How does the temperature ever change? If we assume that temperature is a continuous measurement, then we know that it has an infinite number of potential values. In order for temperature to transition between two values, it must then pass over the infinite set of values that lies between whichever two values the temperature is transitioning between. It now seems that temperature should not be able to change at all because before it may change to a given value, it must first reach a value between the desired and the current. Since we can make this claim infinitely, it would seem that temperature becomes "trapped", in a sense, at its current value, unable to change at all. Of course this problem can be applied to other concepts as well, and we might easily draw comparisons to Zeno's ancient thought experiment of Achilles and the tortoise. But the logic here is slightly different; the desired temperature is not continuously fleeing from the present as the tortoise is from Achilles. I simply raise the question as to how Achilles would be able to change his position at all as long as we observe the concept of infinity.

I'll have to confess that I'm one of those people who was early on seduced by a particular sort of solution to this sort of problem, and since then I've never been able to feel the force of the puzzle. Here's a somewhat fanciful example that conveys the idea.

Suppose that we have a body whose temperature at 12:00 midnite is o° Celsius. And imagine that the body's temperature is increasing at a steady rate as follows. Let Tt be the body's temperature at time t. And let the temperature at any given time over the hour after midnite be given by

Tt = t

where t is the number of minutes after midnite. In other words, the temperature rises steadily at the rate of one degree Celsius per minute. It goes through all the intermediate values in finite time. If the arithmetic of real numbers makes sense, so does this.

And so I can't find the puzzle. Pick any time you like over that one hour period. There's an answer to the question "What is the body's temperature at that instant?" And in this example, the temperature is never the same at two different instants. That means that the body's temperature is changing.

As the question was stated, it assumed that the idea of a continuum makes sense. That's good. In particular, there's no good reason to think that the concept of the field of real numbers is paradoxical or contradictory. But if so, then we can use the real numbers to represent time, we can also use them to represent temperatures, and we can straightforwardly describe relations between the two: at time t, the temperature Tt is thus-and-such. Nothing is missing. At every moment, there's a temperature, and it's no more peculiar that those temperatures need not be the same than that the values of the real numbers differ at different points on the real line.

I'd add: this solution accepts the idea that temperature changes continuously, but even if it didn't, we could use the same kind of solution. So long as there's a unique temperature for each moment, there's no conceptual puzzle in the idea that the temperature changes -- i.e., that it isn't the same at each instant. It could be, for instance, that in the half our from midnite up to but not including 12:30, the temperature is a constant o°, and then at every later instant, it's a constant 1o°. Of course, there's no last instant during which it's still o°, but if there's a puzzle about that, it's a puzzle about the very idea of a continuum -- not a puzzle about change.

And in fact, we could go further. Even if time is granular, we can still make sense of the idea of change in exactly the same way: we need only imagine that the temperature is different at some moments than at others.

As noted, I was seduced by this approach early on. Since I'm thoroughly in the grip of this picture, I can't make myself feel the pull of the problem. Perhaps I'm missing something, but I can't for the life of me see what it is.

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