A Look Back at Where Mathematics Comes From (reconciling the internal and the external)

I wanted to take a closer look at the Lakoff/Nuñez book Where Mathematics Comes From and its relationship to what has come to be called the embodied mind.  It seems to me that the biologists who pioneered embodiment had a more radical view of cognition than many of the cognitive scientists who use the paradigm.  For example, in one of his papers, Humberto Maturana Romesin, an early explorer of a very broadly defined notion of cognition, has said:

In my view the central theme of cognition is the explanation of experience, not reality, because reality is an explanatory notion invented to explain experience.

This might sound vexingly circular, but this is only because we are so locked into our usual point of view.  One of the things that came to mind when I read it, was the difficulty we have with many of the ideas in modern physics  – like, for example, the idea that there is no place and time before the big bang, that there is no such thing as nothing, or that virtual particles can pop in and out of existence.  These realities are the explanatory notions invented to explain a very broadened experience, the experience that physics provides.

The difficulty with the view presented in Where Mathematics Comes From is that, while it associates mathematics with some very fundamental human experiences, it fails to fully appreciate the way living systems are understood as totalities by many of the biologists whose work inspired embodiment ideas.  In particular, it fails to fully appreciate the way living systems exist in relation to their medium and are, in no way, independent of it. As Maturana says:

the behavior that appears is not a feature of the organism, but a condition of its existence in the relational space in which it is a totality, and in which behavior as a relational dynamic involves both the organism and the medium in which it exists.

He goes on to say that one of the reasons we have difficulty with this view:

arises from our cultural training that leads us to think in terms of external causes to explain the occurrence of any phenomenon. This attitude blinds us to the spontaneous nature of all processes in the molecular domain in which we exist. All molecular processes occur spontaneously following a path that arises moment after moment according to the structural dynamics of the different molecules involved.

Where Mathematics Comes From approaches mathematics in a very tool-like way, with an emphasis on how we might be building it.  And this is certainly a useful perspective.  But it likely misses something because it only minimally addresses the spontaneity of cognitive processes.

In a review of the book back in 2001, James Madden takes note of other deficiencies in the Lakoff/Nuñex perspective:

If I think about the portrayal of mathematics in the book as a whole, I find myself disappointed by the pale picture the authors have drawn. In the book, people formulate ideas and reason mathematically, realize things, extend ideas, infer, understand, symbolize, calculate, and, most frequently of all, conceptualize. These plain vanilla words scarcely exhaust the kinds of things that go on when people do mathematics. They explore, search for patterns, organize data, keep track of information, make and refine conjectures, monitor their own thinking, develop and execute strategies (or modify or abandon them), check their reasoning, write and rewrite proofs, look for and recognize errors, seek alternate descriptions, look for analogies, consult one another, share ideas, encourage one another, change points of view, learn new theories, translate problems from one language into another, become obsessed, bang their heads against walls, despair, and find light. Any one of these activities is itself enormously complex cognitively—and in social, cultural, and historical dimensions as well. In all this, what role do metaphors play?

With a critique more centered on mathematics education, Martin Schiralli and Nathalie Sinclair make a related observation:

WMCF is right about the sensory-motor basis of abstract concepts, but their reduction of abstract concepts to more concrete ones through metaphor fails to explain the fundamental processes involved in acts of abstraction. The very phrases ‘abstract thought’ and ‘abstract concept’ are misleading. The expression that needs to be analysed is ‘thinking abstractly’.

An intuitive framework (more Kantian naturalism than Platonic idealism) might be a given, but later ‘intuitions’ might be the result of the ideational being (the thinker) tapping into the embodied ordering principles and categories that the visceral being (the organism) has been subliminally and experientially processing.

I believe a more careful look at this biological view of mathematics has been explored by Yehuda Rav.  Here are some excerpts from his essay on mathematics as seen in the light of evolutionary epistemology.

Thus, Maturna (1980, p. 13) writes: “Living systems are cognitive systems, and living as a process is a process of cognition”. What I wish to stress here is that there is a continuum of cognitive mechanisms, from molecular cognition to cognitive acts of organisms, and that some of these fittings have become genetically fixed and are transmitted from generation to generation. Cognition is not a passive act on the part of an organism, but a dynamic process realized in and through action.

When we form a representation for possible action, the nervous system apparently treats this representation as if it were a sensory input, hence processes it by the same logico-operational schemes as when dealing with an environmental situation. From a different perspective, Maturana and Varela (1980, p. 131) express it this way:  “all states of the nervous system are internal states, and the nervous system cannot make a distinction in its process of transformations between its internally and externally generated changes.”

Thus, the logical schemes in hypothetical representations are the same as the logical schemes in coordination of actions, schemes which have been tested through eons of evolution and which by now are genetically fixed.

As it is a fundamental property of the nervous system to function through recursive loops, any hypothetical representation which we form is dealt with by the same ‘logic’ of coordination as in dealing with real life situations. Starting from the elementary logico-mathematical schemes, a hierarchy is established. Under the impetus of socio-cultural factors, new mathematical concepts are progressively introduced, and each new layer fuses with the previous layers.  In structuring new layers, the same cognitive mechanisms operate with respect to the previous layers as they operate with respect to an environmental input. …..The sense of reality which one experiences in dealing with mathematical concepts stems in part from the fact that in all our hypothetical reasonings, the object of our reasoning is treated by the nervous system by means of cognitive mechanisms which have evolved through interactions with external reality.

Mathematics is a singularly rich cognition pool of mankind from which schemes can be drawn for formulating theories which deal with phenomena which lie outside the range of daily experience, and hence for which ordinary language is inadequate.

