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My friend and I had an argument about this topic. He says that the cold air in the mattress will stay at the ambient temperature, so if it's -20 outside the air in the mattress will stay at -20 so I'll keep freezing as opposed to sleeping on the ground ()both with a sleeping bag). I think that this is false since the air is trapped in the mattress and that air is an insulator. The ground will keep on being cold though. I'm not sure which one is right here.

Effectively, it isn't much better to sleep on an air mattress that isn't specifically made to insulate from the ground than on the ground itself although it does make a slight difference.

The main takeaway is that you should always avoid sleeping directly on the ground so even a poorly insulated air mattress will be better than nothing.

Your friend isn't right in assuming the air in the mattress will stay at ambient temperature as you are constantly heating it and it doesn't mix with the rest of the air. It should remain a bit warmer than the air around although it would lose heat both to the air and the ground.

There are three mechanisms involved in losing heat to the ground when camping (provided the tent is setup on the ground).

• Radiation
  
  
  • Your body is the heat source and will radiate heat away. That's why we use sleeping bags to capture that heat and slow that process down.
  
  


• Conduction
  
  
  • When a hot body is in contact with a cold body, heat will be transferred to equilibrium. Putting your hand on a metal pole in the winter is a good illustration. Your hand will feel cold as its heat is transfered in the metal. In the case of sleeping on the ground, since you keep producing heat to maintain temperature, this is a constant process. Considering the ground is essentially the entirety of the planet, your body will never be able to heat it up enough so that you become comfortable directly on the ground.
  
  


• Convection
  
  
  • If your air mattress isn't insulated, or if it doesn't include a mechanical barrier to convection, any air that you heat up will eventually reach the bottom near the ground and then slowly lose heat to it. You will still feel cold.
  
  

Here's a very rough sketch of the process. Your body (red) heats the air at the top of the mattress. This air mixes with the colder air from the bottom and any heat that gets to the ground gets transferred to it.

There are a few ways to combat the heat exchange in an air mattress:

• Baffles
  
  
  • The idea is to restrict air movement with physical barriers. The longer it takes for the air to reach the bottom, the slower it will cool.
  
  


• Insulating fibers
  
  
  • Similarly to baffles, adding an insulating material like down, synthetic or not, will restrict air movement.
  
  


• Thermal reflector
  
  
  • It is possible to add a reflector, like aluminium foil, somewhere in the mattress. Just as with an emergency blanket, heat will be reflected.
  
  


• A foam structure
  
  
  • Self-inflating mattresses like Therm-a-Rest are composed of a loose open cell foam that gets filled with air. This is a relatively effective insulating material as it greatly restricts convection currents.
  
  

Some brands combine those solutions (mainly baffles and reflectors) to reduce bulk and weight although you then have the issue of mattress integrity. If not airtight, an air mattress is useless.

edit per comments with my experience:

There are many field experiments that have been conducted (intentionally or not) where the end result is that a plain air ballon mattress isn't insulating at all. In my 8 years of outdoors retail, I had plenty of angry customers coming back to the store with a summer mattress saying they hadn't been able to sleep all night when in the intermediate seasons when the ground freezes or sometimes winter when camped on snow, even if they had a sufficient sleeping bag.

The blue foam sleeping pads will maybe fare better than an air mattress but I can't confirm first hand. On frozen ground, open cell foam like the blue stuff is usually not considered enough and closed cell EVA is preferred (like Evazote yellow pads from Zote Foams).

Regarding foam pads, I had the ability to do a field test with a friend. We both had -20°C sleeping bags, a 1" self-inflating Therm-a-Rest, and EVA foams. Mine was 15mm, his was 10mm. The first night's air temps was around -15°C but the next morning, he complained of having chills in his back all night. Since we had a few other pads with us, I proposed he use a 15mm that night and even though the air temp went as low as -25°C, the additional 5mm was enough to cut the cold from his back. There was no other changed variable.

There is an old adage

One blanket underneath is worth two on top.

but I can find little support for this idea except a mention on mumsnet.

The reasoning is, that the floor or the ground underneath is solid, and has a much higher thermal capacity than the air above, and so will draw a lot of heat away from your body. It's not just the temperature of the surroundings which matters, but its ability to draw heat from you too.

If you have ever worn boots or shoes with air-cushion soles, you'll know that your feet stay warmer than in those with solid soles, which conduct heat better.

So any kind of insulation between you and the ground is worth having.

Of great benefit too, is to sleep on a waterproof ground sheet, which prevents moisture travelling up to your bedding and body from the ground. Dry bedding is a better thermal insulator than damp bedding.

Air circulation in the mattress causes some heat loss by convection, but will be much less than by direct conduction to the ground. If taking an air mattress with you is an option, then you will be more comfortable, but warmer than sleeping on the ground.