Just curious-where have you seen the “tao of mathematics” been written down?

As Chuang-Tze said, “The Tao that can be spoken is not the eternal Tao”.

I knew Sylvester is often credited with having invented graph theory, but I haven’t read his original work on that subject. I imagine the concept of ‘set’ may have been a bit murky back then, but I don’t know the early history of that subject either!

The first link you gave doesn’t work here in Singapore. That’s true of many Google Books links.

I believe that understanding category theory helps me ‘follow the tao of mathematics’

Just curious-where have you seen the “tao of mathematics” been written down?

But speaking of historical connotations – John, you probably saw this already – but concerning your chemical reaction networks – you may eventually dig out also something interesting in: here.

Since I don’t have a library account I can’t easily follow the journal links so in particular I couldn’t really understand Sylvester’s definition of a graph in particular I couldn’t figure out whether Sylvester used the word “graph” as being defined on a set or on a multiset. He seemed to have derived the word graph from the word “chemicograph.”

(A multiset is a set where elements can appear with finite mulitpicities; according to W.D. Blizard this term seems to have been discussed first in Dedekind’s treatise on numbers (last chapter nr. 172) however the term multiset is not by him in particular if I understand him correctly Dedekind called a set “ein System” but the german word for set is “Menge”, so here the expressions have changed, )

Frankly speaking I am still not so enthusiastic about category theory as you, so the above argumentation is for me less of an argument.

I believe that understanding category theory helps me ‘follow the tao of mathematics’ and lets me do things better than I would otherwise. But I’ve spent a lot of time writing stuff that advocates the virtues of category theory and I’m pretty much done with that now; at this point it makes more sense for me to come up with good results rather than talk about how I go after them.

Also by the way, the definition of ‘deficiency’ makes it impossible that the deficiency could be negative: it’s the dimension of the vector space

Right. For the case I was talking about, one would thus actually need to count multiplicities and and in the the turn wouldn’t get negative deficiencies and and hence wouldn’t run into this what had disturbed me.

Thanks for giving me an excuse to talk about this stuff! There are lots of ‘design decisions’ going into these posts, which I normally don’t talk about, because I think it would be distracting.

Thanks for explaining these details. I understand now that you really want this definition. Maybe I have read too many fuzzy papers and thus got too suspicious about definitions.

It could be helpful if your graphical examples (the arrow pictures) would contain double arrows in the same direction and selfdirected arrows. That would sort of confirm your definitions.

Given the reflections and not too big resolutions of my laptop monitor, my glasses etc. it seems I had actually read the symbols:

in your comment above as

and had falsely interpreted it as some kind of sloppiness.

It’s worth noting that of all types of graph, this is the only one we can define internal to any category: that is, with objects of that category replacing the sets E and V, and morphisms replacing the functions s and t. All the other 7 kinds of graph can only be defined in restricted kinds of categories.

Frankly speaking I am still not so enthusiastic about category theory as you, so the above argumentation is for me less of an argument. The discussion is however interesting for me because as I had said above I try to infiltrate the RDF community to talk about the graph you get when you identify parts of triples (i.e. (subject, predicate, object) with “identical” URI’S (labels) (in your language the triples are (complex, transition, complex), where the directions should be taken into account). Canonically one would there start with identifying the subject and objects with identical URI’s and then as you pointed out above eventually also think about identifying transitions.

The problem is that for RDF triples there are often subjects which have a different URI but have the same meaning.
There is a project called SILK here in Berlin where (as I understood) these “synonyms” are being tracked (partially by genetic algorithms) and interlinked by a “same as” transition. These efforts are important for reducing unnecessary (and storage intensive) dublicates and as I think eventually also for the serialization of the RDF triples.

So far the RDF community speaks about the triples themselves as “the RDF Graph” (i.e. a highly disconnected graph) (that’s what I understood) and not about the graph you have when you look at the corresponding sets. But the discussion here had let me to think that may be they would agree to the term “RDF multigraph.”

Frankly speaking I still think that you want to add in the definition that for every ordered pair there should be at most two transitions, which are the reverse of each other.

No, I don’t want that—but this is an interesting issue, which I’ve been thinking about quite a lot. I addressed it at the end of this post, but let me say more.

There are at least 2³ = 8 kinds of graphs, depending on our answers to three questions:

• Do the edges have arrows on them?

• Can more than one edge can go from a vertex to a vertex ?

and

• Can an edge can go from a vertex to itself?

So: which kind of graph should we use for reaction networks?

We need the edges to have arrows on them to be able to write down the rate equation, so we’re down to four options.

An edge going from a vertex to itself means a chemical reaction that turns some complex into itself. Such reactions don’t affect the rate equation, so we can exclude them if we want. But, we can also include them if we want!

If we have several edges going from one vertex to another, we have several chemical reactions that turn one complex into another. We can combine these into one reaction without changing the rate equation, as long as we add their rate constants. So, we can exclude the possibility of multiple edges going from one vertex to another if we want. But, we can also include it if we want!