I would like to add one further note.  Near the beginning of his essay, Rav says this:

mathematics and objective reality are related, but the relationship is extremely complex and no magic formula can replace patient epistemological analysis.

And later:

The nervous system is foremost a steering device for internal and external coordination of activities.

For me the most provocative thing about mathematics is what it may be telling us about the connectedness of the internal and the external – experiences that we often have great difficulty reconciling.

7 comments to A Look Back at Where Mathematics Comes From (reconciling the internal and the external)

  • david pinto

    I like what you are saying, in as much as it corresponds to my explorations.

    What we seem to differ on is our methodology… how we go about making further explorations. I will keep mathematics (or indeed arithmetic) as the object of thought, and I shall not replace it with a meta-narrative of words. Thus, I shall continue conducting thought experiments without the results becoming “objects of thought” in their own right. The lightest of touches is required, far lighter than the weight of thought, and certainly the mass of the written word.

    Nevertheless, I enjoy reading your posts, and I am encouraged that there are others who are exploring this material.

  • Joselle

    I understand what you say, and soon we’ll just be going around in circles, but I am actually inclined to break down the distinction between object and process in such a way that a physical object can be understood as some process (of our living experience) while also taking note of the fact that a process (like mathematics) creates objects (which are then explored). I think my bridge between internal and external is actually some kind of blend.

    But I’ll stop there!

  • david pinto

    Hmmm… I don’t like to add extra terms or concepts… something about Occam’s Razor is in play, or buddhist minimalism.

    We definitely “ground” things in physical experience. Even if this is our experience, it does have some bearing to physical objects in existence.

    I also think that we “manifest” thought internally. With regards to our experience, this has less to do with the physical objects of nerves and so on, and more to do with the processes which these physical substrates enable. That is, our experience is processural, of the substance of time.

    So, I am wary of using words like the “space” or even the “intersection” because it biases the first, implying a physical environment. I think the “intersection between internal and external experiences” is the interaction of external “objects” and internal “processes”. Thus, the reason why maths is powerful is because it corrects our bias for material, as you say. Math is a means by which we can describe process, not only of processes of physical objects (it is named “physics” after all), but of mental processes.

    And further, and rather contentiously, we do not need to model our mental processes with math, because the math itself is the model of our mental processes.

    Nevertheless, however much I enjoy finding a mind which is concerned for this “material” too, I am conscious we can warble on with these variations of words endlessly. If what we are talking about here is true, I believe we can approach mathematics in a completely different way. I have made some fresh developments with regards to simple arithmetic and wonder if you’d be interested in exploring them.

  • Joselle

    What I’m trying to get at could go something like this:

    Counting can be said to be grounded in physical experience. But while the action is bound to physical things (like the objects we count), the action itself emerges from within. And once it’s there, the action of counting is as real to us as the action of swimming, or running. But when we count, we’re not acting on the water or the ground. Yet the body is acting on something.

    There is a space (an intersection if you like) that internal and external experience shares. And this has been obscured by the proliferation and the effectiveness of scientific analysis. It is for this reason that I believe mathematics in particular has the potential to correct our bias for material. And I think this would have implications for what we call spiritual experience as well.

  • david pinto

    This has taken some time for me to face squarely. I spent a year in Madeira recently, and WMCF was the main book that stimulated my cogitation. As a teacher of high school maths, I appreciated the conceptual blends they used to explain euler’s beautiful equation. The notion that math is metaphoric carried much validity, and I can see how the various authors you have cited here match sense-experience with the entirely feedback loop of internal processes. As far as I can tell from your composition, Joselle, there is a strong suggestion that mathematics is the purely internal loop which parallel’s the evolutionary loop that engages external conditions. The meta of the metaphor that L&N suggest and well put by Jason’s #2 above. So mathematics requires no “object” upon which to base its mental processing, however useful objects may be. That is, triangles and circles are more to do with the mind than to do with objects out there.

    I suspect I can not see the cause of altercation, the fine detail perhaps. There is much insight supplied into this phenomena by all these contributing authors. No need for polemic. And I rather like your conclusion, that maths may allow us to reflect on the link between the internal and external worlds. And I prefer reflection and contemplation than epistemological analysis, but that is something of taste, the shape one favours to form of one’s mind as a tool.

    I must say, I like the attempt by L&N to actually use math, rather than rely on words to expose the ideas, and perhaps the relationships and processes going on internally. I think we need to move on, and consider math as thought experiments. And the simpler, the better, it seems to me. Have you any math based questions that you think reflect the link between internal and external “worlds”?

    Thank you again for this digest. Most appreciated.

  • I think the Lakoff/Nunez book agrees with you.

  • All sorts of things about the Lakoff/Nuñez book bother me, but my two big ones are:

    1. There is a smell of the Whorfian hypothesis, which for a while believed that people who did not have individual words to express particular things could not express or understand them at all. For the Lakoff/Nuñez thesis to be true there would need to be mathematics that it would be literally impossible for us to comprehend. While there are some parts that require contortions of thought (like inconsistent mathematics) and while I could even see an “alien mathematics” being rather different from ours, I reject the idea of total cognitive incompatability. Just as the Whorfian standpoint has now been softened into being a subconscious impulse that can be fought against, I think Lakoff/Nuñez is salvagable in the same way (certainly people will tend to develop the easily concretized and conceptualized and mathematics first) but it makes the view much less radical.

    2. Arguably the mind is a total product of environment, and thus these metaphors created through the mind are really created through evolutionary responses to environment. Seeing things as discrete objects, for instance, seems to be a survival necessity. Although I was a little vague on their standpoint it appears Lakoff and Nuñez believe cognition is mystically “above” reality and not shaped in response, thus allowing them dismiss any philosophical claims about the unreasonable effectiveness of mathematics in reality.

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