Nill

A quantity is said to be nill, or negligible, if it is small enough that it can be ignored in the present context.

Nill is closely related to the standard mathematics concept of an infinitesimal, but it is not the same. Nill is also closely related to the concept of zero, but is not the same.

[TODO: I think that the concept nill should mean it's so small that it's not measurable at all. This means you can't even see it. It's smaller than just whatever margin of error is on your ruler. I need more concepts than just this though, because there's also an idea of something which is perceptible but within "margin of error" of 0. There's also an idea of something which perfectly measurable, but is negligible in the present context, like the width of a piece of paper.]

Examples

Examples of negligible nill

If a piece of paper is being considered in a context where one is measuring skyscrapers, its thickness is nill. The amount by which a piece of paper placed on top of a skyscraper augments its height is completely negligible.

If a scale measures weight in units of 0.1 pounds, then any object weighing much less than 0.1 pounds has nill weight.

Gas stations sometimes charge amounts of USD which are more precise than 1 cent, despite the fact that 1 cent is the lowest denomination of US currency. E.g. gas might be worth $3.782 per gallon. What this really means is that, when the customer has finished filling his car with gas, the amount charged to the customer is rounded to the nearest cent. For example, at the aforementioned price, if the customer purchases 13 gallons of gas, he will be charged $49.17, rather than $49.166 = 13 * $3.782. In this context, we say that any amount charged which is below $0.01 is nill.

Examples of unmeasurable nill

I cannot tell the difference between a 120Hz display and a 240Hz display, so in the context of human perception, the nill interval for time should be about 1/120 seconds, or 0.0083 seconds.

In a context where one is using a 1000x visible light microscope, I expect that the nill interval for length should be about 0.1 micrometers.

At 1m distance, I cannot see things which are much smaller than the width of a human hair, so in that context, the nill interval for length should be about 0.1 millimeters.

I can definitely hear the difference between a tone of 450Hz and a tone of 455Hz, but I definitely cannot hear the difference between a tone of 450Hz and 451Hz. Therefore, I estimate that in the context of human perception of tone, the nill interval should be about 1Hz.

Non-examples

In the context where a piece of paper is considered together with the hundreds of other pages of a book, the thickness of a piece of paper is not nill.

Nill cutoff

One way of using the concept of nill in mathematics to introduce nill cutoff, i.e. a concrete number $\nu :\mathbb {Q} _{>0}$, such that any $x:\mathbb {Q}$ such that $|x|<\nu$ is considered to be nill.

Contrast with infinitesimal

If a large number of nill quantities are added together, the resulting quantity will not necessarily be nill. This is unlike infinitesimals; a finite sum of infinitely small things is still supposed to be infinitely small.

References

The name nill for this concept was coined by Harry Binswanger in his lecture Saving Math from Plato. [TODO rephrase or put this elsewhere in the document]