Why not just shrink the cognitive system to brain-based systems? Is there a way to bridge the impasse between moderate and strong embedding? One argument concerns whether it makes any difference to cognitive science to consider, for example, the manipulation of public symbols to be cognitive processes (Sprevak 2010). Ultimately, to give a decisive answer to that question we would need to change our conception of cognitive processes to on-going dynamical interactions with the environment that loop through brain, body, and environment. However, weak and moderate embedded approaches do not work with such a conception of cognitive process; they work with an input-process-output style sandwich model, where processes supervene on bodily states and processes. For them, there is no reason to accept strong embedding, and much of the discussion has been based around thought experiments or abstract definitions rather than concrete examples.
However, even on a scaffolded view of cognition we can’t deny the difference-making role the manipulations of symbols make to the completion of cognitive tasks. Manipulating public symbols is unique; there is a difference between internalised strategies for completing mathematical tasks and strategies for manipulating mathematical inscriptions. Our cognitive capacities cannot cope with long sequences of complex symbols and operations on them. This is why we must learn strategies and methods for writing out proofs. Symbol manipulation makes a unique difference to our ability to complete mathematical tasks, and we cannot simply ignore their role. If we take the approach of CI, then mathematical cognition is constituted by these bouts of symbol manipulation, and we cannot simply shrink the system back to the brain. The case for a strongly embedded approach to mathematical cognition depends upon the novelty and uniqueness of mathematical practices and dual component transformations. Our evolutionary endowments of numerosity are not up to the task of exact symbolic arithmetic and mathematics. Without symbolic number systems and sequential algorithms there would be no mathematical innovation. Mathematical innovation includes representational novelty: negative numbers, square roots, zero, etc., but also novel functions: multiplication, division, etc. Novelty comes about from the ability to abstractly combine symbols and functions that apply to the symbols.
Uniquely, symbols represent quantities discretely, but there is also the unique human capacity of manipulating symbols in public space. We learn to manipulate symbols in public space and we continue to do so when completing cognitive tasks.
The entire system of mathematics is not contained in a single brain. Symbol systems are public systems of representations and practices for their manipulation. Mathematical practices are part of the niche that we inherit—they are part of our cultural inheritance.
Another objection concerns the impermenance of the scaffolding required for mathematical cognition. Once we have internalised the scaffolding of symbolic number systems, we have no further need for it, except for communication purposes. This claim would be proven if we did not continue to manipulate numerals when completing cognitive tasks. Even if we think that transformation only results in new internal representational resources, and that this just amounts to moderate embedding/scaffolding, we must also concede that most mathematics is conducted on the page.
Scaffolding theorists, like Sterelny, can endorse this idea; indeed they can agree with the bulk of the framework provided by CI whilst avoiding the constitutive claim. What they cannot do is deny that mathematical practice and the manipulation of physically laid-out symbols on the page is a difference maker for mathematical cognition. If you remove it, the ability to complete mathematical tasks drops considerably. To do so is to fly in the face of the empirical evidence from psychology (Landy & Goldstone 2007) and cognitive neuroscience (Dehaene & Cohen 2007; Ansari 2012). Consequently, it is clear that cognitive practices transform our mathematical abilities, lending weight to the CI approach.
The case I have presented in this paper is that symbols are not simply impermanent scaffolds, they are permanent scaffolds. They become part of the architecture of cognition (and not simply through internalisation). Mastery of symbol systems results in changes to cortical circuitry, altering function and sensitivity to a new, public, representational system. However, it also results in new sensori-motor capacities for manipulating symbols in public space. The case can be made in terms of what a symbol system is:
A symbol is a physical mark (or trace), either in physical space, or as a digital trace. Symbol systems contain rules and practices for interpreting symbols, for combining them, and for ordering and manipulating them. A large body of often tacit practices for interpreting and manipulating symbols is acquired. Scaffolding is not simply an amodal symbol with an abstract designation that needs to be learnt (or mapped onto some innate symbol); scaffolding is also how the symbols are physically arranged, how symbols are pushed from one place to the next in a regular fashion. Finally, scaffolding is also how we use our own bodies, eyes, ears, and hands to create and manipulate symbols.