A naive notion of efficiency urges us to leave out anything that’s not required, but in mathematics we eventually learn that leaving things out is often more work than leaving them in. Leaving out possibilities often requires extra clauses in our definitions that explicitly exclude these things. This makes it more tedious to reason with these definitions. It also makes the resulting category of mathematical gadgets have worse properties, and be definable only in a more restricted context.

So, I have chosen to study reaction networks using the type of graph that we get when we answer YES, YES and YES to the above three questions. Some people call this kind of graph a directed multigraph, where ‘multi’ means that we can have more than one edge from one vertex to another. This may sound complicated, but it’s incredibly simple: it’s just a pair of functions

It’s worth noting that of all types of graph, this is the only one we can define internal to any category: that is, with objects of that category replacing the sets and , and morphisms replacing the functions and . All the other 7 kinds of graph can only be defined in restricted kinds of categories.

It’s a fun exercise to see exactly how this works. For example, suppose we’re working internal to a category and we want to define graphs with the extra property that there’s at most one edge from any vertex to any other. Then we need to require that

be a monomorphism. This makes things more complicated, but also it demands that have binary products, so we can define Or at the very least, we need the product to exist.

It’s also worth noting that any category gives a directed multigraph with objects as vertices and morphisms as edges. To study categories, we definitely need to allow more than one morphism going from one object to another. And we definitely need to allow morphisms going from an object to itself!

So, this subject will mesh with category theory most easily if we use directed multigraphs. And it’s better for chemistry too. Directed multigraphs, being more general, allow for more operations. For example, we can add new edges to a directed multigraph without having to check if we’re violating a rule saying that at most one edge can go from one vertex to another.

One more thing: when our reaction network has a transition going between two complexes, adding more transitions (going either way!) between these complexes will not change the deficiency. Why? Because it won’t change the number of complexes, or the number of components, or the dimension of the stoichiometric subspace.

Similarly, adding a transition going from a complex to itself will not change the deficiency.

Thanks for giving me an excuse to talk about this stuff! There are lots of ‘design decisions’ going into these posts, which I normally don’t talk about, because I think it would be distracting. But this particular one is more interesting than most.

You’re not asking about the definition itself: you’re asking about various consequences of the definition.

Yes because I had the feeling that something was missing and I wanted to find out what you were having in mind.

All the rules are in the definition above.
…
Let’s try to answer this question starting from the definition. Can you, or anyone here, see what the definition says about this issue?

Frankly speaking I still think that you want to add in the definition that for every ordered pair there should be at most two transitions, which are the reverse of each other.

I think the definition doesn’t rule out that there could be e.g. three “different arrows” in the same direction between two complexes
or in other words by the definition you could have e.g. three different transitions between two complexes (s and t could be surjective).

On the other hand what I understood from what you had said here in the comments you want to have only at most two types of arrows between complexes and thus you intrinsically seem to want to have this in the definition.

But it is 35°C here and may be I just don’t have my brain in gear.

If there’s anything about that definition that seems mysterious, maybe you should ask.

Nad replied:

but that’s what I did!

You’re not asking about the definition itself: you’re asking about various consequences of the definition. The definition is this:

Definition. A reaction network consists of:

• a finite set of species,

• a finite set of transitions,

• a finite set of complexes,

• source and target maps

Maybe I’m just too much of a mathematician, but it seems to me that the beauty of definitions is that they answer an infinite number of questions in a very short time. After one understands this definition, it’s pretty easy to answer all the questions you’ve been asking. Before understanding it, there’s an infinite list of questions one could ask about what counts as a reaction network, without ever finding out for sure what all the rules are. All the rules are in the definition above.

Of course there could be ways in which this definition needs clarification: for example, it only makes sense to people who know that means the set of functions from to , and that means the natural numbers, and that I define the natural numbers to be the set . So if anything like this is unclear, I should definitely explain it.

So if I understood you now correctly you assume that there is for every pair of complexes at most two transitions, one is denoted by the symbol “rightarrow” and the other (which is as you wrote is to be interpreted as the reverse transition) with one with the symbol “leftarrow”, respectively.

Let’s try to answer this question starting from the definition. Can you, or anyone here, see what the definition says about this issue?

I don’t know whether you have the usual administration possibilities which come with a usual wordpress package. So maybe check out whether you have on your administration page under “appearances” (bar on the right side) the topic “widgets”, if yes then go there and check wether you can activate a so-called text widget (by dragging it to the right side). If yes then double click on this and you should be able to insert text, which should appear in the side bar.

I have all the usual powers people get with a free WordPress blog. A lot of fun ‘plugins’ require paying extra. At some point I will start paying and improve this blog in a bunch of ways. But for now, let me see if I can get a text widget. Thanks.

“Mathematicians would call $\partial$ a boundary operator.”

using normal LaTeX rather than wordpress LaTeX. The second is where you’re reminding us what the image of the boundary map is. The sentence containing the displayed equation still uses a “d”, namely, “The image of d is…”.

And I just have to say, if those are the only two places you missed, I for one am really impressed.