How do we build a math-finance setup when we do not have a probability space but we do have market prices?

In this talk I want to ask how to create a coherent mathematical framework for pricing and hedging which starts with the information available in the market and does not assume a given probabilistic setup. This calls for re-definition of notions of arbitrage and trading and, subsequently, for a ``probability-free first fundamental theorem of asset pricing". The new setup should also link with a classical approach if our uncertainty about the model vanishes and we are convinced a particular probabilistic structure holds. I explore some recent results but, predominantly, I present the resulting open questions and problems. It is an ``internal talk" which does not necessarily present one paper but rather wants to engage into a discussion. Ideas for the talk come in particular from joint works with Alex Cox and Mark Davis.