Reasoning with Machines
How do you reason well with, and about, machines that reason fluently?
Starts with how computing and modern AI came to be and why a fluent answer is not evidence of truth, then meets the hard limits proved by Cantor and Gödel, and the myths that grew around them. It ends with working rules: separate an output from a proof, review every diff, judge an agent against a simple baseline, and cross-examine your own trust in the tool.
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Technology, AI and Human Judgment Liberal Arts & Reasoning
Start with the map: how Boole, Turing and Shannon led to neural networks and language models, what the critics argue, why a fluent answer is a fact about training rather than evidence of truth, and four tests (hallucination, calibration, bias, prompt sensitivity) for cross-examining a model.
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Cantor's diagonal argument Logic & Proof · interactive
Then a hard limit in three lines: Cantor's diagonal argument shows the real numbers cannot be listed, and because names, formulas and computer programs can be, almost every real number has no name, no formula and no program.
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Gödel's incompleteness Logic & Proof · interactive
A little further down the same page, Gödel sets a limit on proof itself (a consistent, effectively axiomatised system strong enough for arithmetic has a sentence it can neither prove nor refute), and the page warns that this licenses no mysticism about consciousness.
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The Canon The Canon · interactive
Search the Canon for 'machine' to find Gödel's own remark that either mathematics is too big for the human mind or the mind is more than a machine, graded Reported and offered as a dilemma rather than a theorem, with an exercise that asks you to state what the theorems prove without mentioning consciousness.
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Discrete Mathematics and Graphs: Counting, Relations and Networks Mathematics, Systems & Languages
Back to working code: running an algorithm is not proving it, since Dijkstra's shortest-path method is correct only for non-negative weights, and the four-colour theorem, first proved with a computer in a form no one could check by hand and later verified formally, is the page's case for asking what a machine's result is worth.
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Working with AI Coding Agents: A Disciplined Loop Engineering & Career
Now the daily discipline with a coding agent: acceptance checks written first, a plan before any edit, the smallest coherent change, a diff you can explain line by line, a check you have seen fail, and a prediction written before the agent answers.
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From Assistant to Agent: Tools, MCP and Evaluation Engineering & Career
Climb from assistant to one bounded tool-using agent only as far as measurement justifies: write its limits down first because of prompt injection, and score it on a fixed set (including unanswerable, stale and injected cases) against a plain keyword search.
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The Elenchus Logic & Proof · interactive
Close by turning the Academy's six-step cross-examination on your own trust in a tool: state 'this model is reliable for this task' so it could be wrong, define reliable, name your grounds and their kind, face the strongest objection, fix the test before you run it, and record a verdict.