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.

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9 thoughts on “Mathematics, things, objects and brains”

  1. There has been a lot of confusion about “reification” in recent years, much of it instigated by some not-too-literate people in the OOPS and Ontology communities. I recommend checking a good dictionary to get a sense of the original meaning, before it got warped all out of shape.

    “Things we talk and think about” and “objects of discussion and thought” are pretty much synonymous phrases. The Greeks had a word for them — pragmata.

    Operations like personification and reification refer to figures of speech that take an abstract quality, process, or thing and treat it as a concrete subject.

    Another name for such operations is “hypostatic abstraction”, which C.S. Peirce emphasized as being heavily involved in the formation of mathematical objects.

    Hypostatic Abstraction

  2. If you take a pragmatic stance (which I don’t) then your things are my objects, and hypostatic abstraction is something which mimics a kind of a reverse of reification, but which stays at the level of objects.
    But I don’t think all this is relevant for the post, otherwise that it gives me the oportunity to say again that I am very well with Plato.
    There was an inversion (one of many) which occured historically, concerning things and objects. Instead of “pragma”, better is to use the latin “res” for thing (as Heidegger, but discarding the further confusion between things and objects he makes). For the moderns, “real” (which is a word which comes from “res”) is made by objects. If the res publica was an alive, self-sustaining, ongoing discussion, then the modern corpus of laws is a recording of the res. An object, which lost all conversational properties. If you take the opinion that reality is objective then you have the inversion. Things are not objects.

  3. Like I said, there has been a lot of confusion in recent years.

    I watched some of it happen when a class of people in programming went from talking about Object Code Modules, where the adjective “object” referred to the fact that the code module was the output of a compilation process, to calling OCMs simply “objects”, thereby instituting a hopeless confusion between bodies of code in the sign domain and semantic objects in Plato’s Heaven and more Earthly Spheres both.

      1. I am not happy with calling 2 a ‘thing’. 2 is a concept or abstraction that can be handled formally, as can most concepts in mathematics. It is more a relationship than a thing or an object. Note that some concepts closely related to mathematics do not have any formal counterpart in some peoples’ systems. For example, the concept of a set is not a set and so has no formal counterpart in most set theories! Nevertheless,it is the concepts that make up the content of mathematics. We barely know how to map the concepts to our formal systems and it is actually quite wild speculation to think that there are ‘brain states’ corresponding to concepts.

  4. I stand by calling 2 a thing. A thing is an ongoing, self-sustained discussion. “Thing” is a germanic name which means an assembly who gather for discussing. The latin equivalent is “res”. Res publica is the public thing.
    So, what I say is that 2 is a thing and the set { { }, { { } } } is an object of that thing.
    An object is something derived from a thing, is what you get from the discussion when it stops by reaching an agreement. More generally, an object is a “reification” of a thing.
    But here, as Jon Awbrey points rightfully, there is lot of room for confusion.
    Indeed, “real” comes from “res” so the original sense of reality is that it is the thing of all things.
    But the moderns inversed objects with things, or they even confounded objects with things.

    When I say that reality is objective, what I want to say is derived by hypostatic abstraction (see the previous commnet by Jon Awbrey) from “reality is made by objects”.
    Then, “reification” becomes a very strange word. But as long as it is explained, can be used as any other word, like “duck”.

    What about “concept”? Something is a concept if it has been conceived by somebody or something. Something has been pregnant with the concept, and by discussion about the status of concept of something, one really discusses about this conceiving-conceived relation.

    Let’s see now:
    – a thing (discussion) can be a concept, seen from the point of view of being conceived by somebody or something.
    – an object can be a concept, for example in the sense that it has been conceived by a thing,
    – a concept can be the object of a discussion-thing
    – a thing can be the object of another thing.

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