You look at the market. Calls with the same expiry but different strikes have different implied volatilities. The Black-Scholes model says this should not happen. Constant volatility means one number for all strikes. But the market does not care what Black-Scholes says.
Before we tear Black-Scholes apart, Wilmott wants to make something clear. This model is a triumph. It changed finance forever. Two of its three creators won the Nobel Prize. Everyone in derivatives uses it, from salesmen to traders to quants. Option prices are often quoted not in dollars but in volatility terms, with the understanding that you plug that number into Black-Scholes to get the price.