yesterday” or “you are the same person as yesterday”. There’s not just one thing you might mean; there are lots.

In reality, any mathematical concept of isomorphism seems insufficiently vague to capture ordinary language. We’d really need a notion of “approximate isomorphism”, like “isomorphism up to $\epsilon$”.

And that’s not the only problem.

The project of fully formalizing ordinary language usage is hugely complicated, and I don’t really want to try. I really just want to understand things like the passage of time. In a specific theory of physics, the passage of time is typically an isomorphism in some precisely defined category. This makes it possible to say that we have ‘the same’ physical system at a different time, even though change has occurred. And that’s what I’m actually interested in: confronting Heraclitus’ puzzle in the context of our best current scientific understandings of the world.

But my draft here did not make that clear.

Posted by: John Baez on February 28, 2015 2:01 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I guess what really threw me was a couple of changes of perspective required to take the Hamiltonian mechanics approach to the river.

First, we have to agree a priori that $Nile_0$ and $Nile_1$ are both points in a common phase space – the phase space of the river! So in order to get started, we have to presuppose in some sense that the two rivers are the same. It seems to beg the question. This is just the issue John discussed when he said

To me, this poses the great mystery of time: we can only say an entity changes if it is also the same in some way — because if it were completely different, we could not speak of an entity.

On the other hand, perhaps “$Nile_0$ and $Nile_1$ are points in a common phase space” should be considered a pretty solid observation on which to base an explication of their sameness.

There is a second level to the sameness of $Nile_0$ and $Nile_1$, namely the fact that a time-translation automorphism of phase space brings one to the other. This level is strange as well. For one thing, in this picture $Nile_0$ and $Nile_1$ are points - in some sense they have no “intrinsic” properties to be respected by an isomorphism of the structure-preserving-map variety. Rather, it’s an automorphism of all of phase space – an auxiliary object – which relates them, and is relevant to their sameness. Maybe this shouldn’t be too surprising, though. We’ve already agreed that an identical copy of the Nile located on the other side of the galaxy is not the same river.

For another thing, isomorphism is not the whole story. $Nile_0$ and $Nile_1$ are the same because there is a time-translation automorphism of phase space that brings $Nile_0$ to $Nile_1$. There are lots of automorphisms of phase space which are not time-translation automorphisms. And in order to say whether a particular automorphism is a time-translation automorphism, you have to say something the whole automorphism flow – about all the other time-translation automorphisms between this one and the identity, and the fact that they are related by the Hamiltonian vector field. Again, though, maybe this shouldn’t be surprising. It validates the intuitive idea that the “sameness” of $Nile_0$ and $Nile_1$ does have something to do with all the intervening $Nile_t$’s between them, leading from one to the other.

Posted by: Tim Campion on March 10, 2015 3:35 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Another issue is that our choice of category depends on whether we are studying a system by means of passive observations or with active interventions. For example, there are four chopsticks sitting on top of the book to the right of my computer at the moment. They’re not getting up and changing places of their own accord; there’s no dynamical process changing their current ordering into other permutations. But I’m entirely comfortable decategorifying and saying that there are four of them, because with different dynamics we would have the appropriate isomorphisms.

Mundane enough for eating utensils, I suppose, but similar problems can pop up in scientific practice. Consider how an introductory biology textbook typically defines a species: two organisms belong to different species if interbreeding is impossible. Sounds fairly simple. But what if we have two populations of similar-looking organisms in widely separated geographical regions, such that in the wild there is no opportunity for them to try to interbreed? We bring specimens into the lab and find that we can cross them and produce fertile offspring. Do we count them as the same species? Or, suppose we can’t get our captive specimens to mate in the lab, because their mating seasons fall in different times of year. By controlling the temperatures, light levels and so on, we fool one set of specimens and trigger their mating habits “out of season.” If we find successful interbreeding now, do we count the populations as separate species or not?

I suspect that a good theory of “practical isomorphism” would have to make use of a theory of resources. And, interestingly enough, the paper by Coecke, Fritz and Spekkens, which obtains a resource theory from the category-theoretic study of processes, says the following in its conclusion:

A major problem that we have not yet touched upon is epsilonification. The idea here is that in many applications, such as in the resource theory of communication (Example 2.6), it may be sufficient to turn a given resource $a$ into some $b'$ close to a target resource $b$, i.e. turning $a$ into $b$ “up to $\epsilon$”. Typically, one would like the $\epsilon$ to become arbitrarily small, possibly as the number of copies of $a$ and $b$ increases. We are currently investigating how our definitions should be modified in order to be able to deal with this kind of question as well.

Posted by: Blake Stacey on February 28, 2015 5:10 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Blake wrote:

I suspect that a good theory of “practical isomorphism” would have to make use of a theory of resources.

I agree, although I’d rather phrase it the other way around: theories of resources and their convertibility also require a notion of “practical isomorphism”!

