Some underlying senses we need before
we can
begin to understand and use mathematics
on
our route from infancy to sciencesee:
Negation
. . . in mathematics
and
elsewhere
.
Jerry Manheim tells this story from his course
in Differential Equations:
"I asked him to help me work through a problem (since I firmly believe Math is not a spectator sport). No matter what question I asked he was silent. "Finally, in desperation, I said: "Look, this is basically subtraction; what if the temp is 4 degrees F outside (O.K., so it 's never 4 degrees in CA....pretend you're in Alaska), and then the temperature drops 7 degrees, what is the new temperature? SILENCE. I decided to wait him out on this one and eventually he told me that the temperature is now zero. B's and C's in College Alg, Calc and Analyt (3 terms) and doesn't know about neg numbers! Who in hell's name are we doing a favor for....surely not this ELECTRICAL ENGINEERING student. 
. Similarly, "He could even care less," and "He couldn't care less." “It got to minus ten degrees below zero,” and "It got to ten degrees below zero." "Do unto others as they would do unto you." and "Do unto others as you would have them do unto you." All of these share a common error. That error involves a simple abstraction—of the kind that is the heart and soul of modern science and mathematics. ...What
is that error?

The
error in common to the first three statements is the failure to see the
reversal
of
the negative by another negative—a
multiplicative
negative.
Negation is a slightly tricky concept. Negation of negation is even trickier, tricky enough to frequently trip up a lot of people. And a lot of science requires a good sense of negation of negation. Mutual reciprocity (exemplified by the Golden Rule) is a symmetry concept which conceptually stems from a system of potential negations and negations of negations. In physics, the concept of irreversibility is usually misunderstood, demonstating the difficulty of understanding negation of negation. The usually misunderstood concept of Newton's third law of motion (action and reaction) demonstrates the difficulty of understanding mutual reciprocity. 
Some common (mis)uses of words demonstrate that negation is a subtle (abstract) concept at the edges of human comprehension. 
We have ways to use language to express negation.

(not sensing what a multiplicative effect is in such a use of words) 
He hasn't got food on his plate
Our confidence will not easily be restored. Only over my dead body will they raise your taxes. 
He hasn't got no food
on his plate.
Our lack of confidence will not easily be restored. Not over my dead body will they raise your taxes.* 
...Sometimes we want to say that the negation is not.

Here the speaker doesn't realize he wants negation of negation. 
A second one? This guy couldn't
care less about the first.

A second? This guy could even care less about the first.* 
And language can express negation in mathematics, too.
Mathematics is careful to mean just what it says.  The negative of a negative is a positive. 
It got to ten degrees below zero.  It got to minus ten degrees below zero. 