There have been shows on TV where the owners of dogs, birds and horses have come out to proudly tell us that their pets can count. For example, "two plus two," a pet owner says, and the dog barks four times. All these have been investigated and to make a long detective case short, experts found that their owners, inadvertently signal the answer to their pets. Calvin Clawson’s book The Mathematical Traveler (Perseus Publishing, MA, 1994, paperback, 2003) cites experiments with chimps, where these hairy fellows turn out to be able to differentiate between a pattern of up to four- and five-dot patterns but none beyond that. Dolphins are able to memorize the order of eight abstract symbols. But Clawson maintains that experts say that this is not "counting" because this skill that animals exhibit could not be extended as a logical operation, i.e., there is no reason why it should just stop at "5" when there is clear evidence that there are things out there in the world more than "5" and, if the same mechanism in their brain makes them recognize "5" is really "counting," then they should be able to count further.
However, this skill among animals is no less impressive and mathematicians gave it a name – "subitizing" which means they can identify a small number of objects they see by immediate sensing, i.e., with no conscious calculation needed. Clawson cites that animals and humans alike are able to do this. Animals can visually assess what is enough, especially if there is one or more missing among their offspring or their food quantity. Humans exhibit this, too. I am always impressed when mothers with many kids can tell in a flash when one of their seven or more kids are not in the room and even know where the missing toddler may be hiding. I cannot "subitize" beyond "3" when it comes to children. I guess it is because I am the oldest of three and when I was, once in a while, handed temporary reins by my parents to take charge, it was just a world for "3" I kept together. Nowadays, it seems I cannot even subitize "3" since I have two nephews and a niece, all of whom I am so in love with but they seem to be always much more than "3" when I am with them in terms of their effect on my world.
Animals, evolutionary scientists say, did not develop brainpower for "counting" because they were able to survive without the need for it since all they needed was to have a sense of what was "enough." I think that makes sense. I think it is a mighty wise and neat power in itself to have a sense of "enough" that seems to be lacking in the modern human. Just look around and you will see human families wearing out their bodies, their sanity, and their own children, trying to raise a number of kids beyond the number they could support. Look around and see unnecessary waste of resources or energy just to make a telling case for personal worth. I think it is silly at the very least and we should strengthen this skill in this business of "subitizing" so we humans can approach a good sense of what is "enough" for our common economic and ecological futures.
Then I also wonder if indeed all humans can do Math in a common way, i.e., would the way to teach Math to a city kid be the same as that you would use to teach other groups? I guess not. I once observed a group of indigenous women being taught by an environmental organization to do basic addition and subtraction for their livelihood operations. When the instructor explained the concept of "borrowing from the next digit" in matters of subtraction, one indigenous woman in the group asked: "Borrow. Would this involve ‘interest?’" And from there, they just could not move on because the concept of "borrowing" was associated with something else. I was stunned but it was true, their concept of simple Math had been overtaken by usurious economics.
Math, being universal, should not stop after you have counted "3" or "5" because it is how scientists do the Math for optics to make the most powerful telescopes for the heavens should be able to communicate with scientists doing the Math for the galaxies in a language that they both can make sense of, whatever their language is, so that the instrument for observation (telescope) and what is being observed (galaxies) would yield understanding and have the power to predict. And the Math of the heavens is an undertaking that undoubtedly involves more than the numbers "3" and "5." The language of nature is Math and it is how we communicate our intelligence across all kinds of natural phenomena and still come out understanding each other. Math is not only exactitudes and precision; it is also the mysteries of "tendency," "probability," "0" and "infinity." While only a few can do calculations in Math, many can appreciate it and even more can learn to appreciate it if teachers will make it more meaningful and exciting and allow us to look at our hands when we think.
I am still not good in Math but seem to be lucky to have found myself in the company of wonderful minds who could do the Math and showed me the principles and revealed what is so elegant and exciting about it. With the Math they require, I still hide my hands behind my back, not anymore out of fear but out of futility.
So this column in a way, has been brought to you by mathematical minds whoever they are, wherever they lurk, by way of incalculable inspiration, and by the numbers "3" and "5" and by a seething curiosity I had lately about this business of counting.
Count your blessings. Good numbers to you all.