A rundown of "Immaterial Aspects of Thought", by James Ross, which can be read here
I. The Indeterminacy of the Physical
Let's consider an exotic color called "bleen". This color is blue before January 1st, 2050, and green thereafter:
Consider now something that is colored blue:
It is impossible to tell if it is blue or bleen if the current date is before January 1st, 2050. Nothing in the physical state of affairs gives us the correct answer. The physical states of affairs is consistent with both blue and bleen, and both states of affairs are mutually exclusive.
Next, consider these physical points:
There are any number of curves that are compatible with these points, each of which are incompatible with each other:
Which curve is the "correct" one, among these incompatible curves? None. They are all "correct", as they are all compatible with the physical points.
Now consider an alien adding machine that was discovered on, say, Mars. We figured out that it is an adding machine, because you can input two digits in the form of binary, and out pops the correct answer:
It appears to give the correct answer every time, so you deduce that it is an adding machine of some kind. However, when you input 257 and 429, suddenly the answer is 2:
Huh? That can't be right. The answer is wrong. Eventually, you find out that every number input between 250 and 500 gives the output "2", but any numbers above that again begin giving the correction answer as if it were addition. Clearly, the machine must be performing some exotic mathematical function other than addition. So it was never performing addition at all; it just appeared to be doing so. So perhaps it is performing the following function: if x, y >249 and <501, then 2, otherwise x + y. So now you've at least figured out what function it is performing, right?
No. Because it could be that it gives some other output, say "3", when numbers between 400,000 and 600,000 are input. Or it may be that it gives some completely different answer yet again when the input numbers are so large that it would take longer than the lifetime of the universe to type them in. There is no way to know what function the machine is performing.
There are many different functions, incompatible with one another, that are consistent with the physical process that is occurring.
And this supports Ross's premise: no physical process is determinate as to what function it is performing. Because, as shown, there are many mutually exclusive functions that are compatible with what the physical facts are that any physical system is doing.
II. The Determinacy of Formal Human Thought
But now consider human formal thought processes. "Formal" thought is thought involving math, logic, and like concepts.
When you are adding two numbers in your head, you better know what function you are performing, otherwise your answers may be incorrect. To be sure our conclusions in science are true (based on our best ability to judge the evidence, anyway), when we perform a logical function such as modus ponens ("if we share genetic info with chimps then we share a common ancestor, we share genetic info with chimps, therefore, we share a common ancestor"), we need to be actually performing that function and not some other function, otherwise we don't know if our conclusion really follows from our premises or not, and we know nothing we think we know.
When you add 2 and 2 to get 4, you really are adding and not performing some other exotic mathematical function. Or conversely, if you are performing some exotic function (such as the one the alien machine might be performing), then you really are performing that function and not addition. If we are not really performing the mathematical and logical functions we think we are, then everything we think we know goes out the window. Our reasoning abilities
In fact, this premise cannot be coherently denied, because if you are denying it then you are reasoning in the form of an argument, but whatever function you are reasoning with may not actually be the function you think it is.
This supports Ross's other premise: formal thought is determinate. Because if it isn't, then everything we think we know, science, math, everything, is gone.
III. Therefore, no Formal Thought is Physical
And the conclusion follows logically: no formal thought process is a physical process.