Here’s one way to do this, very much in the spirit of Lawvere. Suppose we have a category $C$ whose morphisms include all sorts of badly behaved and non-structure preserving maps; let me call these approxomorphisms (or approximorphisms?). To every approxomorphism $f$, we associate a number $|f|\in\mathbb{R}\cup\{\infty\}$ which measures by how much $f$ deviates from being structure-preserving. This norm should satisfy the following properties: $|id| = 0,\qquad |g\circ f| \leq |g| + |f|.$ Those who know about Lawvere metric spaces will immediately recognize that I’m introducing a kind of enrichment! The approxomorphisms with norm $0$ form an all-object-including subcategory, which plays the role of the “standard” category of structure-preserving maps. (If the norm is allowed to be negative, then one should probably consider all $f$ with $|f|\leq 0$ instead.) Typically, the underlying category $C$ will be rather dull, like Set or Vect. All the interesting structure will be encoded in $|\cdot|$.

Right now I don’t have a nice example in this exact spirit, but the following is a well-trodden example of such a “normed category” of a similar flavour. Take the objects of $C$ to be normed spaces. Take the approxomorphisms between normed vector spaces $V$ and $W$ to be all the linear maps $f:V\to W$. Thus as a category, this is equivalent to the category of vector spaces. However, we can now assign to each $f$ the logarithm of its norm in the usual sense, $|f| := \log \: \sup \:\{\: ||f(x)|| \:|\: ||x||\leq 1 \:\}.$ Since the usual norm of a linear map is submultiplicative, this logarithmic “norm” indeed satisfies the required subadditivity inequality. An invertible map $f$ is a “practical isomorphism” if both $|f|$ and $|f^{-1}|$ are reasonably small. And in fact, $|f|+|f^{-1}|$ is a quantity which Banach space theorists have studied extensively under the name Banach-Mazur distance.

I’ve been trying to apply this idea to information theory with the intention of proving some abstract versions of Shannon’s theorem. The problem is that the standard distance measures used in information theory don’t give rise to a normed category whose norm interacts well with the monoidal structure. But I’m getting carried away, so I’ll stop here unless somebody wants to know the gory details.

Posted by: Tobias Fritz on February 28, 2015 3:08 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

The keyword for the organism-classification problem is “biological species concept.” See, e.g.,

Posted by: Blake Stacey on February 28, 2015 5:33 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Tim wrote:

But as we try to specify what exactly a morphism in this category is, it seems to me that we’re reduced back to the original question, “what does it mean for $Nile_0$ to be the same river as $Nile_1$?”, and that introducing the term “isomorphism” really hasn’t bought us anything.

It’s probably obvious, but when I polish it up, this section about Heraclitus will lead up to simpler question: “how could things possibly be the same but not equal?”

And the answer will be simple, too: “they could be isomorphic!”

It’s simple stuff for everyone here… but I just want all philosophers to know that while set theory is a simple formalism for ‘things’, category theory builds a concept of ‘process’ into mathematics at an equally primitive level: namely, the concept of ‘morphism’. This allows things to be the same yet unequal, and it gives the simplest known way to think about ‘time’.

Posted by: John Baez on March 1, 2015 1:20 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I tried to give a solution to the mystery of time in my essay “Temporal Platonic Metaphysics”, arxiv:0903.1800. The idea is to generalize the notion of temporal evolution by using the canonical formulation of GR and field theories: we have initial and final spatial data (fields on a 3-manifold $S$) and temporal evolution is described by fields on a 4-manifold $S \times [0,1]$ (history). The fields at a moment of time t are given by their values at the corresponding spatial slice $S(t)$. A generalization of this is to have an ordered string of sets $(S_1,S_2,...,S_n)$ with relations $(R_1,R_2,..,R_n)$. Also there is a temporal relation $(S_k,S_{k+1})$ such that for each element $x_k$ of $S_k$ there is an element $x_{k+1}$ of $S_{k+1}$. Not changing in time would correspond to $S_1, S_2, ..., S_n$ isomorphic to each other and the same for $R_1,...,R_n$. However, what I tried to argue, is that the difference between this abstract time evolution and the real word one is that there is a concept of time flow, which cannot be expressed mathematically (basically, watching a movie is not the same of having the CD of that movie).

Posted by: Aleksandar Mikovic on February 23, 2015 3:26 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I am looking forward to putting a bit of money in your pockets. Any timeline on when it will be in print form?

Posted by: babyDragon on February 24, 2015 3:13 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

babyDragon wrote:

I am looking forward to putting a bit of money in your pockets.

Thanks! But the authors of chapters in this sort of book never make any money; only the editor will, and even she’ll just get 10% of the profits or so… so you’ll mainly be enriching Cambridge U. Press. But that’s okay; I’m not really in this for the money.

Any timeline on when it will be in print form?

We’re supposed to hand in our chapters by April, which (knowing the ways of academia) means that everyone will be done sometime after April, and then presumably referees will read what we wrote and demand changes, and then when those are done the publisher will wait a few months and then suddenly demand that we correct all the typos they’ve introduced in just a few days, and then an indefinite amount of time will pass before the book is actually printed.

