192 Normat 60:4, 192 (2012)

Summary in English

Christer O. Kiselman, Euclid’s

straight lines (English)

What is a straight line? Primitive

notions are of course notoriously in-

tractable as to formal deﬁnitions, but

need to be understood through the way

they are actually used. The author ta-

kes as point of departure Euclid’s pro-

positions 27 and 16 in his ﬁrst book.

The ﬁrst gives a suﬃcient criterion for

two lines to be parallel, the second sta-

tes that the exterior angle is larger than

any of the two opposite angles in a tri-

angle. From a strict logical point of view

the propositions do not follow from the

axioms, as one can give a model (the

projective plane) for which they do not

hold. Clearly Euclid made some implicit

assumptions. As Hilbert e t al. pointed

out about a century ago, Euclid ma-

de many implicit assumptions, which do

not, however, detract from his achieve-

ment, so the focus of interest is to try

and pinpoint more exactly how Euclid

really thought (as opposed to what he

wrote down). This leads the author

on an historical and linguistic odyssey,

with special emphasis on the notion of

’eutheia’ which can be understood as

meaning a line segment, a ray, or a line

indeﬁnitely extended, and used in all

three meanings by Euclid. The notion of

inﬁnite extension leads to philosophical

questions about potential versus actual

inﬁnity, and how we in retrospect can

through the notion of equivalence clas-

ses speak about the inﬁnite line without

specifying it as a concrete geometric ob-

ject unlike the ﬁnite line segment. Anot-

her is sue discussed, if brieﬂy, is the le-

gitimacy of relying on visual diagrams

in formal deductive proofs. To use them

as support, be it for the imagination or

memory, is one thing, but to draw actu-

al conclusions from them, quite another

thing. Figures can be misleading espec-

ially when one draws planar diagrams of

geometrical conﬁgurations on a sphere

not to mention a non-orientable surface.

Was Euclid aware of the latter possibili-

ty? According to the author we can only

speculate.

Leif Önneﬂod, Comparison and

Change (English).

There is a discrepancy between m athe-

matical symbolism and what mathe-

matical concepts really me an. The for-

mer is a poor substitute for the lat-

ter. The problem is most acute in ele-

mentary instruction concerning the ba-

sic operations of arithmetic, le ading to

much confusion among pupils. Miscon-

ceptions may never be clariﬁed, thus

sustaining into adulthood, becoming a

potential obstruction for using mathe-

matics in everyday situations. The ar-

ticle can be seen as an introduction to

a longer opus by the author in which he

gives a detailed and systematic explana-

tion of what it really means to perform

elementary arithmetic operations with

special emphasis on the didactic confu-

sions involved.

The former is a poor substitute for

the latter. T he problem is most acute in

elementary instruction concerning the

basic operations of arithmetic, leading

to much confusion among pupils (and

educators). Misconceptions may never

be clariﬁed, thus sustaining into adult-

hood , becoming a pote ntial obstruc-

tion for using mathematics in everyday

situations. The article can be seen as

an introduction to a longer opus by the

author in which he gives a detailed and

systematic explanation of what it really

means to perform elementary arithme-

tic operations with special emphasis on

the didactic confusions involved.