The Development of Mathematical Rigor from Intuition to Axiomatic Method

Abstract

It is proverbial to place one’s trust in the infallibility of mathematics: “as sure as two plus two makes four”, one says about something when there is not a shadow of doubt about it. We, mathematicians delight in basking in the luster of trust and confidence, even though we know that most often, that light shining on us is due not so much to convictions of certainty, but instead to the mighty influence of the proverb. In addition, we might even have created a reputation of formidable impenetrability for our scholarly field — so much so that non=professionals would rather believe everything just to keep us from explaining it all. And we show aristocratic disdain for the “everyman”, because his mode of expression lacks precision, and his views are not as enduring as ours: often we even scorn philosophers and theoretical physicists, even though they are striving to implement the methods they had learned from us, in order to make their own field of inquiry more rigorous. Yet we balk at mentioning that mathematical rigor is also the product of development. Whence the reluctance? Are we perhaps concerned about losing outsiders’ trust? But what is honest in this trust can only grow stronger once we make its grounds explicit. As for trust unaccompanied by conviction, let it crumble so its place can be taken over by honest trust. Instead of sheer (often unstated) criticism leveled at the scholars striving for rigor, we would be of better service to them, too, if we explained how we, mathematicians have attained the level of rigor that they also have come to admire and regard as a paradigm. Lastly, it could well be of use to us, mathematicians as well, if we confronted the fact that there is such a process of development; and if, in addition, we draw the didactic consequences of this fact, we can even advance the cause of teaching mathematics.

Kalmár’s lecture was first published in the 1942 yearbook of the Exodus Working Group (a group of young Calvinist intellectuals influenced by the ideas of Sándor Karácsony a professor of pedagogy). It was reprinted in a volume of Kalmár’s popular and philosophical writings entitled Integrállevél [Letter on Integral] (ed. Antal Varga, Budapest: Gondolat, 1986). The present translation is based on the 1986 edition and is published here with the kind permission of Eva and Zoltán Kalmar. One of Kalmár’s remarks, concerning Hungarian mathematical terminology, has been omitted and marked with ... Note by the editor.