2008 Woodie Awards
Can "1+1+1" ever be true?
Professor Fred Bauer, Special to Le Provocateur
Issue date: 10/31/04 Section: Viewpoint
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First, What Does "1 + 1" Mean? Before we learned the rules for addition, we had to memorize the string of sounds-signified by "one, two, three, four, etc.-that we call "numbers." That took time, just as it took time to memorize the sounds called "letters." Both memorizings were the same. "One, two, five, three, eight" was the same kind of mistake as "A, B, C, G, F, D." So far, so good, but in time we had to match the sounds ('names') with ideas. Most debates about how to teach math are about the best way to teach pupils the right ideas. The major problem is "Which are the right ideas?" Should learners begin by matching 'one' with a single apple, 'two' with a pair of apples, 'three' with a trio of apples, etc.?
No, that's too simple. The idea of 'one thing' is not synonymous with the idea of 'one apple,' because 'one' also goes with 'one pair' and 'one trio,' even with 'one universe' and 'one reality.' Can one pair and one universe be one thing? As soon as we notice such obvious complications, we are 'off to the races' regarding, "What real things, if any, do mathematicians have in mind when they say that 1 + 1 = 2?"
Plato thought mathematics was about invisible, intangible realities which we can learn about but which do not in any way depend upon us. Einstein claimed that number-concepts are 'free creations of our mind' which cannot exist without us. Those who find all talk about intangible minds and ideas irksome have reduced thought to language and mathematical language to a useful tool; for them the meaning of "1 + 1 = 2" consists, not in nonphysical entities (Plato's Forms), nor in mental concepts (Einstein), but in the formula's use. Bertrand Russell provided a dodge for anyone uncomfortable with the question, "What real things do mathematicians have in mind?" It goes like this. "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
Suppose we return to the idea that each of us was very gullible as a child just beginning to find our way around in this vast, complicated universe. In the same way that we believed we saw Santa at the mall, we believed that we could see numbers in the book. But, just as it was not Santa we saw, it was not numbers, either. Otherwise 5, V, 0101, five, cinq, would all be different numbers, since not even a magnifying glass is needed to see that those are different. The fact that we use numbers when we refer to 'the one-hundred-year-old Assumption College' or to 'this one universe' proves only that number-concepts are indispensable conceptual tools. But if a rose by another name is still a rose, pragmatic fictions are still fictions.
No, that's too simple. The idea of 'one thing' is not synonymous with the idea of 'one apple,' because 'one' also goes with 'one pair' and 'one trio,' even with 'one universe' and 'one reality.' Can one pair and one universe be one thing? As soon as we notice such obvious complications, we are 'off to the races' regarding, "What real things, if any, do mathematicians have in mind when they say that 1 + 1 = 2?"
Plato thought mathematics was about invisible, intangible realities which we can learn about but which do not in any way depend upon us. Einstein claimed that number-concepts are 'free creations of our mind' which cannot exist without us. Those who find all talk about intangible minds and ideas irksome have reduced thought to language and mathematical language to a useful tool; for them the meaning of "1 + 1 = 2" consists, not in nonphysical entities (Plato's Forms), nor in mental concepts (Einstein), but in the formula's use. Bertrand Russell provided a dodge for anyone uncomfortable with the question, "What real things do mathematicians have in mind?" It goes like this. "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
Suppose we return to the idea that each of us was very gullible as a child just beginning to find our way around in this vast, complicated universe. In the same way that we believed we saw Santa at the mall, we believed that we could see numbers in the book. But, just as it was not Santa we saw, it was not numbers, either. Otherwise 5, V, 0101, five, cinq, would all be different numbers, since not even a magnifying glass is needed to see that those are different. The fact that we use numbers when we refer to 'the one-hundred-year-old Assumption College' or to 'this one universe' proves only that number-concepts are indispensable conceptual tools. But if a rose by another name is still a rose, pragmatic fictions are still fictions.