Construction Theorems and Constructive Proofs in Geometry

From Firenze University Press Journal: Journal for the Philosophy of Mathematics

John B. Burgess, Princeton University

I will be concerned with comparing and contrasting three types of mathematical assertions, illustrated by these:

(1) Existence claim: There exists a regular heptadecagon.

(2) Constructibility claim: It is possible to construct a regular heptadecagon.

(3) Construction claim: It is possible to construct a regular heptadecagon in the following way . . .

where the ellipsis in the last item would be completed with the specification of Gauss’s construction or someother (as in Weisstein, 2021).

The kind of constructions meant here are those familiar from the early books of Euclid’sElements, traditionally called straight edge and compass constructions. What would have to be meant are anunmarkedstraightedge, not usable as a ruler, and acollapsingcompass, not usable as dividers, since a ruler or dividers would make it a trivial matter to transfer a length, while Euclid takes doing so to require the sort of substantivematerial one finds in his Proposition 2. Actually, what Euclid assumes is that a line can be drawn through two points A and B in the plane (in two steps, first joining the points to form the segment AB as provided for by Postulate 1, then extending the segment indefinitely in a straight line at either end, as provided for by Postulate 2), and also that a circle can be drawn of given center and given radius (as per Postulate 3). And he does notspecify any method or tools for doing such things; in particular, he does not mention the use of straightedgeand compass. Let us nonetheless for convenience retain the traditional label for the class of constructions inquestion.

Many propositions of Book I, beginning with the very first, are “problems,” whose solutions end withwords amounting to “which was to be done” or QEF, in contrast to “theorems,” whose demonstrations end with“which was to be shown” or QED. Both kinds of propositions have in Euclid a very stylized form of exposition, according an analysis of Proclus (see Netz 1999) consisting of the same sequence of a half-dozen parts: protasis, ekthesis, diorismos, kataskeve, apodeixis, symperasma. In a problem, the kataskeve does what is to be done,and theapodeixisshows that it has successfully done it. In a theorem, thekataskeveintroduces auxiliary points,lines, circles, or whatever, and theapodeixisuses them to show what was to be shown. The term “construction”gets used in two ways in discussing these matters, first as a term for problems as opposed to theorems, secondas a translation of kataskeve. Philosophers of mathematics have often contrasted the ancient style of axiomatic geometry representedby Euclid with the modern style represented by David Hilbert (1902). Paul Bernays (1964, p. 275) explicitlydraws the contrast as one between statements of type (2) and statements of type (1):

Euclid postulates: One can join two points by a straight line; Hilbert states the axiom: Given anytwo points, there exists a straight line on which both are situated.

Thomas Heath, however, in his classic translation (1968), renders Euclid’s formulations as infinitival phrases (“to join two points…”) rather than complete sentences (“one can join two points…”) of type (2).But probably it would not matter much to Bernays if his formulation of Euclid had to be confessed to be notquite accurate, since a closer look at the context shows that Bernays is much less concerned with differencesbetween Hilbert and Euclid (who indeed is mentioned only briefly very near the beginning of the paper) thanwith differences between Hilbert and twentieth-century “constructivists.”

Chief among these was L.E.J. Brouwer, the founder of mathematical intuitionism and Hilbert’s mainadversary in theGrundlagenstreitor foundational dispute of the years between the world wars, which was justbeginning to wind down in 1934 when Bernays produced the original French version of the paper just quoted.And Euclid and Brouwer differ from Hilbert in different ways.

DOI: https://doi.org/10.36253/jpm-2932

Read Full Text: https://riviste.fupress.net/index.php/jpm/article/view/2932