Mathematics, things, objects and brains

This is about my understanding of the post Mathematics and the Real by Louis Kauffman.

I start from this quote:

One might hypothesize that any mathematical system will find natural realizations. This is not the same as saying that the mathematics itself is realized. The point of an abstraction is that it is not, as an abstraction, realized. The set { { }, { { } } } has 2 elements, but it is not the number 2. The number 2 is nowhere “in the world”.

Recall that there are things and objects. Objects are real, things are discussions. Mathematics is made of things. In Kauffman’s example the number 2 is a thing and the set { { }, { { } } } is an object of that thing.

Because an object is a reification of a thing. It is therefore real, but less interesting than the thing, because it is obtained by forgetting (much of) the discussion about it.

Reification is not a forgetful functor, though. There are interactions in both directions, from things to objects and from objects to things.

Indeed, in the rhino thing story, a living rhinoceros is brought in Europe. The  sight of it was new. There were remnants of ancient discussions about this creature.

At the beginning that rhinoceros was not an object, not a thing. For us it is a thing though, and what I am writing about it is part of that thing.

From the discussion about that rhinoceros, a new thing emerged. A rhinoceros is an armoured beast which has a horn on its back which is used for killing elephants.

The rhino thing induced a wave of reifications:  nearby the place where that rhino was seen for the first time in Portugal, the Manueline Belém Tower  was under construction at that moment. “The tower was later decorated with gargoyles shaped as rhinoceros heads under its corbels.[11]” [wiki dixit]

Durer’s rhino, another reification of that discussion. And a vector of propagation of the discussion-thing. Yet another real effect, another  object which was created by the rhino thing is “Rinoceronte vestido con puntillas (1956) by Salvador Dalí in Puerto Banús, Marbella, Spain” [wiki dixit].

Let’s take another example. A discussion about the reglementations of the sizes of cucumbers and carrots to be sold in EU is a thing. This will produce a lot of reifications, in particular lots of correct size cucumbers and carrots and also algorithms for selecting them. And thrash, and algorithms for dispensing of that trash. And another discussions-things, like is it moral to dump the unfit carrots to the trash instead of using them to feed somebody who’s in need? or like the algorithm which states that when you go to the market, if you want to find the least poisoned vegetables then you have to pick them among those which are not the right size.

The same with the number 2, is a thing. One of it’s reifications is the set { { }, { { } } }. Once you start to discuss about sets, though, you are back in the world of things.

And so on.

I argue that one should understand from the outset that mathematics is distinct from the physical. Then it is possible to get on with the remarkable task of finding how mathematics fits with the physical, from the fact that we can represent numbers by rows of marks |  , ||, |||, ||||, |||||, ||||||, … (and note that whenever you do something concrete like this it only works for a while and then gets away from the abstraction living on as clear as ever, while the marks get hard to organize and count) to the intricate relationships of the representations of the symmetric groups with particle physics (bringing us back ’round to Littlewood and the Littlewood Richardson rule that appears to be the right abstraction behind elementary particle interactions).

However, note that   “the marks get hard to organize and count” shows only a limitation of the mark algorithm as an object, and there are two aspects of this:

  • to stir a discussion about this algorithm, thus to create a new thing
  • to recognize that such limitations are in fact limitations of our brains in isolation.

Because, I argue, brains (and their working) are real.  Thoughts are objects, in the sense used in this post! When we think about the number 2, there is a reification of out thinking about the number 2 in the brain.

Because brains, and thoughts, are made of an immensely big number of chemical reactions and electromagnetic  interactions, there is no ghost in these machines.

Most of our brain working is “low level”, that is we find hard to account even for the existence of it, we have problems to find the meaning of it, we are very limited into contemplating it in whole, like a self-reflecting mirror. We have to discuss about it, to make it into a thing and to contemplate instead derivative objects from this discussion.

However, following the path of this discussion, it may very well be that brains working thing can be understood as structure processing, with no need for external, high level, semantic, information based meaning.

After all, chemistry is structure processing.

A proof of principle argument for this is Distributed GLC.

The best part, in my opinion, of Kauffman’s post is, as it should, the end of it:

The key is in the seeing of the pattern, not in the mechanical work of the computation. The work of the computation occurs in physicality. The seeing of the pattern, the understanding of its generality occurs in the conceptual domain.

… which says, to my mind at least, that computation (in the usual input-output-with-bits-in-between sense) is just one of the derivative objects of the discussion about how brains (and anything) work.

Closer to the brain working thing, including the understanding of those thoughts about mathematics, is the discussion about “computation” as structure processing.

UPDATE: A discussion started in this G+ post.


Clocks, guns, propagators and distributors

Playing a bit with chemlambda, let’s define:

  • multipliers
  • propagators
  • two types of distributors

described in the first figure.


