QuestionsCan a machine think?Let us calculate: a writing of thoughts to settle disputes
Can a machine think?Thinking is computing

Let us calculate: a writing of thoughts to settle disputes

Calculemus! (the universal characteristic)

Gottfried Wilhelm Leibniz 1646 · 1716We live in the best of all possible worlds.
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What it claims

When a controversy arises, there will be no more need for dispute between two philosophers than between two accountants: it will be enough to say 'let us calculate'.

Why

If thoughts are written as signs, reasoning becomes calculating, and a faulty argument is just a miscalculation.

The same idea through history
Raymond Lulle1305Ramon Llull imagines an 'art' that combines basic ideas, written on rotating discs, to produce truths mechanically (Ars generalis ultima, c. 1305-1308).
Gottfried Wilhelm Leibniz1666In his Dissertation on the Art of Combinations, the young Leibniz dreams of an 'alphabet of human thoughts' whose letters could be combined.
George Boole1854Boole realises part of the dream: in The Laws of Thought he treats logic as an algebra in which one calculates with symbols.
In short

"Let us calculate: a writing of thoughts to settle disputes" is an argument by Gottfried Wilhelm Leibniz (Scientia generalis, Characteristica (Gerhardt VII, pp. 200, 204-205), 1666) on the question "Can a machine think?" (answer: thinking is computing). Its claim: When a controversy arises, there will be no more need for dispute between two philosophers than between two accountants: it will be enough to say 'let us calculate'. According to AskPhilosophy's logic check (Z3 solver), the argument is valid only if one grants a premise the author leaves unstated (P4, P5). AskPhilosophy presents 2 objections, 1 of them with a documented reply, and the original text behind each premise, checked word for word.

Scientia generalis, Characteristica (Gerhardt VII, pp. 200, 204-205)
Valid with an implicit premise

Leibniz dreams of a writing in which each idea would be marked by a character, like numbers in arithmetic. Then a dispute between two philosophers would be settled like an account: sit down, calculate, check. This project is the ancestor of formal logic and of the idea that a machine could handle thought as a calculation. Leibniz himself built a calculating machine, but he does not say that a machine thinks.

P1All human thoughts can be resolved into a small number of primitive thoughts. By giving a character to each, one can form the characters of all derived notions.
P2With such characters, reasoning consists in handling them as one does in arithmetic or algebra, where the mind's error is the same as an error of calculation.
P3In this writing, every false reasoning is nothing but an error of calculation, which can be found and corrected.
P4 · implicit premisePhilosophers' disagreements are about what can be drawn from shared data (not about chance facts or choices of values). Leibniz specifies this: the art yields only what follows from the data.
P5 · implicit premiseIf a disagreement is about what follows from shared data, and every false reasoning is a calculation error, then redoing the calculation is enough to settle it.
So when a controversy arises, there will be no more need for dispute between two philosophers than between two accountants: it will be enough to say 'let us calculate'.

click a premise to reject it

If you accept every premise, logic forces you to accept the conclusion. So the debate is about the premises.
Checked by a logic solver: the premises cannot all be true while the conclusion is false.
Without the implicit premise: Without P4 and P5, the computer finds a case where the writing exists and every false reasoning is a calculation error, but the disagreement is about something else (values, starting definitions): the calculation is right on both sides and the dispute remains.
Valid with P4 and P5. P1 to P3 describe the writing Leibniz dreams of. To conclude that disputes are settled by calculation, disagreements must also be of the kind calculation can settle (P4) and checking the calculation must then suffice (P5). Leibniz assumes them more than he proves them. The objections target exactly these points.
Analogy. Two pupils disagree on the result of a long multiplication: they redo it line by line and one finds his mistake. Leibniz would like the same to hold for metaphysics or morals.

Objections

2

Each objection is a conversation: click to see who replies and where the debate stands.

Kurt Gödel 1931 · attacks the conclusion

Gödel proves that in any consistent formal system rich enough to contain arithmetic, there are statements the system can neither prove nor refute. No universal calculus can therefore settle every question, even in mathematics.

a defender replies
see the argument
No reply possible: objection raised in 1931, after the author's death (1716).
Alan Turing defender of the argument · 1950
Turing grants that any given machine has limits, but notes that it has only been stated, without proof, that the human intellect has none.
John Lucas 1961
Lucas argues on the contrary that for any given machine, a mind can see the truth of that machine's Gödel statement, which the machine itself cannot prove: so the mind is not a machine.
see the argument
Where the debate stands (our summary)Our summary: it is accepted that the dream of a calculus that settles everything is impossible. What remains disputed is what follows: do these limits also apply to the human mind (Lucas, Penrose) or only to machines?
Leibniz himself (the mill, 1714) 1714 · attacks the conclusion

There is a tension within Leibniz's own work. If reasoning is only calculating on characters, parts could do it. Yet in the Monadology he writes that walking into an enlarged machine as into a mill, one would see only parts pushing one another, never anything explaining a perception.

no reply
see the argument
We have not yet found a direct reply by Leibniz in the texts.
The debate goes on (our summary) Leibniz does not link the two texts himself. Our reading: for him, calculation is the rule-governed part of reasoning and may be handed to a machine, but perceiving or understanding takes more than a mechanism.
Where the debate stands (our summary)Our summary: the two theses are compatible if calculating is not thinking. But then 'let us calculate' settles disputes without saying who understands anything. An open point.