Introducción

As we have seen in Chapter 2, basic cognitive capacities for mathematical ability is not special among animals, generally speaking. However, what has been shown in depth from the many studies presented is an evolutionary lineage of for the capability of non-verbal arithmetic. To recall, non-verbal arithmetic can be characterized as the ability to recognize differences in magnitudes given a pair of nonidentical sets of, say, dot arrays. So closely are we tied evolutionary to our cognitive brethren that rhesus monkeys perform admirably similarly to undergraduate students. However, it is not until we make the leap to humans that advanced abilities in mathematics – what we can rightfully call advanced mathematics – becomes possible. This advanced ability may be attributed to a set of cognitive capacities that outstrip the capacities of non-human animals. This section investigates which capacities are crucial to the development of conceptual metaphor in mathematics.

decade and a half. Some of these capacities include (but are not limited to) grouping, subitizing, pairing, and symbolizing. So, the grouping capacity allows us to distinguish what is being counted. Subitizing allows us to quickly recognize small numbers, pairing allows for the sequential pairing of numbers and objects, and symbolizing allows for associating symbols with numbers (or other mathematical objects. These capacities and others all underlie our more advanced mathematical capacities, namely metaphorical and conceptual blending capacities. The ability to group, subitize, and pair are all evident in non-human animals. It is with the rise of the ability to symbolize that we can use metaphorical reasoning and blending. The metaphorical capacity is a kind of generalization of symbolizing since metaphorical thinking is the conceptualization of our symbolizations of things like numbers and arithmetic operations onto higher order structures such as groups or distance functions. While these capacities have been identified in more or less generalization in previous chapters, it is our capacity of conceptual blending that will be of utmost importance for what is to follow. Conceptual blending is our capacity to form correspondences across various metaphorical domains with the intent to develop more complex metaphorical domains. As an example of what it means to form complex metaphors consider the conceptual blending example of the Space as a Set of Points metaphor from the previous chapter. This metaphor takes space as a set of elements, which are denoted as points. The notion of point blends container schemas with the classes are container metaphor to form a correspondence to elements in a set, which we can generalize as a space. In developing a notion of space we have effectively blended lower metaphors (in a sense to be defined below) in service of the correspondence to a set that contains elements. We need not stop here, of course. In order to gain an understanding of a geometric figure such as a circle, say, we blend our capacity

for grouping with the Space as a Set of Points metaphor to identify a correspondence to a so-called distance function. This newly formed distance function then gives rise to notions like distances between two points or the set of all points equidistant from a given point. The blends of conceptual metaphors give rise to the ability for understanding circle, which exists in space.

When our conceptual blends consist of previously developed conceptual metaphors, we call them metaphorical blends. Once we move beyond ideas such as number, point, or basic arithmetic, we become engaged in an activity of metaphorical blending. As it turns out, understanding mathematics can then be understood as the mastering of the complex networks of metaphorical blends. We are getting ahead of ourselves somewhat. While it is the case that conceptual blending, and then metaphorical blending is what grounds our ability to do higher mathematics, we still need an explanation of what it is that grounds our most basic mathematical abilities, arithmetic and number theory. Well before the recent cognitive scientific exploration of mathematical cognition, Moritz Pasch – in 1882 – predicted the grounding of our most basic geometric and arithmetic understandings in commonplace physical phenomena. According to Pasch, when we symbolize a proof we develop basic axioms from semantic content. This is to say that he distinguished between the empirical and the deductive aspects of proof. Pasch held that we must start with empirical core ideas of mathematical objects before being able to rigorously develop mathematical theories. This is to say that before we can deductively reason about polygons, for example, we must first have experience of (some) polygons. The same goes for any other simple geometric or arithmetic object. It is only from a grounded experience of physical phenomena that we can develop our geometric and arithmetic ideas in the first place.

“In fact, if geometry is genuinely deductive [given its empirical core], the process of deducing must be in all respects independent of the sense of the geometrical concepts, just as it must be independent of figures; only the relations set out between the geometrical concepts used in the propositions ... concerned ought to be taken into account.” (Pasch, 1882a)

A rigorous proof, for Pasch, involved developing the fewest core empirical ideas that are the simplest possible so that the deductions that proceed from axioms built upon these ideas retain the epistemological status of the core ideas themselves.3 _{Pasch explains that we retain}

understanding of mathematical ideas through a list of features that the core ideas must have. Moving away from Pasch’s terminology, mathematical proof – on this picture – occurs in the move from observation to propositions. We repeatedly observe the objects whose contents fully characterize our core ideas, and from the iterated viewing we express propositions that describe the relations between core ideas. Pasch holds that any regularity that occurs must be kept track of using non diagrammatic or intuitive syntactic means, and then no recourse to empirical notions is necessary for the remainder of a proof. As the above quote highlights, proofs then carry on independently of core concepts of mathematics.

Of course, I here part ways with Pasch. My thesis is that we can’t but involve core mathematical concepts in mathematical proof. However, this historical aside very nicely illustrates the current understanding of the grounding of some of our simple mathematical ideas. A current hypothesis (from Lakoff and N´u˜nez) is that we begin conceptualizing various arithmetic (geometric) operations at a very early age without the use of symbolization. For

3_{Of course, there may be issues surrounding the way in which Pasch developed his empirical conceptual}

example, when given a collection of three objects in front of a child, the hypothesis implies that the child correlates addition with adding objects to the collections. As this and similar correlations aggregate regularly, neural connections between sensory information becomes connected to basic arithmetic operations like addition.

Pasch predicted such a connection in his position that empirical core ideas of mathe- matical objects were necessary before being able to perform rigorous mathematical proof. So, for example, before we can reason about geometric figures, we must first correlate the manipulations of blocks or triangles with fitting blocks into a square grid game. As these correlations aggregate, the hypothesis above implies that neural connections between these kinds of sensory information are then connected to geometric operations like rotation. Such neural connections constitute the most primitive conceptual metaphor at the neural level, namely, in the case of arithmetic anyway, the Arithmetic is Object Collection metaphor. There is much evidence for this kind of neural-level conceptual metaphor given how we teach mathematics at a very low level. When we first learn to add and subtract a standard method for teaching the operations are through an application of adding to and taking away from a given group of objects. We assume that a physical corollary exists before any formal training, and then we build upon the concept to develop higher level concepts.

Now that we have these two resources, namely capacities for the development of concep- tual metaphor in mathematics and evidence that our most primitive metaphors are reliant on sensory-motor physical phenomena, we can move to the next section where I take the important capacity for conceptual blending and impose a gradually increasing structure on our higher level metaphors. By gradually increasing structure I mean that there will be levels of conceptual metaphor corresponding to the method of proof for which they are re-

sponsible. The conceptual metaphors will be increasing along the dimensions of complexity – where complexity may be a measure of the number, say, of other metaphors used in the development of a given novel metaphor. The connection between physical phenomena and conceptual metaphor will not be lost in this structure, which will ultimately ground EBP.