Tee for Two
I have this tee shirt. Okay, I have a lot of tee shirts but one in particular I’m going to talk about now. The shirt is emblazoned in large characters with the formula “2 + 2 = 5” and then, in considerably smaller print, it adds “for extremely large values of 2.”
God scoffs at my shirt.
God says that “2 + 2 = 4” and will brook now further discussion. So I’m bringing the discussion to you.
I’d actually go further than the shirt, I’d say that “2 + 2 = 5” merely for sufficiently large values of 2, no need to go to extremely, and certainly no need to push both values as far as extreme.
Really, what it comes down to is just a matter of defining your terms. Or I guess my terms, in this case. And I can do that. The first thing to do is to define what is “a value of” everything else flows from that. So here goes, “a value of 2” is any number expressed in base ten whose first character is the digit two and who either has no additional characters or whose second character is a decimal point and whose third and higher characters are all digits. I don’t think this is an outlandish definition. Obviously “5” in the equation can then be understood to be “a value of 5” and the definition of that would be the same as for “a value of 2” but substituting “five” as the first character.
Now we just need to define “sufficiently large values” and I’m afraid this is going to end up sounding a little tautological but so be it. Given that the equation has two values of two (that would be “2” and “2”), and given our somewhat expansive definition of what “2” is, which I’ll point out is actually necessitated by the later reference of them as “values of 2” rather than as “just” two, we can readily guess that for them to be “sufficiently” large they are likely to be values greater than merely two. “Merely two” being the same as “just two” which is the same as “exactly two.” In point of fact, we know by simple addition that if they are both “just” two, they will only add up to four, so at least one of them must have a decimal portion. We likewise know, by simple subtraction of two from five that if even one of them is “just” two, than they are in aggregate insufficiently large, because five minus two yields three and three is not “a value of” two. So now we know that both values must be greater than “just” two, so we know that they are each large enough to require the decimal point and some number of digits to its right.
Given all of that, I’m going to now abandon hard numbers and go into the territory of “thought experiment.” We know, and now my knowledge of math is too weak to tell you “how” we know, but I know I understood it in some class somewhere in my youth for at least a number of seconds greater than two, we know that five minus “a value of 2” is going to yield another “value of 2.” I think I can explain this by telling you that we know that exactly five minus exactly two is exactly three, so if you increase the size of the “two” by any amount small enough to still leave it as “a value of 2” then you’ll end up decreasing the “three” by exactly that amount, which will make it less than “three” but not as low as ” just” two so it will be “a value of 2,” but not the value of two that is “just” two. Got that? Not too tautological?
Back to defining “sufficiently large.” So given any number that is not exactly two but is a value of two, subtracting that number from five will yield another value of two, but not exactly two. So sufficiently large is any two values of two where one value is large enough to require the decimal point in order to express it and the other value is at least as large as the remainder of subtracting the first value from five.
Which is all a very long way of getting to the point that the “weasel” word here is “sufficiently.” “Sufficiently” is not “a value,” it’s not even quantifiable in the absence of saying what it is meant that it is “sufficient” for. All in all, I think it’s a word that Humpty Dumpty would like very much. And “extremely” which the shirt uses, is no better. “Extremely” pushes the undefined further but still fails to define it.
But it does make for a good tee shirt.
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