1. Dear Tim,
Your essay was for me the occasion to discover your theory of linear structures, and it was an enchantment. I think it won't be long till I buy your book.
Your text is particularly clear, too.
If I really like your style and your contributed linear structures, I have not really found an answer as to the effectiveness of mathematics, or, as you aptly rephrase it, how mathematics meet the world.
2. I refrained from commenting Wigner's article, because I see too many irrecoverable conceptual errors in it, for instance: ``elementary mathematics [...] entities which are directly suggested by the actual world''.
- The actual world suggests nothing.
I'd like to quote Ferán Adria, the famous Spanish chef; in answer to journalist saying that he was finding inspiration on the Costa Brava, he stood on the shore, staring at the sea and cliffs before a camera, and after a while said: ``No veo nada.'' There is nothing in clear in the landscape. Where is that inspiration?
- Wigner makes what I think is a huge mistake; implicitly, he suggests that there are things in the world that we can access in an absolute way. So, in the objects that I talk about, some are `for sure', and some are hypothetical. If you really try seriously to draw the border between the latter and the former, you come to understand that it does not exist, and that all perceptions are hypothetical. Only, some of them are usually, frequently confirmed, and we deem them solid ground. But, read Karen Blixen report of a earthquake in Out of Africa, and you will have a vivid case of a solid ground becoming moving and alive. Witness how people can be in a state of shock after a large earthquake; Haroun Tazieff used to say that their whole mental system is broken into pieces after experiencing the ``disappearance of space'', in the most concrete sense. Read Gregory Bateson's report of completely loosing ground after his experiments with sensorily puzzling effects at Adelbert Ames Jr. laboratory (in Steps to an ecology of mind). After that, read Richard Gregory's Perception as hypotheses (In The Oxford companion to the mind for instance). And read about Bach-y-Rita's experiment (guess where). You should convince yourself that the most fruitful representation is to consider uniformly all perceptions as hypotheses.
By the same token, you cannot talk about the `actual world', if you mean by that things you have a privileged, certain, absolute access to. All that you know about the world is relative to your perceptions, and is built through interaction with it, in action-perception loops: you impulse transformations into the world, and record the effects, and build pragmatically a whole world of hypotheses. You act, to abstract.
So, when you note after Wigner about the concepts of elementary geometry that ``those very concepts were developed via interaction with the world'', there is no solid way to circumscribe only them, because all your concepts, you eventually develop via interaction with the world. (You certainly agree: ``claims about [...] must be considered just as conjectural and fallible as all other physical claims.'')
(By the way, defining exactly what `elementary geometry' should be is not trivial, and finally not univocal because it poses a problem of foundation, and what we think is the historical, or natural genetic approach is not obviously better. Shall we just say that rays of light are our concept of a straight line? Shall we say that sets and groups are at the root? Shall we say that topos is a more elementary foundation?)
3. In the words of a common metaphor, Wigner sins by confusing the map for the territory.
(Korzybski for instance has repeated much this dictum.)
All what we do in our theories is language, and the world is not language.
The trains of spikes in your neuron chains are not labelled `vision', 'touch', `warmth', etc. These concepts are built afterwards. They do not exist substantially in the world, ready to be channeled to your perception. If you excite the retina by an electrical impulse, the subject reports a flash of light --the usual hypothesis for what comes from the retina. The most striking demonstration is in Bach-y-Rita experiment (refer to my essay for a pointer).
These facts should make clear that all we operate on is within the `maps' world. All we build in our maps is hypothetical constructs, that we try to fit to the interactions we engage with the territory that we know is out there.
So to be accurate, we must always maintain that we talk indirectly about the territory, through a representation. Any statement made directly about the territory, the world is problematic because it cancels this relativity to the representations of the subject. A statement about the territory itself, being absolute, must be completely right. Therefore, it cannot be scientific. It fails to satisfy Popper's criterion. If a statement fits Popper's criterion, then it is hypothetical.
It has nearly always been that progress has occurred when the operation of representation was made explicit and stated clearly. For instance, a correct definition of what a measure is, in mathematics, could not be derived until after there was a clear separation between the measurement, and the scale used to represent measurement, that is, a usable theory for the real numbers. I took an even simpler example in my essay, with the confusion between numbers-of something, and numbers `pure', and how until the late 19th century (!) not being able to separate clearly numbers from what they could be used for --counting objects or representing quantities in this case-- was impeding comprehension and progress in using them.
The conclusion is that to be able to address properly the question of the effectiveness of mathematics in representing the world, we have to include the stage of representation of the world in our framework. That is, we have to include the stage of perception and cognition in our representation, and that means including explicitly the cognitive subject.
4. I have the feeling that you have fallen in the same trap as Wigner, when you write:
``claims about the geometrical structure of physical space or space-time must be considered just as conjectural and fallible as all other physical claims.''
``A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented.''
In both cases, if I am not misinterpreting, by `physical space' and `physical world', you are not talking about the space or the world, as seen by physics, but about the world itself. You have no final word about the structure of the world. You have all latitude to propose hypothetical physical structures that prove fruitful for various uses.
5. Your title is particularly accurate and promising, because the key issue is exactly the meeting of mathematics and the world. I completely agree with it, and it is precisely what I have just said above: the issue is on perception. Perception is fundamentally a meeting, at any considered scale: Subject and event, sensory organ and object, sensory neuron and stimulus, etc.
If you think you talk about perception, and see no meeting, then you are certainly talking about things that take place during perception, but not the fact of perception itself, properly regarded.
Regards