1.1.2 - Number Sense

1.1.2

Number Sense

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Number: Reality or Fiction?

Before we begin discussing large numbers, it may be well to first ask: what is number? Although this question may seem elementary, ask yourself whether numbers are real or imaginary, fundamental or a simple mechanical convenience. For while we may have three apples, three chairs, or three dollar bills, do we ever have simply “three”? The question is of some importance to us here, because once we pass beyond the common numbers, into the transcendent, we won’t have our familiar reality to guide us. At such a point, one might well ask: are such large numbers even real, or just absurd fictions? Or maybe they are just as real, just as immediate as the number three!

I would venture to say that either all numbers are real or they are all mere fictions of the human mind. There can be no arbitrary dividing line between numbers which exist and those which don’t, merely because they have not been concretely realized. For if numbers are merely convenient labels, then a “three” is no more real or fictitious than any other number, and it does not matter whether or not there are in fact “three” of anything. On the other hand perhaps all numbers are equally real in that any number could potentially be realized in the concrete, if only we knew how. There is also the possibility that reality, on its grandest scale, is in fact infinite, and our universe is just an infinitesimal speak in this grand scheme.

Given that there is no way to know the full extent and potential of reality it is best to assume that all numbers are fictions. Yet when one removes the mask of number (the written symbols and names) one finds something that is so nameless and shadowy that I can scarcely imagine it isn’t real. It is this fundamental form which I will try to assert is absolutely real, with or without a concrete example. It is only the written and verbal form of number that we can call a “mechanical convenience” or “arbitrary human construct”. What those forms and phrases represent: ah, now there is a terrifying thing to contemplate!

With that in mind, let us now begin to find a foundation for number. For such ideas do not come to men merely because they are strange minded, but because the world is stranger than it seems.

Intuition Precedes Inspiration

If you are now expecting some technical pronouncement on what “number” is, prepare to be pleasantly surprised. Most people have gotten the mistaken impression that mathematics is an inhuman calculating device, whose methods are as arcane as they are unfathomable, and must simply be followed without question in order for their magic to work. Nothing could be further from the truth. Despite the very formal character of mathematics on the surface, buried deep in its belly lays the beast of human intuition. This is a truth that few are aware of, and mathematicians are loath to admit. The formalism of mathematics is actually an attempt to tame this beast, and bring the dark recesses of our minds into the light. If you like, there is an unconscious kind of mathematics that comes before the formal one most are familiar with.

Number is something that must be understood intuitively, if it is to be understood at all. “Number” did not come to man in some mathematical dream, or phantasmal abstraction! No, number came through his eyes and ears! It came in the form of the sun and the moon, the man and his tribe, the sheep in his flock, the days of the year, and the caws of the raven! Where ever there was something to count, there were the counting numbers. We invented the counting numbers because there were things to count...

Number-Sense

It has been suggested that humans are born with some innate grasp of number, sometimes referred to as our "Number-sense". Even animals it seems, possess a rudimentary number-sense.

A number-sense does not require the ability to count, or a familiarity with any form of numeration. Instead, it is our innate ability to recognize and compare small quantities. To illustrate what I'm talking about consider the picture below:

If I ask how many apples are there in the picture you can certainly answer this easily. Is it necessary to count them in order to know, or do you instantly "see" the number without counting? If you saw it for even an instant could you still tell how many? Probably. Clearly there is not enough time to count them out then. So if no counting is occurring how does our mind know how many apples there are?

Now consider this image:

If I ask which box contains more marbles, do you need to count to figure this out, or can you instantly perceive which is more? Can we not "see" which is larger without the aid of counting, language, or even an understanding of number? It is this raw sense of "number" that we can call number-sense.

The ability to identify numbers and compare them isn't something we learn; it's kind of hard-wired. This is why we can make these kinds of comparisons and evaluations without counting.

However, the capacity of our number sense is very limited. Most people can't perceive more than four to six objects simultaneously. When larger groups are involved we either perceive the group as a whole, or we break it up into groups of four or less. Most people will perceive five as a group of three and two, six as a group of three and three, seven as a group of four and three, and eight as two groups of four. Beyond this, it becomes increasingly difficult to simply "see" how many there are.

To show you how grouping works, look at the image below for only an instant and then close your eyes. Try to recall the image and figure out how many there were:

Difficult? Now perform the same task for the next image:

Even if you looked for only an instant you should have been able to figure out it was nine. If on the first you guessed ten, you were close! However it also contained nine pin-balls. The reason the 2nd image is easier for your number-sense to grasp is because it is arranged as a three by three grid. You number sense can handle three by three, but when confronted with a row of nine it is unable to perceive them all at once.

Try to guess the next number with only a glance:

If you guessed ten or twenty, that's pretty good. An estimate of twelve or eighteen would be better. In fact, if you count them out you'll find that there are exactly fifteen pin-balls in the image. Although our number-sense is not able to perceive the fifteen directly, we do seem to have some ability to estimate the amount. This estimation ability however diminishes with scale.

Recall the jelly bean counting contest. The contest relies on the fact that without the aid of counting, our number-sense can only estimate the number of jelly beans. Most people tend to underestimate the number. There could easily be as many as a thousand jelly beans in this can.

Now consider a baseball field like the one pictured below:

How many people do you think this stadium can house? If you guess something in the hundreds or thousands, your guess is way too small. If you guess something in the hundred thousands or millions your guess is way too big! This is an image of San Diego Petco Park, which is said to contain 42,445 seats! Eventually all scale is lost and we are liable to be off by a few orders of magnitude!

