Smaller vs. less vs. lesser

I am confused as to some of the vocabulary that can be used to compare numbers and quantities, and would very much appreciate some clarification.

• I suppose it is safe to say that 1 is smaller than 2.
• Can I also say that 1 is less than 2, and if so, which form is preferable?
• It seems that saying that 1 is lesser than 2 is uncommon - but is it incorrect?
• Finally, -2 is clearly less than -1, but is it true that -2 is smaller than -1?
  (If I have 2$ of debt, then I would say that I have less money than if I had 1$ of debt, but I would also say that my debt was larger, not smaller).

The < symbol is known as less than so less than will probably be better in mathematical contexts. In other contexts smaller than may also be acceptable: My salary is smaller than my wife's.

Lesser can only be used attributively. So you can say: I earn a lesser salary than my wife, but not: My salary is lesser than my wife's.

Small

Small is an adjective (and sometimes an adverb or noun) meaning

not big; less in size or amount than is average

a small business

small lunches

He’s small for his age

Those jackets come in small, medium, and large sizes

It does not require an explicit comparison to any other item for its meaning, although such comparison may be understood or implied.

I ordered a small cup of coffee. (I did not order a large cup.)

Smaller

Smaller is the relative adjective (or sometimes adverb or noun) of small, indicating that referenced term it has more of the characteristic of smallness

I ordered the smaller of the two cup sizes offered.

When using small, no other size needs to be explicitly considered. When using smaller, there must be a point of comparison, and generally only two choices. If there were more than two, referring to the item having the least of the characteristic, you would use smallest.

Less

Less is a determiner or pronoun that means

a smaller amount of; not as much:

[as determiner]: the less time spent there, the better

[as pronoun]: storage is less of a problem than it used to be

they returned in less than an hour

Less requires an explicit comparison. Even When you say I want less, you are asking for an amount that is not as large as the specific one being offered.

Note that less is generally used only with uncountable things. I want less sugar in my coffee. When referring to countable things (things with discrete indivisible units), fewer is used. I want fewer people at my dinner party.

Lesser

Lesser means

Smaller in amount, value, or importance, especially in a comparison between two things: chose the lesser evil.

Of a smaller size than other, similar forms: the lesser anteater.

Lesser refers to something discrete and is a specific comparison between two things. Less is also comparative, but does not refer to an explicit amount.

I want less sugar. [How much? Not clear.]

I want the soda with the lesser amount of sugar. [How much? The exact amount in the diet drink.]

The Questions

1 is smaller than 2

1 is less than 2

1 is the lesser number when compared to 2

Which you use depends totally on the context. Whether a higher integer means more depends on what you mean when you say more - as you suggest more debt means less cash.

This is even more complex when considering negative numbers. Which is greater, -1 or -2. If you are discussing temperature -2 is colder (more cold) that -1.

I'll chime in as a scientist to give a different perspective.

Everyone would agree that "less" and "smaller" are synonyms for positive numbers. In other words, we would agree that "5 is less than 10" and "5 is smaller than 10" are both true.

The difference between "less than" and "smaller than" is more subtle when negative numbers are included. In terms of pure math alone, many find it acceptable to say both that "-6 is less than -4" and "-6 is smaller than -4". (This is sometimes taught in primary school classrooms in the USA.) This makes some kind of sense: if you're comparing a person with a debt of $4 to a person with a debt of $6, the person with the debt of $4 has less money and therefore a smaller net worth. But as @bib suggested in their answer above, a debt of $4 is smaller than a debt of $6.

However, as many have pointed out, "smaller than / larger than" compares size. The "size" of (-4) is smaller than the "size" of (-6). For this reason, many mathematicians, engineers and scientists prefer to use "smaller than/larger than" for comparing absolute values -- but this is not universal.

So, in mathematical language, there are sometimes ambiguities in what somebody says. What word to use depends on the kind of comparison you're making. If you aren't sure what to say, perhaps you could be more precise (e.g. give values) or say it a different way (to avoid confusion). If you aren't sure what somebody else means by it, you can always ask for clarification.