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Bhāskara drew a figure and wrote “Behold!”. Can a picture ever be a proof?

Vikram Sethi Fellow
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Philosophy of mathematicsIndianGreek & RomanDavid Hilbert#diagrams#indian-mathematics#proof#upapatti
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Its coordinates: 1.1.4 Abstract Objects · 6.9 Philosophy of Mathematics

Every year I tell my Class 10 students the story: the twelfth-century mathematician Bhāskara II gave the theorem we call Pythagoras's as a single figure, four right triangles arranged inside a square, with one word underneath: Behold!

Historians tell me the real text of the Bījagaṇita has some calculation alongside, so the legend is tidier than the truth. But the legend survives because mathematicians like what it says. When I draw the figure on the board, the students see it. Something clicks that the algebra never produced.

My questions:

  • Is seeing the figure a proof, or only a reason to believe there is one?
  • A figure shows one square with one set of triangles. What makes us confident it holds for every right triangle?
  • We also know diagrams that lie: the "missing square" puzzle convinces children that area can vanish.

Indian mathematicians had a word, upapatti, for the demonstration that justifies a rule, and commentators wrote them out at length. Euclid leaned on his diagrams more than he admitted, and Hilbert's axioms were partly written to remove that reliance. So was Bhāskara's "Behold!" a shortcut, or a different and legitimate kind of proving?

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3 answers

  1. Ana Clara Ferreira

    Fellow

    Your second question is the old problem of generality, and it isn't special to pictures. Euclid's very first proposition constructs an equilateral triangle from two circles and simply assumes they intersect. Nothing in his postulates guarantees it. The diagram does the work, and Hilbert had to add axioms of order and continuity to make the gap explicit.

    So I'd say: a figure can be a proof if what you see in it does not depend on the particular figure. In Bhāskara's square, the argument works whatever the proportions of the triangle. In the missing-square puzzle it depends on a slope you cannot see. The skill is knowing which kind you are looking at.

  2. Tomasz Szewczak

    Fellow

    From the proof-assistant side: I can't type "Behold!" into Lean. But I've found that a good diagram is very often a recipe for the formal proof. Rearranging the triangles becomes two calculations of the same area, and the algebra falls out.

    So maybe the honest answer to your first question is that the picture is not a proof, but a reliable instruction for producing one, and people who can follow the instruction in their heads are entitled to say they have proved it. That is what your students are doing when it clicks.

  3. Zsófia Bodnár

    Reader

    A mathematician's answer, Vikram: a picture is a proof when you can say exactly what in it is general.

    The Bhāskara figure works because the argument does not depend on the particular shape you drew. Any right triangle, rearranged the same way, gives the same equal areas. The missing-square puzzle fails because it depends on something that only looks true at that scale: the long "hypotenuse" is not actually a straight line.

    So I would answer your first question with a test I use in exercise classes. After "Behold!", ask the student: what would have to be different about this figure for the argument to fail? If she can answer, she has the proof. If she can only point at the picture again, she has a strong reason to believe there is one. Lakatos would add: then go looking for the counterexample.