# Model schema and the ‘structuralist’ interpretation of ‘for all’: the uniformity of what?

Disclaimer
A short note mentioning an update of a previous post, as well as an additional comment.

Overview
I have re-written a number of parts of my first post ‘For all’ is not ‘catch all’: closure, model schema and how a Bayesian can be a Falsificationist. I’ve added some brief references to Jaynes, for one, but also have tried to clarify the nature of the ‘for all’ statement and its domain of application. This came up in the comment section. I’ve copied this section below (in blue), as well as added an additional comment after.

What is the domain of the ‘for all’?
A further clarification is needed [see the comment section for the origins of this]: the closure conditions are schematic/structural and only implicitly determine the domain of validity B for a given theory. That is, in the general scheme, b and B are placeholders; for a particular proposed theory we need to find particular b and B such that the closure conditions are satisfied. This has an affinity with the ideas of mathematical structuralism (without necessarily committing to endorsing the entire position, at least for now). For example, Awodey (2004, An Answer to Hellman’s Question), describes:

the idea of specifying, for a given…theory only the required or relevant degree of information or structure, the essential features of a given situation, for the purpose at hand, without assuming some ultimate knowledge, specification, or determination of the ‘objects’ involved…The statement of the inferential machinery involved thus becomes a…part of the mathematics…the methods of reasoning involved in different parts of mathematics are not ‘global’ and uniform across fields…but are themselves ‘local’ or relative…[we make] schematic statement[s] about a structure…which can have various instances

This lack of specificity or determination is not an accidental feature of mathematics, to be described as universal quantification over all particular instances in a specific foundational system as the foundationalist would have it…rather it is characteristic of mathematical statements that the particular nature of the entities involved plays no role, but rather their relations, operations, etc. – the ‘structures’ that they bear – are related, connected, and described in the statements and proofs of the theorems.

This can be seen as following in the (in this case, algebraic) ‘structuralist’ tradition of Hilbert (1899, in a letter to Frege):

it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes…

…the application of a theory to the world of appearances always requires a certain measure of good will and tactfulness: e.g., that we substitute the smallest possible bodies for points and the longest possible ones, e.g., light-rays, for lines. At the same time, the further a theory has been developed and the more finely articulated its structure, the more obvious the kind of application it has to the world of appearances

So, here we are defining a model schema capturing the idea of the ‘closure of a model’ or, alternatively, a ‘closed model structure’, and meant to capture some notion of induction ‘within’ a model structure and falsification ‘outside’ a model structure. Hilbert’s last paragraph captures this second point.

Suppose we have a background of interest for which we want to create a theory. It may be/almost certainly is the case that there are (many) possible contexts/backgrounds for which we cannot find ‘good’ theories satisfying the closure conditions – e.g. the theories are either much too general or much too specific. This is why psychology is in some ways ‘harder’ than physics – it is very difficult to partition the large number of possibly relevant variables for predicting ‘target’ variables y into a small number of invariant theoretical contructs x, a small set of ‘controllable’ variables b’ and a large set of ‘irrelevant’ variables b”. If we wish to retain an ability to ‘realistically represent’ the phenomenon of interest captured by y, then most things will be ‘explanatory variables’ needing to be placed in x and/or controlled in b’. That is, we will have a very descriptive theory, as opposed to a very ‘causal’ theory. Note that the division (3) into ‘controlled’ and ‘irrelevant’ variables b’ and b”, respectively, tries to help with this, to some extent, but means that controlled lab experiments can be both quite reproducible within a lab but can fail to generalise outside it.

The closure conditions mean that we still know what a theory should look like, if it exists, though and this helps with the search.

A further comment – the uniformity of what?
It’s also worth noting (as I did in the comment section on the original post) that when confronted with complex phenomena we have a choice of

– developing a (probably ad hoc) theory for unusual events by moving some b” variables into b’ or x and having a more complicated theory structure (in terms of number of theoretically-relevant variables)

– having no theory for unusual events (for the moment) and focusing on those which satisfy the closure. These are the ‘simple but general’ theories like the ideal gas.

Thus different theories have different divisions of x, b’ and b”.

We are hence attempting to avoid the problem of requiring an assumption about the uniformity of nature by only making assumptions about the uniformity of our models and accepting that they may not ‘cover’ the entire real world (see also here). This explains why we are ‘inductive’ ‘inside’ the model – uniformity applies here – but ‘falsificationist’ ‘outside’ our model, i.e. in assessing whether its assumed uniformity holds when held up against the real world.

Whether the closure conditions are satisfied depends on your willingness to accept the closure conditions in a given situation which depends on how you define your observable y (for example). I have more to say on this measurement issue at some point in the future, but suffice to say a ‘coarse’ y makes it easier to accept that the conditions are satisfied, as Hilbert implied.