Matpic
mathematics, in pictures
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Decided by exhaustion

Questions with finitely many cases, settled by going through all of them — and what changes when a claim about every argument becomes a count.
000001010011100101110111(p ∨ q) ∧ ¬r on the 3-cube of assignments — 3 of 8 cornerscorners next to each other differ in one variable, which every edge here was checkedagainst Logic

A formula is a corner of a cube

A formula about three letters is a set of eight rows. Written as a table that is a list; drawn on a cube it is a shape — and the shape is what almost every later question in this field turns out to be about.

 keeps 0keeps 1monotoneself-dualaffineenough alone?∧ and✓✓✓··no∨ or✓✓✓··no¬p not p···✓✓no↑ nand·····yes⊕ exclusive or✓···✓no→ implication·✓···no6 connectives against Post's five classes — a tick means the connective stays insidenand escapes all five, and is therefore enough alone Logic

One connective is enough

Of the sixteen ways to combine two truth values, exactly two can build all the others by themselves. Which two is not obvious, and the reason turns out to be five properties that a connective either has or escapes.

 rspq00011110000111101110111011110010((p ∧ q) ∨ (r ∧ s)) ∨ (¬p ∧ ¬r) covered by 3 of its 6 primeimplicantsr∧s ∨ ¬p∧¬r ∨ p∧q — checked against the formula on all 16assignments Logic

The map that puts neighbours side by side

Reorder the rows of a truth table so that neighbouring squares differ in one letter, and finding a short formula stops being algebra and becomes the problem of covering a shape with rectangles.

 ABCDABABCBCDADCDABCD4 circles cut the plane into 14 pieces — Euler's count is 1414 of the 16 patterns appear; missing: A¬BC¬D, ¬AB¬CD Logic

Four circles cannot do it

Three overlapping circles cut the plane into exactly the eight regions three sets need. Four circles cut it into fourteen, and sixteen are required — so the diagram everyone draws stops working at four, and the reason is a count.

 AAAEAIAOEAEEEIEOIAIEIIIOOAOEOIOO1·A1·E1·I1·O2·A2·E2·I2·O3·A3·E3·I3·O4·A4·E4·I4·Ovalid: AAA-1 AII-1 EAE-1 EIO-1 AEE-2 AOO-2 EAE-2 EIO-2 AII-3 EIO-3 IAI-3 OAO-3 AEE-4 EIO-4 IAI-4valid only with existential import: AAI-1 EAO-1 AEO-2 EAO-2 AAI-3 EAO-3 AAI-4 AEO-4 EAO-4256 forms — 15 valid outright, 9 more if every term is assumed to have memberseach cell is one mood in one figure, and the verdict was reached by trying all 256 occupancies Logic

Twenty-four out of two hundred and fifty-six

Aristotle's syllogisms are four sentence forms in four arrangements, which makes 256 patterns of argument. Fifteen of them are valid. Nine more become valid if you assume the things being talked about exist, and the gap between those numbers is a two-thousand-year-old disagreement.

 ji123456123456∀i ∃j : trueevery row carries at least one mark∃j ∀i : falsesome one column is marked all the way downthe relation "j is one more than i, counting round", on 6 rows and 6 columnsevery row has a mark: yes · some column is all marks: no Logic

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

 every node has at most 3 children and the tree reaches level 416 nodes on the bottom row; the marked walk takes a surviving child at every step Logic

An infinite tree has an infinite path

A tree that goes on forever, in which every node has only finitely many children, must contain a single branch that goes on forever. The proof is a rule for walking, and the rule is the whole of why finite information can decide an infinite question.

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