- 3351 - TIME - how short is instantaneous? How short is an “instant” in time? Is it a second? A tenth of a second? A microsecond? What about 100 years? That certainly doesn’t seem like an instant, and to a human being, it isn’t. But to a giant sequoia 100 years is no big deal. And in geological terms it’s practically nothing.
--------------------- 3351 - TIME - how short is instantaneous?
- When Newton and Leibniz developed the math called calculus they were forced to confront the infinitely small. The goal was to understand the idea of the “instantaneous velocity” of an object. That is the speed at which something is moving at a particular ‘instant in time“.
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- We know how to compute average speed over some time interval. Simply take the total distance traveled and divide by the total time. If the object travels 1 meter in 1 second then the average speed is 1 m/s. But what if you have a better measuring device? Say instead you can discover that the object really traveled 20 cm in the first 10th of a second.
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- Then the average speed over that interval is 2 m/s and you’d probably agree that is a better approximation to what we mean by the instantaneous velocity of the object at that point.
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- But it’s still just an approximation. To get the real value, you would need to take smaller and smaller time intervals and have increasingly accurate measuring equipment.
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- The way mathematicians got around this was to talk about infinitesimals, quantities that were not zero yet were smaller than any positive number you can think of, including really tiny fractions like:
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---------------------------- 1/1,000,000,000,000,000,000,000,000,000.
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- Some scientists of the day rejected this idea as nonsense. Indeed, the idea that one could divide things forever flew counter to the Platonic ideal of indivisibles (also called atoms) and therefore did not sit well with the Renaissance embrace of ancient Greek philosophy.
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- There’s a great book about this called “Infinitesimal: “How a Dangerous Mathematical Theory Shaped the Modern World”. Still, this is how calculus was done until Cauchy introduced the formalism of limits, thereby pushing infinitesimals out of the picture.
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- Roughly speaking, a function “f” has limit “L” as “x” approaches a if the values of f(x) can be made arbitrarily close to “L”by taking “x” sufficiently close to “a“. The precise mathematical definition of this idea obviates the need for the old-fashioned use of infinitesimals.
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- Infinitesimals help us deal with the infinitely small, a microsecond (1 millionth of a second) is a short amount of time, but it’s huge relative to a picosecond (10^-12) of a second). In mathematical terms, if “dx” denotes a small amount (like a microsecond) then its square (dx)² (a picosecond) is negligible. So when you’re working on time scales in the seconds you don’t really care about microseconds, and when you’re working on microsecond scales you don’t really care about picoseconds.
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- Our words for time are based on these relative notions of smallness. A minute is so named because it was considered small relative to an hour. Seconds were once called “second minutes” to indicate their relative insignificance.
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- Compared to Earth’s existence, yours doesn’t even look like a blip in time.
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- The Earth is roughly 4.5 billion years old. Even if we decided this new epoch began 3,000 years ago, that is still effectively “now” in geological terms. There have been a million and a half 3,000-year periods in the planet’s life.
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- Climate change presents another example. Sea levels are rising, but the change is not immediately noticeable. Still, by the end of the 21st century, even the most conservative estimates suggest a three- or four-foot rise, with some scientists predicting it will be double that amount.
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- In human terms, these changes are not instantaneous, but in the Earth’s climate cycle they effectively are. We are waiting for some catastrophic event to clearly tell us the climate has officially changed, but it simply takes longer than that.
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- We’re looking for a sign on our human timescale, which is just infinitesimal from a geological viewpoint. But once a few more billion years have passed, some future entity will be able to spot the turning point though not down to the year or century which is a geological instant.
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- As far as the planet is concerned, with its phases measured in the tens or hundreds of millions of years, things are moving pretty quickly. A 1 degree increase in global temperature in 100 years is very fast. If we use this to approximate the future, we quickly see that the planet would be virtually uninhabitable within a few hundred years.
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- In the absence of catastrophic planetary events, such as a large meteor collision, significant change to the Earth takes time. But it’s important to keep in mind that our relatively short lifespans distort our perception of “instantaneous” events.
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- The real dynamics are complicated, of course, but perhaps we should keep this simple calculus in mind as we attempt to craft sustainable solutions. Scale is everything and our idea of small doesn’t necessarily align with reality.
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November 21, 2021 TIME - how short is instantaneous? 3351
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