Posted by: John Baez on February 24, 2015 5:06 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

In Tom Leister’s book “Basic Category Theory”, he starts by talking about equality of elements, isomorphisms of functors, equivalence of categories, and at each stage, he acts like you have an object, and that object can change, and then, lo and behold, the “change” itself could be viewed as object that could itself change. This indicates a way of thinking, where you think primarily as one level, and as you go up each level, it gets increasingly difficult to get your mind around.

I think a better way to think about it to to say that in a set you have elements, in a category, you have morphisms between elements, in a 2-category, you have morphisms between elements, and also have 2-morphisms between morphisms, and you can keep on going to n-categories, with arbitrary n, with n-morphisms between (n-1)-morphisms. That way, there is no implied sense that is amazing that there is a level higher than the current level you are talking about.

In Tom Leister’s book, there is an implied sense that in each example, there is a “main thing” you are talking about, and then you could talk about how that thing could change, and then possibly how the change itself could change, but you don’t go beyond that because it’s to difficult to get your mind around.

For example, in classical physics, we talk about position, velocity, acceleration, but they hardly ever go beyond that, talking about putting x to higher derivatives of t.

However, in different examples, people can disagree as to what should be considered “the main thing”. Someone could say something is the “main thing” and talk about that main thing changing, and someone else could say that what you are calling “the main thing” is actually technically something else changing.

Talking about “sameness” you could say that in symmetry, a square looks the same if rotated by 90 degrees. By this definition, it has to look exactly the same to be the same. However, most people would still consider a square to be “the same” if rotated by 10 degrees. They would say it’s still the same shape. However, most people would say a square would not be the same shape if turned into a triangle. On the other hand, a topologist would say a square and triangle are the same shape, which by convention, they draw as a circle, and designate S^1. So, these are different definitions of “sameness”. You could say that a cube and octahedron are, in some sense, the same, because they have the same symmetry group, while a dodecahedron would not be. By a weaker definition of sameness, a cube, octahedron, and dodecahedron, would all be the same, because they are all topological represented by S^2. A plane, R^2, is topologically distinct from a cylinder, S^1 x R^1, however, they are both “the same” under parallel transport, as opposed to a sphere, S^2, which would not be. Broader definitions of sameness include homotopy, homology, and cohomology.

Posted by: Jeffery Winkler on February 24, 2015 10:16 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Let’s say you have a forgetful functor from a group to a monoid.

Grp -> Mon

Let’s say you have a forgetfulm functor going from a group to the underlying set.

Grp -> Set

You could say that the second forgetful functor is forgetting more than the first one, and so in the first example, the two things are more the same than in the second example.

Here are some examples of differing amounts of sameness.

A square rotated by 90 degrees is the same as an unrotated square, since it looks exactly the same, but a square rotated 10 degrees is not.

A square rotated by 10 degrees is the same as an unrotated square but not the same as a rectangle where not all sides are equal length.

A square is the same as any rectangle but not the same as a triangle.

A square and triangle are the same since they are both topologically S^1.

A cube is not the same as an octahedron.

A cube is the same as an octahedron since they have the same symmetry group but not the same as a dodecahedron.

A cube, octahedron, and dodecahedron are all the same, since they are all platonic solids, but not the same as a cuboctahedron.

A cube, octahedron, dodecahedron, and cuboctahedron are all the same, since they are all topologically S^3.

A plane, R^2, and an infinite cylinder, S^1 x R^1, are not the same because they are topologically distinct.

A plane, R^2, and an infinite cylinder, S^1 x R^1, are the same because they are the same under parallel transport, but a sphere, S^3, is not.

A plane, R^2, infinite cylinder, S^1 x R^1, and a sphere,S^3, are all the the same because they are all Riemannian manifolds.

Posted by: Jeffery Winkler on March 2, 2015 11:00 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Possibly relevant piece on the topic by Barry Mazur:

When is one thing equal to some other thing?

Posted by: RA on February 26, 2015 9:52 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

Thanks for reminding me of that! I’ll have to refer to that.

Posted by: John Baez on February 28, 2015 2:12 AM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I’ve been unable to get myself to write my paper for Landry’s volume. I thought it would be a good chance to express, in a more organized way, some thoughts about equality and isomorphism that I used to enjoy telling everyone all the time. But it turns out that I’ve moved too far from my old interests.

Posted by: John Baez on April 30, 2015 9:59 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

That’s too bad! I was looking forward to reading it, and I’m sure the volume will be the poorer for it.

Posted by: Mike Shulman on April 30, 2015 11:22 PM | Permalink | Reply to this

Re: Concepts of Sameness (Part 1)

I have gravitated towards the notion that only projections of real things can be equal. No two things entirely the same, may be two distinct things. But if we project them down… to some reduction in their full set of observable properties, then that projection can be the same for two or more things. Counting apples, we project down their properties in such a way that their atoms, cells, structure, the boundary created by their skin, are reduced by some criteria to ‘apple’. Two apples may be equal in the respect that they are each an apple, where ‘apple’ has some imprecision in its definition. By this means each apple can be reduced down to simply the number ‘one’ by removing all of its appleness so we may count two sets of apples and declare them to be equal in number.

Posted by: Robert Frost on August 18, 2016 8:21 PM | Permalink | Reply to this

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