Wikipedia:Reference desk/Archives/Mathematics/2010 July 18

= July 18 =

Hyperbolic geometry
It's been proved equiconsistent to Euclidean geometry, but has anyone successfully proved that Euclidean geometry is consistent? --138.110.206.101 (talk) 16:00, 18 July 2010 (UTC)


 * Euclid's axioms are rather imprecise. Hilbert rephrased some of them and proved the consistency of his axiomitization (Hilbert's axioms) in 1899. 67.122.211.208 (talk) 02:51, 19 July 2010 (UTC)


 * I don't think Hilbert actually proved consistency, but using analytic geometry the consistency of geometry can be regarded as consequence of the consistency of arithmetic.--RDBury (talk) 03:54, 19 July 2010 (UTC)

Ancient Greek rejection of the fourth power
I think I've read the following somewhere: Since the Greeks thought of numbers geometrically, they viewed exponentiation—in particular, squaring and cubing—as geometrical processes, forming a square or a cube out of a line segment, respectively, and so they rejected the concept of a fourth power (or in fact the multiplication of any four quantities) as geometrically absurd. Heron's formula involves multiplying four lengths together in an intermediate step, which is remedied by the square root at the end to obtain a geometrically meaningful result, but is itself "meaningless". Either the Greeks accepted this as an "imaginary" quantity (similar to the reaction in the 16th century to Tartaglia's method of solving cubic equations, which occasionally requires the extraction of the square root of a negative number), or they rewrote the formula to avoid the problem:
 * $$A=\sqrt{s(s-a)}\sqrt{(s-b)(s-c)}$$.

This is an interesting piece of mathematical history if true, but now I cannot find a reference for it. Does anyone know of a source for this story? —Bkell (talk) 20:10, 18 July 2010 (UTC)
 * Heron's formula is from 300 years after Euclid and Diophantus who lived another century later certainly was quite happy to do arithmetic without reference to geometry and used fourth powers. So perhaps they had given up their inhibitions by then. I believe you're right about the ancient Greeks and that even the Pythagorean's who were more arithmetically inclined liked to make their numbers exist in the real world as figures. My guess is there probably is some study around saying something like what you say. Dmcq (talk) 20:51, 18 July 2010 (UTC)

It seems rather bigoted to speak of "inhibitions". I don't think Euclid et al. were "inhibited" about fourth powers, nor that the "viewed exponentiation geometrically". Rather, they had a concept of a square whose side had a particular length, and similarly a cube, and the square and the cube and an area and a volume respectively. They didn't have an "inhibition" about fourth powers; rather, the concept simply didn't make sense in the context in which they worked. They had no concept of real number. When we do Euclidean geometry today, we often think of lengths, areas, and volumes as real numbers. Euclid did not have that concept and did not use it in doing geometry. He did have a concept of what it meant to say that the ratio of areas of two polygons is the same as the ratio of lengths of two segments, etc.

Off hand, I don't know exactly what Heron did; I'll see if I can find it...... Michael Hardy (talk) 00:36, 19 July 2010 (UTC)
 * Could you be a bit more careful with the 'bigoted' word please? Thanks. Dmcq (talk) 08:57, 19 July 2010 (UTC)

The Greeks did not use modern notation, such as the expression
 * $$A=\sqrt{s(s-a)}\sqrt{(s-b)(s-c)}. \, $$

My guess would be that they spoke of a line segment that is the side of a square whose area is the same as that of a rectangle whose sides have lengths s and s &minus; a, etc. Except that they wouldn't have written those last two expressions in modern notation either. Michael Hardy (talk) 00:45, 19 July 2010 (UTC)
 * The Greeks had terminology that would be understood by few mathematicians today, so it would be a mistake to underestimate how flexible their terminology was even though they did not use modern formulas. Heath is probably the best authority on Greek mathematics. I have to disagree about them having no concept of real numbers; they had ratios which played much the same role in their math as real numbers do in ours. In fact, if you read Euclid's definition of the equality of ratios you'll find a striking similarity with Dedekind cuts.--RDBury (talk) 03:41, 19 July 2010 (UTC)
 * That's not identical to the real numbers. In particular, they didn't multiply them.  They could, however, in effect multiply two lengths and get an area. Michael Hardy (talk) 17:07, 19 July 2010 (UTC)
 * They did compound ratios to give another ratio. I guess really what they had was dimensional analysis. As a matter of interest, did they have anything like velocity as a ratio of distance over time, i.e. where two different things were used? Dmcq (talk) 17:39, 19 July 2010 (UTC)
 * Did the Ancient Greeks use maths to describe physical quantities like velocity? Pretty much all the Ancient Greek maths I've seen has been abstract. --Tango (talk) 19:00, 19 July 2010 (UTC)
 * Archimedes did concrete physical mechanics, hydrostatics, and optics. Bo Jacoby (talk) 20:51, 19 July 2010 (UTC).