## A mathematician’s take on phenomena

This is, in part, a response to the recent John Wilkins blog post “More on phenomena.”  It is based on my ideas on human cognition and human perception.  Unavoidably, this will be a tad mathematical.  However, I will avoid getting into technical terminology to the extent that I can, though I’ll give enough as hints to the mathematical literature for those who want to pursue the underlying mathematics.

Think of the world as a topological space, call it W.  (For the mathematicians, I am taking W to be a normal Hausdorff space).  Because W is a topological space, we can think about continuous functions over that space.  So for a point x in W, and a continuous function f, there is a value f(x) for that function at that point.  For technical reasons, mathematicians usually take their continuous functions to have values that are complex numbers.  However, I suggest thinking about them as functions with values that are real numbers.

If we look at this in terms of science, then we can think of the function f as a method of measuring, and we can think of the value f(x) as an actual measurement (or as a datum).

Taking this as a basic mathematical model, I want to look at John’s concerns about phenomena.  A scientific theory is, at least in part, a theory of measurement.  That is to say, it gives a method for getting data.  From that perspective, the continuous functions such as f correspond to theories, and a value f(x) corresponds to a particular datum (or measurement).  To say that data is theory laden, is just to say that the value f(x) depends on the function f as well as on the point x in W.  This should be unsurprising and no reason at all for concern.  In terms of John’s post the data constitute the phenomena.

In addition to studying a topological space W, mathematicians like to also study the space C(W) consisting of all bounded continuous functions over W.  The topological space W completely determines the space C(W).  Conversely, it turns out that the space C(W) tells us a great deal about W and in suitable cases C(W) actually determines W.  So we can think of that as a kind of duality between W and C(W).  This is standard functional analysis.  For a classic text on the topic, try Gillman and Jerison, Rings of Continuous Functions.

The main concern of science, is in finding C(W), and from that making inferences about the structure of W.  That makes theories the particular concern of science.    The importance of the theories, are that they connect the data to reality, and in so doing the address the intentionality question (or the question how can data be about something?)  In my opinion, philosophy has overemphasized the data and paid too little attention to the methods of collecting that data.

I have tried to discuss some of these ideas in previous posts, where I used an analogy with the camera.  With this analogy, the camera plays the role of the scientific theory while the photograph plays the role of the data.  To say that the data is theory laden is about equivalent to saying that a photograph taken with a Minolta camera is Minolta laden.  There really isn’t any reason for concern about that.  Sure, if I take a picture with a Minolta camera, and another picture with a Nikon camera, the two pictures will not exactly align with one another due to differences in the optics.  But it is still clear that both are pictures of the same scene.  That the two do not exactly align amounts to the “meaning incommensurability” that Kuhn discussed.  However, as the camera analogy suggests, there is no need for concern about such incommensurability.

John asked particularly about phenomena that do not derive from a theory.  However, as best I can tell, the same sort of thing has to be going on in the brain.  We might not use the term “theory” for neural structures and acquired perceptual abilities, but those are surely analogous to theories.  I suggest J.J. Gibson’s direct perception as the account of perception that best fits.  Gibson suggested that there are transducers that are tuned to particular things in the world, and I see those transducers as being a kind of neural counterpart to a scientific theory.  Gibson’s wife, Eleanor Gibson, wrote a book on perceptual learning which I take to be related to how we learn about the world by building and tuning those neural transducers.

It is likely that my system of neural transducers is different from yours, and that there is some consequent meaning incommensurability.  We see this in the apparent subectivity of meaning, in the disagreement of some questions of meaning.  But that should be no more a reaason for concern than your use of a Canon camera while I use a Nikon camera.