Mathematics is more than the memorization and application of various rules. Although the language of mathematics can be intimidating, the concepts themselves are built into everyday life. In the following excerpt from

A Brief History of Mathematical Thought, Luke Heaton examines the concepts behind mathematics and the language we use to describe them.

There is strong empirical evidence that before they learn to speak, and long before they learn mathematics, children start to structure their perceptual world. For example, a child might play with some eggs by putting them in a bowl, and they have some sense that this collection of eggs is in a different spatial region to the things that are outside the bowl. This kind of spatial understanding is a basic cognitive ability, and we do not need symbols to begin to appreciate the sense that we can make of moving something into or out of a container. Furthermore, we can see in an instant the difference between collections containing one, two, three or four eggs. These cognitive capacities enable us to see that when we add an egg to our bowl (moving it from outside to inside), the collection somehow changes, and likewise, taking an egg out of the bowl changes the collection. Even when we have a bowl of sugar, where we cannot see how many grains there might be, small children have some kind of understanding of the process of adding sugar to a bowl, or taking some sugar away. That is to say, we can recognize particular acts of adding sugar to a bowl as being examples of someone ‘adding something to a bowl’, so the word ‘adding’ has some grounding in physical experience.

Of course, adding sugar to my cup of tea is not an example of mathematical addition. My point is that our innate cognitive capabilities provide a foundation for our notions of containers, of collections of things, and of adding or taking away from those collections. Furthermore, when we teach the more sophisticated, abstract concepts of addition and subtraction (which are certainly not innate), we do so by referring to those more basic, physically grounded forms of understanding. When we use pen and paper to do some sums we do not literally add objects to a collection, but it is no coincidence that we use the same words for both mathematical addition and the physical case where we literally move some objects. After all, even the greatest of mathematicians first understood mathematical addition by hearing things like ‘If you have two apples in a basket and you add three more, how many do you have?’

As the cognitive scientists George Lakoff and Rafael Núñez argue in their thought-provoking and controversial book *Where Mathematics Comes From*, our understanding of mathematical symbols is rooted in our cognitive capabilities. In particular, we have some innate understanding of spatial relations, and we have the ability to construct ‘conceptual metaphors’, where we understand an idea or conceptual domain by employing the language and patterns of thought that were first developed in some other domain. The use of conceptual metaphor is something that is common to all forms of understanding, and as such it is not characteristic of mathematics in particular. That is simply to say, I take it for granted that new ideas do not descend from on high: they must relate to what we already know, as physically embodied human beings, and we explain new concepts by talking about how they are akin to some other, familiar concept.

Conceptual mappings from one thing to another are fundamental to human understanding, not least because they allow us to reason about unfamiliar or abstract things by using the inferential structure of things that are deeply familiar. For example, when we are asked to think about adding the numbers two and three, we know that this operation is like adding three apples to a basket that already contains two apples, and it is also like taking two steps followed by three steps. Of course, whether we are imagining moving apples into a basket or thinking about an abstract form of addition, we don’t actually need to move any objects. Furthermore, we understand that the touch and smell of apples are not part of the facts of addition, as the concepts involved are very general, and can be applied to all manner of situations. Nevertheless, we understand that when we are adding two numbers, the meaning of the symbols entitles us to think in terms of concrete, physical cases, though we are not obliged to do so. Indeed, it may well be true to say that our minds and brains are capable of forming abstract number concepts because we are capable of thinking about particular, concrete cases.

Mathematical reasoning involves rules and definitions, and the fact that computers can add correctly demonstrates that you don’t even need to have a brain to correctly employ a specific, notational system. In other words, in a very limited way we can ‘do mathematics’ without needing to reflect on the significance or meaning of our symbols. However, mathematics isn’t only about the proper, rule-governed use of symbols: it is about *ideas *that can be expressed by the rule-governed use of symbols, and it seems that many mathematical ideas are deeply rooted in the structure of the world that we perceive.

*Featured image credit: “mental-human-experience-mindset” by johnhain. CC0 via **Pixabay.
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Mathematical ideas?

Surely there’s something odd here.

If we accept that mathematics is a language, then are we to admit that some ideas are English? Others Burmese?

Are we not in danger of conflating a concept with its meta-concept?

I suppose that by “mathematical ideas” the writer in this context meant not ideas obtained by putting together signs of a certain language according to its syntax, but rather the syntactical rules themselves. Perhaps the word “principles” would have been more suitable in the above context.

I’d agree that similar processes have taken place in mathematics and human (national) languages over time. Of course at some point it took a leap into the abstract to create mathematics. A leap which made such a big difference over time that at present it can indeed appear as if some ideas came to us from “on high”. But there seems to be complex dialectics at play in such processes. It’s natural to suppose that originally the concept of addition sprang up from the usual, “everyday” manipulations with objects, i.e. getting them within reach, arranging them conveniently or changing their ownership. What at present, from the point of view of pure physics would appear as simple change of position in space, within the framework of common, “down-to-earth” understanding appeared as grouping and regrouping of objects significant for the observer, and thus gave rise to the mathematical concepts of addition and subtraction. In a similar way in the language names were assigned to phenomena not according to what they are scientifically, which was not known at the time, but according to how they appear to a common observer, e.g. the words for “sunset” or “sunrise”. Or, to take another aspect of this process, it required a considerable degree of abstraction to call all water bodies of a certain type “rivers”, “steams”, “lakes” etc. In other words, what originally was a mix of breakthroughs into the abstract on the one hand with a lack of information about the world or consistency of approach on the other hand, over time has given rise to a highly complex system of human language, which at present is capable of operating with concepts of such order of abstraction, whether in the domain of sciences, philosophy or poetry, that in many cases it’s hardly possible to indicate any counterparts, any observable building blocks of the physical world which might be considered their constituent parts. Essentially, if not in strictly logical terms, a similar process appears to have taken place in the language of mathematics. And both types of language have been able to evolve due to some abilities of human brain such as categorization, which have been found manifest even in infants and therefore can be considered inherent.