The notion of number line was formed in XX c. We consider the generation of this conception in works by M. Stiefel (1544), Galilei (1633), Euler (1748), Lambert (1766), Bolzano (1830-1834), Meray (1869-1872), Cantor (1872), Dedekind (1872), Heine (1872) and Weierstrass (1861-1885).My first thought -- is "XX c" some Russian name for an ancient Greek or Egyptian city? Didn't Euclid have the number line?

No, "XX c" means the twentieth century, from 1900 to 2000. (Or 1901 to 2001, maybe if you quibble about the invention of the zero.) It will always be the greatest century in the history of intellectual thought, and a term like "XX c" gives it a dignified respect. Just like WWII was the greatest war.

I have been using "XX century" to denote that century for a while. I will consider XX c.

The paper is quite serious about arguing that pre XX c concepts of the number line are all defective. It does not explain who finally got it right in the XX c. I would have said that Cauchy, Weierstrauss, and Cantor all had the concept in the 19th century. Surely Bourbaki had the modern concept in the XX c.

I would have said that early XX c great mathematical concepts (related to the number line) were the manifold and axiomatic set theory. But maybe the number line itself is from the XX c.

The paper argues that Cantor's formulation of the reals was deficient, but the references are in Russian, and I do not know whether it is correct.

I posted below about a famous modern mathematician who did not understand axiomatic set theory. The concept of Lorentz covariance was crucial to the development of special relativity by Poincare and Minkowski, but Einstein did not understand it until many years later.

Physicists will be especially perplexed by this. My FQXi essay discusses how mathematicians and physicists view random and infinite numbers differently. Non-mathemathematicians are endless confused about properties of the real numbers. See for example the Wikipedia article on 0.9999 or Zeno's paradoxes.

If it is really true that 19th century mathematicians did not have the modern concept of the number line, then I should assume that nearly all physicists do not either today. I just listened to physicists Sean M. Carroll's dopey comments on Science Friday. He said that we should retire the concept of falsifiability, because it gets used against untestable theories like string theory. He also argued that space may not be fundamental. I wonder if he even accepts the number line the way mathematicians do.

Another new paper says:

The novice, through the standard elementary mathematics indoctrination, may fail to appreciate that, compared to the natural, integer, and rational numbers, there is nothing simple about defining the real numbers. The gap, both conceptual and technical, that one must cross when passing from the former to the latter is substantial and perhaps best witnessed by history. The existence of line segments whose length can not be measured by any rational number is well-known to have been discovered many centuries ago (though the precise details are unknown). The simple problem of rigorously introducing mathematical entities that do suffice to measure the length of any line segment proved very challenging. Even relatively modern attempts due to such prominent figures as Bolzano, Hamilton, and Weierstrass were only partially rigorous and it was only with the work of Cantor and Dedekind in the early part of the 1870’s that the reals finally came into existence.The paper goes on to give a construction of the reals, based on a more elementary version of Bourbaki's. It also outlines other constructions of historical significance.

As you can see, the construction is probably more complicated than you expect. And it skips construction of the natural numbers (usually done with Peano axioms) and the rational numbers (usually done as equivalence classes of ordered pairs of integers).

This puts the number line before the XX c, but there is still the problem that set theory was not axiomatized until the early XX c.

"The paper goes on to give a construction of the reals, based on a more elementary version of Bourbaki's."

ReplyDeleteSet theory is highly overrated and so let me call the bluff. Show me a real number! Just one!