Do you mean that one even cannot predict the price of the next data point with probability ? For example, 60% of chances, price will go up ?
Having an idea of math is a great basis for trading. But think of pattern recognition: The math you need to identify the shape of a letter in pixel data is pretty simple. Sophisticated mathematical methods or models are not suited for this goal. Trading is pattern recognition.
The trading rulesets I use, don't try to predict price. They just try to predict the sign of the first derivative of price. Lots of times they have no actionable prediction and so they say "Do Nothing - take no action." Are they linear? Well, no, not really. In the same sense that the absolute value function is not linear: ABS(x + y) != ABS(x) + ABS(y). Some of my rulesets use functions involving exponents other than one (for example: some of them calculate the standard deviation, which includes the exponent "2" so it cannot be linear). And other rulesets use the max() and min() functions, which are certainly not linear. "The Highest High of the past ten price bars" is not a linear function at all.
That's more doable. Most try to do it from the prices alone, some with price series and volume and most systems, by far, work fine in backtesting and fail in real trading because of curve fitting. There are some mathematical realities that one has to be fully cognizant of before one can devise a working system. Look at Collective 2, thousands of systems, and not many are profitable for more than a month or two... everybody, I mean nearly everybody is not cognizant of the mathematical realities and are forging ahead crazily.
You can't ever prove that a data stream is non-predictabe. However you can prove otherwise - if you find the underlying law.
It's funny because even a math hobbyist should remember godel's incompleteness theorem... certain things can't ever be proven... it seems logical enough that you can't prove that a market is non-predictable... they usually try that by intermixing time series, volatility or different instruments / equities altogether, essentially spitting out a pile of rubbish that proves their naive thesis right. But certain people on long island know this and move on...