Intuition and Rigor

One of the torch bearers of the formalization attempts in the 19th century was the Czech analyst Bernhard Bolzano (1781-1848.) In his critique of the attempts to prove the Fundamental Theorem of Algebra, he wrote

The most common kind of proof depends on a truth borrowed from geometry, namely, that every continuous line of simple curvature of which the ordinates are first positive and then negative (or conversely) must necessarily intersect the x-axis somewhere at a point that lies in between those ordinates. There is certainly no question concerning the correctness, nor indeed the obviousness, of this geometrical proposition. But it is clear that it is an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e., arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely, geometry ...

By which he of course meant that reliance on the geometrical intuition is an unacceptable tool in deriving analytic truths. He clearly accepts the statement as true but objects to the fact of its being used offhandedly, as a self-evident truth. In the article, Bolzano proceeds to justify the statement that is variably now known as Bolzano's or the Intermediate Value Theorem.

His proof depends on the definition of continuity by Cauchy from which he derives the Sign Preserving Property of Continuous Functions. Assuming that at the left end of an interval the function is negative, he observes that it stays negative on a certain bounded set but not for the points near the second end of the interval. He continues

Now the theorem holds that whenever a certain property M belongs to all values of a variable quantity i which are smaller than a given value and yet not for all values in general, then there is always some greatest value u, for which it can be asserted that all i<u possess property M.

I am not critical of Bolzano; for it's easy to see faults from the distance of 180 years. I only want to demonstrate how painful the labors were in delivering the present day mathematics. Bolzano ends his proof with the following remark:

It is now only a question of the proof of the theorem mentioned. The theorem is proved by showing that those values of i of which it can be asserted that all smaller values possess property M and those of which this cannot be asserted can be brought as near one another as desired. Whence it follows, for anyone who has a correct concept of quantity, that the idea of a greatest value i of which it can be said that all below it possess property M is the idea of a real, i.e., actual, quantity.

I have a feeling that the actual quantity is somehow related to a point on a (geometric) line. In any event, Richard Dedekind (1831-1916) might not have had the "correct concept of quantity" for in 1872 he published the first rigorous introduction into the theory of real numbers. The claimed theorem has been indeed proven on the foundation of Dedekind's Cuts.

The property claimed by the theorem is known as completeness of the set of real numbers. Depending on how deep one wants to make a presentation of mathematics, the theorem may as well be declared an axiom - the fact that requires no proof in that particular setting. However, in the footsteps of Bolzano, it would be "an intolerable offense against correct method" to refer to somebody's concept of quantity in a proof claimed to be rigorous.


  1. The History of Mathematics, ed J.Fauvel and J.Gray, The Open University, 1987

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