This is about the limit of our built-in number-sense. Our abilities in this respect are probably not much better than other animals. If man had no mathematics and no language, any discussion about large numbers could never really move beyond qualitative descriptions like "many" a "great multitude", "a vast empire" etc. As we will see, it is our propensity for abstraction and symbolic representation that will allow us to blast off to the stars!

I first read the term "number sense" in an interesting little book on the origin of numbers entitled:

" Number The Language of Science : A Critical survey written for the Cultured Non-mathematician "

by Tobias Dantzig , Ph.D. Professor of Mathematics

Here is a quotation from the books introduction :

" Man , even in the lower stages of development , possesses a faculty which , for want of a better name , I shall call 'Number Sense'. This faculty permits him to recognize that something has changed in a small collection when , without his direct knowledge , an object has been removed from or added to the collection. "

The author further notes that " Number sense should not be confused with counting ... ". Counting is a step by step process, and involves many higher mental functions such as memory and language. Number sense on the other hand occurs as an "immediate visual impact" as the result of perceiving a collection of distinct objects and mentally assigning a "numberness" to the collection (Note: Non-italized quotes are my own emphasis, not citations ).

The book cites many cases where animals use this number-sense. Here is another quotation :

" Many birds , for instance, possess such a number-sense. If a nest contains four eggs one can safely be taken, but when two are removed the bird generally deserts. In some unaccountable way the bird can distinguish two from three. But this faculty is by no means confined to birds. In fact the most striking instance we know is that of the insect called the 'solitary wasp.' The mother wasp lays her eggs in individual cells and provides each egg with a number of live caterpillars on which the young feed when hatched. Now , the number of victims is remarkably constant for a given species of wasp: some species provide five, others twelve, others again as high as twenty-four caterpillars per cell. But most remarkable is the case of the Genus Eumenus, a variety in which the male is much smaller than the female. In some mysterious way the mother knows whether the egg will produce a male or a female grub and apportions the quantity of food accordingly; she does not change the species or size of the prey, but if the egg is male she supplies it with five victims, if female with ten. "

While these are fascinating cases to consider, they may not seem too impressive, as they are related to instinctive behaviors, deeply ingrained in the creature.

The Book then goes on to give a much more relevant example, which not only helps to clarify the term "number-sense" but allows us to better distinguish it from counting. Here is the story as Tobias tells it...

" A squire was determined to shoot a crow which made its nest in the watch-tower of his estate. Repeatedly he had tried to surprise the bird, but in vain : at the approach of man the crow would leave its nest. From a distant tree it would watchfully wait until the man had left the tower and then return to its nest. One day the squire hit upon a ruse : two men entered the tower, one remained within, the other came out and went on. But the bird was not deceived : it kept away until the man within came out. The experiment was repeated on the succeeding days with two , three , then four men , yet without success. Finally , five men were sent : as before , all entered the tower , and one remained while the other four came out and went away. Here the crow lost count. Unable to distinguish between four and five it promptly returned to its nest. "

Lastly, Tobias goes on to critique his own case. The final conclusion is ...

" carefully conducted experiments lead to the inevitable conclusion that the direct visual number sense of the average civilized man rarely extends beyond four , and that the tactile sense is still more limited in scope . "

A little thought and you may find that these findings seem to have a ring of truth in them.

The point is that humans, even before civilization, and even language, would have possessed some kind of concept of number, however vague. If humans had not progressed beyond immediate experience we would not have gotten much further than the animals. Humans however are able to "conceptualize" their world, and therefore make sense of it. Out of a vague sense of number, humans developed something more concrete, but also more abstract. That abstraction is "number"...

The Number by itself

So even a child has some sense of "how much" or "how many" of something is out there. If we are to move beyond vague impressions however we need to understand what number actually is.

One way to think about number is that it is our minds way of grouping things together. The earliest traces of number in our language is not words for oneness, twoness, and threeness. Instead, we can see traces of an early understanding of number in the fact that many nouns have a singular and a plural form. For example we can have a "goose", or we can have "geese". There is also many names for groups of animals. For example: a herd of goats, a pride of lions, a flock of sheep, a murder of crows, a dray of squirrels, a brood of hens, a flight of doves, etc.

What is interesting about these examples, is that each group has a different group name. Why not simply say a bunch of goats, lions, sheep, crows, squirrels, hens, doves, etc. Why is there a unique group name for each type of animal? Early people probably had not completely divorced the concept of number from the concrete. That is to say three sheep was another thing entirely from three goats. To some extent, numbers were context sensitive. It didn't make sense to speak of "three", you had to have "three" of something.

If you asked someone what the number "three" was, and they pointed to a group of three stones, would you think they had really answered the question? Are the stones the "three", or do they merely embody it. The problem with pointing to an example to define threeness, is it is just an example. Is the person to understand that "three" means, only these three stones, any set of three stones, or any set of three things?

From a certain point of view then, numbers do not exactly rest in the world. That is, there is no "three" or "five" floating somewhere in the air. There are only "examples" of three and five in the world, but the concept itself is ephemeral.

Eventually man was able to divorce the number concept from its qualitative aspect, to obtain its purely quantitative sense. In the physical world, "three" has no inherent meaning. You must have three of something for that statement to mean something. It is this something that is the qualitative aspect that a number takes on, and depends on context. Remove this however, and one is left with only the concept. This bare form of "three" stands in for any set of three things. It is merely a quantity without a quality.

Once man had extracted this raw concept of number he had a very useful and versatile tool. One that could be applied to anything which required quantification. While this gets to the heart of what number is, it doesn't really give us a sound definition. In the next article we will take what we know about numbers to come to a conceptual definition that will serve us well in our journey to the infinite.

Next Article:1.1.3 A Definition of The Counting Numbers