This has nothing to do with math, actually. It’s instead a conceptual and linguistic problem because of two different metaphors we use in English. One is that the increase in value of numbers from zero to infinity is modelled as a vertical scale. Thus zero is at the bottom, one is above zero, two is above one, and so forth. The other is that the *decrease* in value of numbers from infinity to zero is *also* a vertical scale. Thus zero is at the top, one is below zero, two is below one, and so forth. So we have two metaphors:
1. Numbers are vertical. Zero is the top.
2. Numbers are vertical. Zero is the bottom.
Note that neither of these is actually valid in any physical sense. Numbers have no physical relationship with vertical alignment in a space. We use these sorts of metaphors because they map abstract concepts to our perceptions of the physical world, thus making it easier for us to visualize them – to “see” them mentally. Unfortunately for us, metaphors may conflict between people, and then our communication about these abstract concepts becomes confused.
A similar situation arises with time, which is another abstract concept that we can’t perceive (we have no perceptual apparatus for time itself, only for physical changes over time). Suppose I have a party scheduled on Tuesday. A friend can’t make it, so he wants to reschedule it. He says to me “Can we move the party ahead?” Does this mean the party should be moved to Monday, or to Wednesday? It turns out there are two competing metaphors involved.
1. Time moves forward.
2. Events in the future move toward us.
If you apply the metaphor in 1 then the party should be moved to Wednesday. This is because, since time moves forward, “ahead” means a point in the future in the direction of time’s movement. But if you apply the metaphor in 2 then the party should be moved to Monday. This is because, from where we “stand” in this vision of time, if an event moves “ahead” of its position then it will move toward us. In effect the events “face” us. The party then occurs *earlier* in time, hence on the day before Tuesday. Now that you’re aware of this difference, you may discover that it depends on some physical properties of our experience. In fact, people who are moving – say walking or riding a bike – are more likely to use metaphor 2 above. People who are sitting still are more likely to use metaphor number 1. So if you walk into someone’s office, you’re primed for 2 and the seated person is primed for 1. You agree together to move a meeting “ahead” and then later discover the misunderstanding.
These sorts of metaphors are typical across the world’s languages because they handle perceptual limitations common to all humans. The need for these metaphors is universal, but the precise metaphors are not necessarily the same. For example, there is evidence that Aymara – a language indigenous to the northern Andes of South America – has a metaphor for time quite unlike what English speakers are used to. In Aymara, people have a metaphor that amounts to “Time is visible”. Events that occurred in the past are visible, and thus lie ahead of the speaker. Events that occur in the future are not visible, and hence lie behind the speaker. Time then moves backward in conceptual space, exactly the opposite of what we’re accustomed to in English. This isn’t the same as “Events in time move toward us”, but it’s similar.