Metaphysics, Metacognition, Language and Number
by Mark English
The logical positivists took a very hard anti-metaphysical line. They were right, in my view, to see traditional metaphysics as being futile and pointless. The essential problem with metaphysics is epistemic. How (given a basically scientific view of the world) can purely metaphysical statements be justified? Rudolf Carnap and most of his Vienna Circle colleagues didn’t think they could and consequently saw no place for metaphysics as a serious discipline.
There is no denying that fundamental, foundational and relational questions arise naturally in the course of scientific and other forms of rigorous inquiry. These kinds of questions are not only worthwhile, they are necessary and inescapable, and to call them philosophical or (in certain cases) metaphysical is not out of line with common usage. Problems arise, however, when philosophical or metaphysical thinking becomes detached from empirical reality and begins to feed on itself.
In the 1940’s and ’50s, Carnap articulated a nuanced account of ontological claims in the context of mathematical and scientific inquiry. He saw such claims as being either trivially true or false (if considered within the theoretical framework in question) or nonsensical (if not). The former were associated with “internal” questions, the latter with “external” questions. Internal questions are asked with a particular framework in mind. Do numbers exist? Within the framework of arithmetic, (trivially) yes. But do numbers really exist in some absolute sense? The question, arguably, is meaningless.
This approach works for formal disciplines and strictly scientific theoretical concepts but the sciences are not entirely formal. They have their origin in our interactions with, and natural curiosity about, the world. It is a mistake to imagine that we are ever entirely locked into specific and rigid linguistic or theoretical frameworks.
Ordinary thinking is an element of our engagement with the world and is never entirely mechanical or formal. It is not formal because interpretation of one kind or another is always involved, in the sciences and elsewhere. And it is holistic in the sense of not being comprised of discrete levels or self-contained modules. Levels and modules are useful abstractions in dealing with human cognition. And, to the extent that they are useful, you could say that they are real, they exist. My point is simply that they are not discrete or self-contained. Not only are various parts of the brain interconnected in complex ways, the broader physical, social and cultural matrix within which neural processing occurs and upon which it depends is also holistic and massively interconnected. A broader context, in other words, always obtrudes.
An ability to conceptualize and deal in a practical way with a wide range of contingencies involves special forms of thinking – or meta-thinking. ‘Metacognition’ is a term applied in the area of cognitive neuroscience to a form (or forms) of higher-order thinking exclusive to humans and strongly associated with the prefrontal cortex. It involves both monitoring (involving an awareness of sensory input and stored information) and control (deploying this information for decision-making and intelligent action). My focus here is on aspects of meta-thinking which relate to language and number.
Alfred Tarski developed the notion of metalanguage, though he was concerned mainly with formal rather than natural languages. Karl Popper explicitly drew on Tarski’s concept of metalanguage to defend a form of the correspondence theory of truth. The linguist Roman Jakobson appealed to the same basic idea when, late in his career, he outlined what he saw as the functions of language. One of these was the metalingual (or metalinguistic) function. It applies when a language is used to talk about itself.
The notion of metalinguistic awareness relates closely to metacognition and is often utilized in explaining such phenomena as code-switching and language alternation. But metalinguistic awareness also applies to phenomena which occur in strictly monolingual environments. As noted above, languages are routinely used in a reflexive way (i.e. to refer to themselves). A speaker’s awareness of implied (as distinct from literal) meaning also requires a certain level of metalinguistic awareness; likewise using and understanding metaphors. Using and understanding irony requires a relatively high level of metalinguistic awareness. Sarcasm is less subtle than irony but provides a clearer illustration that what is literally being said is not always what is actually being said, the intended sense being (in the case of sarcasm) the converse of the literal sense.
The general idea that a broader context always obtrudes applies not just to ordinary life and language use but also to specialized scientific and scholarly work. No significant area of study is self-contained. Not even formal disciplines, such as arithmetic.
Gödel’s work demonstrated the limitations of formal axiomatic systems. He showed that no such system is capable of proving all truths about the arithmetic of natural numbers. He also demonstrated that no formal system which is complex enough to model basic arithmetic can prove its own consistency.
Formal systems then (at least those beyond a certain level of complexity) are not self-contained, not sufficient unto themselves. They are necessarily situated in – and in a real sense are dependent on – a broader context. And any expanded system is dependent on a yet broader context in the same way.
Gödel was a Platonist and saw his incompleteness theorems as vindicating his position on the power of the human mind. The main lesson I take from his work, however, is that productive thinking is necessarily contingent rather than self-contained; that it necessarily engages with a wider world.
What this wider world consists in or of is open to debate. It comprises seemingly very different kinds of things and/or processes: the processes studied by mathematicians; the physical processes studied by physicists and biologists; social and cultural processes; etc..
But on what basis – other than practicality or convenience – do we draw dividing lines between these different kinds of process and, by extension, between disciplines?
Because of the problems of justifying metaphysical statements I prefer to remain metaphysically agnostic and avoid making claims about the nature of these various processes which go beyond common sense, common usage and the relevant disciplines. Neither ordinary thinking nor rigorous intellectual inquiry requires an explicit metaphysical foundation. Effective thinking, speaking and research do have prerequisites, but such a foundation is not one of them.
Sure, language and our natural habits of thought constantly throw up or betray largely implicit metaphysical assumptions or commitments. This is something to be aware and wary of, however, not something which should form a basis or foundation for serious metaphysical claims or systematizing.
The philosophical arguments against realism with which I am most familiar (including Carnap’s view) have, as I see it, a fatal flaw. They give too great a prominence to language and fail to give sufficient recognition to non-linguistic aspects of human thought and understanding.
In respect of the existence or non-existence of entities postulated by scientific theories, Carnap’s approach works well enough because the theories in question are clearly identifiable and distinguishable one from another. If you move beyond particular theories, however, and focus on more mundane concepts which we can approach from many directions and in many ways, there is no single language or system or theory to which we can appeal. This applies to concrete things that we might touch or eat or bump into and also to more complex or abstract things like an ambush or a stock market correction or a political movement or the passing of a bill into law.
Even something like the concept of number can be approached and conceptualized in different ways: using set theory, for example; or the lambda calculus developed by Alonzo Church. Alternatively, numbers and counting can be dealt with via psychology or via linguistics (where the focus is on the ways people calculate and conceptualize number in the course of ordinary social life).
Though there are certainly links between language and counting etc., counting and number clearly take us beyond the bounds of natural language and linguistic processing. It is notable in this connection that savants with advanced computational abilities may be linguistically impaired, just as many linguistically gifted people can’t handle mathematics.
And what are we to make of birds that keep track of the comings and goings of their potential prey by counting and remembering how many entered or exited the burrow they are spying on? These predators would not be much interested in questions about the concept of number, but their counting abilities derive from a pattern of neural processing which necessarily represents or instantiates the concept in some form. Arguably, some such primitive, pre-linguistic and pre-theoretical notion of number underlies even our most sophisticated mathematical ideas and capacities.