You can add apples and oranges.
The opposite assertion is a well known maxim we learn in school and carry in memory for the rest of our lives our every day experience notwithstanding. What do 2 apples and 5 oranges add up to? 7 fruit, of course. (7 fruits, I think, is also acceptable.) Putting 2 apples and 5 oranges into a single bag does not amount to addition but may be suggestive that the objects at hand are not so distant from each other as to make the concept of counting them together completely incomprehensible. Both apples and oranges belong to the fruit category. So 2 apples are legitimately called "2 fruit" and analogously 5 oranges amount to "5 fruit". So shifting the category, or, if you will, the view point, immediately makes the problem trivial.
I was reminded that many people take the impossibility of adding objects of different categories by a small book Mathsemantics incidentally written by a business consultant, not a professional mathematician. The book is eminently readable with stories from the author's childhood and business practice. I never suffered from math anxiety but can easily imagine how the book may help those who do.
Semantics is the science of meanings (and the relationship of symbols to objects they represent) often juxtaposed to syntax which is the study of sentence structures and rules for forming correct sentences. Mathsemantics is a neologism coined to indicate the author's intention to uncover the meaning behind our every day number usage.
Common difficulties we experience with mathematical concepts are traced back to the research by Jean Piaget in 1930s. Piaget observed that children do not distinguish between the outside world objects and their names. "By the sound of it," apples and oranges are different objects and thus could not be added. Piaget said that children's beliefs resist attempts to change them before their time. The author gives several examples of how his father, an accountant and a number lover, helped him learn and appreciate the numbers and their internal structure. Parents will do well to learn and follow these examples with their children.
Since the concept of a number as opposed to a more tangible notion of a number of objects ("5" vs. "5 cows") is an abstraction with no real world prototype, Piaget and his followers denied young children any numerical abilities. I learned from another book (The Number Sense) that from early 1950s Piaget's theories have been successfully challenged. The book looks into the faculty possessed not only by humans but by some animals, of making quantitative assessments. The author repeatedly expresses the view that the ability to evaluate quantitatively (the strength of a predator or the number of predators, the amount of food, the number of available mating partners, etc.) should have been a propitious factor in the natural selection of species.
Following is a Description of one set of experiments cited in the book:
However, Russell Church and Warren Meck have shown that rats represent number as an abstract parameter that is not tied to a specific sensory modality, be it auditory or visual. They again placed rats in a cage with two levers, but this time stimulated them with visual as well as with auditory sequences. Initially, the rats were conditioned to press the left lever when they heard two tones, and the right lever when they heard four tones. Separately, they were also taught to associate two light flashes with the left lever, and four light flashes with the right lever. The issue was, how were these two learning experiences coded in the rat brain? Were they stored as two unrelated pieces of knowledge? Or had the rats learned an abstract rule such as "2 is left, and 4 is right"? To find out, the two researchers presented mixtures of sounds and light flashes on some trials. They were amazed to observe that when they presented a single tone synchronized with a flash, a total of two events, the rats immediately pressed the left lever. Conversely, when they presented a sequence of two tones synchronized with two light flashes, for a total of four events, the rats systematically pressed the right lever. The animals generalized their knowledge to an entirely novel situation. Their concepts of the numbers "2" and "4" were not linked to a low level of visual or auditory perception.
Consider how peculiar the rats' behavior was on trials with two tones synchronized with two light flashes. Remember that in the course of their training, the rats were always rewarded for pressing the left lever after hearing two tones, and likewise after seeing two flashes of light. Thus, both the auditory "two tones" stimulus and the visual "two flashes" stimulus were associated with pressing the left lever. Nevertheless, when these two stimuli were presented together, the rats pressed the lever that had been associated with the number 4! To better grasp the significance of this finding, compare it with a putative experiment in which rats are trained to press the left lever whenever they see a square (as opposed to a circle), and to respond left whenever they see the color red (as opposed to green). If the rats were presented with a red square - the combination of both stimuli - I bet that they would press even more decidedly on the left lever. Why are the numbers of tones and flashes grasped differently from shapes and colors? The experiment demonstrates that rats "know," to some extent, that numbers do not add up in the same way as shapes and colors. A square plus the color red makes a red square, but two tones plus two flashes do not evoke an even greater sensation of twoness. Rather, 2 plus 2 makes 4, and the rat brain seems to appreciate this fundamental law of arithmetic.
Neither book is about mathematics. Mathsemantics touches on the borderline in common language that deals with numbers and number manipulations. Like several other books in the same category (J.Paulos, Innumeracy, for one), this book helps one to overcome number incomprehension. It could be a good stepping stone for self-study or a pool of consistent ideas for a parent or a teacher. The Number Sense (which I only just started) tells the story of how our brain forms a notion of number and other math abstractions. Its positive message is that children are capable of forming abstract math concepts very early in life. Taking this for granted, it's a thought provoking matter why most of us lose this ability by the time we enter grade school.
(By the way, there is a research paper describing the results of comparison of varieties of apples and oranges.)
- E. MacNeal, Mathsemantics, Making Numbers Talk Sense, Penguin Books, 1995
- S. Dehaene, The Number Sense, Oxford University Press, 1997