### 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.

## 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

isomorphicobject, 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 stateofthe 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?)