The blue arrows are compositions of moves from chemlambda. For instance, referring to  the picture from above,  a graph (or molecule) A is a multiplier if there is a definite finite sequence of moves in chemlambda which transforms the LHS of the first row into the RHS of the first row, and so on.

For example:

  • any combinator (molecule from chemlambda) is a multiplier; I proved this for the BCKW system in this post,
  • the bit is a propagator
  • the application node is a distributor of the first kind, because of the first DIST move in chemlambda
  • the abstraction node is a distributor of the second kind, because of the second DIST move in chemlambda.

Starting from those, we can build a lot of others.

If A \rightarrow  is  a multiplier and \rightarrow B \rightarrow  is a propagator then A \rightarrow B \rightarrow is a multiplier. That’s easy.

From a multiplier and a distributor of the first kind we can make a propagator, look:


From a distributor of the second kind we can make a multiplier.


We can make as well guns, which shoot graphs, like the guns from the Game of Life.  Here are two examples:


We can make clocks (which are also shooting like guns):


Funny! Possibilities are endless.


Tibit game with two players: trickster and webster

Here is a version of the tibit game with two players, called “trickster” and “webster”.

The webster is the first player. The trickster is the second player.

The webster manipulates (i.e. modifies) a “web”, which is a

  • trivalent graph
  • with oriented arrows
  • with a cyclic orientation of arrows around any node (i.e. locally planar)
  • it may have free arrows, i.e. ones with free tail or free tip.

Tokens.   The webster  has one type of token, called a “termination token”.  The trickster has two types of tokens, one yellow, another magenta.

Moves.  When his turn comes,  any player can do one of the moves listed further, or he may choose to pass.


Some of the webster moves are “reversible”, meaning that the webster may do them in both direction. Let’s say that the “+” direction is the one from left to right in the graphical moves and also means that the webster may put a termination token (but not take one). The “-” direction is from right to left in the graphical moves and also means that the webster may take a termination token (but not put one).


The loop rule.  It is possible that, after a move by one of the players, the web is no longer a trivalent graph, because a loop (an arrow which closes itself, without any node or token on it) appears. In such a case the loop is erased before the next move.


This is a collaborative game.

There are two webs, with tokens on them, the first called the “datum” and the second called the “goal”.

The players have to modify the datum into the goal, by playing collaboratively, in the following way.

The game has two parts.

Preparation.     The players start from a given web with given tokens placed on it (called the “datum”).  Further,   the webster builds a web which contains the datum and the trickster places tokens on it, but nowhere in the datum .

Eventually they obtain a larger web which contains the datum.

Alternatively, the players may  choose an initial   web, with tokens on it, which contains the datum.

Play.   Now the webster can do only the “+” moves and the trickster can’t put any token on the web.  The players try to obtain a web, with tokens on it, which contains the goal by using their other moves.


Peer-review, is good or bad?

I shall state my belief about this, along with my advice for you to make your own, informed, opinion:

  • Peer-review as a bottleneck on the road to legacy publication is BAD
  • Peer-review as an authority argument (I read the article because is published in a peer-reviewed journal) is UNSCIENTIFIC
  • Open, perpetual peer-review offers a huge potential for scientific communication, thus is GOOD.
  • It is though the option of the author to choose to submit the article to public attention, this should not be mandatory.
  • Moreover, smart editors should jump on the possibility to exploit open peer-review instead of
  • the old way to throw to the wastebasket the peer-reviews, once the article is accepted or rejected, which is BAD.
  • Finally, there is NO OBLIGATION for youngsters to peer-review, contrary to the folklore that it is somehow their duty to the community to do this. No, this is only a perverse way to keep the legacy publishing going, as long as the publishers use them only as an anonymous filter. On the contrary, youngsters, everybody honest in fact, should be encouraged to use rewarded for using any of the abundant new means of communication for the benefit of research.

This post is motivated by the Mike Taylor’s Why peer-review may be worth persisting with, despite everything and by comments at the post Two pieces of all too obvious propaganda.

See also the post Journal of uncalled advices (and links therein).

An experiment in open writing and open peer-review

I shall try the following experiment in open writing/open peer-review which uses only available soft and tools.

No technical  knowledge is needed to do this.

No new platform is needed for this.

The idea is the following. I take an article (written by me) and I copy-paste it as text + figures in a publicly shared google document with comments allowed.

On top of the document I mention the source (where is the article from) , then I add a CC-BY licence.

This is all. If anybody wishes to comment the article, it can be done precisely, by pointing to the controversial paragraphs.

In the comments are allowed links, of course, therefore there is no limit to the quantity of data which can be put in such a comment.

There could be comment replies.

In conclusion, this is a very cheap way to do both a (limited) way of open writing and to allow open peer-review.

For the moment I started not with articles directly, but with edited content from this open notebook.  I made until now two “archives”

Even better would be to make a copy of the doc and put it in the figshare, to get a DOI. Then you stick the DOI  link in the doc.