The Scientific definition of heat is the change of internal energy. Something doesn't possess heat. However, when you bring two objects into thermal contact energy spontaneously moves from the object with more thermal energy to the object with less thermal energy. The rate of transfer of this energy is what we dub 'heat'.
This explains why a wooden floor and a marble floor at the same temperature feel as though they contain differing amounts of "heat". They do not! What is different is that the marble floor whisks away thermal energy from your body faster than the wood. This is due to its internal structure.
So, it depends on whether we are viewing things from the floors point of view or our foots point of view. From the floor it seems as though heat flow is positive, from the foot it seems as though heat flow is negative. This is actually the case.
What a layman considers 'cold' is a negative heat flow. However, as heat is defined as the rate of change of internal energy and internal energy can range between zero and infinity (well technically not, but close enough...) then the change needs to be defined on the same scale.
For example - you don't say that speeding up and slowing down in your car are two different things. They are the same thing -- the change of speed, which may be positive, negative or zero. Same with heat.
Anyway, on to the PHILOSOPHY.
If one concedes that there exists an infinite set of attributes then it should be quite clear that you cannot define something by the lack of one particular attribute. Of course, you may do so for a finite set. Example if you have {A,B,C}, B=~{A,C}. The negation of infinite is just finite; it is not some specific finite thing.
Hope that helps!