The standard mathematical definition is a set is finite provided that its cardinality is a natural number. But what are the natural numbers? Think about all the non-standard models of the natural numbers, models of arbitrarily high cardinality, and note that if we use one of these high-cardinality models as what we mean by "natural number", we will get a different extension for "finite". So how do we manage to pick out one particular meaning for "finite" or "natural number"? I want to offer four options.
1. Physics. The world gets described by a lot of equations. In these equations, mathematical objects (numbers, Hilbert spaces, etc.) represent features of the real world. We then constrain our interpretation of mathematical terms like "finite", "natural number" and "real number" by the requirement that the mathematical objects correspond as closely as possible to the features of the real world. In other words, we privilege those models of the natural numbers which fit the physical world. Note that this very much requires a significant dollop of scientific realism.
2. The future. We use the infinity of future days to define "finite": a natural number is any number that we will ever reach by counting once per day. This requires the actual world to have an infinite future.
3. Causal finitism. According to causal finitism, no object has an infinite causal history. But, very plausibly, any finite number of causes can be the causal history of an event. We can use these modal claims to constrain the interpretation of "finite".
4. God. Maybe God simply chose one extension for "finite", "natural number", etc., and made our words correspond to that. Cf. this.