Read these only after you have written your own three sentences. Two of the paragraphs are good, one is not. The question is not which side is right — the lesson does not settle that — but what a paragraph has to do to count as an argument at all.

Model A — with Lucas and Penrose

The mathematician does something no formal system can. I accept premise 1 as Gödel’s theorem, and I locate my reason at premise 2: given F and the assumption that F is consistent, the mathematician sees that G(F) is true, and I grant Putnam that the seeing is conditional — but the condition is one mathematicians in fact meet, since they hold the consistency of PA and ZFC with rational confidence, not as a proof inside those systems but as insight. The price I pay is Putnam’s charge of equivocation: I owe an account of why that insight is not simply an axiom F could have been handed — and my answer is that whenever the confidence is added as an axiom, the enlarged system acquires a new Gödel sentence and the mathematician sees that one too, so the insight is never exhausted by any single formalisation. What I cannot yet say is what kind of thing an insight is that no formalisation catches, and I admit that Penrose’s microtubules are a speculative attempt to say it.

What it does: names the side; keeps premise 1 and locates the reason at premise 2, stating the consistency condition; names the price — Putnam’s charge — and pays it with the iteration reply, then admits what is still owed.

Model B — with Putnam and Feferman

Nothing has been shown. Premise 1 is Gödel’s theorem and I keep it; my reason is at premise 2: the mathematician sees that G(F) is true only on the assumption that F is consistent, and the Second Theorem says no strong consistent system can prove its own consistency — humans are not exempt, and for arbitrary strong systems they cannot reliably check it. Formalise the conditional, “if F is consistent then G(F)”, and a formal system proves exactly what the mathematician saw; the argument equivocates between an unconditional insight it needs and a conditional one it has. The price I pay is that I must not wave Penrose away: his microtubule proposal is a bad answer to a real question the standard reply leaves open — what the mathematician’s confidence in consistency rests on if it is neither a proof inside the system nor an arbitrary assumption — and my side owes a better answer to that question than a shrug.

What it does: the same three moves, on the other side. Notice that both good paragraphs agree on premise 1 and on the form of premise 2 (the seeing is conditional); they disagree about whether the condition is met in a way a formal system could not itself be given. That is where the case lives.

Model C — a paragraph that does not count

Obviously the human mind is more than a formal system, because a mathematician can just see that G is true while the computer cannot prove it, and Gödel proved that computers can’t think. The MCAS crashes show the same thing: formal systems fail and human pilots had to try to save the plane. So Position A is right, and Putnam is only playing with words.

What goes wrong:

  • The seeing is stated flat, without its condition. “Can just see that G is true” leaves out “given the assumption that F is consistent” — and that assumption is the whole dispute. A verdict that omits it has not yet begun.
  • “Gödel proved that computers can’t think.” The lesson lists this among the things the theorems do not show; whether incompleteness bears on minds is a separate, contested question. This is one system read as all knowledge.
  • The case is used as evidence for a conclusion it cannot reach. MCAS is the lesson’s picture of a system that cannot see from inside that it has left its specification — the shape of the Second Theorem — not evidence that humans transcend formal systems. The pilots, told nothing of MCAS, did not save the aircraft.
  • No price. “Putnam is only playing with words” is not an answer to the charge of equivocation; it is a refusal to hear it. Position A’s price is to say why the seeing is not an axiom F could have been handed.

The checklist

A paragraph counts as an argument when it:

  1. names a side;
  2. gives a reason that names a premise of the standardised argument — premise 2 in every good case — and states the consistency condition the seeing depends on;
  3. names the price the side pays, in the writer’s own words: Putnam’s charge (A) or what Penrose is trying to supply (B);
  4. answers, or at least states, the strongest objection from the other side.

Model C fails 2, 3 and 4. Go back to your own paragraph and mark it 1–4.